Aspects of Noisy NeuralInformation Processing

vorgelegt vonDiplom-PhysikerGregor Wenning

aus Munster

von der Fakultat IV - Elektrotechnik und Informatikder Technischen Universitat Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften- Dr. rer. nat. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. F. WysotzkiBerichter: Prof. Dr. K. ObermayerBerichter: Prof. Dr. A. Herz

Tag der wissenschaftlichen Aussprache: 11.11.04

Berlin 2004D 83

Aspects of Noisy NeuralInformation Processing

Gregor Wenning

- PhD Thesis -2004

Technische Universitat BerlinFachgebiet Neuronale Informationsverarbeitung

Fakultat fur Elektrotechnik und InformatikFranklinstr. 28/29D - 10587 Berlin

Zusammenfassung:

Aspects of Noisy Neural Information Processing, Gregor Wenning

Neuronale Informationsverarbeitung scheint zu einem wesentlichen Teil auf stochas-tischen Prozessen und deren nichtlinearen Transformationen zu beruhen. Im Rahmendieser Arbeit werden verschiedene Aspekte der stochastischen neuronalen Informations-verarbeitung anhand von theoretischen Modellstudien untersucht. Von besonderemInteresse ist dabei das Ph anomen der stochastischen Resonanz. Im Kontext der neuro-nalen Kodierung hat es folgende Auspr agung: Schwache Eingaben (Signale), welchealleine nicht intensiv genug sind, um ein Neuron zu einer Antwort zu treiben, k onnenmit Hilfe von Membranpotentialfluktuationen(Rauschen) stochastisch verst arkt werden.Die wichtigste Quelle der Membranpotentialfluktuationen ist die Variabilit at der neuro-nalen Eingaben. Es gibt eine optimale Rauschintensit at. Die optimale Rauschintensit atist aber keine Konstante bez uglich der Signalintensit at.Es wird eine biologisch plausible M oglichkeit vorgestellt, wie einzelne Neurone fest-stellen k onnen, was die optimale Rauschintensit at ist, und wie sie ihre Eigenschaftenadaptieren k onnen, um diese einzustellen.Es ist davon auszugehen, dass in biologischen neuronalen Systemen die metabolischenKosten wesentlich die Art der Kodierung bestimmen. Der Einfluss metabolischerKosten auf optimale Signalverteilungen zur Informations ubertragung wird untersucht.Dazu muss insbesondere ein quantitativer Zusammenhang zwischen neuronaler Ak-tivit at und metabolischen Kosten angenommen werden. Es stellt sich heraus, dass indiesem Szenario ein Großteil der zu verwendenden Signale optimalerweise so schwachsein sollte, dass stochastische Resonanz auftritt. Metabolische Kosten und Zuverl assig-keit in der Signal ubertragung k onnen gegeneinander ausbalanciert werden.Kortikale Neurone f uhren Ihre Aufgaben innerhalb von Verb anden, Populationen, aus.Die optimalen Rauschintensit aten, welche f ur das einzelne Neuron gelten sind i. Allg.nicht identisch mit der optimalen Rauschintensit at f ur eine Population von Neuronen.Es werden Resultate vorgestellt die demonstrieren, dass durch Adaptation, basierendauf der Information welche in einem einzelnen Neuron zur Verf ugung steht, auch dieInformationstransmission einer gesammten Population maximiert werden kann.Die Detektion von transienten Signalen wird untersucht. Auch hier tritt das Ph anomender stochastischen Resonanz auf. Mit zunehmender zeitlicher Korrelation in der ver-rauschten Hintergrundaktivit at wird die Detektion immer robuster. Die Anzahl derFehldetektionen nimmt ab.H ohere Momente in der Statistik der Eingaben haben aber auch weitere Effekte auf dasneuronale Verhalten. Es werden die Konsequenzen von zeitlichen Korrelationen undKoinzidenzen in den Eingaben auf das neuronale Verhalten auf einer gemeinsamenBasis miteinander verglichen. Aus den Beobachtungen folgt, dass sich transienteAnderungen in der Anzahl der koinzidenten Eingaben gut zur Signal ubertragung eignen,transiente zeitliche Korrelationen aber kaum einen Effekt erzielen. Diese haben jedocheinen großen Einfluss auf die Detektion von Koinzidenzen.Abschliessend werden analytische Ausdr ucke zur Berechung der Response StimulusCorrelation in einem abstrakten Neuronenmodell vorgestellt. Diese beschreiben dieKorrelation zwischen stochastischen Eingaben und der neuronalen Antwort, dies sindwichtige Aspekte stochastischen neuronalen Verhaltens. Die analytischen Ausdr uckeerleichtern das Verst andnis von dem Wechselspiel zwischen Neuroneneigenschaften,Eingabestatistik und neuronaler Antwort.

Acknowledgments:

First of all I would like to thank my parents for their support that greatly eased mystudies.

The work presented in this thesis was done in the Neural Information (NI) Processinggroup of Prof. Dr. Klaus Obermayer at the Technical University of Berlin. I would liketo thank him for giving me the opportunity to do the research for this thesis in his groupand his supervision. He provides a stimulating scientific environment. The possibilityto present my work on many international events accelerated my research and enrichedthis thesis. I enjoyed large freedom in electing and developing my subjects of interestand methods of choice. As a consequence I have learned a lot during my time in hisgroup.

I am also grateful to my colleagues who are also responsible for this fruitful scientificenvironment.

Thanks to the members of the Graduiertenkolleg (GK) Signalketten in lebenden Systemenfor an interesting time. I enjoyed it a lot.

With the help of the GK (student members and Professors) Peter Wiesing, Lars Schwabeand me organized the International Neuroscience Summit (INS2002), an internationalneuroscience conference. This was an enriching experience. I would like to thank allpeople who made it possible. The members of the NI group spent a lot of time andeffort.

Special thanks to Lars Schwabe who helped me with getting started with the strangeworlds of theoretical neuroscience and computing.

I owe great thanks to Andre Gaudnek, Thomas Hoch, Philipp Kallerhoff, Jacob Kanevand Roberto Lopez-Sanchez (alphabetical order). They contributed with a studentproject or their master thesis to my studies. It is a special pleasure for me, that some ofthem continue doing research in theoretical neuroscience.

Very special thanks to Beate Caspers who gave me strength and provided support inmy private life.

Thanks to my friends for their understanding.

My work was supported by DFG (GK 120-3, SFB 618) and Wellcome Trust (050080/z/97,061113/z/00).

Contents

1 Introduction and Scope 1

2 Stochastic Neuron Models 82.1 The Inter-Spike Interval . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 The Leaky-Integrate-and-Fire (LIF) Neuron . . . . . . . . . . . . . . 122.5 Euler Integration of the Ornstein-Uhlenbeck Process . . . . . . . . . 132.6 Moments of the Firing Time . . . . . . . . . . . . . . . . . . . . . . 13

2.6.1 Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . 142.6.2 Integral Expressions . . . . . . . . . . . . . . . . . . . . . . 15

2.7 First Passage Time (FPT) and Spike Count . . . . . . . . . . . . . . . 172.8 Multiplicative Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 Colored Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10 Hazard Function Approximation of the LIF Neuron . . . . . . . . . . 192.11 Hodgkin-Huxley (HH) Model Neurons . . . . . . . . . . . . . . . . . 21

2.11.1 Action Potentials and HH Models . . . . . . . . . . . . . . . 212.11.2 Point-Conductance Model of in Vivo Activity . . . . . . . . . 23

3 Information Transmission 253.1 Entropy and Mutual Information . . . . . . . . . . . . . . . . . . . . 253.2 Fisher Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Adaptive Stochastic Resonance 284.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Adaptation, Hazard Function Approach . . . . . . . . . . . . . . . . 294.4 Adaptation in a LIF Neuron . . . . . . . . . . . . . . . . . . . . . . . 334.5 Adaptation in a HH Neuron . . . . . . . . . . . . . . . . . . . . . . . 344.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.7 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Energy Efficient Coding 395.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Capacity of Abstract Model Neurons . . . . . . . . . . . . . . . . . . 405.4 LIF Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.5 Information Transmission . . . . . . . . . . . . . . . . . . . . . . . . 43

i

5.6 Metabolic Constraints on Information Transmission . . . . . . . . . . 455.7 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . 48

6 Population Coding 506.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 The Binary Threshold Model . . . . . . . . . . . . . . . . . . . . . . 52

6.3.1 Architecture of the Model . . . . . . . . . . . . . . . . . . . 526.3.2 Approximation of the Mutual Information . . . . . . . . . . . 536.3.3 Analysis of Information Transmission . . . . . . . . . . . . . 54

6.4 Populations of Spiking Neurons . . . . . . . . . . . . . . . . . . . . 566.4.1 LIF Framework . . . . . . . . . . . . . . . . . . . . . . . . . 566.4.2 HH Framework . . . . . . . . . . . . . . . . . . . . . . . . . 566.4.3 Quantification of Information Transmission . . . . . . . . . . 586.4.4 Results of Numerical Simulations . . . . . . . . . . . . . . . 59

6.5 Energy Efficient Information Transmission . . . . . . . . . . . . . . . 616.5.1 Optimal Input Distribution . . . . . . . . . . . . . . . . . . . 626.5.2 Information Rate per Metabolic Cost . . . . . . . . . . . . . 63

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7 Detection of Pulses in a Colored Noise Setting 687.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.3 The Pulse Detection Scenario . . . . . . . . . . . . . . . . . . . . . . 697.4 The LIF Model with Colored Noise . . . . . . . . . . . . . . . . . . 707.5 HH Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.6 LIF Framework, Results . . . . . . . . . . . . . . . . . . . . . . . . 727.7 Analysis of the LIF Model . . . . . . . . . . . . . . . . . . . . . . . 76

7.7.1 Second Order Moments . . . . . . . . . . . . . . . . . . . . 767.7.2 Probability of Correct Detection . . . . . . . . . . . . . . . . 767.7.3 The Gaussian Process Approximation . . . . . . . . . . . . . 787.7.4 Quality of Pulse Detection . . . . . . . . . . . . . . . . . . . 78

7.8 Simulation Results for the HH Framework . . . . . . . . . . . . . . . 817.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8 Higher Moments In Neural Integration 848.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.3 LIF Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.3.1 Explicit Interpretation of Coincidences and Temporal Correlation 868.3.2 Differential Equations for the Moments and their Solution . . 87

8.4 LIF Framework, Results . . . . . . . . . . . . . . . . . . . . . . . . 888.5 HH Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928.6 HH Framework, Results . . . . . . . . . . . . . . . . . . . . . . . . 938.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

ii

9 Response Stimulus Correlation 969.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.3 Model, Materials, Methods . . . . . . . . . . . . . . . . . . . . . . . 98

9.3.1 General Framework . . . . . . . . . . . . . . . . . . . . . . . 989.3.2 Membrane Potential . . . . . . . . . . . . . . . . . . . . . . 1019.3.3 Synaptic Conductance . . . . . . . . . . . . . . . . . . . . . 1059.3.4 Reversing Time . . . . . . . . . . . . . . . . . . . . . . . . . 1109.3.5 Response-Stimulus Correlation . . . . . . . . . . . . . . . . 116

9.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229.4.1 General Performance of the Model Equations . . . . . . . . . 1229.4.2 Single Excitatory Conductance . . . . . . . . . . . . . . . . . 1229.4.3 Excitatory versus Inhibitory Synapses . . . . . . . . . . . . . 1269.4.4 Negative Conductances . . . . . . . . . . . . . . . . . . . . . 128

9.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299.5.1 Ito or Stratonovitch? . . . . . . . . . . . . . . . . . . . . . . 1299.5.2 RSC Analysis of Neural Behavior . . . . . . . . . . . . . . . 129

9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

10 Summary and Discussion 131

11 Appendix 13311.1 HH Framework, Slope of the fI-Curve . . . . . . . . . . . . . . . . . 13311.2 Information Measures, Small Signals . . . . . . . . . . . . . . . . . . 13311.3 LIF in the Population Coding Chapter . . . . . . . . . . . . . . . . . 13311.4 Hodgkin-Huxley Type Neuron . . . . . . . . . . . . . . . . . . . . . 13411.5 Synaptic Background Activity . . . . . . . . . . . . . . . . . . . . . 13511.6 Optimization of IF with respect to the Noise Level . . . . . . . . . . . 13611.7 Discriminability in Populations of Neurons . . . . . . . . . . . . . . 13711.8 Moments of the Membrane Potential, LIF . . . . . . . . . . . . . . . 13911.9 Pulse Detection in Populations of Neurons . . . . . . . . . . . . . . . 14011.10Strictly positive Conductances with White Noise . . . . . . . . . . . 14111.11Notes On The Simulations . . . . . . . . . . . . . . . . . . . . . . . 142

11.11.1 General Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 14211.11.2 Simulations in Part One . . . . . . . . . . . . . . . . . . . . 14311.11.3 Simulation illustrating mutual synaptic relations . . . . . . . 14311.11.4 Single Excitatory Conductance . . . . . . . . . . . . . . . . . 14411.11.5 Simulation showing Excitation and Inhibition . . . . . . . . . 14411.11.6 Poisson Inhibition Simulation . . . . . . . . . . . . . . . . . 144

11.12Details of the calculations in the RSC Chapter . . . . . . . . . . . . . 14511.13Ito Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

11.13.1 Ito Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 15211.13.2 The Quadratic Covariation Process . . . . . . . . . . . . . . . 15311.13.3 The Ito Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 15311.13.4 Stratonovitch Integrals . . . . . . . . . . . . . . . . . . . . . 15311.13.5 Ito Isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

11.14Publications, Gregor Wenning . . . . . . . . . . . . . . . . . . . . . 15511.14.1 Publications, Journals and Proceedings . . . . . . . . . . . . 15511.14.2 Manusscripts, Submitted . . . . . . . . . . . . . . . . . . . . 155

iii

References 156

iv

Chapter 1

Introduction and Scope

The apparent irregularity of neural activity is traditionally declared noise. Figure 1.1(a) shows intracellular recordings of a cortical neuron in the primary visual area of ananesthetized cat. It shows the strongly fluctuating membrane potential which reflectsthe properties of the input and of the neuronal properties. The main reason for a fluc-tuating membrane potential are the inputs each neuron receives. Recent experimentsand modeling studies have shown that cortical neurons receive a large number of si-multaneously active inputs that give rise to the so-called high conductance state andthat induces strong fluctuations of the neurons’ membrane potential (Destexhe et al.,2003). As a rule of thumb one can say that outputs are generated, whenever the mem-brane potential exceeds a critical value - the threshold voltage - as a consequence aspike is initiated, the signature output of a neuron. The irregularity of the membranepotential is reflected in a high variability of the neural spiking activity. Thus a statis-tical description of neural activity is appropriate. In a statistical description the notionof noise can be quantified, e.g. as the variance of the inter-spike-intervals or of themembrane potential fluctuations.

The declaration as noise implies that no relevant function is related to the apparentirregularity of neural activity, even worse, the word noise implies, that irregular ac-tivity should be considered as a nuisance. However, there is increasing evidence thatfar from being noisy and unreliable neurons can handle individual spike timings veryprecisely (de Ruyter van Steveninck et al., 1997; Berry et al., 1997). Considering thisand the astonishing capabilities and efficiency of existing nervous systems, it seemshardly plausible that nervous systems would employ noise if it would not be usefulor necessary. To the contrary, one can be inclined to assume that nervous systems doeven benefit from irregularity. This idea has been gaining more and more acceptancein recent years.

Some interesting concepts related to the role of irregularity in neural coding havebeen put forward. In previous studies it has been suggested that noise may facilitatefast information transfer through a population of neurons (Tsodyks & Sejnowski, 1995;Silberberg et al., 2004), may tune the gain of a neuron’s transfer function (Chance et al.,2002), or may allow the transmission of otherwise sub-threshold inputs, for examplethrough the phenomenon of stochastic resonance.

Large parts of this thesis are devoted to the phenomenon of stochastic resonance.Stochastic resonance is likely to appear in information transmission whenever weaksignals, noise and an inherently non-linear information transmission channel are in-volved. This is exactly the case in a neural coding context, since neurons might be

1

0 1000 2000 3000 4000 5000 6000−1

−0.5

0

0.5

1

1.5

2

2.5x 10

4

time [ms]

−65

volta

ge [

mV

]

20 40 60 80 100

−60

−55

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

noise level

noise leveloptimal

info

rmat

ion

tran

smis

sion(a) (b)

time [ms] noise level

volta

ge[m

V]

info

rmat

ion

tran

smis

sion

20 100

-65

-60

Figure 1.1: (a) Membrane potential (intra-cellular recording) of a single cell within catvisual cortex. Note the strong fluctuations around −65mV . If the membrane potentialexceeds a value of roughly−63mV , the generation of a spike, the prototypical responseof neurons, is likely. Here, spikes go up to roughly −40mV and last about 1ms. Thedata was recorded in the laboratory of Maxim Volgushev. (b) Quintessential stochasticresonance curve in the context of this thesis: A performance measure, which is relatedto information transmission is plotted versus the amount of present noise. The shape ofthese curves can roughly be explained as follows: An information transmission systemwill produce an output every time its internal state reaches some barrier. If the inputsignal is too weak to induce crossings, the system will be silent in the presence ofnoise. Weak noise will induce crossings, related to the input, but seldom. Strong noisewill induce frequent but more random transitions. At an intermediate noise level anoptimum exists.

2

described as threshold elements, which transmit information only when their mem-brane potential - driven by their inputs - crosses a certain threshold value. Input sig-nals, which are too weak to sufficiently increase the membrane potential, are lost. Oneway to amplify weak inputs, however, is to add noise with a proper variance to thesignal: Fluctuations of the membrane potential may lead to threshold crossings and tooutput spikes. Any modulation of the otherwise sub-threshold input signals then leadsto a modulation of the probability of generating these spikes and can be observed as achange in the neuron’s output rate. If the fluctuations are too strong, signal transmissiondeteriorates. These characteristic performance changes of information transmission incase of stochastic resonance are sketched in figure 1.1 (b).

The use of low intensity signals in a noisy context might be advantageous for bio-logical systems, since neural activity is costly in metabolic terms, and energy consump-tion and dissipation becomes a concern, for example for the densely packed centralnervous system of higher animals. The mammalian brain is a metabolically expen-sive tissue whose evolutionary development has been influenced by the availability ofmetabolic energy, as can be deduced from comparative studies, (Aiello & Wheeler,1995; Martin, 1996). The adult human brain accounts for roughly 20% of the restingenergy consumption (Rolfe & Brown, 1997) in children these costs can reach 50%.Energy consumption in grey matter is suggested to be due to transmission of electri-cal signals along axons and across synapses (Ames, 2000; Attwell & Laughlin, 2001).Several researchers have suggested that the overall energy consumption constrains in-formation transmission, and it has been argued that neurons try to achieve a balancebetween information transmission and energy consumption, leading to energy efficientcodes (Levy & Baxter, 1996; Laughlin et al., 1998; Balasubramanian et al., 2001;de Polavieja, 2002). As is concluded by Laughlin et al. (Laughlin et al., 1998) bio-logical neural systems prefer information transmission via many parallel low intensitychannels, compared to few high intensity ones, since metabolic cost grow super-linearwith the information rate. One can infer from the above studies that energy efficientcodes favor low spike rates and input distributions with mainly weak signals. Fromthese results one would expect that the strive for energy efficient codes creates condi-tions in which stochastic resonance is of tremendous relevance. Physiological studiesdo indeed support the idea of low average activities. Baddeley et al. (Baddeley et al.,1997) find an average firing rate of 4Hz in an ensemble of cortical neurons from anes-thetized cat and awake macaque monkeys. Olshausen et al. (Olshausen & Field, 2004)point out that typical intracellular measurements introduce a bias. The process of hunt-ing for neurons with a single micro-electrode will almost invariably steer one towardsneurons with higher firing rates. Based on statistical considerations Olshausen et al.note that the actual average firing rate might be much lower for cortical neurons.

However, serious concerns exist in the literature of whether stochastic resonanceis actually of functional importance. J. Tougaard (Tougaard, 2000) demonstrates in asignal detection framework, employing different kinds of abstract model neurons, thatthe improvement of detection through the addition of noise can never improve detectionbeyond that of a corresponding adaptive system. Taking the astonishing capabilities ofnervous systems to adapt to changing conditions seriously, this is indeed a noteworthycomment.

The study of stochastic resonance is well established in a neural context. Longtin,Bulsara and Moss (Longtin et al., 1991) were the first to explicitly relate stochasticresonance to neural behavior. They compared results from a very abstract model (a pe-riodically driven bistable system) to the interspike interval distribution of real neurons.Stochastic resonance has also experimentally been observed in neural systems. Most

3

studies are related to early sensory information transmission. Stochastic resonancehas been demonstrated in crayfish mechanoreceptors (Douglass et al., 1993), cricketcercal system (Levin & Miller, 1996), rat cutaneous receptors (Collins et al., 1996),human muscle spindles (Cordo et al., 1996), hair cells of the inner ear (Jaramillo &Wiesenfeld, 1998), and the human visual system (Simonotto et al., 1997). It has alsobeen shown, that stochastic resonance can be of behavioral significance. Russel et al.(Russell et al., 1999) have found that stochastic resonance enhances the normal feedingbehavior of paddlefish. They make use of passive electroreceptors to detect electric sig-nals from their prey (i.e. plankton). A noisy electric field improves the performance ofthe fish. The authors do also demonstrate that plankton (Daphnia) is a natural sourceof electric noise. Kitajo et al. (Kitajo et al., 2003) provide evidence that stochasticresonance within the human brain can enhance behavioral responses to weak sensoryinputs. They asked human subjects to adjust handgrip force to a slowly changing, sub-threshold gray level signal presented to one eye. Behavioral responses were optimizedby presenting randomly changing gray levels separately to the other eye. Their resultsindicate that the observation of stochastic resonance was mediated by neural activitywhere the information from both eyes converges.

In theoretical work, stationary (Stemmler, 1996), sinusoidal (Longtin, 1993), orgeneral continuous time-dependent inputs (Bulsara & Zador, 1996) have been consid-ered. Early theoretical work on stochastic resonance in neurons was based on two-stateneurons as in (Bulsara et al., 1991), more realistic models as the Fitz-Hugh Nagumoneuron relied on simulations (Longtin, 1993; Wiesenfeld et al., 1994), the analyticstudy of more realistic models like the integrate-and-fire neuron suffer from technicaldifficulties (Bulsara et al., 1994) and (Plesser & Tanaka, 1997). Studies on more com-plex and realistic neuron models rely on simulations as in Destexhe et al. (Destexhe &Pare, 1999). In their contribution they have employed a biophysically and anatomicallyhighly realistic and very detailed model neuron. They observed noise aided informationtransmission.

Whether an optimal noise level exists or not does not only depend on the system, thesignal and noise characteristics, but in addition to that also on the performance measurewhich quantifies information transmission. Several measures haven been employed sofar, among them: The power spectral density (Benzi et al., 1981), signal-to-noise ratio(McNamara et al., 1988), the correlation to the input signal (Collins et al., 1995a), themutual information (Levin & Miller, 1996) and the residence time distribution (Gam-maitoni et al., 1989). Several common measures are systematically compared in astudy from M. Rudolph et al. (Rudolph & Destexhe, 2001b) in a highly realistic andvery detailed model neuron. The mutual information is especially important, it mightbe considered as a benchmark measure for information transmission.

Stochastic resonance has been studied in many systems outside of neuroscience,like bistable electric circuits (Fauve & Heslot, 1983), lasers (McNamara et al., 1988)or level crossing detectors (Gingl et al., 1995). Originally the concept of stochasticresonance was introduced to explain the periodicity of the earth’s ice ages (Benzi et al.,1981; Benzi et al., 1982). Based on geophysical observations Benzi et al. (Benzi et al.,1981) suggested that minute periodic variations of the orbit of the earth might regularlyinduce ice ages by virtue of noise-induced transitions, an effect they called stochasticresonance. For a general review on stochastic resonance see (Gammaitoni et al., 1998),

Apparently we live in an ever-changing world. The environment changes, animalsmove within this world and thus change their perspective. Intrinsic properties change,e.g. while aging, or on a shorter timescale, changing the state of alertness. Manytheoretical and experimental studies suggest that nervous systems adapt their properties

4

to changing conditions and sensory inputs (Laughlin, 1981; Carandini & Ferster, 1997;Stemmler & Koch, 1999; Kim & Rieke, 2001; Brown & Masland, 2001; Adorjanet al., 2002). Adaptation of neural properties might thus allow reasonable behavioralperformance under changing conditions. The omnipresence of change suggests thatthe ability to adapt is a general property of nervous systems, especially for sensorysystems. Though studying the role of noise in neural coding is well established by now,only very few studies consider adaptation of the optimal noise level to changing inputintensities (Collins et al., 1995b; Wenning & Obermayer, 2003; Hoch et al., 2003b).In this thesis a special emphasis will be put on adaptation of the optimal noise level tochanging inputs. However, adaptation of the optimal noise level has been investigatedin a signal processing, but not a neuroscience related, setting before (Mitaim & Kosko,1998).

In real neural tissue neurons are organized in anatomical, physiological or compu-tational assemblies, called populations (Kandel et al., 2001). Optimal properties forneural information transmission do not only depend on the single neuron properties,but also on the properties of the population. Thus adaptation of the neural propertiesfor optimal information transmission might not work with single neuron adaptationrules. Corresponding investigations are presented and discussed in this thesis.

Many studies related to the role of noise in a neural context focus on the role ofnoise in information transmission. An analysis of plain information transmission is aprerequisite for understanding, how noise may influence - and improve - transmissionof information, after neural computation has been accomplished by dendritic integra-tion. Some studies, as in (Panzeri et al., 2003) or (L ubke et al., 1999), indicate thatoptimal information transmission itself can play a major role in natural neural systems.In (L ubke et al., 1999) it has been suggested that a major task of spiny neurons in thebarrel field in layer 4 of rat somatosensory cortex is to amplify the weak thalamic inputin order to transmit it to the different regions of the cortical column.

The detection of transient inputs might be considered as a basic neural computation.It is important for coincidence detection as well as for the detection of synchronousspiking events in neural systems. The presence and the properties of background noiseinfluence the ability of a neuron to detect transient inputs. Noise enhances the detectionof weak transient inputs, the phenomenon of stochastic resonance occurs. Experimen-tal findings (Destexhe et al., 2003; Destexhe et al., 2001) demonstrate, that colorednoise, rather than white noise, provides the best model for the background input. Onlylittle work has been devoted to understand the impact of these temporal correlations onneural information transmission and processing (Nozaki et al., 1999b; Capurro et al.,1998; Mato, 1998). The color of the noise has a significant influence on the ability ofneurons to detect transient inputs, as will be demonstrated in this thesis.

The response-stimulus correlation (RSC) is a function of the time-distance to aneuron’s observed response spike. It expresses the correlation of response and stimulus,as well as the expected number of stimulus spikes which have led to a response. TheRSC thus provides an important characterization of neural behavior in a stochasticsetting.

The general goal of the studies presented in this thesis is to elucidate noisy neuralinformation transmission and processing. In order to proceed in that direction I focuson the following aspects:

• The relation between neural information transmission, processing and modelproperties,

• adaptation to optimal noise levels,

5

• consequences of metabolic cost of neural activity for neural coding,

• and the interplay between the statistics of the input and neural spiking activity.

Scope

Mathematical descriptions of noise are introduced in chapter 2, together with theircorresponding neuron models employed in this thesis.

Quantifying information transmission is of major importance in studying neuralinformation transmission, the most relevant measure, the mutual information, is intro-duced in chapter 3.

Chapters 4 to 9 present original research, the chapters are written such that each oneof them can be read separately. Models and their mathematical descriptions, thoughintroduced in detail in chapter 2, will be introduced briefly in each chapter for betterreadability.

Chapter 4 focuses on the question: Is there a simple way for a single neuron tocalculate the proper noise level for a given ensemble of input signals and would it befeasible for a neuron to adjust the noise level if the input is changing? A hypotheticalprocedure is suggested and demonstrated in two abstract and one biophysically moreplausible model neuron.

In chapter 5 metabolic cost is related to neural activity and thus information trans-mission. By investigating the transfer properties of model neurons the relevance ofweak signals in neural information transmission is discussed. Optimal signal distribu-tions are calculated. It is found that metabolic cost has indeed a considerable effect onoptimal neural codes and that low intensity communication is highly relevant, thoughlow signal intensities imply a large unreliability. These results confirm the existingtrend in the literature.

In chapter 6 the optimal noise level for the transmission of low intensity signals isinvestigated in a simplistic model of a population. Implications for learning rules basedon single neuron properties are discussed. The work presented in this chapter is donein close collaboration with Thomas Hoch.

In chapter 7 it is investigated how the presence and properties of membrane po-tential fluctuations influence the ability of a neuron to detect transient inputs. In anabstract as well as in a biophysically more plausible model neuron it is demonstratedthat temporal correlations in the membrane potential fluctuations yield a more robustperformance in the detection of transient inputs. An approximate theory describes theresults.

In chapter 8 the neural response to coincident and temporally correlated inputs is in-vestigated. Both aspects of the input statistics have a significant influence on the mem-brane potential variance. An abstract model neuron is employed to study the effects ofcoincident and temporally correlated inputs. In the presented framework these can becompared on a common ground. From studying the dynamics of the neuronal responseto coincidences and temporal correlations it can be deduced, that temporal correlationshave an important modulating effect on the neuronal response, but that transient co-incidences are much more appropriate for neural coding than transient changes in thetemporal correlations. The main results are verified in a biophysically realistic modelneuron.

In chapter 9 an abstract model neuron is investigated, and the response-stimuluscorrelation is obtained analytically. Fits of the results of this analysis are compared

6

to simulations, and various involved measures as well as resulting consequences arediscussed. This has been done in close collaboration with Jacob Kanev.

A brief summary is provided in chapter 10. The appendix contains supplementarymaterial which did not really fit into the main body of text, as equations and parameters.

7

Chapter 2

Stochastic Neuron Models

A neuron will typically fire an action potential when its membrane potential reachesits threshold value. During the action potential, called spike, the membrane potentialfollows a rapid, more or less stereotyped trajectory and then returns to a value belowthe threshold potential. These mechanisms are well understood and can be modeled ac-curately. Neuron models can be simplified and simulations can be accelerated dramat-ically if the biophysical mechanism responsible for action potentials is not explicitlyincluded in the model. In this thesis different abstract and effective models of neuralbehavior are employed. Among them is the so called leaky integrate-and-fire (LIF)neuron model as the main workhorse. This important model class eases computationby neglecting the intricate dynamics of spike generation, as soon as the membrane po-tential reaches the threshold, a spike is supposed to occur. The membrane potential isreset to a certain value below threshold.

Various versions of the integrate-and-fire model describe neural behavior at dif-ferent levels of rigor. In the simplest version of these models, all active membraneconductances are ignored. The entire membrane conductance is modeled as a sin-gle passive leakage term. The integrate-and-fire model was originally proposed byLapicque (Lapicque, 1907) in 1907, long before the mechanism that generates actionpotentials has successfully been modeled by Hodgkin and Huxley (Hodgkin & Huxley,1952). Though simple, the integrate-and-fire model is an extremely useful descriptionof neural activity. For some deterministic and simple inputs the response of the LIFneuron can be solved analytically in simple closed form expressions. Stochastic in-puts impose a serious problem on an analytic treatment. Only for very few scenariosanalytic expressions are available, and these tend to be challenging in their numericalevaluation.

For investigating noisy neural information processing not only appropriate neuronmodels are necessary, but also models of the noisy synaptic input. A typical corticalpyramidal cell (Kandel et al., 2001) receives input from 103 to 105 different synapses.In a statistical description this can be considered as a huge amount of incoming parallelstochastic processes. If the individual spike train is not of interest the whole input canbe modeled as one single stochastic process, which captures the most essential featuresof the cumulative effect of many spike trains. A popular model of this kind is theWiener process, also known as Brownian motion. The Wiener process is the mostessential noise model employed for the studies in this thesis.

The notion of the inter-spike-interval (ISI) is introduced in section 2.1. Many ofthe subsequent sections rely on this concept.

8

One of the most fundamental stochastic processes and a simple model of a spiketrain is the Poisson process, in section 2.2. The Poisson process is introduced and asimple way to generate a Poisson spike train in a simulation is sketched.

The cumulative effect of many Poisson spike trains might be modeled by a Wienerprocess, this process is introduced and defined in section 2.3. The most relevant prop-erties are introduced as well. The time derivative of the Wiener process is sometimescalled Gaussian white noise.

The Ornstein-Uhlenbeck process, section 2.4, is basically a low-pass filtered Wienerprocess. If the Wiener process is taken as a model of the neuronal input, low-pass fil-tering can be considered as an abstract model of neural integration. Thus the Ornstein-Uhlenbeck process resembles the sub-threshold dynamics of a leaky passive integrator.Adding a threshold and reseting the process after a threshold crossing makes it a ver-satile stochastic version of the LIF neuron.

Many numerical schemes exist to integrate the equation of the stochastic LIF neu-ron. The most fundamental one is introduced in section 2.5, the Euler scheme.

There is no analytic closed form expression for calculating the ISI distribution of aLIF neuron with Gaussian white noise inputs, even in the case of constant mean input.Fortunately, in that case, the moments of the ISI distribution are analytically available.One known solution is expressed in terms of an infinite series expansion of the LaplaceTransformation, the other is expressed in terms of recursively related integrals. Bothapproaches are sketched in section 2.6.

For many practical purposes the moments of the spike count distribution are moreappropriate than the moments of the ISI distribution. Their relation is briefly explainedin section 2.7.

Within the framework of the stochastic LIF neuron one can differ between additiveand multiplicative noise. These concepts are explained in section 2.8. However, thestudies presented in this thesis employ additive noise only.

Synaptic background activity in neural tissue is temporally correlated (”colored”).Low-pass filtering a Wiener process causes temporal dependencies in the resultingstochastic process. Such a process realizes a special version of colored noise. In sec-tion 2.9 the Onrstein-Uhlenbeck process is introduced as colored noise input to a LIFneuron.

The LIF neuron with additive noise maps stochastic inputs to stochastic outputs,whereas the moments of the ISI distribution depend in a deterministic fashion on thestatistical properties of the input. In section 2.10 a simplistic approach of such a map-ping is introduced, though this model can only be considered as a qualitative model ofthe LIF neuron it is a valuable tool to build up some intuition. Using this model, it iseasy to gain some analytic insights.

The leaky integrate-and-fire neuron is widely used because of its simplicity. How-ever, this model does not account for two properties of real neurons which may turnout to be important in the context of this thesis. Firstly, it does not include the changesin the membrane conductance caused by the synaptic input. Secondly, the membranepotential is reset to a certain fixed value after each spike, an assumption which may notcapture the actual changes in the membrane potential. The simplest model which ac-counts for the above mentioned phenomena is a single compartment Hodgkin-Huxleymodel, see section 2.11. A model for fluctuating synaptic inputs is introduced as well.

9

2.1 The Inter-Spike Interval

The notion of an inter-spike-interval (ISI) is best introduced by a definition, as in (Tuck-well, 1998):

Definition:

Let tk,k = 0,1,2, . . .be a sequence of times at which a neuron emits action potentials,with t0 = 0 and t0 < t1 < t2 < .. . . The kth interspike interval (ISI) is

Tk = tk− tk−1, k = 1,2, . . . . (2.1)

A possible way to describe the response properties of a neuron is the ISI distribution.Later in this chapter, in section 2.6, explicit expressions for the moments of the ISIdistribution of a LIF neuron with stationary stochastic input are given.

2.2 Poisson Process

The Poisson process is a versatile stochastic process. In the context of this thesis it isimportant for several reasons:

1. It is much studied process and can be used as a standard against which otherprocesses can be compared.

2. It is a useful approximation for the inputs to a neuron when these are many andnot synchronized.

3. It is easy to generate.

A Poisson process is defined as follows, see (Tuckwell, 1998) :

N(t), t ≤ 0 is a simple Poisson process with intensity or mean rate λ if:(a) N(0) = 0(b) given any 0 = t0 < t1 < t2 < · · ·< tn−1 < tn, the random variables N(tk)−N(tk−1),k =1,2, . . . ,n, are mutually independent; and(c) for any 0 ≤ t1 < t2, N(t2)−N(t1) is a Poisson random variable with probabilitydistribution

PN(t2)−N(t1) = k=(λ (t2− t1))ke−λ (t2−t1)

k!, k = 0,1,2, . . . (2.2)

The rate λ corresponds to the number of events per unit time. Property (a) is thestarting condition, (b) tells us that the increments are independent and (c) says thatthese increments are stationary (only time differences matter) and Poisson distributed.N(t), t = t2− t1 is a Poisson random variable with mean and variance equal to λ t.Setting t1 = t and t2 = t +∆t yields

10

PN(t +∆t)−N(t) = k =(λ∆t)ke−λ ∆t

k!,

= 1−λ∆t +o(∆t), k = 0,

= λ∆t +o(∆t), k = 1,

= o(∆t), k = 2,

(2.3)

where o(∆t) means terms that, as ∆t→ 0, approach zero faster than ∆t itself. Thus forsmall ∆t, the process is most likely to stay unchanged (k = 0) or undergo an increaseof unity (k = 1). Actually this Taylor expansion is a recipe for numerically generatinga Poisson spike train. The time has to be divided in small bins ∆t, such that the prob-ability of having two events within ∆t is negligible. For each bin a random numberbetween zero and one is drawn, if this number is smaller then λ∆t an event occurs inthis bin, otherwise not.

2.3 Wiener Process

The Wiener process, or Brownian motion, belongs to the class of stochastic processescalled diffusion processes. Diffusion processes have trajectories that are continuousfunctions of time t, in distinction to a random walk whose sample path are discontinu-ous (Gardiner, 1985). Among the reasons for studying the Wiener process as a modelfor stochastic neural input are:

1. It is a well studied process and many of the relevant mathematical aspects areknown.

2. Based on the Wiener process more realistic models of neuron activity can beconstructed.

As in (Tuckwell, 1998) the Wiener process is introduced here as a limiting case of aPoisson process. Consider a process defined by

Wa(t) = a[Ne(t)−Ni(t)], ≥ 0, (2.4)

where a is a constant, and Ne and Ni are excitatory (positive weight) and inhibitory(negative weight) independent Poisson processes with mean rates λe = λi = λ . Theprocess Wa displays jumps of magnitude a. It applies

E[Wa(t)] = a[λe−λi]t = 0

Var[Wa(t)] = a2[Var[Ne(t)]+Var[Ni(t)]] = 2a2λ t.

(2.5)

E[.] and Var[.] denote the expectation and variance. Assume λe = λi, as a→ 0, thesequence of random variables Wa(t) converges in distribution to a normal random vari-able with mean zero and variance t. Let the limiting variable be W (t)

W (t) = lima→0

Wa(t). (2.6)

11

The process W (t), t ≥ 0 is called a standard Wiener process, an exact definition ofthe Wiener process is the following one:

(a) W (0) = 0(b) given any 0≤ t0 < t1 < t2 < · · ·< tn−1 < tn, the random variables W (t)−W (tk−1), k =1,2, . . . ,n, are independent(c) for any 0≤ t1 < t2,W (t2)−W(t1) is a normal random variable with mean zero andvariance t2− t1.

Wa(t) satisfies (a)-(c) asymptotically as a→ 0. The density of W (t) is

fW (x, t) =1√2πt

ex22t . (2.7)

The paths of W are smooth enough to be continuous, their derivative in time, calledwhite noise, is not. Whenever white noise dW(t)

dt appears in an equation, an integrationis implied. New processes may be constructed from W by multiplying it by a constantand adding a linear drift. Thus

X(t) = x0 +σW(t)+ µt, t > 0, (2.8)

where X(0) = x0, defines a Wiener process with variance parameter σ and drift pa-rameter µ . Linear operations on Gaussian processes, such as W , produce Gaussianprocesses. Since

E[X(t)] = x0 + µt

Var[X(t)] = σ 2t, (2.9)

the density of X(t) is

fX (x, t) =1√

2π σ 2te− (x−x0−µt)2

2σ2t , −∞ < x < ∞, t > 0. (2.10)

2.4 The Leaky-Integrate-and-Fire (LIF) Neuron

Consider many Poisson spike trains, Ne excitatory with rate λe and Ni inhibitory withrate λi. The corresponding diffusion approximation yields a Wiener process with drift,and with an infinesitesimal mean

µ = ∑e

weλe−∑i

wiλi, we, wi > 0, (2.11)

and infinitesimal variance

σ2 = ∑e

w2e λe +∑

iw2

i λi, (2.12)

see (Tuckwell, 1998). The differential equation for the leaky integrator with the aboveinputs is

dV (t) = (−V (t)τ

+ µ)dt +σdW (t), (2.13)

12

which is that of an Ornstein-Uhlenbeck process (OUP). Mean and variance of V (t) inthe absence of a threshold are given below:

E[V (t)] = µτ +(V (t = 0)−µτ)exp(−t/τ) (2.14)

Var[V (t)] =σ2τ

2(1− exp(−2t/τ)) (2.15)

The above equation, eq. (2.13), describes the sub-threshold dynamics of a LIF neuronstarting from V at t = 0. To model the spiking activity of a neuron, a threshold conditionhas to be imposed. The firing of an action potential is identified with the first crossingof Vt through a voltage threshold θ . The time origin is the moment of the last firing.At these moments, the membrane potential is repeatedly reset to its initial value, calledthe reset potential. For simplicity it is chosen V (t = 0) = 0. The reset of V (t) afterfiring introduces a strong nonlinearity into the model.

Mean and variance of the input describe completely the behavior of the membranedepolarization in the above framework because the OUP is Gaussian and at steady stateit is normal distributed V (t = ∞)∼ N(µτ , σ 2τ

2 ).The mean defines sub- and supra-threshold stimulation of the model neuron. If

E[V (t = ∞)] = µτ < θ , then the stimulation is called sub-threshold and in the absenceof noise the neuron never fires. If the stimulation is such that µτ > θ , the stimulationis called supra-threshold and the neuron fires even in the absence of noise.

2.5 Euler Integration of the Ornstein-Uhlenbeck Pro-cess

Different numerical schemes exist for the integration of the Ornstein-Uhlenbeck pro-cess, see e.g. (Gillespie, 1996). The so called Euler scheme is an approximate method,easy to implement and very reliable. As the width of the time step approaches zero,the approximation is more and more exact. The infinitesimal increment of the Wienerprocess dW can be approximated as N(t)

√dt within a short time bin dt. Thus equation

(2.13) becomes

V (t +dt) = V (t)+ µdt− 1τ

V (t)dt +σN(t)√

dt. (2.16)

2.6 Moments of the Firing Time

The moments of the ISI distribution characterize the response of the model neuron to itsinputs. The LIF neuron from section 2.4 is considered. For supra-threshold stimulationand σ = 0 (deterministic model), the length of the ISI is unique and can easily becalculated from eq. (2.13) by searching for the time when the mean depolarizationreaches the threshold θ . The solution for the mean ISI is

µISI = τ ln(µτ

µτ−θ), µτ > θ . (2.17)

For µτ < θ no spike occurs. Determining the ISI distribution of a model neuron withstochastic input is closely related to the so called first-passage-time (FPT) problem.In FPT like problems the time it takes from a starting point within an interval to the

13

boundary of this interval is of interest. Calculating the ISI distribution of a LIF neuronwith additive white noise is equivalent to calculating the FPT distribution of an OUPwith upper absorbing boundary (Gardiner, 1985), i.e. the LIF neuron with stochasticinput as introduced in the previous section.

For σ > 0 no closed form expression exists for the OUP with upper absorbingboundary. However, exact expressions for the moments of the FPT distribution exist(Ricciardi & Sacerdote, 1979) if the input statistics are stationary. Computing thesemoments, although straightforward, is a considerable effort.

Two different types of expressions are presented, a series expansion in subsection2.6.1 and an integral expression for the moments of the FPT distribution in subsection2.6.2.

2.6.1 Series Expansion

In this section a sketch of the derivation of the moments of the ISI distribution is given.It basically follows the work of Ricciardi et al. (Ricciardi & Sacerdote, 1979) and itspresentation in a paper from M. Stemmler (Stemmler, 1996).

The random noise fluctuations will lead to a probability distribution of voltages Vt

at time t for a LIF neuron that was initially at Vi. This probability distribution is givenby a Fokker-Planck equation associated with eq. (2.13)

τ∂P(V, t,Vi,0)

∂ t=

σ2(t)2

∂ 2P(V, t,Vi,0)

∂V 2 +∂

∂V((V −µ(t))P(V, t)), (2.18)

which describes the evolution of the probability that the voltage is V at time t, given thatthe voltage was Vi at time t = 0. The two terms on the r.h.s. of eq. (2.18) correspond to adiffusion term with diffusion constant σ , and a drift term with mean µ . The probabilityof first crossing the threshold θ in the time [t, t +dt), starting from Vi, is

F(Vi, t) =ddt

[1−

∫ θ

−∞P(V, t;Vi,0)dV

]. (2.19)

The above expression satisfies the adjoint Fokker-Planck equation

∂F(Vi, t)∂ t

=− Vi

τ+ µ

∂F∂V

+12

σ2 ∂ 2F

∂V 2i

. (2.20)

In terms of the Laplace transform, F(Vi,s) =∫ ∞

0 exp(−st)F(Vi, t)dt, the solution is

F(Vi,s) =U(

sτ2 , 1

2 , (µτ−Vi)2

σ 2τ

)

U(

sτ2 , 1

2 , (µτ−θ)2

σ 2τ

) , (2.21)

where U(a,b,z) are confluent hypergeometric functions of the second kind and thesolution has been constrained to satisfy the following boundary conditions

F(θ ,s) = 1, limVi→−∞

F(Vi,s) = 0. (2.22)

Unfortunately, inverting the Laplace transform, eq. (2.21), in an analytic fashion hasnot been done yet. However, the mean and variance of the first passage time can bedirectly derived from the Laplace transform (higher moments as well):

14

µFPT =−dF(Vi,s)ds

|s=0, σ2FPT =

d2F(Vi,s)ds2 |s=0−

(dF(Vi,s)ds

|s=0

)2. (2.23)

Evaluating the above expressions yields the first moment of the FPT distribution µFPT :

µFPT =τ2

∞

∑n=1

x2n

n( 12)n−2√

πx∞

∑n=0

( 12 )nx2n

( 32)nn!

|µτ−θ√

σ2τµτ−V0√

σ2τ

, (2.24)

and for the second moment

σ2FPT =

τ2

4

2(γ +2ln(2))∞

∑n=1

x2n

n( 12 )n

+2∞

∑n=2

(n−1

∑m=1

1m

)x2n

n( 12 )n

−2√

πx[2γ∞

∑n=1

( 12 )nx2n

( 32 )nn!

−2ex2∞

∑n=1

(n

∑m=1

1m

)(−x)2n

n( 32)n

]

−[∞

∑n=1

x2n

n( 12 )n−2√

πx∞

∑n=0

12 )nx2n

32)nn!

]

|µτ−V0√

σ2τµτ−θ√

σ2τ

, (2.25)

where V0 is the reset potential and γ is the Euler-Masheroni constant, defined as

γ = limn→∞

(n

∑k=1

1k− ln(n)), (2.26)

its numerical value is approximately 0.5772 . . . . The above formulas use the Pochham-mer notation:

(a)n = a(a+1)(a+2) . . .(a+n−1), n≥ 1

(a)0 = 1. (2.27)

2.6.2 Integral Expressions

As in the previous section the starting point is the Fokker-Planck equation, eq. (2.18).This introduction follows (Tuckwell, 1998) and (Brunel, 2000). Equation (2.18) can berewritten as the continuity equation

∂P(V, t)∂ t

=∂S(V, t)

∂V, (2.28)

where S(V, t) is the probability current through V at time t (Risken, 1984):

S(V, t) =−σ2(t)2τ

∂P(V, t)∂V

− (V −µ(t))τ

P(V, t), (2.29)

The boundary conditions have to be specified at V = −∞, the reset potential V0, andthe threshold θ . The probability current through θ gives the instantaneous firing rateat t, f (t) = S(θ , t). The membrane potential density must vanish at θ , the correspond-ing boundary condition is P(θ , t) = 0. Inserting P(θ , t) = 0 in eq. (2.28) yields theboundary condition for the derivative of P at θ :

15

∂P(θ , t)∂V

=−2 f (t)τσ2(t)

. (2.30)

At the reset potential V = V0, P(V, t) there is an additional probability current, due toneurons that just crossed the threshold and are reset. Thus, the difference between theprobability currents above and below the reset potential at time t must be proportionalto the fraction of cells firing at t. This is expressed by S(V +

0 , t)−S(V−0 , t) = f (t), whichyields the following derivative discontinuity

∂P(V+0 , t)

∂V− ∂P(V−0 , t)

∂V=

2 f (t)τσ2(t)

, (2.31)

As a boundary condition for V =−∞ it is chosen that P should tend sufficiently quicklytowards zero to be integrable,

limV→−∞

P(V, t) = 0, limV→−∞

V P(V, t) = 0. (2.32)

Since P(V, t) is a probability distribution, it should satisfy the normalization condition

∫ θ

−∞P(V, t)dV = 1. (2.33)

In a stationary solution, P(V, t) = P0(V ), the time independent solution of eq. (2.18)satisfying the above boundary conditions is given by

P0(V ) = 2f0τσ

exp(− (V −µ)2

σ2 )∫ θ−µ

σ

V−µσ

Θ(u−V0)eu2

du, (2.34)

in which Θ(x) denotes the Heaviside function, Θ(x) = 1, for x > 0 and Θ(x) = 0 oth-erwise.

The first moment µFPT = 1/ f of the FPT distribution is then:

µFPT = 2τ∫ θ−µ

σ

V0−µσ

dueu2∫ u

∞dvev2

= τ√

π∫ θ−µ

σ

V0−µσ

dueu2(1+ er f (u)), (2.35)

where er f is the error function. Note that the above expression is equivalent to theanalytical expression for the mean first passage time of a LIF neuron with randomGaussian inputs (Sato, 1978). Equation (2.35) has been obtained by Amit and Brunel(Amit & Brunel, 1997).

The higher moments µk of the inter-spike intervals can then be computed be arecurrence relation, see (Tuckwell, 1998):

σ2

2d2µk

dx2 +(µ− x)dµk

dx=−kµk−1. (2.36)

From eq. (2.35) the first moment of the stationary FPT distribution is known. Thecomputation of the stationary second moment yields

σ2FPT = µ2

FPT +2π∫ θ−µ

σ

V0−µσ

dx ex2∫ x

−∞dy ey2

(1+ er f (y))2. (2.37)

16

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

µ µ

f in

Hz

σN(a) (b)

0 1 0 1

80

10

0.4

1

Figure 2.1: (a) Output rate f [Hz] as a function of µ (fI-curve) for increasing values ofσ = 0.0,0.1,0.2,0.3,0.4,0.5,0.6. µ and σ are made dimensionless by multiplying µwith τ

θ and σ with√

τθ , θ = 30mV , τ = 20ms (b) Second moment of the spike count

distribution, σN . Same parameters as in (a).

2.7 First Passage Time (FPT) and Spike Count

If the mean input µ is constant and the additive noise is uncorrelated (in time), thenthe spike intervals are statistically uncorrelated. Thus, the mean µN and variance σ 2

Nof the spike count distribution are completely determined by the moments of the ISIdistribution. A spike train from a LIF neuron is observed for a time interval T . Astandard result from statistical renewal theory (Cox & Lewis, 1966), states that, in theasymptotic limit, the mean and variance of the spike count become

µN =1

µFPTT, (2.38)

σN =σ2

FPT

µ3FPT

T, (2.39)

where µFPT is the mean ISI (FPT) and σFPT is the standard deviation of the ISI dis-tribution. The first equation is strictly true under the equilibrium assumption, in whichthe start of the process occurred long before the period of observation. Both expres-sions are exact if the inter-spike interval probability distribution is exponential, whichwill approximately be true if the mean inter-spike interval, µFPT , is much longer thanthe membrane time constant τ .

2.8 Multiplicative Noise

In most scenarios related to stochastic resonance, the source of noise is assumed tobe additive and independent of the input signal. Nevertheless, input signals might beirregular as well. In a neural coding context this is obvious. E.g. Ns ”signal” Poissonspike trains with rate λs and weight ws contribute with their own mean and varianceto the statistics of the input. The ”noise” processes have rate λn and weight wn. Meaninput is

17

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t in ms

V in

mV

0 1000 2000-10

0

10

0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

(a) (b)

µ t in ms

f in

Hz

V in

mV

10 1000 200

10

40

- 10

10

Figure 2.2: (a) Output rate f [Hz] as a function of µ (fI-curve) for increasing values ofσ = 0.0,0.1,0.2,0.3,0.4,0.5,0.6. µ and σ are made dimensionless by multiplying µwith τ

θ and σ with√

τθ . Solid lines: fI-curves according to eq. (2.24). Dashed lines:

f-I curve including signal dependent noise, σ 2 = w2s λs + w2

nλn = ws µ + w2nλn with

ws = 0.02, as in (Lansky & Sacerdote, 2001). µ = 1 corresponds to the threshold, τ =20 ms, θ = 30 mV . (b) Trace of the membrane potential of a LIF neuron with colorednoise input with the two different time constants τX = 0.1ms (left) and τX = 10ms(right). The diffusion coefficient D was adjusted to keep the variance of the membranepotential constant, D = 320 mV2

ms (left) and D = 3.2 mV 2

ms (right). Reset potential V0 = 0.Threshold is at 20mV.

µ = ∑s

wsλs, (2.40)

and the total variance of all incoming spike trains is then

σ2 = ∑s

w2s λs +∑

nw2

nλn. (2.41)

In all chapters of this thesis, the variance is chosen to be determined be the noise inputsonly, i.e. ∑s w2

s λs ∑n w2nλn. Figure 2.2 displays the effect of multiplicative noise

in a LIF neuron, as presented in the previous section. Parameters were chosen as in(Lansky & Sacerdote, 2001). Multiplicative noise basically increases and modulatesthe effect of additive noise.

2.9 Colored Noise

In section 2.4 the LIF neuron with additive noise is introduced. The noise inputs aremodeled as Brownian motion, the dynamics of the sub-threshold membrane potentialcorrespond to the dynamics of an OUP. Experimental and modeling studies (Destexheet al., 2003) suggest that the synaptic activity has a significant temporal structure. Des-texhe et al. (Destexhe et al., 2001) suggest to model the effect of the temporal cor-relations by a low-pass filtered Brownian motion, i.e. an OUP. More details on thisapproach in a biophysically more realistic setting will be given in section 2.11. Ac-cording to this approach the noise input Xt is given by an Ornstein-Uhlenbeck process

18

0 0.5 10

0.2

0.4

0.6

0.8

Err

or

Exponential kl.MSE

0 0.5 10

0.2

0.4

0.6

0.8

Err

or

Exponential Ew

0 0.5 10

0.2

0.4

0.6

0.8

Err

or

Exponential approx.

0 0.5 10

0.05

0.1

0.15

0.2

Err

or

Weibull Ew/Var

0 0.5 10

0.02

0.04

0.06

Err

or

Weibull kl.MSE

0 0.5 10

0.05

0.1

Abstand zum Threshold

Err

or

Gamma Ew/Var

0 0.5 10

0.02

0.04

0.06

Abstand zum Threshold

Err

or

Gamma kl.MSE

0 0.5 10

0.5

1

1.5

2

Err

or

Arrhenius−Modell

Var=0.81Var=0.49Var=0.25Var=0.09

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1pr

obab

ility

of

inst

ant f

irin

g b)

subthreshold potential ( = 1)θ

σ

(a) (b)

µ 1 – V0

hArr EISI

0 1 0 10.5

1

0.41

2

Figure 2.3: (a) Arrhenius type hazard function, eq. (2.45), as a function of the averagemembrane potential V0. Dimensionless units are employed, as in figure 2.1. The noiselevel σ = 0.1,0.2,0.3,0.4,0.5 is increasing from bottom to top. Threshold is at 1.(b) Integrated quadratic error, eq. (2.46), between an ISI distribution calculated froman OUP with upper absorbing boundary, ISIOUN , and an ISI distribution according toeq. (2.44), using an Arrhenius type Hazard function, eq. (2.45), ISIH . The effectivedistance, 1−V0, between threshold and the average membrane potential is given on thex-axis. Constant current injections are employed. Dashed-dotted line: σ = 0.1. Dottedline: σ = 0.2. Dashed line: σ = 0.3, Solid line: σ = 0.4.

with a time-constant τX and a diffusion coefficient D, dWt are the infinitesimal incre-ments of the Wiener process (see (Tuckwell, 1998) for an introduction)

dX(t)dt

=− 1τX

X(t)+√

DdW (t)

dt. (2.42)

The above noise process then serves as an input to a leaky integrator

dV (t)dt

=− 1τV

V (t)+X(t). (2.43)

Changing the time-constant τX (the ’color’) has two effects on the membrane potential,it changes the variance of the membrane potential and it changes the temporal cor-relations of the membrane potential. Figure 2.2 (b) displays the membrane potentialaccording to eqns. (2.42, 2.43). On the left a small time constant τX is employed, onthe right a large one. The diffusion coefficient D has been adjusted to keep the mem-brane potential variance constant. Stationary moments (the first two) of the membranepotential of a LIF neuron with colored noise input are given in chapter 7. The dynam-ics of these moments are required in chapter 8 and the corresponding expressions aregiven in the appendix 11.8.

2.10 Hazard Function Approximation of the LIF Neu-ron

A simple, intuitive and qualitative solution to the FPT problem of the LIF neuron canbe based on the idea of an escape probability, see (Plesser & Gerstner, 2000). At eachmoment in time the neuron may fire with an instantaneous rate h, which depends on themomentary distance between the average (deterministic) membrane potential and the

19

threshold and on the variance of the membrane potential fluctuations. The firing time ofthe last output spike is denoted as t∗. For a deterministic input I(t ′) for t∗ < t ′ < T themembrane potential V (t) (t > t∗) can be found be a standard Riemann integration. Thenoise free (average) membrane potential is denoted as V0(T |t∗, I(.)). For dealing withthe stochasticity of the input, Plesser et al., see (Plesser & Gerstner, 2000), introduce ahazard function h(T |t∗, I(.)) that describes the risk of escape across the threshold anddepends on the last firing time t∗ and on the input I(t ′) for t∗ ≤ t ′ ≤ t∗+ T . Once thehazard function is given, the ISI distribution is given by

ρ(T |t∗, I(.)) = h(T |t∗, I(.))e−∫ T

0 h(s|t∗,I(.))ds, (2.44)

see either (Cox & Lewis, 1966) or (Plesser & Gerstner, 2000). The exponential termaccounts for the probability that the neuron survives from t∗ to t∗+ T without firing.The factor h(T |t∗, I(.)) gives the rate of firing at t∗+ T , provided that the neuron hasnot fired thus far. In (Plesser & Gerstner, 2000) Plesser et al. discuss four models ofsuch simplified dynamics, all of them aim to approximate the diffusion model by anescape noise ansatz. The various models differ in their choice of the hazard functionh(τ |t∗, I(.)). In chapter 4 of this thesis the Arrhenius type Hazard function will be used.It is the most intuitive one and allows simple analytic considerations

hArr(T |t∗) = e− (1−V0(T |t∗))2

σ2 . (2.45)

Here, 1−V0(τ |t∗) is the voltage gap between threshold and the average membranepotential that needs to be bridged to initiate a spike. The stochasticity of the membranepotential is reflected in the parameter σ . Figure 2.3 (a) shows hArr for different noiselevels σ . Note that for strongly super-threshold stimuli (V0 1) the hazard functionvanishes exponentially. This is of little concern as long as the input I(t) contains noδ -pulses and V0 reaches the threshold only along continuous trajectories. Then, super-threshold levels of the potential are accessible only via periods of maximum hazard, sothat the neuron will usually have fired before V0 becomes significantly super-threshold.In (Plesser & Gerstner, 2000) Plesser et al. test the hazard ansatz, using different hazardfunctions, for a special choice of time-dependent stimuli. They find that the approachcan be a useful qualitative model. In chapter 4 constant inputs are of special interest.For this class of inputs the quality of the approximation eq. (2.45) has been tested (byThomas Hoch in a student project). The ISI distribution is determined according to eq.(2.44), ISIH , and via numerical evaluation of eq. (2.16) with upper absorbing boundary,ISIOUN . The integrated quadratic deviation between both ISI distributions is the errormeasure

EISI =

∫ ∞0 dT ′(ISIOUN(T ′)− ISIH(T ′))2

∫ ∞0 dT ′ISI2

OUN(T ′). (2.46)

The total error is normalized by the integral over the squared ISIOUN . Figure 2.3 (b)displays EISI as a function of 1−V0(τ |t∗), the voltage gap between threshold and theaverage membrane potential. The error is large near the threshold and smaller for largedistances to the threshold. For small membrane potential fluctuations the error is large,for increasing variances it decreases. As a consequence the Arrhenius ansatz might beemployed for qualitative investigations and intuition building w.r.t. the behavior of aLIF neuron. The ansatz is not appropriate for a quantitative analysis. Nevertheless itdoes prove to be useful in chapter 4.

20

0 200 400 600 800 1000 1200 1400 1600 1800 2000−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

(a)

time in ms

V in

mV

0 2000 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

45

(b)

mean input

f in

Hz

100 5 10 15 20 25 30 35 40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c)

mean input

σN

0 1

45

10

0.8

0.2

1.6 1.6

Figure 2.4: (a) Voltage trace of a HH type model neuron with fluctuating conductanceinputs, eqns. (2.52, 2.54). Parameters are the default parameters given in the appendix(section 11.4 and 11.5), except for the standard deviations of the synaptic inputs, σe,σi,which are increased by a factor a four, for demonstration purposes. (b) Firing rate ofthe HH type model presented in section 2.11.2 as a function of constant current forfive different levels of noise. Different noise conditions are modeled as a change of theconductances by a (gain) factor α , α× (ge0,gi0), and by a corresponding change in thediffusion coefficients α× (

√De0,√

Di0), α = 0.0,0.075,0.125,0.175,0.225. The val-ues of ge0,gi0,De0,Di0 correspond to the standart values given in the appendix (section11.5). The current threshold is denoted as 1 and corresponds to 0.5 nA. (c) Standarddeviation of the spike count distribution. Same model and parameters as in (b).

2.11 Hodgkin-Huxley (HH) Model Neurons

In subsection 2.11.1 a biophysical realistic model of membrane potential dynamicsand spike generation is introduced. Since A.L. Hodgkin and A.F. Huxley (Hodgkin& Huxley, 1952) have introduced this model class in 1952 it has proven to be highlysuccessful in many studies. In subsection 2.11.2, a special realization of a Hodgkin-Huxley type model and a recent model of synaptic input is introduced. The model issuggested by A. Destexhe and M. Rudolph et al. (Destexhe et al., 2001) and it modelseffectively the conductance fluctuations due to many synaptic events in parallel.

2.11.1 Action Potentials and HH Models

A classic and important result of modern neurobiology is the analysis by Hodgkin andHuxley (Hodgkin & Huxley, 1952) of the initiation and propagation of the action po-tential in the squid giant axon. Their description accounted for two ionic currents, asodium current INa and a potassium current IK . Analyzing action potential generationin terms of ionic currents has proven to be highly successful. A well written intro-duction to Hodgkin-Huxley type models can be found in (Dayan & Abbott, 2001) or(Koch & Segev, 1998). Since the landmark paper by Hodgkin and Huxley in 1952(Hodgkin & Huxley, 1952), numerous ionic membrane currents have been described.These differ in ion type, voltage and time dependence, dependence on internal states,and susceptibility to modulation by synaptic input and second messengers. Here thegeneral methodology used to describe ionic currents is briefly reviewed, focusing onthe representative fast sodium current. It is assumed that all ionic current flow occursthrough channels and that the instantaneous current-voltage relation is linear (Ohm’sLaw). The ionic current, I(t), is then related to the voltage across the membrane, V (t):

I(t) = g(V, t)(V(t)−E), (2.47)

21

where E is the Nernst potential (also known as reversal potential) for the ionic currentunder study and g(V, t) is the conductance associated with the channel. Conductancesdepend on time and on the membrane potential, additional dependencies as on chemicalmediators are possible. It is assumed that all ionic movements across the membrane isvia channels permeable to a single ionic species and having two states, open or closed.

The total conductance of a particular population of ion-channels can be expressedas the maximal conductance of the particular membrane patch under investigation, g.This is the conductance of a single channel in the open state times the channel densitytimes the area of the membrane patch, times the fraction of all channels that are open.This fraction is determined by hypothetical activation and inactivation variables m andh, raise to some integer power. Thus the total time and voltage dependent conductanceis modeled as:

g(V, t) = g m(V, t)ih(V, t) j, (2.48)

where i and j are positive integers. The dynamics of the variables m and h obey first-order kinetics of the form

m(V, t)dt

=m(V,∞)−m(V, t)

τm(V,t)V

, (2.49)

where the steady-state value of m, m(∞) and the time constant, τm, are functions ofthe membrane potential V . Usually m(∞) and τm are expressed in terms of rate con-stants, αm and βm, that can be thought of as the forward and backward rates governingthe transition of the channel between hypothetical open and closed states. The ratesthemselves depend on the potential across the membrane in a well specified manner.The rate constants are usually determined experimentally on the basis of voltage clampexperiments.

Patch clamp recording techniques have revealed ionic channels to be all-or-nothingpores that behave very much in a probabilistic manner (Sakmann & Neher, 1983). In aprobabilistic interpretation the macroscopic activation (also known as gating particle),m(t), is equivalent to the open-channel probability. If the channel is closed, the proba-bility that the channel gate will remain closed for a time T , is eαT , and if it is open theprobability that the channel will remain open is eβ T . If several hundred or more chan-nels are simulated consistency between the macroscopic and the microscopic domaincan be demonstrated (Strassberg & DeFelice, 1993). In other words, fluctuation areaveraged out. For this reason stochastic channel opening and closing events are not amajor source of membrane potential fluctuations. Spike generation is best sketched in amembrane patch with leak, sodium and potassium conductances only. The membranecurrent is then given by:

Im = IL + INa + IK . (2.50)

The sodium and potassium currents are determined by their activation m(t) (sodium),n(t) (potassium), and inactivation variables h(t) (sodium only).

INa = gNam3h(V −ENa),

IK = gKdn4(V −EK).

(2.51)

22

An initial rise of the membrane potential, e.g. due to the integration of a positivecurrent, causes the Na+ conductance activation m to jump suddenly from nearly 0 toa value near 1. This causes a large influx of Na+ ions. The inward current pulsecauses the membrane potential to rise rapidly to higher membrane potentials, towardsthe Na+ reversal potential. This rapid increase is due to a positive feedback effect,since depolarization of the membrane potential causes m to increase, and the resultingactivation of the Na+ conductance makes V increase (towards depolarization). Assoon as the membrane potential experiences a significant depolarization, the rise in themembrane potential causes the Na+ conductance to inactivate by driving h towards 0.This shuts off the Na+ current. In addition, the rise in V activates the potassium K+

conductance by driving n toward 1. This increases the K+ current, which drives themembrane potential back down to hyperpolarized values. The final recovery involvesthe readjustment of m,h and n to their initial values.

2.11.2 Point-Conductance Model of in Vivo Activity

In this section a special realization of a Hodgkin-Huxley type point neuron is intro-duced. The membrane potential V changes in time according to the differential equa-tion

Cm∂V∂ t

= −gL(V −EL)− INa− IK− IM− Isyn. (2.52)

The left hand side of this equation describes the influence of the membrane’s capaci-tance Cm, while all ionic currents through the cell’s membrane, including the synapticnoise (Isyn) and the synaptic noise are summed on the right hand side. Cm is the specificmembrane capacitance, gL is the leak conductance density, and EL is the leak reversalpotential. Following (Destexhe et al., 2001) the following intrinsic currents in additionto the leak current are considered: INa is the voltage-dependent Na+ current and IKd isthe ’delayed-rectifier’ K+ current. IM is a non-inactivating K+ current, responsible forspike frequency adaptation.

INa = gNam3h(V −ENa),

IKd = gKdn4(V −EK),

IM = gM p(V −EK).

(2.53)

Details of the intrinsic currents and model parameters are listed in the appendix 11.4.They were chosen according to (Destexhe et al., 2001; Destexhe & Pare, 1999) andare consistent with available experimental evidence in cat cortical pyramidal neurons.The total synaptic noise current Isyn is generated by fluctuating synaptic conductances.These conductances are thought to be induced by stochastic spike trains which arriveat the excitatory (e) and the inhibitory (i) synapses of the neuron. Following (Destexheet al., 2001) the synaptic currents are modeled as

Isyn = ge(t)(V (t)−Ee) + gi(t)(V (t)−Ei), (2.54)

where ge and gi are the conductances of the excitatory and inhibitory synapses, and Ee

and Ei are the corresponding reversal potentials. The time-dependent conductances areeffectively described as an Ornstein-Uhlenbeck process

23

dge(t)dt

= − 1τe

[ge(t)−ge0] +√

De0dW (t)

dt, (2.55)

dgi(t)dt

= − 1τi

[gi(t)−gi0] +√

Di0dW (t)

dt, (2.56)

where τe, τi are the time constants, and ge0 gi0 are average values of the synaptic con-ductances, and De0, Di0 diffusion coefficients. Different noise conditions are modeledby changing the synaptic conductances and the square root of the diffusion coefficientby a common gain factor α , α× (ge0,gi0) and α× (

√De0,√

Di0). Since the Ornstein-Uhlenbeck process models the cumulative effect of many stochastic processes this cor-responds to a simplistic model of the effect of increasing the synaptic peak conduc-tances.

The parameters of the Ornstein-Uhlenbeck processes are chosen in such a way thatthey resemble in vivo like activity (Destexhe et al., 2001). The average synaptic noisecurrent is balanced, i.e. it is zero just below threshold (roughly 3.5mV, for the parame-ters given in the appendix 11.5) because the inhibitory and the excitatory currents haveopposite sign but equal strength. If the membrane potential is at its “balanced” value,a change in the parameter α leads to a change in the variance of the fluctuations of themembrane potential only. For other values of V , however, a change in α also induces ashift of the average value of V .

24

Chapter 3

Information Transmission

Information theory in terms of the influential work of Shannon (Shannon, 1948) makesessential contributions to theoretical neuroscience. Figure 3.1 shows the fundamentalinformation transmission scenario employed in this thesis. A neuron model receivessignal and noise inputs. The response reflects the properties of the input, the model andthe noise characteristics. Information transmission is measured between the responseand the ”signal” input. From a formal point of view, information theory offers a frame-work for characterizing the statistical structure of a group of random variables. Thereare two essential concepts for information transmission in terms of information theory:Entropy and mutual information. For a well written text book see (Cover & Thomas,1991). After introducing entropy as a measure of information or uncertainty, the mu-tual information between two random variables is presented. An alternative notation ofinformation is given in the framework of the Fisher information. It is briefly introducedin section 3.2 and put into relation with the mutual information.

3.1 Entropy and Mutual Information

Entropy is introduced as a measure of uncertainty w.r.t. a random variable. The caseof discrete random variables is considered, The discrete random variable X takes ondifferent discrete values x from an alphabet ℵ. Let p(x) be a probability for all x ∈ℵ.A measure of the uncertainty of the probability distribution p(x) is then given by the

signal response

noise

Figure 3.1: A neuron model receives signal and noise inputs. The response reflects theproperties of the input, the model and the noise characteristics. Information transmis-sion is measured between the response and the ”signal” input.

25

entropy H(x). It is defined as

H(x) =−∑x∈ℵ

p(x)log(p(x)), (3.1)

where E(.) denotes the expectation. The entropy provides a measure of sharpness ofthe distribution, which is nothing else than the degree of uncertainty corresponding tothe random variable X . If H(X) = 0, the variable X describes a deterministic process.In other words, zero entropy implies that there is an absolute certainty that only oneoutcome of X is possible. On the other hand, the maximum value of H(X) is reachedwhen the distribution p(x) is uniform, i.e. when the uncertainty about the randomvariable X is maximal.

Let X ,Y be a pair of random variables over discrete alphabets ℵ,ℑ, respectively.The joint probability will be denoted by p(x,y) and the conditional probability of y fora given outcome x by p(y|x). Then, the joint entropy H(X ,Y ) is defined by

H(X ,Y ) =−∑x∈ℵ

∑y∈ℑ

p(x,y)log(p(x,y)) (3.2)

and the conditional entropy H(Y |X) by

H(Y |X) = ∑x∈ℵ

p(x)H(Y |X = x) =− ∑x∈ℵ

∑y∈ℑ

p(x,y)log(p(y|x)). (3.3)

that is, by the average of the degree of uncertainty of Y over all the concrete outcomesof X .

An important problem often encountered in statistics is to quantify the differencebetween two distributions. The Kullback-Leibler (KL) distance K(p,q) can be con-sidered as a measure of the distance between two distributions p(x) and q(x) and it isdefined as

K(p,q) = ∑x∈ℵ

p(x)log(p(x)q(x)

). (3.4)

The KL distance is not a true distance due to the fact that it is not symmetric, i.e.K(p,q) 6= K(q, p). However, both expressions can be interpreted as quasidistances thatare always positive, and equal to zero if and only if p(x) = q(x).

In order to measure the statistical independence of the two random variables Xand Y with associated probability distributions p(x) and p(y), respectively, it is usefulto introduce the notion of mutual information I(X ;Y ). The latter is defined as theKullback-Leibler distance between the joint probability and the factorized ones, and itis equal to zero if and only if X and Y are independent. Following Shannon (Cover &Thomas, 1991), the mutual information between X and Y is defined as

I(X ;Y ) = K(p(x,y), p(x)p(y)) = ∑x∈ℵ

∑y∈ℑ

p(x,y)log(p(x,y)

p(x)p(y)). (3.5)

The mutual information is symmetric, i.e. I(X ;Y ) = I(Y ;X). It is a measure of theamount of information that Y conveys about X , or vice versa. When X and Y aredefined as the input and output of a stochastic channel, then I(X ;Y ) is the amount ofinformation transmitted in the stochastic channel.

26

3.2 Fisher Information

The Fisher information is closely related to estimation theory, for an introductory text-book see (Kay, 1993). Consider the problem of estimating a single parameter of amodel from data of some measurements. Let the parameter have a value Θ, and let therebe N data values y1, . . . ,yN =~y. The system is specified by a conditional probabilitylaw p(~y|Θ), what is the likelihood. The data obey~y = f (θ )+~x, where x1, . . . ,xN =~xis additive noise and f is a deterministic function. The data are used in an estimationto form an estimate of Θ which is a function Θ(~y) of all the data. Usually Θ(~y) is abetter estimate on average than is any one of the data observables alone. The noise ~xis assumed to be intrinsic to the parameter Θ under measurement. No additional noiseeffects, such as noise of detection, are assumed to be present here.

Fisher information arises as a measure of the expected error in a measurement.Consider the class of unbiased estimates, obeying < Θ(~y) >= Θ, these are correct onaverage. The mean squared error ε2 in such an estimate obeys a relation (Kay, 1993)

ε2IF ≥ 1, (3.6)

where IF is called the Fisher information. Let p(x) denote the probability density of1-dimensional noise. The Fisher information is then

IF =

∫

dx( d p

dx )2

p(x). (3.7)

Eq. (3.6) is called the Cramer-Rao inequality. It expresses the reciprocal relationbetween the mean-squared-error ε2 and the Fisher information IF . The Cramer-Raoinequality shows that estimation quality increases (ε decreases) as IF increases. There-fore, IF is a quality measure of the estimation procedure. This is the essential reasonwhy IF is called an information.

Both information measures, the mutual information and the Fisher information, arefunctionals of an underlying probability density function. The analytic properties ofthe two information measures are quite different. Whereas the entropy H is a globalmeasure w.r.t. p(x), IF is a local measure.

27

Chapter 4

Adaptive Stochastic Resonance

This chapter is based on (Wenning & Obermayer, 2002b; Wenning & Obermayer,2002a; Wenning & Obermayer, 2003).

4.1 Abstract

Cortical neurons in vivo show fluctuations of their membrane potential in the order ofseveral millivolts. Using simple and biophysically realistic models of a single neuronit is demonstrated that noise induced fluctuations can be used to adaptively optimizethe sensitivity of the neuron’s output to ensembles of sub-threshold inputs of differentaverage strength. Optimal information transfer is achieved by changing the strength ofthe noise such that the neuron’s average firing rate remains constant. In a biophysicallymore realistic model it is shown that adaptation is fast, because only crude estimates ofthe output rate are required at any time.

4.2 Introduction

One basic cornerstone of an understanding of the role of stochastic resonance in neuralsystems is the problem of adjusting the optimal level of noise. Is there a simple way fora single neuron to calculate the proper noise level for a given ensemble of input signalsand would it be feasible for a neuron to adjust the noise level if the input is changing?

Using two abstract as well as a biophysically more realistic neuron model it isdemonstrated that as long as the variance of the noise remains optimally adapted tothe average strength of the inputs, the neuron’s average output rate is approximatelyconstant and independent of the ensemble of input signals. This suggests a simpleadaptation principle for the neuron to properly adjust the variance of the noise: Thesynaptic conductances of the noise inputs are strengthened if the neuron’s average fir-ing rate is below its target value and weakened otherwise. Activity dependent weightregulation of this kind has been advanced in various theoretical and experimental stud-ies (Bienenstock et al., 1982; Katz, 1999; Song et al., 2000; Turrigiano et al., 1998).Rate stabilization using spike-timing dependend plasticity has been investigated in amodel study (Kempter et al., 2001).

In section 4.3 adaptation to changing input distributions is studied in an abstractmodel framework. The description of the membrane potential dynamics is based on a

28

leaky integrator model. Spiking activity is modeled via a deterministic function whichmaps the membrane potential mean and variance to the probability of instantly emit-ting a spike. Given two different constant mean input currents, optimal performanceis achieved if the difference of the corresponding output rates is maximal. For this ap-proach an analytic expression for the optimal output rate can be given. In section 4.4 aleaky integrate-and-fire model neuron is applied, given constant current inputs first andsecond moments of the spike count distribution are determined. Information transmis-sion is based on a measure closely related to mutual information. The results of section4.4 are verified in a Hodgkin-Huxley type model neuron in section 4.5. Consequencesfor a hypothetical learning rule are discussed in section 4.6. This chapter ends with asummary of the results.

4.3 Adaptation, Hazard Function Approach

θ

2 Nnλn

λowssλsN

nw

a)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

prob

abil

ity

of in

stan

t fir

ing b)

subthreshold potential ( = 1)θ

σ

(a) (b)

µ

hArr

1

0 1

Figure 4.1: (a) The basic model set up. A leaky integrator neuron receives “signal” in-puts, namely Ns Poisson spike trains with rate λs and weight ws. Furthermore it receivesNn Possion spike trains as noise input. They have rate λn and weight wn. The outputactivity is determined from a deterministic mapping, eq. (4.2), modeling effectively theneural activity. (b) Arrhenius type hazard function. The figure shows the instantaneousprobability of firing, which is proportional to the output rate λo, as a function of themean membrane potential for different values of the variance σ . The horizontal bardenotes 1/

√e, eq.(4.6). σ = 2Nnw2

nλn with wn ∈ 0.01,0.02,0.03,0.04,0.05 fromright to left (see arrow), N = 100,λn = 7,θ = 1 (no explicit units).

Figure 4.1 (a) shows the basic model setup. A leaky integrate-and-fire neuron re-ceives a “signal” input, which is assumed to be Ns Poisson spike trains with a rate λs.The rate λs is low enough, so that the membrane potential V of the neuron remains sub-threshold and no output spikes are generated. For the following it is assumed that theinformation the input and output of the neuron convey is coded by its input and outputrates λs and λo only. Sensitivity is then increased by adding 2Nn balanced excitatoryand inhibitory “noise” inputs (Nn inputs each) with rates λn and Poisson distributedspikes. Balanced inputs (Shadlen & Newsome, 1998; Tsodyks & Sejnowski, 1995)were chosen, because they do not affect the average membrane potential and allow toseparate the effect of decreasing the distance of the neuron’s operating point to thethreshold potential from the effect of increasing the variance of the noise. Signal and

29

noise inputs are coupled to the neuron via synaptic weights ws and wn for the signaland noise inputs. The threshold of the neuron is denoted by Θ. Without loss of gen-erality the membrane time-constant τ , the neuron’s reset potential V0, and the neuron’sthreshold θ are set to one, zero, and one, respectively. If the total rate 2Nnλn of in-coming spikes on the “noise” channel is large and the individual coupling constantswn are small, the usual diffusion approximation (Tuckwell, 1998) applies (i.e. the cu-mulative effect of all Poisson spike trains might be replaced by a Wiener process). Asa consequence the dynamics of the membrane potential can be approximated by anOrnstein-Uhlenbeck process,

dV = (−Vτ

+ µ)dt +σ dW, (4.1)

where drift µ and variance σ 2 are given by µ = Nswsλs and σ 2 = Nsw2s λs +2Nnw2

NλN ,and where dW describes the infinitesimal increment of the Wiener process (Tuckwell,1998), see also chapter 2 for an introduction. In a leaky integrate-and-fire neuron spikeinitiation would be included by inserting an absorbing boundary with reset. Such asystem is not easy to deal with. Analytic solutions do exist only for special cases, see(Tuckwell, 1998) and chapter 2. In this section an approximation, cf. (Plesser & Gerst-ner, 2000), is chosen which is a great help in building intuition and deducing an analyticexpression for an adaptation rule. In this approximation, the probability of crossing thethreshold, which is proportional to the instantaneous output rate of the neuron, is de-scribed by an effective transfer function. In (Plesser & Gerstner, 2000) several transferfunctions were compared in their performance, from which an Arrhenius-type func-tion, eq. (4.2), is chosen. θ −V0(t) is the distance in voltage between the noise freetrajectory of the membrane potential V0(t) and the threshold θ . The noise free trajec-tory is calculated from eq. (4.1) without its diffusion term. So the approximation forthe instantaneous output-rate is

λo(t) = c exp−(θ −V0(t))2

σ2 . (4.2)

Note that V0(t) is a function of Nswsλs, c is a constant. Figure 4.1 (b) shows a familyof Arrhenius type transfer functions for different noise levels σ .

The quality of the transmission of sub-threshold input signals which are given byinput rates near an “operating point”, determined by λs, is evaluated. A simple scenariois considered, in which a neuron should distinguish between two sub-threshold inputrates λs and λs + ∆s. Optimal transmission is achieved if the difference ∆o of thecorresponding output rates λo is maximal, i.e. if

∆o = λo(λs +∆s)−λo(λs) = max, (4.3)

where λo(./.) is given by eq. (4.2). The above condition implies that the stochasticnature of the problem is neglected - eq. (4.2) is a deterministic mapping between thefirst two moments of the input and the response rate - the next to sections of this chapter,sections 4.4 and 4.5, will catch up on this point.

Figures 4.2 (a) and (b) display stochastic resonance curves according to eq. (4.3)and eq. (4.2) for different mean input intensities. The optimal noise level is then givenby the condition

ddσ 2 (λo(λs +∆s)−λo(λs)) = 0. (4.4)

30

0 50 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 50 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4∆∆o o

∆s ∆sa) b)

subthreshold potential ( = 100 )θ0 50 100

0

10

20

30

40

50

60

70

80

90

100

0 50 1000

10

20

30

40

50

60

70

80

90

100

0 50 1000

10

20

30

40

50

60

70

80

90

100

0 50 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

opti

mal

w i

n pe

r ce

nt

opti

mal

w

n

n

subthreshold potential ( = 100 )θ

c) d) e) f)

µ µ µ µ µ µ0 0 0 0 01 1 1 1 1 1

(c)(b)(a) (e)(d) (f)

0

Figure 4.2: (a) ∆o vs. σ 2 for λs = 2 and (b) for λs = 7 with 10 different values of∆s = 0.01,0.02, . . . ,0.1. σ 2 is given in per cent of the maximum σ 2 = 2Nw2

nλn. Thearrows in (a) and (b) denote the optimal noise levels σ . (c) Optimal wn calculatedaccording to eq. (4.3), (d) according to eq. (4.3), with 2nd oder Taylor expansions ofeq. (4.2), (e) with the additional assumption that the balanced input is the only sourceof membrane potential fluctuations. (f) Optimal wn according to the eq. (4.5), thuscorresponding to the noise level which is determined from the optimal output rate λo,eq. (4.6). Parameters are: N = 100,λn = 7,ws = 0.1, and wn ∈ [0,0.1],λs ∈ [0,10] (noexplicit units)

Using a Taylor expansion to second order in ∆s yields

σ2optimal = 2(1−wsλs)

2 (4.5)

for small values of ∆ (∆s→ 0). Is is assumed that the fluctuations of the membrane po-tential are dominated by the balanced input (Nsw2

s 2Nw2nλn). For the corresponding

output rate follows

λ optimalo = 1/

√e. (4.6)

Figures 4.2 (c)-(f) demonstrate the quality of the applied approximations. Equation(4.6) is a conceptually important result: The output rate is independent of the strengthλs of the input if the noise level σ is at its optimal value, eq. (4.6). Hence - for any inputλs - the output rate indicates, whether the strength of the noise is optimal. Note, that theinflection point of all functions of the family, eq. (4.2), corresponds to an equal outputrate of λ optimal

o = 1/√

e, see figure 4.1 (b). Optimizing the noise level corresponds to achange of the transfer function, such that the average membrane potential correspondsto the point of the steepest slope of the transfer function, i.e. to the point of strongest“amplification” of the input signal.

Information about the output rate of a neuron is available to any synapse on the cell,whereas information about the input rates might be restricted to small sub-volumes.Hence a neuron may utilize the fact that the output rate must always be constant toadapt its internal noise level to changing inputs λs. A simple strategy is given by

∆wn = ε (λ optimalo −λo), (4.7)

i.e. the neuron should increase the strengths of its noise input if its firing rate is too lowand decrease it otherwise. Figure 4.3 shows the application of this strategy to the leaky

31

0 500 1000 1500 2000 2500 30000

0.1

0.2

0 500 1000 1500 2000 2500 30000

5

10

0 500 1000 1500 2000 2500 30000

0.2

0 500 1000 1500 2000 2500 30000

5

number of iterations

λs

wn

∆∆

os

MI[nats](a)

(b)

(c)

(d)

Figure 4.3: Adaptation of the noise level to three different input base rates λs. At eachtime step, the input rate was chosen randomly and independently from the interval[λs− 0.25,λs + 0.25]. (a) Mutual information MI between input and output rates asa function of the changing synaptic coupling constant wn. The solid lines indicatethe maximum mutual information that is possible (3.31, 4.42, 3.04 nats) (b) Dashed-dotted line: The ratio ∆o/∆s computed from eq. (4.3). Solid line: ∆o/∆s computedfrom eq. (4.3), but using the quadratic approximation of eq. (4.2). The quadraticapproximation of eq. (4.2) is the reason for the difference between the solid and dashedline. (c) Solid line: wn as a function of time according to eq. (4.7), dashed dotted line:wn calculated using eq. (4.3), dashed line: value of wn which maximizes the mutualinformation between the distribution of the input and output rates. (d) The three baserates λs.

integrator neuron with its output activity determined by the deterministic mapping fromeq. (4.2). The signal consists of inputs distributed around three different base rates λs.The quality of transmission is assessed in terms of eq. (4.3). It is also compared tothe mutual information (Cover & Thomas, 1991; Bulsara & Zador, 1996) between aprobability distributions p(λs;∆s) for the input and p(λo) for the output rates, which is amore principled measure appropriate for arbitrary distributions of rates. Since eq. (4.2)is a deterministic mapping the mutual information is reduced to the output entropy.With a time constant determined by the learning constant ε , information transmissionis indeed almost optimized.

32

4.4 Adaptation in a LIF Neuron

The results of the last section are conceptually interesting, they give a hint on howadaptation of the membrane potential fluctuations to changing signal intensities mightbe possible. But the deterministic form of the transfer function, eq. (4.2), is unsatisfy-ing. In this section the stochastic nature of neural information processing will be takenmore seriously. As in the previous section Poisson spike trains are chosen as signaland noise inputs. They are coupled to a leaky integrate-and-fire neuron via synapticweights ws (signal) and wn (noise). The membrane potential dynamics are given byeq. (4.1) again. However, in this section spiking is modeled more explicitly, once themembrane potential reaches a given value θ this event is called a spike and the voltageis reset to V0 = 0. A signal input µ , see eq. (4.1), is sub-threshold if µτ < θ and supra-threshold otherwise. The output rate, f [ spikes

s ], of the Ornstein-Uhlenbeck neuron canbe calculated using the expression

1f τ

=

∫ ∞

0du e−u2 e(2yθ u)− e(2yru)

u, (4.8)

where yθ = θ−µτ√σ 2τ

, yr = V0−µτ√σ 2τ

(see (Brunel & Hakim, 1999)). In chapter 2, figure 1 (a),the frequency-current (f-I) curve according to eq. (4.8) is shown. As in the previoussection it is assumed that σ is determined by the noise inputs only. Thus, the influenceof the signal inputs on the membrane potential fluctuations is assumed to be negligible.

In order to quantify information transmission in single neurons, M. Stemmler intro-duced a simple expression for a lower bound J on the Fisher-Information (Stemmler,1996). In spite of it being an approximation it is shown numerically in (Stemmler,1996) that the optimal noise level calculated from

J(µ) =4T

√

(τ3π)

(θ −µτ)3

σ5 e(θ−µτ)2

σ2τ (4.9)

deviates only negligibly from the one that would have resulted from the use of the mu-tual information (Cover & Thomas, 1991; Bulsara & Zador, 1996) between the spikecount distribution and the distribution of small deviations from µ , for all sub-thresholdµ . This property allows to cover the whole parameter regime of interest, using onlythe simple expression eq. (4.9), without the need to calculate the 2nd moment of thefirst-passage-time density near the threshold for low noise, what is notoriously difficult(see e.g. comments in (Lansky & Sacerdote, 2001)).

Figure 4.4 (a) shows stochastic resonance curves for different mean input inten-sities, J is employed as a performance measure. Figure 4.4 (b), solid line, displaysthe output rate f vs. µ using a noise level which maximizes J, eq. (4.9), for everyµ . The dashed lines correspond to output rates which belong to noise levels such thatJ = 0.95 Jmax. Note that the output rate at the optimal noise level is almost a constantwith respect to changing signal intensity. Hence, as noted in the previous section, theoutput rate indicates, whether the strength of the noise is optimal. Due to the widthof the maximum in J performance is only slightly degraded in case of a sub-optimallychosen output rate. A learning rule like eq. (4.7) can easily be applied to optimize thenoise level. This is not demonstrated in this LIF framework, but in the next section anadaptation rule corresponding to eq. (4.7) will be applied in a Hodgkin-Huxley setting.

33

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

f [H

z]

µµ0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

50

60

70

80

90

100

noise level

µ 1

µ 2

µ 3

µ 1 > µ 2 > µ 3

J

σ µ0 0.6 1.0 0 0.6 1.0

J f [H

z]

1

3

6

20

90 (a) (b)

Figure 4.4: (a) J, eq. (4.9), as a function of σ . Three stochastic resonance curvescorrespond to three different mean input currents, µ1 = 0.65,µ2 = 0.5,µ3 = 0.3 µ and

σ are made dimensionless by multiplying µ with τθ and σ with

√τ

θ . Threshold is at 1.(b) Solid line: Optimal output rate f vs. µ . For each µ , σ is adjusted such that J, eq.(4.9), is maximal. Dashed line: f as a function of µ with σ adjusted such that J equals95 % of the maximum possible value for each µ . Parameters as in (a), T = 200 ms.

4.5 Adaptation in a HH Neuron

It is far from trivial whether the above demonstrated property of the LIF neuron modelcarries over to real neurons. In order to provide stronger evidence a Hodgkin-Huxleytype model is employed. It is introduced in chapter 2. The main equations are brieflyreviewed for better readability. The dynamics of the membrane potential V is given by

CmdVdt

=−ILeak− INa− IKd− IM− Isyn + µ . (4.10)

The left hand side of the equation describes the influence of the membrane’s capac-itance, while all ionic currents through the cell membrane are summed on the righthand side. INa and IKd , are the sodium and potassium currents through the membrane,which are responsible for the neuron’s ability to produce a spike. IM is a potassiumcurrent typical for cortical pyramidal cells and ILeak summarizes all other (unspecific)leak currents through the membrane. Any incoming spike also causes a change inthe conductance of the neuron’s membrane. This causes currents flowing through themembrane, which are summarized in the total synaptic current Isyn. µ , finally is theapplied constant signal current. Isyn is responsible for the noise input,

Isyn = ge(t)(V −Ee)+gi(t)(V −Ei), (4.11)

it is mediated by time dependent excitatory (ge(t)) and inhibitory (gi(t)) conductanceswhich change as a result of the incoming spikes. Ee and Ei are the reversal potentials ofthe excitatory and inhibitory synapses. The effect of the incoming spikes of the noiseinputs is not modeled in detail, but effectively described by an Ornstein-Uhlenbeckprocess, as in (Destexhe et al., 2001),

dge(t)dt

= − 1τe

[ge(t)−ge0] +√

De0dW (t)

dt, (4.12)

34

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

I

I

I

1

2

3

I1 > I > I32d

noise level0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

7

8

9

10

µ0 1.0α1 20

d

0.1

0.8

f [H

z]

4

2

10(a) (b)

0.45

µ1 > µ2 > µ3

µ1

µ2

µ3

Figure 4.5: (a) Discriminability d, eq. (4.14), versus noise level α . Three baselinecurrents are injected. µ = 0.35,0.25,0.1 nA. (b) Solid line: Optimal output rate as afunction of the “signal” input current. The level of noise is always chosen such thatinformation transmission is optimal. Dashed lines: output rate at noise levels that give90% of the performance, in terms of d, eq. (4.15), compared to the optimal case.∆µ = 0.02 nA.

dgi(t)dt

= − 1τi

[gi(t)−gi0] +√

Di0dW (t)

dt, (4.13)

with ge0,gi0 are average conductances, τe,i are time constants, and dW is the infinites-imal increment of the Wiener process. De0,Di0 are the diffusion coefficients. Parame-ters are given in the appendix 11.5. Different noise levels are modeled by changing thesynaptic conductances and the square root of the diffusion coefficients by a commongain factor α , α × (ge0,gi0) and α × (

√De0,√

Di0). This corresponds to a simplisticmodel of altering the synaptic strength. The numerical values of ge0,gi0,De0,Di0 aregiven in the appendix 11.5. Figure 4 chapter 2 displays the frequency-current-curve forthe above choice of parameters for different noise conditions.

For the evaluation of signal transmission in the Hodgkin Huxley type model thediscriminability d is chosen,

d = 2µN(µ +∆µ)−µN(µ)

σN(µ +∆µ)+σN(µ), (4.14)

µ is a constant current injection, ∆µ is a small deviation from this baseline current. Eq.(4.14) relates to J, as d = ∆µ

√J for small ∆µ , see (Stemmler, 1996). The discrim-

inability is calculated from the average spike count (µN) and the spike count variability(σ 2

N). In figure 4.5 (b) the output rate at the noise level which maximizes the discrim-inability d is plotted versus the injected current (solid line). Like in the abstract model,the “optimal” output-rate is constant over a wide range of average signal inputs. Thedashed lines indicate the output firing rates in the case of sub-optimally adjusted noise,here the noise is adjusted to yield 90% performance in terms of the optimal value of d.Thus the width of the maximum in d is fairly broad. The resulting optimal frequency is≈ 4Hz, what is close to the typical activity of cortical neurons (Baddeley et al., 1997),but see also (Olshausen & Field, 2004).

If the noise level is optimized for a certain signal strength, it is guaranteed thatthe discriminability for small deviations from this input intensity is maximal. Inputstatistics may change with time, thus the optimal noise level may have to change as

35

0 5 10 150

0.5

0 5 10 15

−50

0

0 5 10 150

5

0 5 10 150

1

2

0 5 10 150

0.5

1

0 5 10 15

0.5

0

-50

0

0

5

0

12

0.5

1

0

time [s]

(a)

(b)

(c)

(d)

(e)relative

deviationfrom

optimal α

α

spikesin

500 ms

V [mV]

I [nA]

Figure 4.6: Demonstration of adapting the noise level. There are three distinct timescales: Fast membrane potential fluctuations according to eq. (4.14), “signal” inputschanging every 50 ms, such that many “signals” correspond to deviations from the samemean, and adaptation of the noise level on the slowest time scale. (a) Brief “signal”current injections, Iapp, of duration 50 ms, randomly drawn from normal distributionswith mean 0.25, 0.35, 0.1 nA and variance 0.0005 nA2, one every 50 ms. (b) Exampleof a voltage trace of the Hodgkin-Huxley model, eq. (4.11), as a consequence of currentinjections and adaptation. (c) Estimated output rate fest , based on a 500 ms average ofthe recent past, fest is evaluated every 25 ms. Thick line: mean over 10 independentruns, thin lines: corresponding standard deviations, straight lines: fopt = 4 Hz. (d)Noise amplification α , with ε = 0.007. The update occurs in the same time step asfest is evaluated, every 25 ms. Thick line: mean over 10 independent runs. Thin lines:corresponding standard deviation. Straight line: optimal noise amplification (w.r.t.mean of stimulus distribution) αopt = 0.8, 0.4, 1.3, changing every 5s according to

stimulation. (e) Relative deviation from optimal noise level: |αopt−α|αopt

.

36

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f)

P

PH

F

number ofneurons

6 Hz signal4 Hz background

Figure 4.7: Signal detection probability PH vs. false alarm rate PF , in an array of Nneurons with independent noise, receiving the same stimulus input. Each neuron emitsa Poisson spike train of 4 Hz if no stimulus is present, the presence of a stimulus makesthe neurons fire with 6 Hz. Integration time is 10 ms. The solid line along the diagonalcorresponds to N = 1, N then increases to N = 1000 in steps of 100.

well. Here a scenario is considered in which the mean input intensity may change withtime. Signals are then considered to be small deviations from this mean, time scale andstatistics (of the changing mean) are assumed to be such that one noise level may beoptimal for many signals. Adaptation to the mean of a signal distribution is commonin nature, see e.g. contrast adaptation (Carandini & Ferster, 1997).

4.6 Discussion

The above results immediately lead to three questions:

1. Can a neuron use such a low output rate to adapt to new stimulus distributionswithout averaging for unrealistically long time intervals?

2. Would a read-out mechanism be able to infer the existence of a signal within areasonable time?

3. How does the learning rule, designed for sub-threshold inputs, perform in thesuper-threshold regime?

Figure 4.6 demonstrates in the HH type framework that adaptation to some meaninput intensity is well possible within biologically realistic time scales. The figureshows the dynamics of adaptation of the noise level for a single model neuron to threedifferent mean input intensities. Adaptation is based on an estimate fest of the averageoutput activity calculated from the number of output spikes in the preceding 500 ms.The synaptic conductances of the noise inputs are strengthened if the neurons aver-age firing rate is below the optimal output activity fopt and weakened otherwise. Thesimplest possible (formal) adaptation rule (update rule) for the noise amplification pa-rameter is given by

37

δα(tn) = ε( fopt − fest(tn)), n is integer (4.15)

where ε determines the time scale for adaptation and α determines the membrane po-tential variations. At times tn, fest is evaluated and the update applies. As can be seenin figure 4.6 (d) the neuron succeeds in adapting the noise level to different signaldistributions well within approximately 2 s, despite a low output rate. An adaptationtime of 2-3 s is well within biological typical time scales (Carandini & Ferster, 1997).Whether a read-out mechanism would be able to infer the existence of a signal dependson the number of neurons receiving correlated (signal) input. The performance for sin-gle neurons would be poor. In figure 4.7 the performance of a readout mechanism in asignal detection task (Wickens, 1991) is quantified. A readout receives Poisson spiketrains from a population of N neurons, each at an average rate of 4 Hz. If the signalchanges there is a transient change in the output rates. The readout has to decide inevery moment, based on the overall spike count of the recent past, whether a changein signal intensity has occurred or not. As can be seen in fig. 4.7 the performance fora single neuron would be very poor, but populations of neurons can perform arbitrarywell. Since cortical populations of neurons might well consist of several hundred sin-gle neurons (Feldmann, 1984) and may receive highly correlated input, fast detectionof signals may indeed be possible.

Until now only the sub-threshold regime has been dealt with, but would an adaptationrule similar to eq. (4.15) make sense in case of supra-threshold stimuli? Supra-threshold stimuli result in output rates that are higher than the optimal rate in thestochastic resonance regime. As a consequence, the noise level would be reduced asmuch as possible such that supra-threshold inputs are not disturbed by - unnecessary -additional noise.

4.7 Summary and Conclusion

In this chapter it is shown, that a simple and activity driven learning rule can be derivedfor the adaptation of the optimal noise level in a stochastic resonance setting. It isbased on the average spiking activity of the neuron and its input/output properties.Approximately optimal information transfer is achieved by changing the strength of thenoise such that the neuron’s average firing rate remains constant. In order to illustratethe general principle two abstract and highly simplified description of the neuron’sdynamics are employed. In order to provide quantitative predictions, a biophysicallymore realistic Hodgkin-Huxley type neuron is used.

38

Chapter 5

Energy Efficient Coding

5.1 Abstract

In this chapter the relevance of weak, i.e. sub-threshold, inputs for neural informationtransmission is discussed. The discussion is based on an inspection of the input-outputproperties of abstract model neurons, especially those influencing the representationalcapacity and the mutual information. Information transmission is related to neuralactivity and thus to metabolic cost. It is hypothesized that optimal information trans-mission is constrained by metabolic cost. Optimal input distributions, constrained bythe amount of available energy, are calculated. The results suggest that weak - sub-threshold - signals become beneficial despite the fact, that their transmission is lessreliable then for strong signals.

5.2 Introduction

In most of the chapters in this thesis the starting point of any further considerationis the assumption, that the transmission and computation with weak neural inputs isbiologically relevant. Considering that neural activity is metabolically expensive, itstands to reason that the evolutionary mechanism of mutation and selection ensures theuse of weak, less expensive, but more unreliable, inputs. In this chapter the relevanceof weak signals to information transmission is investigated in three ways.

A classic view on neural encoding (Barlow, 1969; Adelsberger-Mangan & Levy,1929; Foldiak, 1990; Redlich, 1993) hypothesizes that maximizing the representa-tional capacity is highly desirable and thus is a driving force in the evolution of neuralcodes. Investigating representational capacity can tell us a lot about the representa-tional capabilities of a neuron and maybe, what kind of stimuli should be used, froma representational perspective. However, for information transmission also the noisecharacteristics of the information transmission channel - a neuron in this case - is ofimportance, see chapter 3 for an introduction and (Cover & Thomas, 1991) as a text-book. Maximal information transmission is probably not the only goal neurons try toachieve. Neural activity is costly in metabolic terms, thus energy consumption anddissipation becomes a concern, (Balasubramanian et al., 2001; Levy & Baxter, 1996;Schreiber et al., 2002), hence a subset of possible neural activities which optimize atrade off between information transmission and metabolic cost, should be of interest.

39

Figure 5.1 (a) shows the general set up, a model neuron receives a constant currentsignal and additive white noise as inputs, a corresponding response is produced. Therepresentational capacity is quantified as the entropy of the neural output. Spike-countcoding is assumed. Input symbols are discretized constant current injections, the outputsymbols are the spike count within a given time interval. This scenario has the advan-tage to yield a simple, discrete, low dimensional, probability distribution for the inputand output symbols. In addition to that it is highly plausible, that spike-count codingis indeed very relevant in the transmission and computation in real nervous systems,(Dayan & Abbott, 2001). In the most abstract scenario for investigating the represen-tational capacity it is assumed that the spike count is maximum entropy distributedaround a given mean, as investigated before by (Levy & Baxter, 1996). A more real-istic scenario describes the neural response as a Poisson spike train. The most explicitmodel of neural spiking activity then employs a leaky integrate-and-fire model (LIF)neuron. In the two very abstract models the capacity grows monotone with the averagespike count, for the leaky integrate-and-fire model a maximum appears at intermediatespike counts.

For information transmission not only the capacity is relevant, but also the noisecharacteristics of the neuron model. Therefore the mutual information between a dis-tribution of constant current inputs and the spike count response of the LIF neuron iscalculated as a function of the variance of the noise input, i.e. the noise level.

For investigating how energy consumption constraints information transmission theoptimal input signal distributions are calculated for a LIF neuron for different noiselevels and different amounts of available energy. The metabolic cost is modeled as asum of baseline costs plus an increment for an increase in the average number of outputspikes. It is demonstrated that limited energy resources force the model neuron toachieve a trade-off between high quality information transmission with reliable, strong,but expensive signals and metabolic cost. Thus a reinforced use of weak and unreliablesignals becomes optimal.

In section 5.3, the representational capacity as entropy of the neural response isintroduced. It is investigated for two very abstract model neurons. The LIF neuronis briefly introduced in section 5.4. Furthermore, a method to calculate the first twomoments of the spike-count distribution is introduced and the representational capac-ity employing a LIF neuron is calculated. In section 5.5 the mutual information isdetermined based on the transfer function of the LIF neuron. The metabolic cost func-tion is introduced in section 5.6, optimal input symbol distributions for LIF neuron arecalculated. This chapter ends with a summary and discussion of the methods and theresults.

5.3 Capacity of Abstract Model Neurons

The representational capacity is a measure for the variety of possible neural responses.The more different representations are available and employed, the higher is the ca-pacity. As in (Levy & Baxter, 1996) the representational capacity is quantified as theentropy of the output.

To reduce the representational complexity it is assumed that only the number ofspikes within a given time interval T are relevant for information transmission. Thiscorresponds to a spike-count-code or rate coding, for an introduction to neural codingsee e.g., (Dayan & Abbott, 2001). The firing rate f j is equal to the number of spikes j inT . A maximum of N spikes might be emitted within T , corresponding to a maximum

40

signal response

noise

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

(a) (b)

sign

al

noise

response

f [Hz]20 40 80

C [

nats

]

1

10

σ

Figure 5.1: (a) The basic set up. A neuron model receives signal and noise input. Theresponse reflects the properties of the input and of the model properties. (b) Capacity,according to eq. (5.1) versus spike frequency f j = 1/T,2/T, . . .,N/T, j = 1, . . . ,N,i.e. spike count within T . The probability of counting j spikes within T is denotedas p j. Top curve: Capacity is calculated employing the geometric distribution fordetermining p j, eq. (5.2). Second curve from top: Capacity is calculated employingthe Poisson distribution for determining p j, eq. (5.3). Lower set of five curves: Thecapacity is calculated employing the LIF neuron with additive white noise and constantcurrent input, eq. (5.4). Evaluating the expressions eq. (5.5) and eq. (5.6) yieldthe first and second moments of the interspike interval distribution of the LIF neuron.From these the spike count variance σ 2

N can be determined, see text. The spike countdistribution p j is then chosen to be normal distributed around the first and secondmoment. From bottom to top the variance of the white noise input increases. For theLIF model neuron dimensionless units are applied, i.e. µ is multiplied with τ

θ and σwith

√τ

θ . Standard deviations are σ = 0.2,0.4,0.6,0.8,1.0. x0 = 0mV, θ = 30mV andτ = 5ms. For all curves T = 1s.

41

rate fN . The available discrete firing rates f j = 1/T,2/T, . . .N/T may occur withprobability p j. Representational capacity C is expressed as a function of p j

C = ∑ p j log(p j). (5.1)

In general p j depends on the model neuron and on the input statistics. First a veryabstract setting is employed. Probabilities p j are chosen according to a maximum en-tropy distribution, given the mean output frequency f . The distribution that maximizesthe entropy - in the positive half-space of discrete frequencies - with no additional in-formation about p j is the geometric distribution. A proof is given in the appendix of(Levy & Baxter, 1996). Jaynes introduced the principle of maximum entropy in 1957,(Jaynes, 1957). The geometric distribution is given by

p j =µ j

(1+ µ) j+1 . (5.2)

In figure 5.1 (b) the top curve shows the representational capacity C as a function of themean output rate f . Note that the capacity C grows strictly monotonic with f . For ftowards infinity the capacity will become infinite, it does not converge to some value.

A little bit more constrained, but still very abstract, is another model of neuralactivity. It is assumed that spike trains are Poisson spike trains with rate λ . Still, onlythe first moment of the spike count distribution is relevant. The probabilities p j, ofhaving j spikes within T , are Poisson distributed

p j =(λT ) je−λ T

j!. (5.3)

In figure 5.1 (b) the second curve from top displays the capacity as a function of themean output rate f in case of a Poisson output spike train. The resulting capacity islower then in the case of the geometric distribution. This is due to the fact, that thevariance of the Poisson distribution is equal to its mean, and thus smaller then thevariance of the geometric distribution, given f .

A scenario in which the properties of a model neuron are taken explicitly into ac-count is presented in the next section.

5.4 LIF Framework

In this section the leaky integrate-and-fire model neuron is briefly introduced, a moredetailed description can be found in chapter 2.

The input is modeled as a Wiener process with mean µ and variance σ 2. Thedynamics of the membrane potential is given by an Ornstein-Uhlenbeck process withdrift and upper absorbing boundary. Below the boundary, the evolution of the mem-brane potential is governed by

dV (t) = (−V (t)τ

+ µ)dt +σdW(t), V0 = 0, (5.4)

with τ as a time-constant, and dW (t) is the infinitesimal increment of the Wiener pro-cess at time t. Once the membrane potential reaches a given value θ this event is calleda spike and the voltage is reset to V0 = 0. A signal input µ is sub-threshold if µτ < θand supra-threshold otherwise. The output rate, f [ spikes

s ], of the Ornstein-Uhlenbeck

42

neuron can be calculated using either a series expansion approach or integral expres-sions, an overview is given in chapter 2. Here the integral expressions are employed.The first moment can be calculated from the following integral:

1f τ

=

∫ ∞

0du e−u2 e(2yθ u)− e(2yru)

u, (5.5)

where yθ = θ−µτ√σ 2τ

, yr = V0−µτ√σ 2τ

, see (Brunel & Hakim, 1999). In chapter 2 the frequency-current (f-I) curve of a LIF neuron according to eq. (5.5) can be seen for several dif-ferent noise levels σ . The second moment σ 2

ISI of the interspike interval distribution isthe given by:

σ2ISI =

1f 2 +2π

∫ θ−µσ

V0−µσ

dxex2∫ x

−∞dyey2

(1+ er f (y))2. (5.6)

Once the second moment of the interspike interval, σ 2ISI , is available, the second mo-

ment of the spike count distribution is given by σN =σ 2

ISIµ3

ISIT , as introduced in chapter

2. The probability distribution p j is then chosen to be Gaussian with mean µN = f Tand standard deviation σN of the spike count. Thus, the capacity, eq. (5.1), can becalculated.

Figure 5.1 (b) shows, that including the leaky integrate-and-fire properties yields aqualitatively different behavior then the previous two more abstract models. It can beobserved, that the capacity is no longer a monotonic function of the mean output rate f .A maximum appears at intermediate firing rates. This maximum is very pronounced forlow noise levels. The dots mark the firing rate f ∗ where the mean input just drives theneuron to the threshold. Note that for small firing rates the maximum capacity is belowthis firing rate. For large firing rates the maximum is to the right of f ∗. There is a fastinitial rise in capacity as a function of f for small output spike rates. Different to thescenario with the geometric distribution, top curve, the capacity does not grow withoutbounds with the mean firing rate f . This is due to the fact, that σN does decreasebeyond a critical strength of the mean input. For larger noise levels σ , this happens atlarger mean input intensities µ .

5.5 Information Transmission

For information transmission not only the representational capacity is relevant, but alsothe quality of transmission. This is the case for any neuron, or more general, any infor-mation transmission channel, which has a certain non-vanishing probability of stochas-tically interchanging output signals, given the same input signal. For an introduction toinformation theory in the tradition of Shannon (Shannon, 1948), see (Cover & Thomas,1991) and chapter 3 for a short summary of the concept of mutual information. Thusthe full capacity is not available for information transmission. The mutual informationreflects both relevant terms for information transmission, the output entropy and thedependency on the channel noise characteristics, expressed as a conditional entropy.The mutual information is expressed in terms of two random variables, say Y and Z. Yis chosen to be the discrete signal input variable, the possible inputs are denoted as y j,each input has a probability q j. The elements of Z, zk, have probabilities pk

Y : q j→ Z : pk. (5.7)

43

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

µ

mut

uali

nfor

mat

ion

[nat

s]

σ

Figure 5.2: Mutual information, eq. (5.8), versus mean constant current input signalsfor several values of the standard deviation of the additive white noise σ . The rectangu-lar input distribution of the constant current inputs has a width of 0.1 in dimensionlessunits of µ , see caption of fig. 5.1. Threshold is at µ = 1. The spike count distributionis chosen to be Gaussian, with mean µN and standard deviation σN , for the calculationsee text. From top to bottom (supertreshold regime) the standard deviation of the noiseprocess, eq. (5.4), increases, σ = 0.1,0.2, . . . ,1.0 in dimensionless units. Parametersas in figure 5.1.

44

The mutual information between Y and Z is then

I(Y ;Z) = ∑y

∑z

p(y,z) ln(p(y,z)

p(y)p(z)). (5.8)

Given the combined and marginal distributions of input symbols and output symbols,the mutual information can be calculated. Typically the framework is set up to providethe input symbol distribution and the conditional distribution of the output symbols,given the input symbols. As in the above section it is only dealt with spike countcoding. The LIF neuron is employed and the available discrete output firing ratesf j = 0,1/T,2/T, . . .,N/T may occur with probability q j. Given the mean µ andstandard deviation σ of the input, eq.(5.1), the first and second moment of the spikecount distribution, µN and σN , are calculated as in the above section. The spike countdistribution is then assumed to be Gaussian with mean µN and variance σ 2

N

p j(µ ,σ) = N(µN ,σN). (5.9)

The input signal distribution is chosen to be of rectangular shape, outside of the rect-angle the probability of a signal is zero.

Figure 5.2 displays the mutual information as a function of the mean of a rectan-gular constant current distribution. Different noise levels are applied. For low noiselevels the mutual information is zero far below threshold. For sub-threshold inputswhich drive the neuron close to threshold in the absence of noise the mutual informa-tion displays a steep rise in the presence of additional noise. For super-threshold inputs,it merely grows any further. For large noise levels the mutual information has signif-icant positive values even for sub-threshold inputs, but in the super-threshold regime,the mutual information is much lower then for small noise levels. For all noise levelsone can observe, that the mutual information does not grow without bound with themean input intensity. This is due to the finite width σN of the output symbol distribu-tion, given that the input symbol distribution has a finite width as well.

5.6 Metabolic Constraints on Information Transmission

In this section metabolic cost for neural activity related to information transmission areintroduced. This is done to approach the following question: Given a fixed amountof available energy, what is the input signal distribution which maximizes the mutualinformation?

As in the previous section spike count coding is employed, as well as the notationfor the information channel, see eq. (5.7). The stochastic variables Y and Z take onthe values y j and zk with probabilities q j and pk. The properties of the informationtransmission channel are described by the following transition matrix Q

Qk, j = Pr(z = zk|y = y j). (5.10)

Pr(z = zk|y = y j) is chosen as a conditional Gaussian distribution as in eq. (5.9). Twotypes of energy cost are considered, the cost of generating and transmitting signals andthe cost of keeping a signaling system in a state of readiness.

Laughlin et al. (Laughlin et al., 1998) have argued that the dominant energy costfor a neuron arises in the pumps that actively transport ions across the cell membranes.Because ionic currents are large during an action potential, it is likely that a reason-able model can be given by a term for the baseline metabolic cost plus an additional

45

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

prop

abili

ty d

ensi

ty o

f si

gnal

str

engt

h

0.3

0.1 inte

nsity

bel

ow th

resh

old

0.2

0.9

signal intensity in arbitrary units

increasing energy resources

energy resourcesEµ0 1 2 µ0 1 2 µ0 1 2 µ0 1 2 µ0 1 2 µ0 1 2 5 30

(a)

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

signal intensity in arbitrary units

prob

abili

ty d

ensi

ty o

f si

gnal

str

engt

h

0.2

0.9

0.1

0.3

increasing energy resources

inte

nsity

bel

ow tr

esho

ld

µ0 1 2 µ0 1 2 µ0 1 2 µ0 1 2 µ0 1 2 µ0 1 2 E5 30

(b)

Figure 5.3: Optimal distribution of constant current inputs for the LIF neuron. Subfig-ures (a) and (b) differ in their standard deviation of the additive noise, in (a) σ = 0.1and in (b) σ = 0.6, in dimensionless units, as in figure 5.1. For (a) and (b) the six plotsto the left show the optimal input distributions, according to eq. (5.14). The dashedline indicates the intensity where sub-threshold inputs become super-threshold inputs.From left to right the available energy is increasing E = 5,10,15,20,25,30 (in units of1). The plot to the very right displays the cumulative probability weight of the optimalinput distribution below threshold versus the available energy. Energy is modeled asgiven in eq. (5.10), parameters are: b = 0.

46

increment per spike. The metabolic cost Ek, j of transmitting a single signal y j is nowexpressed in terms of the average number of output spikes zk which are emitted withinT

Ek, j = b+ f (µ ,σ) T. (5.11)

The spike frequency f (µ ,σ), eq. (5.5), times T yields the mean spike count µN asa function of the mean input µ and standard deviation of the Wiener process σ . Aconceptually major assumption is that noise does not contribute to the metabolic costrelated to information transmission, i.e. noise is assumed to contribute as a constantterm to the baseline cost b. This will be discussed in more detail in the last section ofthis chapter. To focus on the activity dependent cost it is chosen b = 0.

Metabolic costs which are due to the application of a given input y j are expressedin terms of the transition matrix Q

E j = ∑j

Qk, jEk, j. (5.12)

The task is to find the input signal distribution which maximizes the information trans-mission with a limited amount of energy consumption. Thus the following maximiza-tion problem has to be solved

L(E) = maxqI(Z;Y ), E = ∑j

q jE j, E < const. (5.13)

The optimal solution can be calculated using the Blahut-Arimoto algorithm (Arimoto,1972) and (Blahut, 1972), here the formulation used in a study from Balasubramanianet al. (Balasubramanian et al., 2001) is adopted.

Eq. (5.13) can be expressed in terms of the transition matrix Q, eq. (5.10) and the

distribution of input signals y j with probabilities q j. Pjk is an abbreviation forq jQk j

∑ j q jQk j,

hence

L(E) = maxq(−∑

j

q jlnq j +∑jk

q jQk jlnPjk),E = ∑

j

q jE j. (5.14)

The properties of the neuron model and coding scenario enter via Q. Here, Q is ap-proximated by a conditional Gaussian distribution. The moments of the Gaussian aredetermined from the first and second moment of the spike count distribution of theoutput as in the previous section.

Blahut (Blahut, 1972) and Arimoto (Arimoto, 1972) rephrased the problem as adouble maximization w.r.t. q j and Pj,k. The advantage is that the maximization can nowbe computed numerically by an iterative algorithm that alternately maximizes L(E)with respect to q j and Pj,k while holding the other variable fixed. The correspondingalgorithm is formulated and implemented as in (Balasubramanian et al., 2001):

• Choose arbitrary nonzero q(0)j .

• For each iteration t = 0,1,2, . . . repeat:

– P(t)jk ←

q(t)j Qk j

∑ j Qk j

47

– qt+1j ← e

−βE j−H(t)j

∑−β E j−H(t)j

,β s.t. ∑ j q(t+1)j E j = E

H j = Qk j lnPjk

– If q(t)j converges, then stop

Figure 5.3 displays the results of applying the above algorithm to the LIF neuron withadditive white noise and constant current inputs. The optimal input distributions areshown. Subfigures (a) and (b) differ in the standard deviation of the additive noise, in(a) a small noise level is applied and in (b) a large one. The six subplots in (a) and (b)show the optimal input distributions with increasing energy resources. For increasingamounts of available energy more expensive and thus reliable symbols are chosen. Forsmall energy budgets low energy symbols and thus more unreliable ones do contributeto information transmission. Employing high noise levels allows the use of very weakand unreliable symbols as in (b). The plots within (a) and (b) to the very right displaythe cumulative probability weight of sub-threshold input signals versus the availableenergy. These indicate, that a large proportion of the optimal input symbol distributionscorrespond to sub-threshold inputs.

5.7 Discussion and Conclusion

In this chapter the existing approach of Levy et al. (Levy & Baxter, 1996) for investi-gating the representational capacity of model neurons has been extended to incorporatemore detailed properties of model neurons. It is found that for the leaky integrate-and-fire model neuron with constant current input and additive white noise, the represen-tational capacity is much lower then in the more abstract model of Levy et al. (Levy& Baxter, 1996). Further, the capacity is not a monotonically increasing function ofthe mean output rate anymore. A maximum appears for intermediate spike rates. Thismaximum appears for sub-threshold inputs at low noise levels and for super-thresholdinputs at higher noise levels. Thus it is demonstrated that weak inputs might exploitlarge parts of the representational capacity. The optimal input symbol distributionswhich maximize the mutual information, given a fixed amount of available energy havebeen determined for different amounts of energy and noise levels. The results supportthe hypothesis that as neural firing becomes energetically more expensive, weak signalsbecome more beneficial, despite the fact that their transmission is less reliable. Severaleffects contribute to the shape of the optimal input symbol distribution: the output en-tropy, the channel noise, the metabolic cost function and the available energy. A toughrestriction on the energy budget enforces the use of weak and unreliable symbols.

The approach described in section 5.6 is quite general and allows to predict theunique symbol distribution that maximizes information transmission given energy con-straints. It might be applied in any system where a suitable discretization of the neuralcode is available, along with a description of noise and cost, as was already pointed outin (Balasubramanian et al., 2001)

Within the metabolic cost function employed in this section the cost of noise is notexplicitly considered, but it is assumed to contribute to the baseline costs, independentof the noise level. An explicit model could be more satisfying. Changing the noiselevel of the membrane potential of a single neuron would correspond to changing one,or some, of the following properties of the nervous system:

• The number of spike trains a neuron receives,

48

• the rate of these spike trains,

• higher moments of these spike trains,

• their temporal statistical structure,

• cross correlation between these spike trains,

• the strength of the synapses,

• and integration on the dendrite.

This list is not meant to be complete, it just gives a hint on some possible mechanismswhich could be involved. Some mechanisms might reasonably be modeled by a con-stant contribution to the baseline cost, as e.g. a change in cross-correlation between theincoming spike trains. As a consequence the net amount of spikes is constant, but themembrane potential variance changes. Other possibilities, like changing the number ofspike trains do certainly contribute significantly to the metabolic cost.

For simplicity a linear relation between the metabolic cost and the average numberof spikes within a certain time interval is chosen. First results employing a HH likepoint model neuron indicate, that this is, in general, not the case. These preliminaryresults were obtained by Roberto Lopez Sanchez during his time in the NI group as amaster student. Studying the metabolic cost, which is related to information transmis-sion might be based upon studying the ion-current turn-around due to neural activity,since large amounts of energy are consumed for the corresponding ion pumps.

49

Chapter 6

Population Coding

The work presented in this chapter was done in close collaboration with Thomas Hoch.This chapter is based on common publications (Hoch et al., 2003a; Hoch et al., 2003b).Most of the work was done during his diploma-project as a computer science student.I had the pleasure to guide and supervise his work.

6.1 Abstract

In chapter 4, a learning rule for adaptation to the optimal noise level in a stochastic res-onance scenario for single neurons has been suggested. In general, the optimal noiselevel depends on the stimulus statistics, the cost function, on the number of neuronsin a population, and on their properties. Using a standard method for quantifying in-formation transmission of time dependent stimuli, see e.g. (Collins et al., 1995b) and(Rieke et al., 1997), the dependency of the optimal noise level on the population sizein an abstract model framework is investigated. This method is not without problems,see discussion. It is demonstrated, that for a large enough number of neurons the latterdependency becomes weak, such that the optimal noise level becomes almost inde-pendent of the number of neurons in the population. First a binary threshold modelof neurons is investigated. An analytic expression for the optimal noise level at eachsingle neuron is derived, which - for a large enough population size - depends only onquantities, which are locally available to a single neuron. Using numerical simulations,then the weak dependence of the optimal noise level on population size in a more real-istic framework using leaky-integrate-and-fire as well as Hodgkin-Huxley type modelneurons is verified. Next a cost function is constructed. The quality of informationtransmission is traded against its metabolic costs. Again it is found that - for sub-threshold signals - that there is an optimal noise level which maximizes this cost. Thisnoise level, however, is almost independent of the number of neurons, even for smallpopulations sizes, as numerical simulations using the Hodgkin-Huxley model show.Since the dependence of the optimal noise level on population size is weak for largeenough populations, local neural adaptation is sufficient to adjust the level of noise toits optimal value.

50

6.2 Introduction

In chapter 4 it has been suggested, that neurons may adjust their individual noise levelbased on quantities, which are accessible to a single neuron and its elements. In thecentral nervous system of higher animals, however, single neurons rarely matter, andinformation is likely to be coded using populations of cells (Pouget et al., 2000). Forthe cerebral cortex of higher animals, population size has been estimated to be between100 and 200 neurons (Feldmann, 1984), but this number may differ between brain ar-eas and species and the method used to estimate this value. Hence the question arises,whether noise may aid the transmission of weak signals also through populations ofneurons and what quantities optimal signal transmission depends on. Recently, Collinset al. (Collins et al., 1995b) and Stocks (Stocks, 2001) have examined the informationtransmission properties of a summing network of FitzHugh-Nagumo model neurons.They showed that stochastic resonance occurs and that there exists an optimal noiselevel for signal transmission. However it turned out, that the optimal noise level doesnot only depend on the input distribution and on neural properties, but also on the sizeof the neuron population. If the dependency on the size is strong, local adaptation rules- as suggested in chapter 4 - might not suffice, because information about populationsize would have to be made available to the single neuron.Another issue to be considered is, that maximal information transmission may not bethe only goal neurons try to achieve. Neural activity is costly in metabolic terms, andenergy consumption and dissipation becomes a concern. One would expect that the de-pendency of the optimal noise level on the input distribution and its dependency on theproperties of the neuron population change as soon as metabolic constrains are added.For a brief review of the relevant literature see chapter 5.In this model study, therefore, the complex relationship between information trans-mission, energy consumption, noise, population size, and the statistics of the inputdistribution is explored. This is done for three classes of neuron models: binary thresh-old neuron (Stocks, 2000; Stocks, 2001), integrate-and-fire neurons (Tuckwell, 1998),and conductance-based point neurons (Destexhe et al., 2001). Binary threshold modelsare simple enough to be analytically tractable and - together with the integrate-and-firemodel - have been widely studied in the context of noisy information transmission. Thisallows to directly compare the results of this study with results already published in theliterature. Conductance-based point neurons (i.e. Hodgkin-Huxley neuron) on the otherhand are biologically more realistic, because the input-, noise- and activity-inducedchanges in membrane conductances and their influence on the neuron’s dynamics aretaken into account. The numerical results are (qualitatively) consistent across the dif-ferent models.This chapter is organized as follows. In section 2 the binary threshold model is in-vestigated and an analytical expression for the optimal noise level is derived. An ap-proximation introduced by Brunel et al. (Brunel & Nadal, 1998), which is valid in thelimit of a large number of neurons is used. It is found that the number of thresholdelements for which the analytical expression holds, coincides with the typical numberof neurons within a cortical subpopulation of neurons (Feldmann, 1984). Comparedto other studies (e.g. (Stemmler, 1996)), the analytic expression for the optimal noiselevel does not only depend on the mean of the stimulus distribution, but also on highermoments. The main results are then verified in section 3 with the biologically morerealistic integrate-and-fire and Hodgkin-Huxley type models of neurons. It is assumedthat the noise inputs are balanced, i.e. that they consist of inhibitory and excitatory in-puts with equal efficacy on average. Using a standard method of estimating information

51

Figure 6.1: A population of binary threshold neurons. Each model receives the sameinput signal and independent Gaussian noise inputs ηi, i ∈ 1, ..,N of zero mean andequal variance. Each noise input is independent of the signal and the other noisesources. The total output Z is the sum over all individual outputs Yi (called poolingin neurophysiological terminology) and is equal to the number of active neurons.

transmission with time-dependent stimuli, it is shown that for populations of spikingneurons, which exceed a certain size, the optimal noise level depends only weakly onthe number of neurons. For small populations, however, the noise level has to be ac-curately adjusted to the number of neurons. In Section 4, metabolic constraints areincluded and it is found that the dependency of the optimal noise level on populationsize changes dramatically. If information transmission is normalized by metabolic costas in (Levy & Baxter, 1996; Balasubramanian et al., 2001; de Polavieja, 2002), theoptimal noise level is almost constant with population size, even for populations withonly a few (> 5) neurons. Calculating the input distribution which is optimal with re-spect to bits per unit metabolic costs it is found, that most of the inputs are actuallysub-threshold.

6.3 The Binary Threshold Model

6.3.1 Architecture of the Model

In this section a population of N binary threshold elements is considered. The totalinput to each neuron i is the sum of a common input signal X and an individual noiseinput ηi, and the output Yi of all these neurons is summed (fig. 6.1, see also (Stocks,2000)).

The output Yi of the single elements is set to one (active), if the total input exceedsa threshold Θ, i.e.

yi =

0, if x+ηi ≤Θ1, if x+ηi > Θ.

(6.1)

The signal X is the same for all threshold elements and is drawn from a Gaussiandistribution PX with mean µX and variance σ 2

X . The noise inputs ηi have a Gaus-

52

sian distribution with zero mean and variance σ 2η , and are mutually independent of

the signal X and the other noise sources. Let Z represent the number n of neuronsthat are set to one for a given realization of X. The distribution of Z is then equal toPZ(n) =

∫ ∞−∞ dxP(Z = n|x)PX(x), where the conditional probability P(Z = n|x) can be

calculated fromP(Z = n|x) = (N

n )Pn1|x(1−P1|x)

N−n. (6.2)

P1|x is the conditional probability, that the output of a neuron is set to one and is givenby

P1|x =

∫ ∞

Θ−x

1√2πση

exp

(

− η2

2σ 2η

)

dη . (6.3)

6.3.2 Approximation of the Mutual Information

The mutual information is an information measure which can be used to quantify theamount of information the output Z of the neural population contains about the input X .A brief introduction is given in chapter 3, for a comprehensive introduction see (Cover& Thomas, 1991). The mutual information IMI between the input distribution PX ofthe signal and the output distribution PZ is given by

IMI = H(Z)−H(Z|X) (6.4)

= −N

∑n=0

P(n) log2 P(n)+ (6.5)

∫ ∞

−∞dxPX(x)

N

∑n=0

P(n|x) log2 P(n|x).

Brunel and Nadal (Brunel & Nadal, 1998) have shown that in the limit of a large num-ber of neurons, the mutual information between input and output becomes equal to themutual information between the input signal and an efficient Gaussian estimator of theinput signal calculated from the population output. Let

x = g−1(nN

) (6.6)

be the maximum likelihood estimator of the signal input x calculated from P(Z = n|x)where g(.) is the error function. This estimator is asymptotically unbiased, efficientand Gaussian distributed around its mean value. Its variance is 1

F(x) , where F(x) is theFisher information:

F(x) = E

[

−∂ 2 log2 P(Z = n|x)∂x2

]

x(6.7)

=

(∂P1|x∂x

)2N

P1|x(1−P1|x). (6.8)

The amount of information the maximum likelihood estimator X contains about thestimulus is then given by:

I(X , X) = H(X) −∫ ∞

−∞dxPX(x) H(X |X = x). (6.9)

53

In the limit of large N, one can approximate the entropy H(X |X = x) with the entropyof a Gaussian distribution with variance 1

F(x) and - because the estimator X is unbiased

- one can replace the entropy H(X) with H(X). One obtains:

I(X ,X) → IF = H(X)− (6.10)∫ ∞

−∞dxPX(x)

12

log2

(2πeF(x)

)

.

Since any processing of signals cannot increase the information content, the mutualinformation between the signal input and the output of the population is at least equalto IF (Brunel & Nadal, 1998),

IMI(X ,Z) ≥ I(X ,X) ≥ IF . (6.11)

IF is used to obtain an analytical expression for the optimal noise level σ Fopt . From the

condition∂ IF

∂ση=− ∂

∂ση

∫ ∞

−∞dxPX(x)

12

log2

(2πeF(x)

)

= 0 (6.12)

one obtains

σFopt '

√(

1− 2π

)

(µ2x +σ 2

x ), (6.13)

where log2(2πeF(x) ) has been replaced by its second order Taylor expansion around x0 = 0

(see appendix 11.6). In this approximation the optimal noise level depends on the firstand second moments of the input distribution, and it is independent of the number ofneurons in the population. For non-Gaussian input distributions higher moments ofthe input distribution can be considered if the Taylor expansion is carried out to higherorder (see appendix 11.6).

6.3.3 Analysis of Information Transmission

In fig. 6.2 (a) the mutual information IMI and its approximation IF are plotted againstthe relative strength σ = ση/σx of the noise for various numbers N of neurons in thepopulation. The threshold Θ was set to Θ = µx = 0. The curves show that for increasingsize N, IF becomes a good approximation of the mutual information IMI .

Note that IF can be negative for small values of noise, because the approximationeq. (6.10) is valid only in the case F(x) 1, when the distribution of the estimatorX is sharply peaked around its mean value. Furthermore, fig. 6.2 (a) shows, thatthe location σ F

opt of the maximum of the Fisher information is independent from thenumber of threshold elements, and approaches the optimum of the mutual informationIMI as the number of threshold elements increases. For finite N and using eq. (6.10),the optimal noise level is overestimated, as can be seen in fig. 6.2 (b). If the numberof neurons in the population is sufficiently large (N ≥ 100), however, the optimal noiselevel calculated by IF , leads to almost optimal information transmission (see fig. 6.2(b)). Figure 6.3 (a) shows the relative deviation (|IMI

max− IMI(σ Fopt)|)/IMI

max of IMI(σ Fopt)

from the maximum IMImax of the mutual information as a function of the population size

N. IMI(σ Fopt) is the mutual information at the optimal noise level of IF , where σ F

optwas obtained from a Monte Carlo integration of eq. (6.10). The figure indicates thatthe relative deviation is less than 5 percent for populations with a size of 100 or moreneurons. In fig. 6.3 (b), σ F

opt calculated from eq. (6.13), is compared to the optimal

54

A

σ

N = 128N = 32N = 8

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

Noise level

Mut

ual i

nfor

mat

ion

[Bit]

(a)

B

Mutual informationApproximation

100 200 300 400 5000

0.25

0.5

0.75

Number of neurons

Opt

imal

noi

se le

vel

(b)

Figure 6.2: (a) Mutual information IMI (thick lines) and its approximation IF according toeq. (6.10) (thin lines) as a function of σ =

σησx

for N = 8, 32, 128 from bottom to top and forΘ = µx = 0. The mutual information was calculated from eq. (6.5), where x was discretized into100 ·N bins. 105 ·N samples are used to estimate the distributions P(Z = n|x),PZ(n) and PX(x).(b) Optimal noise level of the mutual information (solid line) in comparison with the optimalnoise level of IF obtained from eq. (6.10) (dashed line)as a function of the number N of neuronsin the population.

55

noise level of IF , obtained from eq. (6.10). σ Fopt is close to but slightly below the

optimal noise level of IF . Note, that σ Fopt is not zero for supra-threshold input signals

(µx > Θ), rather information transmission is symmetric with respect to the deviation ofµx from the threshold θ . This is due to the intrinsic symmetry of the model. Withoutnoise, the output Z is equal to N for all x > Θ, hence noise is needed to distinguishbetween supra-threshold inputs as well.

6.4 Populations of Spiking Neurons

To verify the results of the previous section in a biologically more realistic framework,the population of binary neurons is replaced by a population of spiking neurons. Theleaky integrate-and-fire as well as the Hodgkin-Huxley type models are employed.

6.4.1 LIF Framework

The membrane potential V of the leaky integrate-and-fire neuron changes in time ac-cording to the differential equation

CmdV (t)

dt=−gL(V (t)−EL)+ Istim(t)+σ

dW (t)dt

, (6.14)

where Cm is the membrane capacitance, gL the leak conductance of the membrane, EL

the reversal potential, Istim(t) the external signal, and dW (t) is the infinitesimal incre-ment of a Wiener process which is used to take into account the effect of the noiseinputs (Tuckwell, 1998). For an introduction to LIF neurons see chapter 2. Equation(6.14) describes the sub-threshold dynamics of the membrane potential, as it ignoresall active membrane conductances, under the assumption that the synaptic current gen-erated by random synaptic inputs (the background activity) can be approximated by aWiener process. σ is chosen to be identical for all neurons within the population. Oncethe membrane potential reaches the threshold a spike is generated and the membranepotential is clamped to a reset value Vreset for an absolute refractory period Tre f . A con-tinuous aperiodic Gaussian input signal Istim(t) is generated by a Fourier transform ofa band-limited white noise power spectrum, which is added to a constant bias currentIbias. The parameters used for the simulations are given in appendix 11.3.

6.4.2 HH Framework

Hodgin-Huxley type models and the point conductance model (Destexhe et al., 2001)are introduced in chapter 2. For better readability the most important terms are intro-duced here as well. The membrane potential V of the Hodgkin-Huxley neurons usedin this study changes in time according to the differential equation:

Cm∂V∂ t

=−gL(V −EL)− INa− IK− IM− Isyn + Istim(t), (6.15)

The left hand side of the equation describes the influence of the membrane’s capaci-tance, while all ionic currents through the cell’s membrane - including the noise andstimulus currents - are summed on the right hand side. Following currents are consid-ered: A leak current IL = gL(V −EL), the spike-generating sodium INa = gNa m3h (V −

56

A

100 200 300 400 5000

5

10

15

Number of neurons

Rel

ativ

e de

viat

ion

[%]

(a)

B

σxσx

µx

= 1 = 2

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

Signal input

Opt

imal

noi

se le

vel

(b)

Figure 6.3: (a) Relative deviation (|IMImax− IMI(σ F

opt)|)/IMImax of IMI(σ F

opt) from the max-imum IMI

max of the mutual information as a function of the population size N. IMI(σ Fopt)

is the mutual information at the optimal noise level of IF . The horizontal line indicatesthe 2% level. (b) The optimal noise level according to eq. (6.13) (thin line) and ac-cording to the maximum of IF directly obtained from eq. (6.10) (thick line) for variousvalues of µx. σx = 1 (solid lines) and σx = 2 (dashed lines), Θ = 0.

57

ENa) and potassium IK = gK n4 (V −EK) currents as well as an additional noninacti-vating potassium current IM = gM p (V −EK). The parameters used for the simulationsare given in appendix 11.4. The total synaptic noise current Isyn is generated by thechanging membrane conductance induced by stochastic spike trains arriving at excita-tory and inhibitory synapses. The aperiodic Gaussian stimulus Istim(t) is generated asdescribed in the previous section. A detailed description of the currents can be foundin (Destexhe & Pare, 1999) and in appendix 11.5.In the model introduced by Destexhe et al. (Destexhe et al., 2001) the total synapticcurrent Isyn is obtained as a superposition of excitatory (e) and inhibitory (i) inputs,

Isyn = ge(t)(V −Ee) + gi(t)(V −Ei), (6.16)

where ge and gi are the synaptic conductances and Ee and Ei the corresponding reversalpotentials. The time-dependent conductances are described as an Ornstein-Uhlenbeckprocess,

dge(t)dt

= − 1τe

[ge(t)−ge0] +√

DedW (t)

dt, (6.17)

dgi(t)dt

= − 1τi

[gi(t)−gi0] +√

DidW (t)

dt. (6.18)

The stationary variance of the Ornstein-Uhlenbeck process expressed in terms of thediffusion coefficients De,Di is then σ 2

e,i =De,iτe,i

2 . The parameter of the Ornstein-Uhlenbeck processes are chosen in such a way that they resemble in-vivo like activity(Destexhe et al., 2001) (see appendix 11.5). Different noise conditions are modeledby changing the synaptic conductances and the square root of the diffusion coeffi-cients of the Ornstein-Uhlenbeck process by a common gain factor α , α ∗ (ge0,gi0)and α ∗ (

√De0,√

Di0). The numerical values of ge0,gi0,De0,Di0 are given in the ap-pendix 11.5.

6.4.3 Quantification of Information Transmission

Information transmission through a population of spiking neurons is much harder to es-timate than information transfer through binary threshold elements, because the amountof data needed to get a reasonable estimate of the probability distributions explodes. Anapproach is used which was recently described in literature (Rieke et al., 1997; Koch& Segev, 1998). Unfortunately this approach is not without problems, first results (byThomas Hoch) indicate, that this approach does introduce a N-dependence in the esti-mate on the transmitted information. See the discussion for more information on thissubject. The main idea of this approach is that a lower bound on the information ratecan be obtained by computing an estimate of the input signal from the observed spiketrain. A sketch of the procedure is given in figure 6.4 (a). The estimate is calculatedwith a method called Wiener-Kolmogorov filtering, and contains no information, thatwas not actually present in the spike train. To get the estimate, the spike train z(t) isconvolved with a filter h, which minimizes the mean square error

ε2(h) =< |Istim(t)−h∗ z(t)|2 > (6.19)

between the stimulus Istim(t) and its estimate Iest(t) = h ∗ z(t). This filter h representsa non causal optimal linear filter of the spike train and can be obtained by solving the

condition dε2(h)dh = 0 for h. This yields h(w) = SIX (−ω)

SXX (ω) , where SXX(ω) is the power

58

spectrum of the spike train and SIX(ω) denotes the Fourier transform of the cross-correlation of the spike train and the stimulus. Next the Fourier components of theeffective noise are calculated ne f f (ω)), via the relation

Iest(ω) = g(ω)[Istim(ω)−ne f f (ω)

], (6.20)

where Istim(ω) and Iest(ω) are the Fourier transforms of the input signal and its esti-mate, g(ω) is the frequency dependent gain introduced to correct for systematic errors,which arise from filtering the spike train (see Rieke et al. (Rieke et al., 1997) for furtherinformation), and ne f f is the effective noise in the estimate. The stimulus and its esti-mate are divided into segments of approximately one second and the Fourier transformof each segment is calculated. Given enough segments, the frequency dependent gaing(ω) and the Fourier components of the effective noise ne f f (ω) can be determined us-ing linear regression (cf. eq. (6.20)). The power spectrum of the effective noise levelNe f f (ω) is obtained by calculating the variance of the noise components ne f f (ω) nor-malized by the time window Tn, Ne f f (ω) = var(ne f f (ω))Tn. Finally, the lower boundRin f o to the information rate is calculated using,

Rin f o =12

∫ ∞

−∞

dω2π

log2

[

1+S(ω)

Ne f f (ω)

]

. (6.21)

which is close to the real information rate, if the effective noise is approximately Gaus-sian distributed (Rieke et al., 1997). This is the case here as shown in fig. 6.4 (b).The procedure is depicted in figure 6.5 (a). The LIF neuron is employed for this demon-stration. In fig. 6.5 (a) the current injection is shown as a function of time. The neuronreceives this current input and additional white noise as inputs, the response is shownin fig. 6.5 (b). The optimal Filter is calculated according to eq. (6.19) and shown insubfigure (c). Subfigure (d) shows the original stimulus current and the correspondingestimate.

6.4.4 Results of Numerical Simulations

Figure 6.6 (a) shows the results obtained with the leaky integrate-and-fire model. Thelower bound RIn f o of the information rate is plotted as a function of the input noisefor different numbers of neurons in the population. The bias current was set to Ibias =0.5nA and the standard deviation was equal to std(Istim) = 0.2nA. Figure 6.6 (a) shows,that the optimal noise level increases with the number of neurons in the population,similar to the binary threshold model. The optimal noise level is plotted as a func-tion of the population size in fig. 6.7 (a) (dashed line). It depends weakly (close tologarithmically) on the number of neurons in a biologically reasonable range.

Figure 6.6 (b) shows the results obtained with the Hodgkin-Huxley model. Rin f o isplotted as a function of the noise level for different numbers of neurons. Because themean firing rate of the Hodgkin-Huxley neurons is smaller than the mean firing rateof the leaky integrate-and-fire neurons for the same stimulus, the information rate issmaller than the rate given in fig. 6.6 (a). Again, the optimal noise level (fig. 6.7 (a),solid line) depends weakly (close to logarithmically) on the number of neurons. Note,that the maxima of the the information rate are broad in both cases (fig. 6.6 (a) and(b)), i.e. the amount of transmitted information degrades only slightly if the level ofinput noise deviates from its optimal value.

Figure 6.7(b) shows, how information transmission depends on the number of neu-rons in the population, for a given level of noise. The noise level was set to its optimal

59

(a)

NeffGaussian

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

Normalized noise amplitude

Pro

babi

lity

dens

ity

(b)

Figure 6.4: (a) A population of neurons receives signal and noise inputs. A corre-sponding spike train is emitted, which is fed into an optimal filter to yield an estimateof the signal. The deviation is used to determine the quality of the transmission. (b)Distribution of the normalized effective noise level Ne f f in comparison to a Gaussiandistribution with the same variance. The histogram is constructed from 5000 segmentsof 819ms duration (dt = 0.2ms). The bias current was set to Ibias = 0.5nA, the stan-dard deviation of the stimulus was set to std(Istim) = 0.2nA and the input noise levelwas equal to σ = 1. Numerical simulations for different levels σ of input noise anddifferent values of Ibias lead to similar results.

value for a population size of N = 100 neurons. The solid line shows the results for theintegrate-and-fire model. The relative deviation (|IIR

max− IIR|)/IIRmax from the maximum

IIRmax of the information rate is plotted for different numbers of neurons. Figure 6.7(b)

shows that the loss in information transmission is less than 2 percent for populationsof neurons with a population size between 80 and 130 neurons. The dashed line in fig.

60

StimulusEstimate

50 100 150 200 250 300 350 400

-0.2

-0.1

0

0.1

0.2

Time [ms]

Pro

babi

lity

of a

spi

ke

Membrane voltage

50 100 150 200 250 300 350 400

-70

-60

-50

-40

-30

-20

-10

0

Time [ms]

Vol

tage

[mV

]

Optimal Wiener filter

50 100 150 200 250 300 350 400-0.05

0

0.05

0.1

0.15

0.2

Time [ms]

Cur

rent

[nA

]

Stimulus

50 100 150 200 250 300 350 400

-0.2

-0.1

0

0.1

0.2

Time [ms]

Cur

rent

[nA

]

(d)(c)

(b)(a)

time [ms]0 100 200

time [ms]0 100 200

time [ms]0 100 200

time [ms]0 100 200

Cur

rent

[nA

]

-0.2

0.2

Vol

tage

[mV

]

-70

0C

urre

nt[n

A]

-0.05

0.2

Cur

rent

[nA

]

-0.2

0.2

Figure 6.5: Estimating the information transmission rate according to eqns. (6.19) to(6.21). The procedure is demonstrated for a LIF neuron. (a) A signal current Istim isapplied to the model neuron. Ibias = 0.5nA and additional noise with std(Istim) = 0.2nA.(b) Corresponding voltage trace of the LIF neuron. (c) The optimal filter, calculatedaccording to eqns. (6.19) to (6.21) is shown. The filter has been calculated using 200samples of 819ms each. (d) Original signal and estimated signal. For the parametersof the LIF neuron see appendix 11.3.

6.7(b) shows the corresponding results for the Hodgkin-Huxley model. Again, the lossin information transmission is small.

6.5 Energy Efficient Information Transmission

Information transmission in the brain is metabolically expensive. If the cost of firingis high, than it is advantageous for the brain to use energy efficient neural codes , seechapter 5. Given a fixed amount of energy, there are several ways to achieve energyefficient information transmission. One strategy is to use an input distribution so thatthe energy constraint can be fulfilled. Another strategy is to maximize the informationrate per metabolic cost to transmit as much information as possible. In the following acloser look on both strategies will be taken.

61

σ

N = 128N = 32N = 8N = 1

0 1 2 3 4 5 6 7 8

0

10

20

30

40

50

60

Noise level

Info

rmat

ion

rate

[Bit/

sec]

α

N = 128N = 32N = 8N = 1

0 1 2 3 4 5

0

10

20

30

40

50

60

Noise level

Info

rmat

ion

rate

[Bit/

sec]

(a) (b)

noise level α noise level α

info

rmat

ion

rate

[bi

ts/s

]

info

rmat

ion

rate

[bi

ts/s

]

10

30

50

10

30

50

1 7 1 5

Figure 6.6: (a) The leaky integrate-and-fire neuron: The information rate calculatedfrom eq. (6.21) is plotted as a function of the input noise for different numbers of neu-rons in the population. The bias current was set to Ibias = 0.5nA. The standard deviationof the stimulus was equal to std(Istim) = 0.2nA and the information rate was calculatedfrom 200 samples of 819ms duration (dt = 0.2ms). N = 1, 8, 32, 128. The thresholdis located at 0.5nA (b) The Hodgkin-Huxley neuron: The information rate calculatedfrom eq. (6.21) is plotted as a function of the input noise for different numbers ofneurons in the population. The bias current was set to Ibias = 0.5nA. The standarddeviation of the stimulus was equal to std(Istim) = 0.2nA and the information rate wascalculated from 200 samples of 819ms duration (dt = 0.2ms). N = 1, 8, 32, 128. Thethreshold is located at 0.5nA

6.5.1 Optimal Input Distribution

In the previous chapter is is shown that the maximum of the information transmissiondepends on the number of neurons in the population and on the input distribution. If thenetwork is forced to use less energy for transmission, than the optimal input distributionis shifted into the sub-threshold regime, as will be shown in the following for the binarythreshold model.The probability distribution PX(x) of the input signal is calculated which maximizesthe information transmission under the constraint that the average cost of transmissionis kept below a given energy Emax:

maxP(x)

I(X ,Z) with E = ∑x

E(x)P(x) < Emax. (6.22)

In order to solve this optimization problem an iterative algorithm is used which isbased on the work of Arimoto (Arimoto, 1972) and Blahut (Blahut, 1972), and whichwas used in several other studies about energy efficient coding (Balasubramanian et al.,2001; Schreiber et al., 2002; de Polavieja, 2002). The procedure is described in moredetail in chapter 5. The input distribution PX(x) was discretized and the cost of trans-mission E(x) for a each x was defined to be equal to the average number of neuronsset to one, i.e. E(x) = b + ∑n nP(Z = n|x) where P(Z = n|x) is defined in eq. (6.2)and b is the fixed baseline cost. Because the range of the input X lies between infin-ity and minus infinity, the optimal input distribution is calculated for the transformedvariable P1|x (lying in the range [0,1], discretized to a resolution ∆P = 0.002) and trans-formed it back to the input space. The number of neurons in the population was set toN = 10000. Figure 6.8 (a) shows the optimal input distributions calculated with the

62

Blahut-Arimoto algorithm. The solid line is the optimal input distribution for unlim-ited energy consumption. In this case the average energy consumption is equal toEmax = N

2 +b.As one can see in fig. 6.8 (a) , the optimal input distribution is symmetric around

the threshold Θ = 0 and very close to a Gaussian distribution (dotted line). Limitingthe amount of metabolic costs (Emax = N

4 + b) destroys the symmetry and leads to aninput distribution which is mainly sub-threshold (dashed line). How much the optimalinput distribution is shifted into the sub-threshold regime depends on the value Emax,but clearly, energy efficient codes favor low rates and sub-threshold inputs.

6.5.2 Information Rate per Metabolic Cost

As a measure of metabolic efficiency the ratio between the transmitted informationRin f o and the total metabolic cost E is now considered. Is is assumed that the averagemetabolic cost E per unit time is a sum of a term proportional to the average rate r ofthe neurons and a term which contributes a fixed baseline cost b. It is set

E = cN(b+ r), (6.23)

where N is the number of neurons and c is a proportionality constant, which can beinterpreted as the average cost per spike. Both information rate and average cost dependon the number of neurons in the population and change with the actual noise level(Levy & Baxter, 1996; Laughlin et al., 1998). To investigate the relationship betweenthe noise level and the information rate per unit cost, the noise level in the populationof neurons is varied for a given population size N.Figures 6.8 (b) shows the results obtained from the Hodgkin-Huxley model. In fig. 6.8(b) the optimal noise level for the unconstrained case (dashed line) is compared withthe optimal noise level of the information rate per unit cost for different baseline cost(b = 0 solid line, b = 5 dotted line) for a sub-threshold input.

Surprisingly, if metabolic cost are taken into account, the dependency of the optimalnoise level on the population size N almost vanishes even for a small population size. Ifbaseline costs increase so does the optimal noise level, because the smaller dependencyof the total cost on the output activity r allows for higher output rates. In the limit oflarge baseline costs, the activity dependent costs ( rN) can be neglected, and the optimalnoise level of the information rate per unit cost is equal to the optimal noise level forthe unconstrained case. For less dominating baseline cost, the information transmissionbecomes more and more robust against changes in the population size even for smallpopulations. This flattening of the curve is a result of the steady increase of the cost,which yields a shift of the optimal noise level to lower values. The optimal noise levelof the information rate per cost is determined by the derivative of the information ratewith respect to the noise and the derivative of the cost with respect to the noise. Thedependency of the optimal noise level on the population size N almost vanishes becausethe slope of the information rate curve is nearly the same in the small noise regime fordifferent numbers of neurons and because the derivative of the cost with respect to thenoise is independent on the number of neurons.

63

6.6 Discussion

In this chapter it is examined how background activity affects the information trans-mission in a population of neurons. Using an abstract framework based on binarythreshold elements it is shown that the optimal noise level of the mutual informationIMI between an input distribution and the corresponding output can be approximatedby the Fisher information IF , calculated from eq. (6.10), for large enough populationsizes, the quality of this approximation increases with the number N of neurons in thepopulation. In general, the optimal noise level σ MI

opt for IMI depends on N, thus a neu-ron that should adapt to the optimal noise level must have some knowledge about thesize of the population. It is shown that for all N, σ F

opt is larger than σ MIopt and that with

increasing N this difference decreases. Due to the broad maxima and the asymme-try of the stochastic resonance curves (transmitted information vs. noise) a moderateoverestimation of the optimal noise level does not degrade the amount of informationtransmitted in a dramatic way, provided that the number of neurons in the population issufficiently large. Thus adaptation to σ F

opt instead of σ MIopt is reasonable for population

sizes N ≥ 100. Furthermore the analytic expression for the optimal noise level, in termsof IF , does not only depend on the mean of the input distribution, as in previous chap-ters, but also on the variance. More general, evaluating the integral in eq. (6.10) usingmore terms in the Taylor expansion yields a weighted sum of the moments of the inputdistribution.For the spiking neuron models the information rate is estimated by a reconstructionmethod. In the leaky-integrate-and-fire respectively Hodgkin-Huxley framework theoptimal noise level also depends strongly on N for small populations, but the depen-dency becomes weak, if N is large enough. These results have consequences for apossible adaptation of the neurons’ noise input to changing distribution of signal in-puts. If N is large, adaptation is local in the sense that it does require only quantitieswhich are locally available at the single neuron. On the other hand, the results indicatethat for small populations the background activity can have a wide influence on theinformation transmission properties of the population and therefore should be adjustedaccurately in each single neuron.

As already mentioned, not only the input and model properties have an influenceon the optimal noise level, but also the method to estimate the transmitted informa-tion. E.g. employing the discriminability as used in chapter 4, with correspondingconstant current inputs does yield the surprising result that the optimal noise level doesnot depend on the population size at all. This is demonstrated in the appendix 11.7.First results (by Thomas Hoch) indicate, that the approach employed in this thesis (assuggested in section 6.4.3) does introduce a N-dependence in the estimate on the trans-mitted information. A conclusion w.r.t. the quality of this method has not been reachedby now (July 2004). Since an interpretation as information rate might be a problem,another interpretation, though it is less general, is suggested. The estimation of theinformation rate, as suggested in section 6.4.3, can be interpreted as the best estimatethat is possible with a linear reconstruction method.

The simulations of the binary threshold model have shown that limiting the us-able energy for transmission leads to a shift of the optimal input distribution to thesub-threshold regime. Since application of noise is one way to allow for transmis-sion of otherwise sub-threshold signals, the strive for energy efficient codes may bea justification of stochastic resonance in neural systems. Back in the Hodgkin-Huxleyframework, taking the cost of information transmission into account, the dependency ofthe optimal noise level (information rate is estimated using the reconstruction method)

64

of the ratio between information rate and average cost on the number of neurons Nbecomes very weak. This holds even for small populations, N < 10, provided that thebaseline costs are small compared to the rate dependent costs.

Altogether one can conclude that noise can contribute to the enhancement of infor-mation transmission, and may be adapted via a learning rule, which depends on singleneuron properties only, even when the number of neurons in the population is small.

65

A

Hodgkin-HuxleyIntegrate-and-fire

100

101

102

0

1

2

3

4

5

Number of neurons

Opt

imal

noi

se le

vel

(a)

BHodgkin-HuxleyIntegrate-and-fire

50 100 150 2000

5

10

15

20

Number of neurons

Rel

ativ

e de

viat

ion

[%]

(b)

Figure 6.7: (a) The optimal noise level as a function of the number of neurons inthe population for the integrate-and-fire model (dashed line) in comparison with theHodgkin-Huxley model (solid line). (b) Relative deviation from the maximal informa-tion rate as a function of the number of neurons in the array for the integrate-and-firemodel (dashed line) in comparison to the Hodgkin-Huxley model (solid line). Thenoise level was chosen to be optimal for a population size N = 100. Parameters for (a)and (b) as in figs. 6.6 (a) and (b).

66

Opt. P(x) restrictedOpt. P(x) not restr. Gaussian distr.

-3 -2 -1 0 1 2 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

X

P(X

)

Aunconstrainedb = 0b = 5

0 10 20 30 40 500

0.5

1

1.5

2

Number of neurons

Opt

imal

noi

se le

vel

(a)

X-3 30 number of neurond10 50

P(X)

0.1

0.35op

timal

noi

sele

velα

0

4 (b)

Figure 6.8: (a) Optimal input distribution for the binary threshold model: The solid lineshows the input distribution which maximizes the information transmission throughthe network measured by the mutual information I(X ,Z) and without metabolic con-straints. The distribution is close to a Gaussian distribution (dotted line). The averagemetabolic costs in this case are Emax = N

2 +b. The dashed line shows the optimal inputdistribution under the metabolic constraint Emax ≤ N

4 +b. The distribution is shifted tothe sub-threshold regime. The optimal input distribution was calculated for the vari-able P1|x (discretized to a resolution ∆P = 0.002) and was transformed back to the inputspace (N = 10000, Θ = 0 and ση = 1). (b) Optimal noise level of the information rateper cost plotted against the number of Hodgkin-Huxley neurons in the population fordifferent baseline cost (b = 0 solid line, b = 5 dotted line). The dashed line is theoptimal noise level for the unconstrained case.

67

Chapter 7

Detection of Pulses in a ColoredNoise Setting

This chapter is based on the following manuscript (Wenning et al., submitted).

7.1 Abstract

In this chapter it is investigated, how the presence and the properties of backgroundnoise influence the ability of a neuron to detect transient inputs, a task which is im-portant for coincidence detection as well as for the detection of synchronous spikingevents in a neural system. Using a leaky integrate-and-fire neuron as well as a biolog-ically more realistic Hodgkin-Huxley type point neuron one finds that noise enhancesthe detection of sub-threshold input pulses and that the phenomenon of stochastic reso-nance occurs. When the noise is colored, pulse detection becomes more robust, becausethe number of false positive events decreases with increasing the temporal correlationwhile the number of correctly detected events is almost unaffected. Therefore, theoptimal variance of the noise also changes with the degree of temporal correlationsof the background activity. For the integrate-and-fire model these effects can be ex-plained using simple model assumptions for the distribution of the neuron’s membranepotential. Numerical simulations show that the leaky integrate-and-fire model and theHodgkin-Huxley type point neuron behave qualitatively similar.

7.2 Introduction

A simple computation a neuron can perform is the detection of coincidences of spikesor the detection of events of synchronous neural activity. These computations can beformulated as a special case of the task to reliably detect and respond to a transient inputand have experienced increasing attention in the recent past (see, e.g., (Diesmann et al.,1999; Rudolph & Destexhe, 2003; Rudolph & Destexhe, 2001a; Kempter et al., 1998;Chapeau-Blondeau et al., 1996)). In particular neural synchrony (Engel et al., 2001)is widely believed to be an important element of cortical processing, but recent experi-mental findings also point to the importance of specific spike sequences (Stopfer et al.,2003), which may give rise to transient events if proper delays are present. But howcan transient inputs be optimally detected, if a neuron is embedded in a background of

68

noise, and how can sub-threshold input pulses be transmitted using, for example themechanism of stochastic resonance?

Motivated by these questions, the response of a single model neuron to brief sub-threshold input pulses is investigated. Two approaches are used, first a leaky integrate-and-fire (LIF) neuron is considered, because it is simple enough to be analyzed math-ematically (Amemori & Ishii, 2001; Tuckwell, 1998). Then a Hodgkin-Huxley (HH)type point neuron is investigated, because this way additional effects resulting frominput-driven changes in the membrane conductance can be incorporated (Dayan & Ab-bott, 2001). In both cases, the neuron additionally receives background noise inputs,which are generated by an Ornstein-Uhlenbeck process, and which affect the mem-brane potential for the LIF (current noise) and for the HH neuron model (conductancenoise). Experimental findings (Destexhe et al., 2003; Destexhe et al., 2001) demon-strate, that colored noise, rather than white noise, provides the best model for the back-ground input. Here also the role of those temporal correlations is investigated, becauseonly little work has been devoted to understand the impact of these temporal correla-tions on neural information transmission and processing (Nozaki et al., 1999b; Capurroet al., 1998; Mato, 1998; Brunel et al., 2001).

This chapter is organized as follows. In section 7.3 the pulse detection scenario ispresented and pulse detection performance is quantified. The noisy LIF and HH neuronmodels are described in sections 7.4 and 7.5. Results for the LIF neuron are shown insections 7.6 (numerical results) and 7.7 (analytical results). The corresponding numer-ical results for the HH neuron are shown in section 7.8.

7.3 The Pulse Detection Scenario

A single neuron is considered, which receives a train of sub-threshold pulses and addi-tive colored noise as inputs. The details of the neuron models are described in the nextsections; here the problem of pulse detection is considered.

The input pulse train is regular, but the time interval between two successive pulsesis large compared to the membrane time constants, so that the preceding pulse hasno significant influence on the following one. Figure 7.1 shows a typical trace of theresulting membrane potential for the case of the LIF model. The input pulses aremarked by the vertical lines going downwards; the output spikes are marked by thevertical lines going upwards and crossing the spike threshold of the neuron. The traceshows that there are several incidences where an input pulse is immediately followedby a spike. This pair of events is called a correctly detected pulse. However, thereare also several incidences where the neuron does not respond to the pulse or where aspike occurs in the absence of a signal input. The presence of a spike is interpreted asa signature for the presence of an input pulse, the latter events correspond to the errorsthe neuron makes in the detection task.

In order to quantify the neuron’s response to the pulse train the total error is con-sidered. The total error is a sum of a term which is proportional to the number of thefalse positive events and a term which is proportional to the number of pulses that arenot detected, i.e. that are not immediately followed by an output spike. Let us considera regular pulse train which consists of n equidistant pulses, separated by a time interval∆T . The fraction Pm of missed pulses is then given by Pm = 1−Pc, when Pc is the frac-tion of correctly detected pulses. The total number of false positives, divided by thetotal number of input pulses, is denoted as Pf . (Note that Pf can easily take on valueslarger than one). We then define a total error Q for pulse detection as the weighted sum

69

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

−20

−10

0

10

20

30

40

50

t in ms0 1000 2000

Vtin

mV

-65

-75

-55

-25

Figure 7.1: Typical trace of the membrane potential of a LIF neuron, eq. (7.2), whichreceives colored noise input (τX = 5ms, D = 3.2 mV 2

ms cf. eq. (7.3)) and a train of sub-threshold input pulses (10Hz, pulse parameters: width dt = 0.1ms, height Vp = 18mV).The time of occurrence is indicated by vertical lines going downwards for the inputpulses and by vertical lines going upwards for the output spikes. Threshold and resetpotential are −45mV and −65mV.

Q = Pm + λ Pf . (7.1)

λ is a constant which weighs the cost of a false positive event against the cost whichoccurs when a pulse is missed. The optimal choice of λ depends on the detection taskwhich the neuron should perform. Note that the value of Q depends indirectly on theinter pulse interval ∆T . For longer intervals ∆T but fixed n more false positive eventsare likely to occur, so that the total error grows with increasing ∆T .

In many previous studies information transmission was quantified using the signal-to-noise ratio (SNR) ((Stemmler, 1996) and references therein), cross-correlation mea-sures (Heneghan et al., 1996), or the mutual information (Bulsara & Zador, 1996)between input and output. Pulse detection, however, is best quantified in terms of theprobability of missing a pulse and the number of false positive events. The total errorQ summarizes their contributions.

7.4 The LIF Model with Colored Noise

The LIF with colored noise input has been introduced in chapter 2. However, for abetter readability the most essential aspects are introduced here again. In contrast toprevious chapters the time t is from now on denoted as a subscript. The membranepotential Vt of the neuron changes in time according to the differential equation

dVt

dt= − 1

τVVt + Xt , (7.2)

where τV is the membrane time constant and Xt is the noise input at time t. Throughoutthis chapter τV = 5ms. Once the membrane potential reaches a threshold Vth (Vth =

70

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t in ms

Vtin

mV

0 100 200-75

-65

-55

Figure 7.2: Trace of the membrane potential of a LIF neuron for colored noise inputswith the two different time constants τX = 0.1ms (left) and τX = 10ms (right). Thediffusion coefficient D was adjusted to keep the variance of the membrane potentialconstant, D = 320 mV2

ms (left) and D = 3.2 mV 2

ms (right).

−45mV), a spike is generated and the membrane potential is reset to Vreset (Vreset =−65mV).

The noise input Xt is given by an Ornstein-Uhlenbeck process (see chapter 2 for anintroduction), which corresponds to low-pass filtered white noise with a time-constantτX and a diffusion coefficient D,

dXt

dt= − 1

τXXt +

√D

dWt

dt. (7.3)

dWt are the infinitesimal increments of the Wiener process Equations (7.2, 7.3) aresolved using the Euler integration scheme (see chapter 2). Figure 7.2 shows a typi-cal trace of the membrane potential for colored noise input with two different time-constants τX .

The signal input is modeled as a series of narrow rectangular pulses with width τp

(τp = 0.1ms) which generate a voltage jump of variable height Vp. These pulses areinjected with a repetition frequency of 1

∆T , where ∆T is the time interval between twosubsequent pulses. Different noise conditions are modeled by changing the time con-stant τx and the diffusion coefficient D of the Ornstein-Uhlenbeck process. If the spikeprocess is omitted, i.e. only the unconstrained dynamics of the membrane potential isconsidered, eqns. (7.2,7.3), a change in the parameters τX and D affects the variance ofVt but not its mean. The relation between the variance of Vt and the parameters of theOrnstein-Uhlenbeck process is given in eq. (7.13) in section 7.7. The variance value isused in order to characterize the ”strength” of the noise.

7.5 HH Framework

The HH model is introduced in chapter 2, in this section the most important aspects,which are relevant for this chapter, are briefly summarized. In the HH type point neu-

71

ron model employed in this chapter (Destexhe et al., 2001) the membrane potential Vchanges in time according to the differential equation

Cm∂Vt

∂ t= −IL− INa− IK− IM− Isyn− Istim. (7.4)

The left hand side of this equation describes the influence of the membrane’s capaci-tance Cm, while all ionic currents through the cell’s membrane, including the synapticnoise (Isyn) and the stimulus (Istim) currents, are summed on the right hand side. Fol-lowing (Destexhe et al., 2001) the following intrinsic currents are considered: a leakcurrent IL =−gL(V −EL), the spike-generating sodium (INa) and potassium (IK) cur-rents, and a non-inactivating potassium current (IM) which is responsible for spike fre-quency adaptation. Details of the intrinsic currents and model parameters are listed inthe appendix. They were chosen according to (Destexhe et al., 2001; Destexhe & Pare,1999) and are consistent with available experimental data from neocortical pyramidalneurons.

The total synaptic noise current Isyn is generated by fluctuating synaptic conduc-tances. These conductances are thought to be induced by stochastic spike trains whicharrive at the excitatory and the inhibitory synapses of the neuron. Following (Destexheet al., 2001)

Isyn = get(Vt −Ee) + git(Vt −Ei), (7.5)

where get and git are the conductances of the excitatory (e) and inhibitory (i) synapses,and Ee and Ei are the corresponding reversal potentials. The time-dependent conduc-tances are effectively described as an Ornstein-Uhlenbeck process

dget

dt= − 1

τe[get −αge0] + α

√De0

dWt

dt, (7.6)

dgit

dt= − 1

τi[git −αgi0] + α

√Di0

dWt

dt, (7.7)

where τe, τi are the time constants, ge0 gi0 are the average values of the synaptic con-ductances, and De0, Di0 are the diffusion coefficients. Different noise conditions aremodeled by changing the synaptic conductances and the square root of the diffusioncoefficient by a common gain factor α . Since the Ornstein-Uhlenbeck process mod-els the cumulative effect of many stochastic processes this corresponds to a simplisticmodel of the effect of increasing the synaptic peak conductances. The parameters ofthe Ornstein-Uhlenbeck processes can be found in the appendix 11.5.

The signal input Istim is modeled as a series of narrow rectangular current pulseswith width τp (τp = 0.5ms), with variable strength Ip, and with a repetition frequencyof 1

∆T , where ∆T is the time interval between two pulses.

7.6 LIF Framework, Results

Figure 7.3 shows the total error Q as a function of the variance of the unconstrainedmembrane potential for different values of the time constant τX of the noise. Thediffusion coefficient D was always adjusted to keep the variance of Vt constant whenτX was changed (see eq. (7.13)). The figure demonstrates that stochastic resonancecurves emerge, and that the presence of temporal correlations (finite values of τX ) im-proves pulse detection. If τX becomes larger, the minimum of Q decreases and becomes

72

0 10 20 30 40 50 60 70 80 90 1000.5

0.6

0.7

0.8

0.9

1

1.1

membrane potential variance in mV2

Q

1

0.510 50 100

τx

Figure 7.3: Total error Q, eq. (7.1), as a function of the variance of the membrane po-tential, eq. (7.13), for the LIF model for different values of τX (τX = 1,2,5,10,20ms).D is adjusted to keep the variance of Vt constant when τX is changed (see eq. (7.13)).Vp = 18mV, ∆T = 50ms. Simulation results for Q were obtained by integrating eqns.(7.2,7.3) and including the spike-reset process.

more shallow. The latter effect makes pulse detection more robust against a suboptimalchoice of the variance of Vt .

Figure 7.4 shows the total error Q as a function of the variance of the membranepotential for several pulse heights and for short- (fig. 7.4 (a)) as well as long-range (fig.7.4 (b)) temporal correlations of the noise. If input pulses are strong enough to make theneuron fire, performance is best if no noise is present. For sub-threshold (except for theweakest) pulses, stochastic resonance like curves appear and a finite optimal strengthof the noise exists. Below a critical pulse height, however, no minimum appears. Thisis due to the fact that below a critical pulse height the normalized number Pf of falsepositives changes more rapidly with the variance of the membrane potential than thefraction Pc of the detected pulses.

Figure 7.5 shows the fraction Pc of correctly detected pulses (fig. 7.5 (a)), the frac-tion Pm of missed pulses (fig. 7.5 (b)), and the fraction Pf of false positives fig. 7.5 (c))as a function of the variance of the membrane potential, again for different values ofthe time constant τX . The temporal correlations of the noise inputs do almost not affectthe number of correct detections; improved performance is almost exclusively due to adecrease in the number of false positives.

Figure 7.6 shows the optimal noise level in terms of the variance of the membranepotential as a function of the height of the input pulses. The lower and upper curvescorrespond to τX = 1ms and τX = 10ms. Pulses which induce jumps larger than 20mVare super-threshold, and any noise inputs are detrimental. At threshold, the optimalvalue of the variance of Vt jumps to a positive value. This follows from the specificshape of Pc and Pf as a function of the noise level. For a small variance of the membranepotential Pc grows rapidly with increasing variance of Vt but almost stagnates after thisinitial rise, while the number of false positive events starts rising only for a much largervariance value.

73

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

0

0.5

1.0

10 50 100

Q

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

Q

0

0.5

1.0

10 50 100membrane potential variance in mV2

(a) (b)

Figure 7.4: Total error Q, eq. (7.1), as a function of the variance of the membranepotential, eq. (7.13), for the LIF model for the case of short-, τX = 1 ms, (a) andlong-range, τX = 10 ms, (b), temporal correlations. Different lines correspond to dif-ferent pulse heights Vp = 9,11,13,15,17,19,21,23mV (bottom to top). ∆T = 50ms.Pulses which induce voltage jumps smaller than 20mV are sub-threshold. Simulationresults are obtained by directly integrating eqns. (7.2,7.3) and including the spike-resetprocess.

0 10 20 30 40 50 60 70 80 90 1000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 10 20 30 40 50 60 70 80 90 1000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

membrane potential variance in mV2

(a) (b) (c)

0 100 0 100 0 10050 50 50

0.5

0 0.5

1.0

0

0.5

Pc Pm Pf τXτX

τX

Figure 7.5: Fraction Pc of correctly detected pulses (a), fraction Pm of missed pulses(b), and the normalized number Pf of false positives (c) as a function of the variance ofthe membrane potential, eq. (7.13), for the LIF neuron. Different curves correspond todifferent choices of the parameter τX = 1,2,5,10,20ms. D was again adjusted to keepthe variance of Vt constant. Pulse train parameters as in fig. 7.3. Simulation results areobtained by directly integrating eqns. (7.2,7.3) and including the spike-reset process.

74

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

7 9 19pulse height in mV

optim

al v

aria

nce

of th

em

embr

ane

pote

ntia

l in

mV

2

10

25

40

0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

<V2> in mV20 100

Pc

Pf

0.5

Figure 7.6: Optimal noise level for signal detection, in terms of the total error Q, asa function of the height of the input pulses for the LIF model. The upper and lowercurves correspond to τX = 10ms and τX = 1ms. Data is shown for pulses which inducevoltage jumps of Vp = 7,9,11,13,15,19mV. For smaller pulse heights and for effectivepulse heights >= 20mV the optimal noise level is zero (for λ = 1). The inset showsPc and Pf as a function of the variance of Vt , for Vp = 19 mV. ∆T = 50ms. Simulationresults are obtained by directly integrating eqns. (7.2,7.3) and including the spike-resetprocess.

75

For pulses inducing a jump smaller than approx. 7mV the minimum in Q vanishes(see fig. 7.6). One observes a non-monotonic dependence of the optimal noise levelon the strength of the input pulses and a pronounced maximum for intermediate pulseheights. For larger temporal correlations of the noise input, i.e. for larger τX , the maxi-mum shifts to larger noise levels. These findings are discussed in more detail in section7.7 (e.g., fig. 7.10).

Note, that increasing λ , eq. (7.1), shifts the optimal noise levels to lower values.

7.7 Analysis of the LIF Model

7.7.1 Second Order Moments

In order to derive expressions for the second moments < X 2t >, < XtVt > and < V 2

t >(<> denotes an ensemble average). The discrete version (Euler-integration) of eqns.(7.2,7.3) is considered

Xt+1 = Xt −1

τXXt dt +

√DdWt (7.8)

Vt+1 = Vt −1τV

Vt dt +Xtdt. (7.9)

dWt are the infinitesimal increments of the Wiener process. After multiplication of theleft and right hand sides of eqns. (7.8,7.9) and after taking an ensemble average oneobtains

d < X2t >

dt+

2τX

< X2t >= D (7.10)

d < XtVt >

dt+

1τ

< XtVt >=< X2t >, τ =

τX τV

τX + τV(7.11)

d < V 2t >

dt+

2τV

< V 2t >= 2 < XtVt >, (7.12)

where it is made use of the fact that < Xt dWt >= 0 and all terms of order larger 1 in dtwere neglected as in (Gillespie, 1996). The stationary state is then given by

< X2t >=

DτX

2, < XtVt >=

DτX τ2

, < V 2t >=

DτX τV τ2

. (7.13)

7.7.2 Probability of Correct Detection

Equations (7.13) describe the second moment of the membrane potential in the absenceof pulses. The stationary probability distribution of the fluctuating membrane potentialis described by a Gaussian distribution with its mean set to the reset potential −65mVand with variance σ 2

V = 12 DτX τV τ . If a narrow input pulse arrives, the membrane poten-

tial is instantaneously shifted by an amount Vp, and the probability that the membranepotential becomes super-threshold is given by

N∫ ∞

Vth−Vp√2σV

exp(− V 2

2σ 2V

) dV, (7.14)

which is equal to the probability of correct detection. N is the normalization constantof the Gaussian distribution.

76

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

membrane potential variance in mV2

0 20 40 60 80

PC

0.1

0.5

1.0

Figure 7.7: Fraction of correctly detected input pulses as a function of the varianceof the membrane potential for the LIF model. Wiggly solid line: Fraction of cor-rectly detected input pulses by numerical evaluation of eqns. (7.2, 7.3) includingthe spike-reset process. Smooth solid line: Probability of correct detection accord-ing to eq. (7.14). Crosses: Fraction of correctly detected input pulses by numeri-cal evaluation of eqns. (7.2, 7.3). The spike reset process is not applied in case offalse positives. The variance σ 2

V of the membrane potential was changed by chang-ing τX while keeping < X2

t > constant. The different pairs of lines correspond toVp = 3,5,7,9,11,13,15,17,19,21,23,25 mV (bottom to top). Pulses, which inducechanges that are larger than 20 mV are super-threshold. ∆T = 50ms. τX takes onvalues between 0.1 and 20.0ms.

77

Figure 7.7 shows the fraction PC of correctly detected pulses according to a nu-merical evaluation of eqns. (7.2, 7.3) in comparison with the probability of correctdetection calculated according to eq. (7.14). Eq. (7.14) provides a good approximationof PC for sub- as well as for super-threshold input pulses. If the eqns. (7.2, 7.3) areevaluated including the spike reset process, small deviations from eq. (7.14) occur forlarge values of < V 2

t >. This is a result of an increased spiking activity due to falsepositive events. If the reset is neglected in the simulations eq. (7.14) fits perfectly.

7.7.3 The Gaussian Process Approximation

If Vp is set to 0 mV, eq. (7.14) corresponds to the probability of emitting a false positivespike. If this probability is high, however, the above approach is no longer appropri-ate because it discards the effects of a prolonged probability flux across the thresholdwhich then leads to a distribution of Vt which is strongly non-Gaussian. A better ap-proximation can be obtained using a Gaussian process approach (Amemori & Ishii,2001). According to (Amemori & Ishii, 2001; Tuckwell, 1998) the transition proba-bility density g(vt |vt−dt) of getting from some voltage vt−dt at time t− dt to anothervoltage vt at time step t is given by

g(vt |vt−dt) =1

√

2πσ 2V (1− e−2dt/τ)

× (7.15)

exp

− (vt − vt−dte−dtτ )2

2σ 2V (1− exp−2dt/τ)

.

τ is the correlation time constant of the Gaussian process and σ 2V is the variance of the

membrane potential. The probability density of spike events is then given as the flux

ρt =∫ ∞

θdvt

∫ θ

−∞g(vt |vt−dt)g0(vt−dt )dvt−dt (7.16)

of probability through the threshold. The density g0(vt) is given by

g0(vt) =

∫ θ

−∞g(vt |vt−dt)g0(vt−dt)dvt−dt +ρt−dtδ (vreset ), (7.17)

i.e. probability mass that crosses the threshold in the preceding step is injected at thereset potential without a refractory period. The average number of spikes within theinterpulse interval ∆T , i.e. the average number of false positives is then obtained byintegration of eq. (7.16).

Figure 7.8 shows the density of the membrane potential calculated by integratingeqns. (7.2,7.3), including spike-reset mechanism, in comparison with its approximationusing the Gaussian process. At t = 0ms the density is initialized as a δ -function. Att = 5ms (fig. 7.8 (a)) the density is significantly broader but only slightly skewed. Astime goes on, however, the density becomes more and more non-Gaussian, but theGaussian process model describes the density well, even close to its stationary state att = 20ms (fig. 7.8 (b)).

7.7.4 Quality of Pulse Detection

Figure 7.9 shows the fraction of correctly detected pulses, the number Pf of false pos-itive events per input pulse, and the total error Q as a function of the variance of the

78

−50 −40 −30 −20 −10 0 10 20 30 40 500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

−50 −40 −30 −20 −10 0 10 20 30 40 500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

membrane potential in mV

prob

. den

sity

prob

. den

sity

(a) (b)

-85 -65-45 -85 -65-45

0.01

0.05

0.01

0.04

Figure 7.8: Probability distribution of the membrane potential for the LIF model. TheGaussian process approximation (thin lines) is compared with the results obtained bythe integration of eqns. (7.2,7.3) (thick lines) including the spiking and reset process.The figure shows the profile at t = 5ms (a) and t = 20ms (b) after initialization witha δ -function at t = 0ms. Parameters were τX = 1ms, τV = 5ms, and τ = 7.5ms (cf.eq. (7.17)). The standard deviation σV of the stationary density in the absence ofa threshold is 10mV. τ was adjusted such that the spike rate of the LIF model andits Gaussian process approximation match each other for the stationary state. Thethreshold is indicated by the vertical dashed line at −45mV.

0 20 40 60 80 100 120 140 160 180 2000.5

0.6

0.7

0.8

0.9

1

1.1

0 20 40 60 80 100 120 140 160 180 2000

2000

4000

6000

8000

10000

12000

14000

0 20 40 60 80 100 120 140 160 180 2000

1000

2000

3000

4000

5000

6000

7000

8000

9000

100variance of the membrane potential in mV2

20 50100 10020 2050 50

Pc PfQ(a) (b) (c)

0.05

0.45

0.1

0.7

0.5

1.0

Figure 7.9: Fraction of correctly detected pulses (a), number of false positive eventsper input pulse (b), and the total error Q (c) as a function of the variance σ 2

V of themembrane potential for the LIF model. Thin (and smooth) lines denote the resultsobtained with eqns. (7.14) and (7.16), thick (and wiggly) lines denote the results ob-tained by integrating eqns. (7.2,7.3), including the spike and reset process. Differentpairs of lines, from top to bottom in (a) and (b), correspond to different values of τX ,τX = 1,2,5,10,20ms. The correlation time constant τ of the Gaussian process, eq.(7.15), was fitted to the data using a least-squares method. The values are (for (b)):τ = 7.5,11.5,20,28,35ms. Parameters are: Vp = 18mV, ∆T = 50ms, other parameteras in fig. 7.5.

79

0 5 10 15 20 25 30 35 40 450.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 10 20 30 40 50 60 70 80 900.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

variance of the membrane potential in mV2

(a) (b) (c)

Q Q Pm

Pf

10 10 1020 202030 30 3040 40 400.5

0.30.7

1.0

0.5

0.6

1.0

0.7

Figure 7.10: Total error Q, the normalized number Pf of false positive events, andthe probability Pm of not detecting an input pulse, as a function of the variance σ 2

Vof the membrane potential. Different curves correspond to different pulses with Vp =7,9,11,13,15,17,19mV (top to bottom). Dots indicate the optimal value of σ 2

V ofthe minimum of the total error Q. (a) Simulation results for Q obtained by directlyintegrating eqns. (7.2,7.3) and including the spike-reset process. (b) Results for Q byevaluating eqns. (7.14,7.16). (c) Pm and Pf calculated according to eqns. (7.14,7.16).Parameters are: τX = 1ms, τ = 7.5ms, ∆T = 50ms.

membrane potential and for different values of the correlation time-constant τX . Re-sults obtained by integration of eqns. (7.2,7.3) including spike-reset mechanism andresults obtained by evaluating eqns. (7.14,7.16) are superimposed. The fraction of cor-rectly detected pulses is roughly independent of the temporal correlations of the mem-brane potential, see fig. 7.9 (a). This finding agrees well with the simplified model,eq. (7.14), which predicts, that the fraction of correct detections should not depend onτX for constant σV . The small deviations in the high variance regime is a result of anincreased spiking activity due to false positive events. The number of false positiveevents per input pulse decreases with increasing strength of the correlation, and the ef-fect is nicely predicted by the simplified Gaussian process approach (fig. 7.9 (b)). Thetotal error Q is plotted in fig. 7.9 (c) for λ = 1. It shows a minimal value at an optimalvalue of σV , which - however - depends on the strength of temporal correlations in thenoise inputs. When temporal correlations increase, the number of false positives be-comes smaller and the total error decreases too. The reduction of false positive eventsis caused by a reduction of the number of rapid and large fluctuations of the membranepotential. Note, that the correlation time constant τ , eq. (7.15), of the Gaussian pro-cess was fitted to the results of the numerical simulations given in fig. 7.9 (b) using astandard least-squares method.

In fig. 7.6 it is shown that the optimal noise level, i.e. the optimal value for σV , isa non-monotonic function of the pulse height Vp. This effect is also well described byeqns. (7.14) and (7.16). Figure 7.10 shows the total error Q, the normalized numberPf of false positive events, and the probability Pm of not detecting an input pulse, asa function of the variance σ 2

V of the membrane potential and for different values ofthe height Vp of the input pulses. The optimal level for the variance of the membranepotential is indicated by dots.

Since Pf does not depend on the signal input, the non-monotonic dependence ofthe optimal variance is due to changes in Pm only. A characteristic of the optimal noiselevel is, that the slopes of Pf and Pm versus the membrane potential variance match

80

each other. Pf as a function of the noise level first starts with a very small slope, butgrows vigorously with increasing noise. In the regime where Pf starts growing, Pm hasa small slope for very weak pulses and large sub-threshold pulses. For intermediatenoise levels, the slope of Pm versus the membrane potential variance is larger, thus theoptimal noise level is shifted to higher values.

7.8 Simulation Results for the HH Framework

In this section it is investigated, whether and under what conditions the predictionsof the LIF model carry over to the biophysically more realistic HH framework. Thecorrelation structure of the noise inputs for the HH model is more complicated thanfor the LIF model. It must be described by two time constants, one for the excitatory,τe, and one for the inhibitory, τi, synaptic inputs (eqns. (7.6,7.7)). As a consequenceof this and of the fact that the HH model is mathematically more complicated, no ex-act closed expression can be derived for the variance of the membrane potential as afunction of the other model parameters. The following strategy is adopted. A value ischosen for α , eqns. (7.6,7.7), which is small enough (α = 0.3) so that the membranepotential remains sub-threshold. Then three sets τe, τi, De0, Di0 of parameters areconstructed - by try and error - (see caption of fig. 7.11). First the correlation structureof the noise inputs was changed by changing the values of the excitatory and inhibitorytime-constants. Then the diffusion constants were adjusted so that the variances of themembrane potential - determined by numerically integrating eq. (7.4) - match eachother. Pulse detection performance is then evaluated as a function of the strength α ofthe conductance noise. When α is increased, the variance of the membrane potentialbecomes larger while the average value of V remains (approximately) constant. Whenα becomes large (i.e. as soon as the number of false positive events becomes large),spiking activity strongly contributes to the temporal changes of the membrane poten-tial. Then its variance is no longer a proper measure in a stochastic resonance context,but α remains as a parameter for measuring the strength of noise.

Figure 7.11 shows the results. In figs. 7.11 (a) and (b) the fraction Pc of correct de-tections and the normalized number Pf of false positive events are plotted as a functionof the noise parameter α and for the three sets of parameters described above. The re-sults are qualitatively similar to the results of the LIF model. The fraction of correctlydetected events increases with increasing strength of the conductance noise, and thedependency is almost unaffected by the properties of the temporal correlations. Withhigher noise, the normalized number of false positive events increases, but strongertemporal correlations lead to a reduction similar to what is observed for the LIF neu-ron. Fig. 7.11 (c) shows the corresponding “stochastic resonance” curves for the totalerror Q (λ = 1). The optimal noise level changes with the height Ip of the input pulsesin a non-monotonic fashion (fig. 7.11 (d)), similar to the results for the LIF case.

7.9 Conclusions

In this chapter it is demonstrated, that the structure of the temporal correlations (the“color”) of the membrane potential has a significant impact on pulse detection perfor-mance. Using a weighted sum of missed pulses and false positive events it is demon-strated, that a leaky integrate-and-fire neuron as well as a biophysically more realisticHodgkin-Huxley type neuron behave in a similar way. Increasing the correlation time-

81

constants of the additive noise makes the detection of sub-threshold pulses more robust.The number of false positive events is reduced and the optimal values of the noise pa-rameter (variance of the membrane potential for the LIF and the noise parameter αfor the HH model) is shifted to higher values. In contrast to the standard informationmaximization approach to stochastic resonance in model neurons (see, e.g., (Stemmler,1996) and references therein), the optimal noise level is not a monotonic function ofthe strength of the input signal.

The total error (λ = 1), which is the sum of the fraction of “missed” pulses (falsenegatives) and the number of false positive events per interpulse interval, is alwayslarger than 0.5. This is a consequence of the zero mean noise. The probability thatan otherwise sub-threshold input pulse gets enhanced by a positive fluctuation is equalto the probability that it is further suppressed by its negative counterpart. Therefore,the probability of correct detection can never exceed 0.5. Because of this fact one canquestion the biological relevance of the described pulse detection scenario for singleneurons, but in the case of populations of neurons weak transient excitations can bereliably detected. First results for a simplistic model of a population are shown in theappendix 11.9.

82

5 10 15 20 25 30 35 400

100

200

300

400

500

600

700

800

(a)

0.5 1.0 2.0

Pc

0.05

0.4

0 5 10 15 20 25 30 35 400

200

400

600

800

1000

1200

noise level α

(b)

2.01.00.5

0.1

0.4

5 10 15 20 25 30 35 400.6

0.7

0.8

0.9

1

1.1

1.2

1.3

τe,i

(c)

2.01.00.5

Q1.0

0.6 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

(d)

3pulse height in nA

5 7 9

optim

al α

2.0

0.5

noise level α

noise level α

Pf

Figure 7.11: Characterization of the pulse detection performance of the HH modelneuron. The fraction Pc of correctly detected events (a), the normalized number Pf

of false positive events (b), and the total error Q (c) is plotted as a function of thenoise parameter α . (a)-(c) The three different curves in each figure correspond to threesets of parameters, see text for details. Parameters were: (τe0,τi0) = β (0.27,1.05) ms,

(De0,Di0) = γ(0.0022,0.0016) µS2

ms , with (β ,γ) = (1,2.75), (10,1), (20,0.085). Pulseheight Ip = 8nA, interpulse interval ∆T = 200ms. (d): Optimal value of α for differentheights Ip of the input pulses, Ip = 3, 4, 5, 6, 7, 8, 9nA, and (β ,γ) = (1, 2.75) (bottomcurve) and (20, 0.085) (top curve). Other parameters as for figs. (a)-(c). Pulses aresuper-threshold for Ip >9nA. In (Destexhe et al., 2001) (τe0,τi0) = (2.7,10.5) ms issuggested as realistic time constants for a layer VI pyramidal cell in cat parietal cortex.

83

Chapter 8

Higher Moments In NeuralIntegration

8.1 Abstract

In this chapter the effects of transient changes in the higher-order moments of the inputon neural behavior are studied. The investigations focus on transient changes in thesecond moments of the input. These are closely related to transient coincidences andtransient changes in temporal correlations. The membrane potential response as well asthe spike response are studied. Two model frameworks are applied. A leaky integrate-and-fire model neuron with additive low-pass filtered white noise, i.e. an Ornstein-Uhlenbeck process, and a Hodgkin-Huxley type model neuron. In the biophysicallymore realistic Hodgkin-Huxley type model neuron the input is modeled as stochasticconductance changes, the dynamics are given by the Ornstein-Uhlenbeck process aswell. For the leaky integrate-and-fire model differential equations for the moments ofthe membrane potential distribution and their analytic solutions are given. It is demon-strated that changes in the temporal correlation and the amount of coincidence bothhave influence on the membrane potential variance and thus on the spiking probabil-ity. However, the time evolution of membrane potential changes is different. Transientchanges in the amount of coincidence are effective in driving the leaky integrate-and-fire neuron, transient changes in the temporal correlation are not effective. For largetemporal correlation, in the leaky integrate-and-fire model as well as in the Hodgkin-Huxley type model, large temporal correlations of the input induce bursts.

8.2 Introduction

The membrane potential statistics of single neurons, and thus their behavior, is signifi-cantly influenced by the higher order moments of the input, see e.g. (Silberberg et al.,2004; Rudolph & Destexhe, 2001a; Tuckwell, 1998), and the corresponding chapters7 and 9 in this thesis. In this chapter investigations will focus on the second momentsof the input. Incoming spike trains are correlated with each other, see e.g. (Salinas& Seijnowki, 2000) and references therein. This is of particular interest in phenom-ena like transient synchronization or the transient occurrence of coincidences. In suchscenarios the variance of the input changes with time. Such phenomena are often con-

84

sidered important for neural computations. Spike trains are not only correlated witheach other, they are also correlated in time. The effects of this temporal structure onthe neural response are in many aspects similar to the effects of cross-correlated spiketrains. In other aspects the consequences are very different, as is demonstrated hereand in chapter 7. Experimental and modeling studies show, that the neural membranepotential and the spike input display a significant amount of temporal structure (Des-texhe et al., 2001; Destexhe et al., 2003). However, only few studies investigate theconsequences of the temporal structure of the spike input for neural behavior (Mato,1998; Nozaki et al., 1999b; Nozaki et al., 1999a; Capurro et al., 1998). A reason mightbe, that transient synchronization is well known to exist and to be relevant in a neuralcoding context whereas changes of temporal correlation have not been observed and re-lated to neural computation. Real nervous systems are full of possible mechanisms thatcould alter the temporal correlation of spike trains or the membrane potential. Candi-dates could be: A synapse changing its bandpass properties, a dendrite which changesits overall conductance (time constant) under the impact of the changing intensity ofsynaptic inputs, neuro-modulation by transmitters (not only at the synapses) whichhave an influence on the neuronal conductance. Last but not least the time structure ofthe incoming spike trains do also determine the time-structure of the neuronal outputspike train.

An abstract leaky integrate-and-fire model and a biophysically more realistic Hodgkin-Huxley like model are employed. The spike input is modeled with the help of theOrnstein-Uhlenbeck process. The leaky integrate-and-fire framework allows an intu-itive and quantitative interpretation of the effects of coincidence- and temporal corre-lation changes on the moments of the membrane potential in terms of the parametersof the Ornstein-Uhlenbeck process. This provides the basis for a comparison of co-incidences and transient temporal correlation changes. Starting from the Euler updateequation of the Ornstein-Uhlenbeck process and from the corresponding equation forthe leaky integrate-and-fire membrane potential, analytic results describing the vari-ance of the membrane potential as a function of time are derived. Analytic solutionsdescribing the dynamics of the moments of the membrane potential are provided. Oneresult is that the dynamics of the membrane potential response differs for the responsesto a step like increase of coincidence or temporal correlation. These differences canproduce significant delays in the spike response in the leaky integrate-and-fire model.Very short transient changes in the amount of coincidence do have a significant influ-ence on the response of a leaky integrate-and-fire neuron, but very short changes intemporal correlation do not. These results are compared to results from the Hodgkin-Huxley type model. In the biophysically more realistic model the response is muchmore sensitive to a change in the amount of coincidence then to a change in the tem-poral correlation of the input. The responses to coincidence- and temporal correlationchanges do not only differ in their dynamics, but also in their functional implications.Large temporal correlation of the input induces bursting, in the leaky integrate-and-fireas well as in the Hodgkin-Huxley framework.

This chapter is structured as follows: First, in section 8.3, the leaky integrate-and-fire model neuron is briefly introduced. Here an explicit connection is given betweenthe amount of coincidence in the input, and its effective modeling as an Ornstein-Uhlenbeck process. Differential equations for the moments of the membrane potentialare derived and their time dependent analytic solutions are given. In section 8.4 theresults for the leaky integrate-and-fire neuron are shown. The Hodgkin-Huxley frame-work is introduced in section 8.5, the corresponding results are shown in section 8.6.At the end, the main results are summed up and briefly discussed in the conclusions,

85

section 8.7.

8.3 LIF Framework

First we employ an abstract, but generic model of a single neuron. It is in more detailintroduced in chapter 2.

In contrast to chapter 2 time t is denoted as a subscript in this chapter. The cu-mulative effect of all spike trains is modeled as an Ornstein-Uhlenbeck process Xt .It corresponds to low-pass filtered white noise with a time-constant τX and diffusioncoefficient D

dXt

dt=− 1

τXXt +√

DdWt

dt, (8.1)

where dWt are the infinitesimal increments of the Wiener process. Its time derivativecorresponds to Gaussian white noise (for an appropriate introduction to the Ornstein-Uhlenbeck process see (Tuckwell, 1998)).

The membrane potential Vt of the leaky integrate-and-fire neuron model changes intime according to the differential equation

dVt

dt=− 1

τVVt +Xt , (8.2)

where τV is the membrane time constant. Throughout this chapter we use τV = 5ms.Once the membrane potential reaches the threshold (20mV), a spike is generated andthe membrane potential is reset to Vreset = 0.

8.3.1 Explicit Interpretation of Coincidences and Temporal Corre-lation

In this chapter the notion of coincidence and temporal correlation are interpreted interms of the Ornstein-Uhlenbeck process eq. (8.1). The sum of many Poisson pro-cesses with rate λ j, j = 1, . . . ,M and weights w j, j = 1, . . . ,M yields a Wiener processwith mean ∑ j w jλ j and variance ∑ j w2

j λ . This is true in case of infinitesimal smallweights w j and an infinite number of Poisson spike trains. This is called the usual dif-fusion approximation (Tuckwell, 1998). A deviation from these conditions results in anapproximation of the Wiener process. The variance of the Wiener process determinesthe diffusion coefficient D of the Ornstein-Uhlenbeck process (Tuckwell, 1998).

Let there be many, N +NC, Poisson spike trains. For simplicity, the mean should bebalanced, i.e. positive, we,n and negative weights wi,m are chosen, such that the mean

∑n

we,nλe,n +∑m

we,mλi,m (8.3)

vanishes. Here n is the index for the excitatory population and m is the index for theinhibitory population. NC of the excitatory Poisson spike trains are identical, model-ing coincidence. This leaves N independent Poisson processes. This does still corre-spond to a Wiener process in the sense of the usual diffusion approximation, (Tuckwell,1998)).

Assuming, for simplicity, that all absolute values of the weights and rates of thePoisson processes are identical, with value w, λ respectively. The diffusion coefficientof the Ornstein-Uhlenbeck process D is then

86

LP Σ

cN coincident Poisson spike trains

responseLIF

N independent Poisson spike trains

Figure 8.1: A leaky integrate-and-fire model neuron with an Ornstein-Uhlenbeck pro-cess as input. The Ornstein-Uhlenbeck process is equivalent to the low pass filteredsum of many, i.e. N + NC, Poisson spike trains. NC of these spike trains are identi-cal, N spike trains are independent. The NC identical spike trains model coincidence.In section 8.3 coincidence is explicitly related to the diffusion coefficient D of theOrnstein-Uhlenbeck process. Since a low-pass is a linear filter, filtering the sum of thePoisson processes is equivalent to the sum of low pass filtered Poisson spike trains.

D = ∑j

w2j λ j = (Ncw)2λ +Nw2λ . (8.4)

According to eq. (8.1) the temporal correlation in the Ornstein-Uhlenbeck processcorresponds to a low pass filtering of a Wiener process. Since a low pass filter is alinear filter, the order of summation and filtering can be interchanged (L ucke, 1995).Thus in this framework temporal correlation changes can be interpreted to occur aftersummation, e.g. due to the dendritic and neuronal properties, or before summation, i.e.due to changes in the temporal correlation of the spike trains.

8.3.2 Differential Equations for the Moments and their Solution

We yield the differential equation for < X 2t >, eq. (8.5), by multiplying eq. (8.1) with

itself, average over an ensemble (<>) and neglect all terms of order > 1 in dt. Notethat < XtdWt > is zero. The differential equations for < XtVt >, eq. (8.6), and < V 2

t >,eq. (8.7), are derived in an analog way

d < X2t >

dt+

2τX

< X2t >= D, (8.5)

< XtVt >

dt+

1τ

< XtVt >=< X2t >, τ =

τX τV

τX + τV, (8.6)

< V 2t >

dt+

2τV

< V 2t >= 2 < XtVt > . (8.7)

Note that < V 2t >, eq. (8.7), depends on τX , D, and τV plus a new time constant τ

which is introduced by the covariance between Xt and Vt and depends on τX and τV .For differential equations of the form

87

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5

(a) (b)

time [ms] time [ms]

<V

2 > [

mV

2 ]

<V

2 > [

mV

2 ]

0.2

0.8

1.2

0.5

1.5

0 50 100 0 50 100

Figure 8.2: Membrane potential variance as a function of time in the leaky integrate-and-fire model. At t = 50ms a step like change in either D (a) or τX is applied (b).Solid line: Analytics solution according to eq. (8.7). Dashed line: Numeric simulationaccording to eqns. (8.1, 8.2). (a) Response to a sudden increase of D, below 50msD = 1 mV 2

ms , at 50ms D jumps to D = 2 mV 2

ms , in both regimes τX = 0.5ms. (b) Responseto a sudden step like change of τX , below 50ms, τX = 0.1, above 50ms τX = 1ms,D = 1 mV 2

ms in both regimes.

dyt

dt+gxyt = hx (8.8)

the solution is

yx = e−Gxη +

∫ x

θhte

Gt dt, (8.9)

where Gx =∫ x

θ gtdt and yθ = η .

Using the above expressions the analytic solution is easy to obtain, however the corre-sponding expressions are lengthy. The results for < Xt >,< XtVt >, and < V 2

t > aregiven in the appendix 11.8.The stationary moments are given by

< X2t >=

DτX

2, < XtVt >=

DτX τ2

, < V 2t >=

DτX τV τ2

. (8.10)

In figure 8.2 (a) and (b) the membrane potential variance is displayed as a function oftime. Numerical results from evaluating eqns. (8.1, 8.2) are compared with the ana-lytic solution, eqns. (11.8), of the differential eqns. (8.5 - 8.7) for the moments of themembrane potential. At 50ms either the diffusion coefficient D (a) or the temporal cor-relation τX (b) are increased. These step-like changes causes a change in the membranepotential variance. The analytic solutions match well the numerical evaluation of eqns.(8.1, 8.2).

8.4 LIF Framework, Results

There are differences in the time-development of the moments of the membrane-potentialdistribution, as is demonstrated in figure 8.3 (a) - (d). Here changes of the temporal

88

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

t [ms]

τx0

< V

> in

arb

. uni

ts2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

t [ms]

< V

> in

arb

. uni

ts2

τx0

0 2 4 6 8 10 12 14 16 18 2010

11

12

13

14

15

16

17

response to D pulse

resp

onse

tim

e [m

s]

τ pulsex

τ x0 [ms]0 5 10 15 20 25 30 35 40 45 50

0

2

4

6

8

10

12

14

16

18

t [ms]

< V

> in

arb

. uni

ts2

τx0

time [ms]

time [ms]

time [ms]5 530 30

5 50τX0 [ms]4 10 20

<V

2 > [

mV

2 ]<

V2 >

[m

V2 ]

<V

2 > [

mV

2 ]re

spon

seti

me

[ms]

12

15

10

90

2

18

2

18

(a) (b)

(c) (d)

Figure 8.3: Differences in membrane potential response to changes in D and τX . (a)Membrane potential variance as a function of time, a step like increase of τX or Doccurs at t = 5ms. Solid line denotes response to a τX step. Dashed line denotesresponse to a D step. The step height in τX is ∆τX = 1ms is fixed. D is adjusted,according to eq. (8.10), such that the stationary membrane potential variance on theright-hand-side of the step are identical for the τX and D response. To the left-hand-side of the step D = 1 mV 2

ms . From bottom to top, the baseline temporal correlationτX0 increases, τX0 = 1,3,5,7,9ms. (b) Membrane potential variance as a function oftime, a change in D or τX is applied as a rectangular additive pulse. The solid linecorresponds to the τX response and the dashed line to the D response. As in (a) thebaseline temporal correlation does increase from bottom to top, τX0 = 1,3,5,7,9ms.Pulse width is 5ms, start at 5ms. The height of the pulse is ∆τX = 1ms, as in (a) D isadjusted such that the stationary values of the membrane potential variance match eachother. For t < 5ms D = 1 mV 2

ms . (c) Time delay until the maximum variance as a responseto a rectangular shaped pulse has been reached as a function of the baseline temporalcorrelation. Setting and parameters as in (b). Measurement starts at t = 0ms. Lowercurve: Time delay until the maximum membrane potential variance as a response to aτX pulse has been reached. Upper curve: Same for D pulse. (d) Membrane potentialvariance versus time. Two rectangular pulses are applied. A first pulse is appliedat t = 5ms, width is 5ms. A second pulse is a applied at t = 20ms, width is 5ms.From bottom to top τX0 = 1,3,5,7,9ms. Thick solid line corresponds to the membranepotential response to D changes, the thin solid line to τX changes respectively. Changein τX and adjustment of D as in (b).

89

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

time [ms]

(a) (b)

5 30

<V

2 > [

mV

2 ]

1

8

max

imum

chan

gein

<V

2 > [

mV

2 ]

5

35

τX0 [ms]5 50

Figure 8.4: Response of the membrane potential to a very short pulse. (a) Membranepotential variance versus time. A very short pulse, width 0.1ms, is applied at t =5ms. Solid line: Response to a τX change. Dashed line: Response to a D change.From bottom to top the temporal baseline correlation does increase τX0 = 1,6,11ms.Pulse height is ∆τX = 10ms. D = 1 mV 2

ms for t < 5ms and t > 5ms. For t = 5ms D isadjusted, according to eq. (8.10), such that the stationary membrane potential varianceswould be the same, if the change in τX and D would be permanent. (b) Maximum ofmembrane potential variance as a function of τX0. Parameters as in (a). Upper curve:Maximum membrane potential variance versus τX0 following a short D-pulse. Lowercurve: Maximum membrane potential variance versus τX0 following a short τX -pulse.

450 500 550 600 650 700 750 8000

10

20

30

40

50

60

70

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−40

−30

−20

−10

0

10

20

30

40

50

60

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−40

−30

−20

−10

0

10

20

30

40

50

60

time [ms] time [ms] time [ms]45 80 100100 1000 1000

(a) (b) (c)

spik

era

te [

Hz]

1

6

volt

age

[mV

]

-30

5050

-30

volt

age

[mV

]

Figure 8.5: Spike response to step-like changes in temporal correlation and the amountof coincidence. (a) Post stimulus time histogram (PSTH). A step like change ∆τX =

5ms is applied at t = 50ms, τX0 = 5ms, D = 1 mV 2

ms for t < 50ms. For t >= 50ms Dis adjusted, according to eq.(8.10), to yield the same stationary variance as for the τX

step. Solid curve: Response to D step. Solid curve with dots: Response to τX step.(b) Example of a typical voltage trace of the leaky integrate-and-fire model with anOrnstein-Uhlenbeck process as the only input, τX = 1ms, D = 50 mV 2

ms . (c) Same as in(b) but the temporal correlation is τX = 20ms. The diffusion coefficient is decreased toyield a comparable firing rate, D = 1 mV 2

ms . Bursts occur.

90

correlation or the amount of coincidence are applied to the leaky integrate-and-fire neu-ron model, either in a step like fashion or as a rectangular pulse of a certain width. Thesecond moment is then plotted as a function of time. Figure 8.3 (a) shows the timedevelopment of the membrane potential variance. At t = 5ms a step-like increase ineither the diffusion coefficient D or the temporal correlation τX is applied. The pa-rameters are chosen, such that the membrane potential variance is identical in bothcases in the long run. From bottom to top the baseline temporal correlation τX0 doesincrease. The step height in case of the increased temporal correlation is identical forevery curve. The diffusion coefficient is then calculated from eq. (8.10) such that thestationary variances match each other. For low baseline temporal correlation τX0, seelower curves, the response of the membrane potential variance to the increase of thediffusion coefficient D is slightly faster than to the change of the temporal correlationτX . This is different for large baseline temporal correlation, see upper curves, here themembrane potential variance responds slightly faster to the step increase of the tempo-ral correlation. Figure 8.3 (b) shows the effect of a rectangular shaped change of thediffusion coefficient D and a corresponding change in the temporal correlation τX . Asin subfigure (a) the parameters are chosen, such that the membrane potential varianceswould be identical in their stationary values. Since the applied rectangular pulses are offinite width (5ms), these stationary values need not necessarily be reached. Thus, themaximum membrane potential variance can take on different values for the response toa temporal correlation change and a change in the diffusion coefficient. From bottomto top the baseline temporal correlation does increase. As can be seen in figure 8.3(b) the time delay until the maximum response is reached is different for a change intemporal correlation and a change in the diffusion coeffiecent. Here the response to thetemporal correlation change is faster for every baseline temporal correlation. This isquantified in figure 8.3 (c), the time delay till the maximal variance response is plot-ted as a function of the baseline temporal correlations. For every τX0 the resonse toa temporal correlation change is faster than the response to a change in the diffusioncoefficient.

In subfigure 8.3 (d) the membrane potential variance is plotted as a function oftime again. Here two pulses, rectangular changes in temporal correlation and diffusioncoefficient, are applied. One is applied at 5ms, the other one at 20ms. From bottomto top the baseline temporal correlation does increase. The response of the membranepotential variance is influenced by the application of the two pulses for a significanttime. The larger the baseline temporal correlation, the longer extends this influence.

Figure 8.4 (a) displays the membrane potential variance response to peak likechanges in the temporal correlation τX and diffusion coefficient D. The peak has awidth of 0.1ms, this corresponds to the time step dt, as used for the evaluation of theeqns. (8.1, 8.2). For the duration of the peak the temporal correlation is increased by10ms. As in the above figures the corresponding peak height for the diffusion coef-ficient change in D is chosen, such that the stationary membrane potential varianceswould be the same. This does not happen here, since the dt = 0.1ms is much to short.From bottom to top the baseline temporal correlation does increase. The membranepotential responds much more vigorously to the peak like changes in D, the responseto the temporal changes is less pronounced. In figure 8.4 (b) the amplitude of these re-sponses is plotted as a function of the temporal baseline correlation, both for peak liketemporal correlation changes and diffusion coefficient changes. For small baseline tem-poral correlation the membrane potential variance does not change as a consequenceof the application of a peak like change in the temporal correlation. For larger τX0 themaximum change in membrane potential variance increases, goes through a maximum

91

and decreases again. In case of peak like D changes, the maximum membrane potentialchange occurs for zero τX0, then it decreases.

The evolution for the moments of the membrane potential distribution differs, infigure 8.5 (a) it is demonstrated that these differences are large enough to cause timedelays until the first spikes are initiated. For figure 8.5 (a) step like changes in temporalcorrelation and the diffusion coefficient are applied at 50ms. As above the change inD is adjusted such that the stationary membrane potential variances are identical, bothafter a step increase of τX and D. Baseline temporal correlation is τX0 = 5ms, thetemporal correlation is increased by 5ms. The average number of spikes which occurare plotted versus time. The change in D and τX occurs at 5ms. Though the stationarymembrane potential variance is identical for the step increase in D and τX , the firingrate in case of an increased τX is lower than in the case of an increased D. This isdue to the fact, that higher temporal correlation of the membrane potential increasesthe average inter-spike-interval (ISI). The effect is explained and demonstrated in moredetail in chapter 7. For the given set of parameters the spike response to a D changeis approximately 5ms faster (maximum difference) then to a change in the temporalcorrelation.

Figures 8.5 (b) and (c) show single voltage traces. In figure (b) the temporal corre-lation of the input is rather small (τX = 1ms), a typical noisy voltage trace can be seen.Several spikes are initiated due to the noisy input. In figure 8.5 (c) τX = 20ms, thediffusion coefficient D is reduced, such that the model neuron does not fire to vigor-ously. In subfigure (c) burst can be observed, this a qualitative change compared to thetrace in subfigure (a). These bursts are induced by the very large temporal correlationof the Ornstein-Uhlenbeck process. If the Ornstein-Uhlenbeck process feeds positivevalues (excitation) into the leaky integrate-and-fire model neuron, it is likely to have anexcitatory effect for a long time. Observed in a short time window (some 10ms) thiswould correspond to a strong mean excitatory input.

8.5 HH Framework

A Hodgkin-Huxley type point neuron model is employed. It is introduced in chapter2 in more detail. The membrane potential V of the Hodgkin-Huxley neuron, adoptedfrom (Destexhe et al., 2001), changes in time according to the differential equation:

Cm∂V∂ t

=−gL(V −EL)− INa− IK− IM− Isyn− Istim. (8.11)

The left hand side of the equation describes the influence of the membrane’s capac-itance, while all ionic currents through the cell’s membrane, including the synap-tic noise and stimulus currents, are summed on the right hand side. A leak currentIL = gL(V −EL), the spike-generating sodium INa = gNam3h(V −ENa) and potassiumcurrents IK = gKn4(V −EK) as well as an additional noninactivating potassium cur-rent IM = gM p(V −EM) are considered. The parameters used for the simulations aregiven in (Destexhe et al., 2001; Destexhe & Pare, 1999) and in the appendix 11.4. Thetotal synaptic noise current Isyn is generated by changing the membrane conductanceinduced by stochastic spike trains arriving at excitatory and inhibitory synapses.

To obtain a realistic approximation of the synaptic background activity, a point-conductance model recently described by Destexhe et al. (Destexhe et al., 2001) isused. In this model the total synaptic current Isyn is obtained as a superposition ofexcitatory (e) and inhibitory (i) inputs,

92

0 200 400 600 800 1000 1200 1400 1600 1800 2000−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

0 200 400 600 800 1000 1200 1400 1600 1800 2000−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

(a) (b)

time [ms]1000 2000 time [ms]1000 2000

volta

ge[m

V]

volta

ge[m

V]

-70

20 20

-70

Figure 8.6: Voltage traces of the Hodgkin-Huxley type neuron model. At t = 1000ms,either a step like change in the variance of the Ornstein-Uhlenbeck process (σe,σi)(mediated by a change in the diffusion coefficients) or the temporal correlation (τe,τi)is applied. Curves are shifted by 20mV each. The lowest one corresponds to a non-shifted trace. Parameters are described in multiples of (σe0,σi0) = (0.003,0.00825)µSand (τe0,τi0) = (2.728,10.5)ms. (a) At t = 1000ms a change in (σe,σi) occurs. Curvesfrom bottom to top correspond to 1,1.5,2.0,5.0 times (σe0,σi0). (b) At t = 1000msa change in (τe,τi) occurs. Curves from bottom to top correspond to 1,5,10,20 times(τe0,τi0). Parameters for τe0,τi0,σe0,σi0 can be found in the appendix 11.5.

Isyn = ge,t(Vt −Ee)+gi,t(Vt −Ei), (8.12)

where ge and gi are the synaptic conductances, and Ee and Ei the corresponding reversalpotentials. The time-dependent conductances are described as an Ornstein-Uhlenbeckprocess

dge,t

dt=− 1

τe[ge,t −ge0]+

√De

dWt

dt, (8.13)

dgi,t

dt=− 1

τi[gi,t −gi0]+

√Di

dWt

dt. (8.14)

De,Di are the diffusion coefficients, ge0,gi0 the mean conductances and τe,τi the timeconstants of the OUP. The stationary variance of the OUP and the diffusion coefficientsare related as follows σ 2

e,i =De,iτe,i

2 . Parameters are given in the appendix 11.5.

8.6 HH Framework, Results

In figure 8.6 (a) voltage traces of the Hodgkin-Huxley type model neuron with a steplike change in the standard-deviation (σe,σi) of the Ornstein-Uhlenbeck process areshown. Subfigure (b) displays voltage traces with a step like change in the temporalcorrelation (τe,τi). The change in (σe,σi) occurs at 1000ms. Different than in theleaky integrate-and-fire framework, the temporal correlation and diffusion coefficientsare not adjusted to yield the same stationary membrane potential variances. Here it isnot necessary since the membrane potential responds to changes in temporal correlationand diffusion coefficient qualitatively different. Figure 8.6 (a) displays several voltage

93

0 200 400 600 800 1000 1200 1400 1600 1800 2000−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

0 200 400 600 800 1000 1200 1400 1600 1800 2000−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

(a) (b)

time [ms]1000 2000 time [ms]1000 200000

volta

ge[m

V]

- 65 - 65

volta

ge[m

V]10 10

Figure 8.7: Occurrence of bursts in a Hodgkin-Huxley type model neuron for largetemporal correlation of the input. Parameters are given in multiples of (σe0,σi0) =(0.003,0.00825)µS and (τe0,τi0) = (2.728,10.5)ms. (a) Voltage trace with 1 time(τe0,τi0) and 4 times (σe0,σi0). Spiking occurs, induced by the stochastic conductanceinput. (b) Voltage trace with 5 times (τe0,τi0) and 1 time (σe0,σi0), these values are de-creased to yield a comparable spike rate. Bursts occur. Parameters for τe0,τi0,σe0,σi0

can be found in the appendix 11.5.

traces. For the bottom trace (σe,σi) = (0.003,0.00825)µS is not changed, from bot-tom to top the step corresponds to 1.5,2,5 times (σe,σi) = (0.003,0.00825)µS. Themembrane potential variance does strongly increase with increasing (σe,σi). For thetop trace also significant spiking activity is observed. Other than in the leaky integrate-and-fire framework, the membrane potential variance is not very sensitive to changes inthe temporal correlation. This can be observed in figure 8.6 (b). From bottom to top theinitial values of (τe,τi) = (2.728,10.5)ms are multiplied by a factor of 20. Though themembrane potential does increase with increasing temporal correlation of the conduc-tance input, these changes do not lead (better: with a low probability only) to spikingactivity for biologically reasonable temporal correlations.

In figure 8.5 (c) bursting due to large temporal correlations in the input is demon-strated for the leaky integrate-and-fire framework. Figure 8.7 (a) and (b) demonstratesthe corresponding phenomenon in the Hodgkin-Huxley type framework. For smalltemporal correlations in the conductance input, subfigure (a), a typical voltage tracewith some spiking activity can be observed, no bursts occur. To induce spiking ac-tivity the standard values of the diffusion coefficients have been increased by a factorof four. For figure 8.7 (b) the standard temporal correlation has been increased by afactor of five. The diffusion coefficients have their standard values. As in the leakyintegrate-and-fire framework burst can now be observed.

8.7 Conclusions

For a leaky integrate-and-fire neuron with an Ornstein-Uhlenbeck process as input,an explicit interpretation of coincident input is given in terms of the diffusion coeffi-cient of the Ornstein-Uhlenbeck process. Both, an increase in temporal correlation andof the amount of coincidence of the input increases the membrane potential variance.However, in their time evolution, the membrane potential responses behave differently.Significant time delays between the maximum response to a change in the temporal

94

correlation and to a change in the diffusion coefficient occur. Such delays can alsobe observed in the spike response. The application of short rectangular, pulse-like,changes in the temporal correlation or the diffusion coefficient, has very different ef-fects on the membrane potential variance. The membrane potential variance is muchmore sensitive to short changes in the diffusion coefficient then to short changes in thetemporal correlation. In case of small baseline temporal correlations, the membranepotential variance change drops down to zero for a short temporal correlation change.Assuming a scenario with short transient events as signals the use of transient coinci-dences would be much more effective. In the Hodgkin-Huxley type model neuron achange in the diffusion coefficients has a significant effect on the membrane potentialvariance, and thus on the spike response. Though increasing the temporal correlationdoes increase the membrane potential variance, the model neuron is not very sensitiveto such changes within biologically realistic parameters.

95

Chapter 9

Response Stimulus Correlation

The work presented in this chapter is done in close collaboration with Jacob Kanev.The presented material is based on a common publication (Kanev et al., 2004) andis extended by material from a blueprint for a common manuscript we submitted forpublication (Kanev et al., submitted). Most of the work was done during the diploma-project of Jacob Kanev as a computer science student. I had the pleasure to supervisehis work.Since the calculations are very involved in this chapter, the notation deviates in someaspects from the other chapters.

9.1 Abstract

The response-stimulus correlation (RSC) is a function of the time-distance to a neuron’sobserved response spike, which expresses the correlation of response and stimulus, aswell as the expected number of stimulus spikes which have led to a response. If aneuron is driven by many different synapses (excitatory, inhibitory, slow, fast etc.)such a correlation curve can be measured for each of them. The calculation of theseRSCs provides an intricate problem, especially if the stimuli of various synapses arecross-correlated. In this chapter a conductance based leaky integrate-and-fire neuron isinvestigated, and the response-stimulus correlation is obtained analytically. Fits of theresults of this analysis are compared to simulations, and various involved measures aswell as resulting consequences are discussed.

9.2 Introduction

The response-stimulus correlation (RSC) is a function describing the probability of astimulus spike to occur before an observed response spike and hence is a measure forthe significance of this response.

A neuron has fired a response spike at a certain time t. What is the probability thatthe neuron has received a stimulus spike at τ− t? This probability, expressed as a func-tion of τ , is usually known as spike-triggered averaged stimulus, or reverse correlation.It can be obtained experimentally by averaging the time cause of a measured stimulusbefore a response spike, and it is used to determine the kind of stimulus to which a neu-ron is sensitive. Because the term “reverse correlation” has a fixed connotation hinting

96

−10 −5 0 5 10

0

conditional conductance

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S]

Figure 9.1: Example of the expected conditional conductance. Values of the conduc-tances are averaged over produced response spikes. The neuron is driven by a leakconductance (not shown), by an excitatory conductance (top curve, pointing upwards)and an inhibitory conductance (bottom curve, pointing downwards), each modeled byderivatives of a Wiener process (see the section on noise models). Excitatory and in-hibitory conductance processes are slightly cross-correlated. A threshold hit (and thusa response spike) has happened at τ = 0. To the left is the time before the response, tothe right the time after. More details on the parameters can be found in the appendix,11.11.2.

at an experimental context the term response-stimulus correlation (RSC) will be used.In this chapter a normalized RSC is used which is independent of the stimulus rate.To make the difference to the original RSC clear, the non-normalized version shall becalled conditional stimulus density.

In this framework, obtaining the RSC is equivalent to decoding single spikes. If anormalized measure which is independent of stimulus and response rate is used for theRSC - as will be presented in the next section - the RSC also gives information aboutthe peri-stimulus time histogram (PSTH). This is closely linked to the first passage timeof a neuron. The PSTH is the time-reversed counterpart of the RSC and indicates theprobability change of response in case of a given stimulus. There are other ways theRSC contributes to the analysis of neural behavior. As well as determining the weightchange in spike-timing based learning mechanisms like STDP (Song et al., 2000) it isinvolved in shaping the frequency-dependent transfer function of a neural population(Gerstner & Kistler, 2002). It is a general measure for the significance of a responsespike, and can be interpreted as a measure of the sensitivity of a responding neuron tovariations of its stimulus in amplitude and time (Dayan & Abbott, 2001). Of specialinterest are the mutual relations of different stimuli - like the relation of the RSC ofexcitatory to the RSC of inhibitory synapses. See figure 9.1 for an example. To our

97

knowledge there has been only one notable attempt to obtain the reverse correlationanalytically (Gerstner, 2000), using a so called Spike-Response-Model (SRM) withexcitatory input only, hazard noise and a population approach.

In this chapter an approximation of the RSC and an introduction to some of theinvolved calculations are presented. The suggested approximation is derived in thefollowing way: the Ito stochastic differential equation (SDE) for the free membranepotential is solved and the expectation of the solution is given. The free membrane po-tential is time-symmetric, an important fact that is explained shortly and sustained bysimulations. Using this symmetry the flow of the potential before a spike is followedbackwards in time. The main obstacle is to disentangle the scalar product of synapticnoise, for which a conjecture of the relations of the synaptic influence is given. Usingthis conjecture, the scalar product is de-assembled, and the mean conditional conduc-tance for each input process can be obtained. This is a measure for the number ofstimulus spikes to be expected at the particular synapse i and time distance τ . Theresponse-stimulus correlation is proportional to the expected conditional conductanceand the factor of this proportionality is simply the mean stimulus.

This chapter consists of three main sections titled “Model, Methods, Materials”,“Results” and “Discussion”. The first section - in five sub-sections - gives a generaloverview, discusses the membrane equation, various noise models, how the thresholdis handled, and finally derives the equation for the RSC. Since some parts provide verydetailed and technical explanations, there is a short outline at the beginning of eachsub-section. To get a quick impression of the theory presented in this chapter it servesto read these outlines only. The “Results” section will demonstrate the capability of theRSC-equation in different settings.

9.3 Model, Materials, Methods

9.3.1 General Framework

The investigation is based on the idea of stochastic processes. Notation is based onthe notation introduced by Protter and F ollmer (Protter, 1995), as well as the calculusinvolving Ito integrals. The model which is investigated in this chapter is the con-ductance based leaky integrate-and-fire neuron including reversal potentials. Synapticconductances are driven by incoming spikes, and these conductances drive the mem-brane potential. When the membrane potential reaches a threshold, a response spike isemitted.

Preliminaries and Notation

Probability space. To ground the further analysis on a sound mathematical basis, aprobability space Ω,F ,(Ft )0<t<∞,P, is assumed. Ω is a set of events also calledsample space, P is the probability law, F is a subset of Ω, and Ft a class of σ -algebrasdescribing the set of possible events at time t. The events generating the stochasticbehavior of the model neuron are a set of multi-dimensional stochastic variables dW i

t ,indexed by t and having a Gaussian distribution.

Ito calculus. In this chapter the calculus introduced by Ito is used (Ito, 1944). Abrief description is given in the appendix 11.13.3.

Stochastic processes and sample paths. A stochastic process is basically a collec-tion of random variables X = X(t), t ∈ T on a common probability space and indexed

98

by t which is interpreted as time. It can thus be formulated as a function X : T×Ω→ IR.In this chapter the usual conventions to write this as Xt are applied. Xt(ω) is thus a func-tion which describes one sample path of a stochastic process, where ω is sometimesreferred to as the label of the sample path.

Expectation. For a stochastic variable Xt which is indexed by time t, we define theexpectation at time t as a set of variables which are indexed by t to be the average overall events possible at time t:

〈X〉t =de f Et X=

∫

Ω

Xt(ω)dPX(ω) .

Usually the event ω will refer to a certain voltage or conductance, and Ω is (−∞,∞).Covariation Process. [X ,Y ]t is the covariation process of two stochastic variables

X and Y , and [X ,X ]t is the quadratic variation process of X (see (Protter, 1995)). Thequadratic variation is a measure of the “roughness” of a process. For example thequadratic variation of the Wiener process is [W,W ]t = t.

Variance. Using a similar shorthand notation for the variance, the variance [X ]t atany time t is defined to be

[X ]t =de f Vart X

= Et

X2−E2t X= Et

(X−〈X〉)2 .

Indicator Function. The indicator function of the set A, 1A(ω), is defined as

1A(ω) =

0, ω 6∈ A1, ω ∈ A

.

d-Operator. Throughout this text a symbolic notation for integral equations is used:

dXt = f (Xt )dYt ,

this is equivalent to the integral equation∫

dXt = C +

∫

f (Xt)dYt .

In many parts of the text expressions of the form dXt are used as variables in their ownright, where d... has the meaning of ∆t...,∆→ 0.

h-Operator. To save space, the h-operator is introduced, which indicates the dis-tance of a stochastic process to its mean:

hXt = Xt −〈X〉t .

In many cases stochastic processes have a mean which does not change with time. Inthis case the h-operator has the meaning

hXt = Xt −〈X〉∞ .

Which of the two versions applies can be taken from the context.

99

The conductance based LIF

The model used in this chapter is the conductance based leaky integrate-and-fire neu-ron. Several synaptic conductances gi,t drive the neuron, each of which is influenced bystimulus spikes. A single incoming spike will trigger an opening of the conductance,and shortly afterwards the conductance closes again. To stay consistent with the usualnotation for stochastic differential equations, the conductances will be referred to viatheir cumulative value Gt =

∫gtdt and its increments dGt . The value of a conductance

and the voltage of the neural membrane determine the amount of ions which flow intothe neuron. For each of the conductances there exists a characteristic voltage, at whichthere is no net flow of ions. If the membrane potential is below that special value, anopening of the conductance will raise it, and if the membrane potential is above thatvalue, a conductance opening will lower it. This characteristic voltage is called reversalpotential vi, and determines whether a synapse has an inhibitory or an excitatory effect.As soon as the membrane voltage hits its threshold vθ , it is reset to a starting point V0

and a spike is emmited.

Response-Stimulus Correlation

Topic of this chapter is to derive a measure for the correlation of response and stimulus.What is the meaning of a response spike r, what is the average stimulus 〈si〉 that hascaused an observed response? These values are expressed in three different measures,which are used throughout the text.

The measure r : t→ 0,1 indicates the emittance of a response spike, and si : t→0,1 indicates the arrival of a stimulus spike at conductance i.

The expected conditional conductance d⟨Gi⟩

τ is the average value of a stimulatingconductance i at a time-difference τ to an observed response:

d⟨Gi⟩

τ =⟨dGi

t+τ |rt⟩. (9.1)

Negative values of τ indicate the past which has lead to the response, positive val-ued indicate the future after the response. Figure 9.1 shows the expected conditionalconductance for a neuron with three conductances, one of which is a leak conductance.

The conditional stimulus density dPsi is the probability density of the stimulus incase of a response at τ = 0. Psi(T ) =

∫

T dPsi(τ) gives the expected amount of stimulusspikes in any time-interval T = [a,b]; dPsi(τ)/dτ gives the conditional rate at timedistance τ . In case of no correlation, dPsi(τ)/dτ gives static rate of the stimulus. Theconditional stimulus density can be derived by multiplying the expected conditionalconductance with a constant factor, usually 1/weighti.

The presence of an observed response spike changes the probability of occurrenceof a single stimulus. The (normalized) response-stimulus correlation is a measure forthis change:

Riτ =

⟨si

τ ,r⟩

〈siτ〉 〈r〉

. (9.2)

This measure is independent of the stimulus rate, and is derived by dividing the con-ditional stimulus density by the static rate of the stimulus. In case of no correlation,Ri

τ = 1; probability is unchanged by an observed response spike. If Riτ > 1 probabil-

ity is increased, which indicates an excitatory synapse, and if Riτ < 1, probability is

100

decreased, which indicates an inhibitory synapse. The response-stimulus correlationshows the connection of conditional stimulus density and the peri-stimulus histogram(PSTH). Since for two stochastic variables x and y the joint probabilities behave as〈x,y〉 = 〈x|y〉 〈y〉 = 〈y|x〉 〈x〉, the normalized response-stimulus correlation introducedabove can be obtained by

Riτ =

⟨si

τ ,r⟩

〈siτ〉 〈r〉

=

⟨si

τ |r⟩

〈siτ 〉

(9.3)

=

⟨r|si

τ⟩

〈r〉 . (9.4)

The measure⟨si

τ |r⟩

in eq. (9.3) is the conditional stimulus density, the probability of astimulus to have occurred before an observed response. This curve shows the “mean-ing” of a response spike. The measure

⟨r|si

τ⟩

in (9.4) is the conditional response den-sity, the probability that an observed stimulus will trigger a response. This curve showsthe “impact” of the stimulus. Eqns. (9.4) and (9.3) show nicely that in terms of con-ditional expectations, coding is the reverse of decoding. If the response rate is known,the PSTH (expected conditional response given a stimulus) can be obtained from theRSC. The presence of a stimulus spike changes the probability of a response. The RSCis a measure for this change as well. However, in this chapter all equations, estimationand simulations are run to predict the stimulus in case of a given response, hence thename response-stimulus correlation, in contrast to stimulus-response correlation.

9.3.2 Membrane Potential

The differential equation which describes the voltage of the neural membrane can bea very simple expression, in which synaptic conductances and leak conductances aretreated in the same way eq. (9.6). This equation can be solved using Ito calculus (see(Protter, 1995), (F ollmer & Protter, 2000), (F ollmer et al., 1995), (Ito, 1944)), andwritten down in an explicit expression eq. (9.10). Using methods provided by the samecalculus, mean and variance can be given eqns. (9.12) and (9.18).

Differential Equation

The Model we use is an integrate-and-fire neuron with conductances and reversal po-tentials. Different reversal potentials describe different synapse types like excitatory orinhibitory synapses. The usual differential equation for a standard neuron with excita-tory stimulus, inhibitory stimulus and a leak conductance is

τC ·dVt = (vm−Vt)gmdt +(vexc−Vt)dGexc

t +(vinh−Vt)dGinh

t . (9.5)

where dVt is the membrane potential, τ the membrane time constant, vm the membraneresting potential, gm the leak conductance, vexc the excitatory reversal potential, vinh

the inhibitory reversal potential and dGexct and dGinh

t the noisy synaptic conductances:dGt = conductance · dt. The time constant and capacity are both set to one, and thedifferent synapse types are labeled with a running i. If the resting potential is namedv0 and the leak conductance gmdt renamed to be dG0

t , above equation is simplified

101

substantially:dVt = ∑

i(vi−Vt)dGi

t . (9.6)

Solution

Equation (9.6) can be solved using Ito Calculus and the method of the integrating fac-tor:

1. Solve the homogeneous equation. The homogeneous equation is the equationconsisting only of factors of Vt . Vt is renamed Φt , called the ’integrating factor’:

dΦt =−Φt ∑i

dGit .

First note that

d [Φ,Φ]t = (dΦt )2

= ∑i, j

Φ2t dGi

t dG jt

= Φ2t ∑

i, j

d[Gi,G j]t .

The Ito rule for the logarithm solves this equation (→ 11.12)-

Φt = exp

(

−∑i

Git −

12 ∑

i, j

[Gi,G j]t

)

. (9.7)

2. Express dΦ−1t in terms of Φ−1

t . This is done by using Ito to get the correctexpression for dΦ−1

t (→ 11.12):

dΦ−1t = Φ−1

t

(

∑i

dGit +∑

i, jd[Gi,G j]t

)

. (9.8)

3. Get dΦ−1t Vt and solve. Use integration by parts to get dΦ−1

t Vt . The first step isto know d[Φ−1,V ]t (→ 11.12)-

d[Φ−1,V ]t = −Φ−1t ∑

i, j(vi−Vt)d[Gi,G j]t , (9.9)

which solves the main equation, since the voltage dependent terms will cancel out (→11.12):

d(Φ−1

t Vt)

= Φ−1t ∑

i

vidGit +Φ−1

t ∑i, j

vid[Gi,G j ]t .

Integration -

VtΦ−1t = V0 +∑

ivi

∫

Φ−1t dGi

t +∑i, j

vi

∫

Φ−1t d[Gi,G j]t

102

0 5 10 15 20 25 30−80

−75

−70

−65

−60

−55

−50

−45expectation after reset

time [ms]

expe

ctat

ion

[µS

]

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3variance after reset

time [ms]

varia

nce

[(µS

)2 ]

(a) (b)

Figure 9.2: (a) Development of expectation after reset. The plot shows eq. (9.12) (thickline, on top of a numerically obtained expectation of equation (9.10). (b) Developmentof variance after reset. Note that the variance is not monotonically increasing. The plotshows eq. (9.18) (thick line) on top of the variance obtained numerically for eq. (9.10).For details on the simulations, as parameters, see 11.11.2.

- followed by multiplication with the integrating factor gives the solution of the mem-brane potential -

Vt = V0Φt +∑i

vi

t∫

0

Φt

ΦsdGi

s +∑i, j

vi

t∫

0

Φt

Φsd[Gi,G j]s

with

Φt = exp

(

−∑i

Git −

12 ∑

i, j[Gi,G j]t

)

. (9.10)

The middle term of Vt integrates over the expected conductances, while the right termintegrates over all correlations in the conductances. Φt/Φs = 1 for s = t and decreasesexponentially into the past. In the integral expression the most recent past is weightedstronger than the more distant past.

Expectation

The solution Vt ’s mean 〈V 〉t := EVt can be taken easily by replacing the noisy partsof dVt with their expectations:

d 〈V 〉t = ∑i(vi−〈V 〉t)d

⟨Gi⟩

t . (9.11)

Since the mean is a smooth process, the covariation processes will then vanish ([⟨Gi⟩,⟨G j⟩]≡

0). Replacing all covariation processes in (9.10) with 0 leads to:

⇒ 〈V 〉t = V0 +∑i

vi

t∫

0

〈Φ〉t〈Φ〉s

d 〈G〉is ; (9.12)

〈Φ〉t〈Φ〉s

= exp

(

∑i(〈G〉is−〈G〉it)

)

. (9.13)

103

As a consequence the mean of the membrane potential is independent of the correla-tions in input. See figure 9.2 (a) for an example of the expectation of the membranepotential, which in the example converges to a static expectation after having set offrom V0.

Variance

For any X , the variance satisfies VarX=⟨X2⟩−〈X〉2 =

⟨(X−〈X〉)2

⟩. This makes

the derivation of an explicit expression for the variance [Vt ] := VarVt more easy.Let hXt := Xt −〈Xt〉 be the deviation of a stochastic variable X from its mean at timet, then hVt := Vt − 〈V 〉t is the deviation of the potential from its mean, its variance[Vt ] =

⟨(hVt)

2⟩, and hGi

t = Git −〈G〉it is the deviation of conductance i from its mean.

dhXt = dXt −d 〈X〉t for the increments of some hX is a nice side-effect. The start is toget an equation for the deviation hVt by subtracting the 〈dVt〉 from dVt (→ 11.12):

dhVt = ∑i

((vi−Vt)dGi

t − (vi−〈V 〉t)d⟨Gi⟩

t

). (9.14)

Since

Git = hGi

t +⟨Gi⟩

t ⇒ dGit = dhGi

t +d⟨Gi⟩

t

Vt = hVt + 〈V 〉t ⇒ dVt = dhVt +d 〈V 〉t ,

the V and G can be replaced by expressions of h· and 〈·〉 (→ 11.12)

dhVt = ∑i

((vi−hVt−〈V 〉t)dhGi

t

−hVtd⟨Gi⟩

t

). (9.15)

The increments d(hVt)2 of (hVt)

2 can be found by applying the Ito rule (→ 11.12)

d(hVt)2 = 2∑

i(vihVt − (hVt)

2−〈V 〉t hVt)dhGit (9.16)

−2∑i(hVt)

2d⟨Gi⟩

t +d[V,V ]t .

Taking the expectation on both sides will give the variance. Obviously for any stochas-tic variable X the expected deviation EhX from its mean is zero. This follows fromthe definition of the mean value. From the Ito integral definition follows independenceof the increments dXt and the actual process Xt

d [Vt ] = −2∑i

[Vt ]d⟨Gi⟩

t +d[V,V ]t ,

another missing piece in the puzzle is the quadratic variation of the membrane voltage[V,V ]t , which is

d[V,V ]t = ∑i, j

(viv j− vi 〈V 〉t − v j 〈V 〉t

+E

V 2t

)d[Gi,G j]t , (9.17)

104

and since VarX= E

X2−EX2, E

V 2

t

can be replaced by 〈Vt〉2 +[Vt ]:

d[V,V ]t = ∑i, j

(viv j− vi 〈Vt〉− v j 〈Vt〉

+ 〈Vt〉2 +[Vt ])

d[Gi,G j]t

= ∑i, j

(vi−〈V 〉t)(v j−〈V 〉t)d[Gi,G j]t

+[Vt ]∑i, j

d[Gi,G j]t ,

which leads to

d [Vt ] = −2∑i

[Vt ]d⟨Gi⟩

t

+∑i, j

(vi−〈V 〉t)(v j−〈V 〉t)d[Gi,G j]t

+[Vt ]∑i, j

d[Gi,G j]t .

This is the differential equation which describes the variance. We solve it in the sameway as already described:

1. Solve homogeneous equation.

d [Φ]t = [Φ]t

(

∑i, j

d[Gi,G j]t −2∑i

d⟨Gi⟩

t

)

[Φ]t = exp

(

∑i, j

[Gi,G j ]t −2∑i

⟨Gi⟩

t

)

d [Φ]−1t =

(

∑i, j

[Gi,G j]t −2∑i

⟨Gi⟩

t

)

[Φ]−1t dt

2. Multiply and solve.

d [Φ]−1t [Vt ] = [Φ]−1

t d [Vt ]+ [Vt ]d [Φ]−1t

[Φ]−1t [Vt ] =

∫

[Φ]−1t ∑

i, j(vi−〈V 〉t)(v j−〈V 〉t)d[Gi,G j]t

[Vt ] = ∑i, j

t∫

0

[Φ]t[Φ]s

(vi−〈Vs〉)(v j−〈Vs〉)d[Gi,G j]s (9.18)

Correlations in the stimulus determine the variance. See figure 9.2(b) for an exampleof the variance of the membrane potential just after reset at t0 = 0. In this example thevariance rises and after ≈ 2ms drops again to converge to a static variance [V ]∞

9.3.3 Synaptic Conductance

The membrane equation does not specify explicitly the function which describes thebehavior of a synapse. Various noise models may be inserted into the membrane equa-tion, for example the Poisson counting process or the Wiener process. The Ornstein-Uhlenbeck process can also be inserted, but analysis will only carry as far as getting anexplicit expression for membrane potential and expectation.

105

Introduction

Each synapse is connected to one (or several) ion channel(s), each of which has acharacteristic reversal potential, and which allows influx of ions into the neuron. Afteran arrival of a pre-synaptic spike a channel is rapidly opened and closed again. The totalamount of ions which flow into the neuron depends on the product of the conductanceof the channel and the distance of the membrane potential to the reversal potential. Thebehavior of the conductance - which reacts to the incoming stimulus spikes - can bemodeled in several ways.

In the used framework the spikes in the stimulus sets are produced by variousstochastic processes, and the effect of these events on the working neuron is gener-ated via ion channels, or conductances. The behavior of a conductance over time thenalso follows a stochastic process, several of which are described below. Accordingto conventions, a conductance is named gi

t . To make the membrane equation of theused neuron model more readable, the conductance processes are referred to via theirintegral, Gi

t =∫ t

0 gisds, and its increments, dGi

t = gtdt.To model inter-neural dependencies as found in real life, stimuli may be cross-

correlated - that is⟨sis j⟩

does not equal⟨si⟩⟨

s j⟩. To describe a cross-correlation of a

certain strength between two synapses i and j, correlation factors ci j will be used.

Poisson Process

For simplicity it can be assumed that the inter-spike intervals arriving at stimulussynapses i are independent. The mathematical model following from events of whichthe inter-event time intervals are independent is the Poisson Process. The distributionof inter-event intervals is exponential.

If the ion channel’s behavior is considered to be very sudden - instantaneous in thesense that the governing time-scales are a lot smaller than the membrane time constant,the model of synaptic activity becomes very simple. A single stimulus spike elicits asharp rise and a sharp drop of the conductance which is modeled with a delta peak atevery stimulus time. The spikes arriving at one single synapse are expressed by thevariable si

t (see Intro), which takes values of zero or one. Let wi be the peak conduc-tance of channel - or synapse - i:

Git = wi ∑

tsit ,

dGit = wi

sit

dt,

where the event times t|sit = 1 are generated by a Poisson process with rate λ . The

interesting values we need for calculating the membrane potential are the expectation⟨Gi⟩

t and the covariation process[Gi,G j

]

t . For the Poisson process they are given by

⟨Gi⟩

t = wiλ t[Gi,G j]

t = ∑t

sit s

jt

⟨[Gi,G j]

t

⟩= ci jwiw jλ t.

A Poisson process can be generated easily by using any random variable dX it with a

known distribution PdX it. If time is discretised into bins of width dt, then a Poisson-

process is generated by a succession of random variables, where an event is assumed if

106

dX it > Θ. If an event has taken place, the process will be incremented by the “synaptic

weight” wi. The process will have the rate λ = PdX it(Θ). Several processes which are

correlated can be expressed (and implemented) by multiplying independent multidi-mensional white noise dW i

t with a mixing matrix a:

dGit =

w0

(dXt > Θ)

(dXt ≤Θ);

dX it = ∑

jai jdW j

t .

Since the dW jt obey a Gaussian distribution, dX i

t will have mean µ = 0 and varianceσ2 = ∑ j a2

i j. The rate of such a process will be (→ 11.12)

λ =12

(

erf

(Θ√2σ

)

+1

)

. (9.19)

Therefore for a given rate λ the threshold Θ for Poisson-events is

Θ = erf−1(2λ −1)√

2∑j

a2i jdt.

Expectation and covariation process are:⟨Gi

t

⟩= wiλ t,

[Gi,Gi]

t = wGit ,

⟨[Gi,Gi]⟩ = w2λ t.

It should be noted that the probability density of the Poisson process is asymmetric,and that dGi

t is strictly positive. If the expected conditional conductance d⟨Gi⟩

τ isgiven, the conditional stimulus density can be derived by dividing the conductance byits weight wi:

dPsi(τ) =d⟨Gi⟩

τwi

,

and the response stimulus correlation by dividing this by the unconditional stimulusrate:

Riτ =

dPsi(τ)

λidτ=

d⟨Gi⟩

τwiλidτ

.

Wiener Process

If many synaptic Poisson processes are added, the result again is a Poisson processwith a rate λi which is the sum of the rates of all processes before being added. If theprocesses are independent, the weight will not change. If there is a certain amount ofsynchrony among the processes, some of the stimulus spikes will arrive at the sameinstant and thus increase the weight wi, but decrease the rate λi. In the limit of in-finitely many synapses and small rates a sum of Poisson processes described abovewill become white noise, which is the derivative of the Wiener process. This sum ofPoisson processes is characterized by a total weight wi and a total rate λi for the con-ductance i. If this sum is modeled by a Wiener process, this will have a mean µi anda standard deviation σi, where the mean mainly provides information about the rate of

107

the stimulus signal, and the standard deviation mainly provides information about thestimulus synchrony. The exact influences of synchrony, rate and weight of many Pois-son processes on the Wiener process are beyond the scope of this work. The relationof mean, variance, rate and weight would be

λ =µ2

σ2 , (9.20)

w =σ2

µ, (9.21)

µ = rw, (9.22)

σ2 = rw2, (9.23)

and the Wiener process will be generated by

dGit = µidt +σidWt .

As above, correlated processes are obtained by using a variable dXt , and a correlationcoefficient ci j. Again, several processes which are correlated can be expressed (andimplemented) by multiplying independent multidimensional white noise dW i

t with amixing matrix a:

dGit = µidt +σidX i

t ,

dX it = ∑

jai jdW j

t .

Expectation and covariation process are:⟨Gi

t

⟩= µit,

[Gi,G j

]

t = σiσ jci jt,

with ci j = ∑k

aika jk.

Please note that the “correlation coefficients” ci j are not normalized, and thereforecii need not = 1. In case a neuron receives both Wiener and Poisson processes, theexpected covariation will look a little different. Let Gi be a Poisson and G j a Wienerprocess. The value in question is

⟨dGidG j

⟩, which will be zero whenever the Poisson

process dGit has no event, and = dG j

t /dt in case of an event in dGit . In this case of an

event in the Poisson process, dX it obviously is > λi

√dt. The single values dW i

t are of

interest, and since dX it = ∑ j ai jdW j

t , the conditional expectation⟨

dW jt |dX i

t > λ√

dt⟩

equals ai j · c. Each X i contributes according to its weight a, multiplied by a scaling

factor c =⟨

X it |X i

t > λ√

dt⟩

/[dX i

t

]. This results in

⟨[Gi,G j]⟩ = σ jci js jw jλ j, (9.24)

s j =

√

2π

e−

λ2j dt

2[dXit ]

(

1− erfλ j√

dt√

2[dX it ]

)√[dX i

t

]. (9.25)

108

If the expected conditional conductance d⟨Gi⟩

τ is given, the conditional stimulus den-sity for a Wiener process conductance can be derived by multiplying the conductancewith µ

σ 2 , which is the same as a division by the weight (compare eqns. (9.20-9.23)):

dPsi(τ) =µi

σ2i

d⟨Gi⟩

τ ,

and the response stimulus correlation is achieved by dividing the expected conditionalconductance by the unconditional mean, which is equal to dividing the conditionalstimulus spike density by the unconditional rate:

Riτ =

d⟨Gi⟩

τµidτ

.

The derivative of the Wiener process, and consequently the RSC of a Wiener processconductance may take negative values. This is biologically not feasible, although theformalism is mathematically correct. Unfortunately there is no way of constructing awhite-noise process which is strictly positive. Appendix 11.10 gives some reasons forthis fact.

Ornstein-Uhlenbeck Process

A more realistic noise model for synaptic inputs is a Poisson process with a decay,where each instantaneous opening of a synaptic conductance is followed by an expo-nential decay rather than an instantaneous drop. Adding many of these “decaying”Poisson processes in the same way as was done to yield the Wiener process as sumof many Poisson processes, results in an Ornstein-Uhlenbeck process. In case of astochastic stimulus with mean µi and variance σ 2

i , the Ornstein-Uhlenbeck process isdescribed by the differential equation

dgit = − 1

τigtdt + stimulusi

t dt,

dgit = − 1

τigtdt +(µi +σiW

′t )dt

=1τi

(τiµi−gt)dt +σidWt , (9.26)

which is solvable:

git = τiµi +σi

∫

[0,t]

e− 1

τi(t−s)

dWs

= τiµi +σiξt ; ξt :=∫

[0,t]

e− 1

τi(t−s)

dWs. (9.27)

Using Ito isometry the values which are important for calculating the membrane equa-tion can be found -

⟨Gi⟩

t = τiµit,[Gi]

t = σ 2i

τi

2

(

1− e− 2

τit)

.

109

−10 −5 0 5 10−54

−52

−50

−48

−46

−44

−42membrane voltage

time in relation to threshold crossing [ms]

volta

ge [m

V]

−10 −5 0 5 10−48

−46

−44

−42

−40

−38

−36membrane voltage

time in relation to threshold crossing [ms]

volta

ge [m

V]

−10 −5 0 5 10−70

−60

−50

−40

−30

−20

−10membrane voltage

time in relation to threshold crossing [ms]

volta

ge [m

V]

(a) (b) (c)

time to threshold [ms]-5 50

time to threshold [ms]-5 50

time to threshold [ms]-5 50

volta

ge[m

V]

volta

ge[m

V]

volta

ge[m

V]

-54

-42

-48

-36 -10

-70

Figure 9.3: (a) Expected conditional voltage (thick line, numerical expectation of eq.(9.10) without reset, averaged over the event of a threshold crossing) and voltage sam-ple (thin line, a sample of eq. (9.10)) of a neuron which is driven by two White-noiseconductances (see 11.11.2 for parameter settings.) (b) Same as (a), but the neuronis driven by two Ornstein-Uhlenbeck conductances. (c) Same as (a), but the neuronreceives a single excitatory Poisson process with a large weight. For details on thesimulations, as parameters, see 11.11.2.

Note that the synaptic time constant τ influences the variance of the process. Theunfortunate point is that the dGt are smooth, which results in the following covariation:

[Gi,G j]t = 0 ,

if Gi or G j are modeled by an Ornstein-Uhlenbeck process. In case of a given Ornstein-Uhlenbeck process gi

t and its increments dgit , the value of the stimulus W can be derived

by (→ 11.12):

1τi

gitdt +dgi

t = µidt +σidWt (9.28)

= stimulusit dt.

9.3.4 Reversing Time

Reversing time (i.e. calculating the voltage’s behavior before an observed spike) is acrucial part of deriving the RSC-equation. Under certain circumstances the stochasticprocess describing the free membrane potential is time-symmetric eqns. (9.33 and9.32). This results in a simple expression of the expected free (no reset) membranepotential before and after the crossing of a threshold. Introducing a reset puts themembrane voltage in an absorbing boundary scenario. The effect of the absorbingboundary can be approximated using the free voltage as it is eq. (9.34), a (numeric)Gaussian approximation introduced by Brunel and Hakim (Gerstner & Kistler, 2002)eq. (9.35), and an analytically motivated approximation eq. (9.43)

Time Symmetry

Calculations so far have provided an expression for the flow of the membrane potential,especially its mean and variance. Time starts at t0 = 0, with the voltage at a reset valueV0 = v0, and variance at zero. The voltage then travels to the threshold vθ , which itreaches at some time t.

The next step in the analysis must be to look at the conditional behavior of thevoltage before it has reached the threshold. As a base for analysis the free membrane

110

voltage is investigated; free in the sense that the voltage is not reset after it has exceededthe threshold value. This reset-less voltage shall be called V ∗t . Because all followingvalues will be conditional on the fact that the voltage has hit threshold at some t, andnegative values of t make no sense, the following shorter notation - introducing thedistance τ to the voltage hit - shall be used:

〈V 〉∗τ =def EVt+τ |Vt = vθ . (9.29)

A few thoughts help to understand the behavior of 〈V 〉∗τ .First thought: If the input from the conductance processes is static, then V ∗τ will

exhibit stability and converge to a static expectation for t →±∞, i.e. some constant〈V 〉∗∞ = c. Far away from the point where the threshold is hit, the voltage can beexpected to rest at this static expectation. The influence of the threshold hit decreasesin time.

Second thought: If the voltage can be interpreted as a diffusion process, its incre-ments can be split into a deterministic part a(Vτ) called ”drift” and a stochastic partb(Vτ) called ”diffusion”. The case of a symmetrical distribution of the distributionexpectation and mean of the diffusion are equal:

dVt = atdt +btdt . (9.30)

Drift and diffusion can be imagined as two different forces which conduct Vτ . Afterthe hit, the condition does not affect the diffusion part of Vτ , provided the conductanceprocesses cannot look into the future. Therefore EVτ = EVt - the drift pulls thevalue of V to the static expectation, and the diffusion adds some noise.

In contrast, when approaching the hit (τ negative), the work of getting the voltagenear the threshold is done by the diffusion, while the drift pulls the voltage in the oppo-site direction. The diffusion - via its probability density p(bτ dτ) - might be related tothe notion of energy, where low values of p(bτdτ) mean unlikely increments bτ dτ andtherefore high energy. If at some point the drift aτ dτ is strong, the diffusive incrementsbτ dτ must be strong as well to compensate the drift. If p(bτ dτ) is low for larger incre-ments (as in most cases) this means high energy. A sample path which takes its coursethrough such an area uses highly improbable diffusion increments and therefore investshigh energy here. Since dVt was assumed to have a diffusion with symmetrical noisedistribution. The expectation equals the path with the lowest energy. This is expressedin the condition

−0∫

−∞

p(bτ dτ)dτ = min,

and since (9.30) this equals

−0∫

−∞

p(〈dVτ〉∗−aτdτ)dτ = min. (9.31)

If p is known, eq. (9.31) can be solved using variational calculus. Lets assume thatp is Gaussian, and - as in most cases - aτ(Vτ) is independent of τ . The equation thenbecomes

0∫

−∞

(d

dτ〈V 〉∗τ −a(V∗τ )

)2

dτ = min.

111

A simple thought: If V ∗τ starts at τ = 0 where V ∗τ = vθ and works its way backwards(time arrow pointing to the past, dτ becomes−dτ) in time, the least energy is investedand the above equation becomes minimal if V ∗τ follows the drift a:

d 〈V 〉∗τ = a(〈V 〉∗τ)(−dτ).

If the direction of view is reversed again (time arrow pointing to the future) this resultsin

d 〈V 〉∗τ = a(〈V 〉∗τ)(−dτ)

= −a(〈V 〉∗τ)dτ= −d 〈V−τ〉∗ ,

because for the positive side d 〈V 〉τ = a(〈V 〉τ). The conditional derivatives form anodd function. As a last step follows

d 〈V 〉∗τ = −d 〈V−τ〉∗ (9.32)

⇒ 〈V 〉∗τ = 〈V−τ〉∗ . (9.33)

The derivative of an even function is odd. That means that the voltage for the neuronwithout reset, time-independent drift and a Gaussian voltage distribution is symmetric.The argument works the same way for other distributions where small increments meanlow energy. Simulations sustain the idea of time symmetry. Figure 9.3(a) shows asample path of a free membrane voltage of a white-noise driven neuron, as well as itsconditional expectation 〈V 〉∗τ ; Figure 9.3 (b) shows the same for an Ornstein-Uhlenbeckprocess driven neuron, where the diffusion is symmetric as well. The Poisson processdoes not show a symmetric probability density. The result of such a diffusion can beseen in figure 9.3 (c).

Above thoughts only hold true if the synaptic processes are uncorrelated in time,and the diffusion bτ has a symmetrical distribution. If the synaptic processes are time-correlated, the membrane potential is still time symmetric by the above argument, butan additional integral will have to be introduced. If the diffusion is not symmetric (thisis the case if the neuron is driven by very few synapses which are modeled as Poissonprocesses), then the potential will not be symmetric, but the left side (τ negative) willbe more steep than the right side.

The model which is investigated in this chapter is an integrate-and-fire neuron withreset, after the voltage has reached a threshold value vθ it will be reset to V0. Thereforestarting at t = t0 and recording the first spike at some later time t means there hashappened no spike between t and t0 and the voltage has reached the threshold vθ forthe first (and only) time at t:

〈V 〉τ =def EVt+τ |Vt = vθ ,τ 6= 0→Vt+τ < vθ .

The result of this is that for negative values of τ the threshold acts as an absorbingboundary on the stochastic walk V ∗τ . The effect of such a boundary is that it drainsprobability from its vicinity. The smaller the distance to the threshold, the lower theprobability density gets. Probability vanishes completely at voltages above threshold.There are different ways of how to incorporate this condition into the equation of V ∗τ .Each of these approaches uses the expected free potential 〈V 〉∗τ and distorts it in thevicinity of the threshold.

112

Free Approximation

A typical cortical neuron fires at very low rates (Baddeley et al., 1997; Olshausen &Field, 2004), which means that the potential quite seldomly touches the threshold. Itcan be assumed that in case of an eventual spike there has only been such a brief touch,and even without reset the potential would quickly restrain to its static expectationvalue. For an approximation of the neuron model, the probability that the potentialexceeds the threshold in the vicinity of t0 is neglected totally. V ∗τ is approximated usingthe symmetry assumption only:

〈V 〉τ =

〈V 〉−τ ,V0 = vθ〈V 〉τ ,V0 = V0

(τ < 0)

(τ > 0). (9.34)

This approximation just introduces the jump at τ = 0 into V ∗τ : Before the threshold hitthe voltage behaves according to 〈V 〉∗τ , that is starting at vθ and flowing backwards intime the same way it would flow into the future; and after the threshold hit the voltagestarts according to 〈V 〉t , which is just the normal expectation starting at the reset valueV0.

Gaussian Threshold Approximation

As a better approach, and to be able to deal with higher response rates, a modifieddistribution (Giorno et al., 1992; Brunel & Hakim, 1999; Gerstner & Kistler, 2002)shall be used, which works in two steps. First step: a probability law PVτ (V ) of thevoltage in absence of the threshold gives the probability of being in the interval V . Thechoice here will be a Gaussian with mean and variance from the previous section

dP∗Vτ (v) = gauss(v;〈V 〉τ , [Vτ ])dv.

(See figure 9.4 (a), top.) Second step: finding a factor with which this density willbe multiplied to accord for the fact that the probability above the threshold value iszero: P([vθ ,∞)) = 0. The solution is to multiply the probability (without threshold)of being at a certain voltage with the probability of being between this value and thethreshold. (See figure 9.4 (a), middle.) Let P∗Vτ (v) be the final approximation andp∗Vτ (v) = dP∗Vτ (v)/dv its density, this means

dPVτ (v) = dP∗Vτ (v) ·P∗Vτ ([v,vθ ])

= dP∗Vτ (v) ·vθ∫

v

dP∗Vτ (v) .

(See figure 9.4 (a), bottom.) The mean value then is given by the integral over thedifferent possible events:

〈V 〉τ =

∞∫

−∞

vdPVτ (v). (9.35)

Figure 9.4 (b), shows this mean in a fictive setting, and figure 9.4 (c) shows this meanin a real IF-neuron. Figure 9.4 (a), bottom, shows the final distribution, both for twodifferent thresholds. It comes as a disadvantage that the integral over the Gaussian bellhas to be approximated numerically. The analytically important value of dVτ is evenmore difficult to obtain. If the emphasis lies on an analytical approach - as it does inthis work - simpler approximations have to be used to be able to handle the flow of thevoltage without resorting to numerical methods.

113

distribution without threshold

correction factor for two different thresholds

corrected distribution for two different thresholds

(a)

0 2 4 6 8 10−5

0

5

10

15Adjusted Flow of Expected Membrane Potential

time

volta

ge

(b)

−10 −5 0 5 10−60

−55

−50

−45

−40conditional voltage, gaussian approximation

mean voltagetime

in r

elat

ion

to th

resh

old

cros

sing

[ms]

(c)

time

to th

resh

old

cros

sing

[ms]

distribution without threshold

correction factor for two different thresholds

corrected distribution for two different thresholds

(d)

0 2 4 6 8 10−5

0

5

10

15Adjusted Flow of Expected Membrane Potential

time

volta

ge

(e)

−10 −5 0 5 10−60

−55

−50

−45

−40conditional voltage, rectangular

time in relation to threshold crossing [ms]

volta

ge [m

V]

(f)

Figure 9.4: Figures demonstrating the approximations (Gaussian (a)-(c) and equal dis-tribution (d)-(f)) suggested in the text. (a) Three plots demonstrating the principle ofapproximating the membrane potential distribution in presence of two different thresh-olds, which are plotted as thin vertical lines. Top: The original Gaussian distribution.Middle: The coefficients which make the distribution vanish above threshold. Bottom:The approximated distribution in presence of a threshold is the product of the originaldistribution and the coefficients. Even in case of a large distance to the threshold, theexpectation will get shifted. (b) The flow of the expectation in a fictive setting. Thethin lines are the threshold, as well as expectation ± std.dev. of the undisturbed dis-tribution. The thick line shows the expectation in presence of the threshold eq. (9.35),using a fictive mean and variance. (c) The effect of the Gaussian approximation ina setting of a leaky integrate-and-fire neuron with exc. and inh. synapses (→ 11.12).Thin lines show the experiment (numerical conditional expectation of eq. (9.10)), thicklines show the theory eq. (9.35), with mean and variance according to eqns. (9.12) and(9.18). (d) Same as (a), but using an equal distribution which has been fitted to theGaussian (see eqns. (9.38-9.41)) as starting point. Note that with a distant thresholdthe approximation performs better than the Gaussian approximation. (e) Same as (b),showing the equal distribution approximation eq. (9.43) in a fictive scenario. (f) Sameas (c), thick lines show the equal distribution approximation according to eq. (9.43)using mean and variance of eqns. (9.12) and (9.18). For details on the simulations, asparameters, see 11.11.2

114

Equal Distribution Approximation

To avoid the above disadvantages, the approximation can be simplified even further.Instead of a Gaussian, an equal distribution with parameter µ (mean) and a (borders ofthe probability rectangle) can be used -

p(v) =

c,0,

(µ−a < v≤ µ +a)

otherwise.

There are various ways to choose the a which defines the borders of the equal distri-bution, and in choosing it one must bear in mind that the equal distribution is meantto approximate the Gaussian distribution, therefore the distance to the Gaussian mustbe kept minimal. The distance of two functions in function space is often calculatedby integrating over the square distance at each point - this is not really applicable here,as the functions are distributions and there are better distance measures. The secondchoice would be minimizing Kulback-Leibler divergence. Unfortunately the resultingexpression has the disadvantage of being independent of the variance (calculations notshown), which is quite an important parameter. Another choice left is for the distribu-tions to have equal expectation and variance, which leads to

a =√

3 [V ]∗τ ,

µ = 〈V 〉∗τ . (9.36)

(See figure 9.4 (d), top.) To mimic the effect of the threshold, this distribution will bemultiplied by the distance to the threshold (see figure 9.4 (d), middle ). If the thresholdlies between µ + a and µ − a this is the same as integrating over the distribution, thesame way as in the Gaussian approach. If the threshold lies above µ + a its influencewill be proportional to its distance from µ , which is the behavior one would expectfrom an absorbing boundary. If the threshold lies infinitely high above the Gaussiandistribution, it will not have any effect on it. This is a feature which the Gaussianapproach fails to reproduce. If the threshold lies below µ − a this approximation willnot work; but since the voltage will never cross the threshold this regime does notapply. Figure 9.4(d), bottom, shows the resulting distribution

p(v) =

c(θ − v) ,

0 ,

µ−a < v≤ µ +aotherwise

. (9.37)

The constant c serves to have the integral over the distribution equal one P(Ω) = 1 (→11.12):

c =

12a(θ−µ)

,

1θ22 +(µ−a)( µ−a

2 −θ),

(θ > µ +a)

(θ ≤ µ +a). (9.38)

From this the expectation Vt =∫

Ω vdP(v) and its time derivative dVtdt = d

dt

∫

Ω vdP(v) canbe calculated easily. The two cases where the threshold θ is either below or above theborder a of the equal distribution are handled separately. From θ > µ +a follows:

〈V 〉t = µ− a2

3(θ −µ), (9.39)

d 〈V 〉t =

(

1− a2

3(θ −µ)2

)

dµ− 13(θ −µ)

da2 , (9.40)

115

and θ < µ +a leads to

〈V 〉t =23(µ−a)+

13

θ , (9.41)

d 〈V 〉t =23(dµ−da) . (9.42)

Since θ = vθ , a =√

3 [V ]∗τ and µ = 〈V 〉∗τ (see above) this gives:

〈V 〉τ =

23 (〈V 〉∗τ −

√

3 [V ]∗τ )+ 13 vθ ,

〈V 〉∗τ −[V ]∗τ√

3(vθ−〈V〉∗τ ),

(∗)(∗∗) (9.43)

and

d 〈V 〉τ =

23

(

d 〈V 〉∗τ −d√

3 [V ]∗τ

)

,(

1− [V ]∗τ√3(vθ−〈V〉∗τ )2

)

d 〈V 〉∗τ −d[V ]∗τ√

3(vθ−〈V〉∗τ ),

(∗)(∗∗) (9.44)

with

(∗) = 〈V 〉∗τ −√

3 [V ]∗τ < vθ < 〈V 〉∗τ +√

3 [V ]∗τ

(∗∗) = vθ > 〈V 〉∗τ +√

3 [V ]∗τ

Figure 9.4 (e), (f) show the behavior of eqns. (9.43) and (9.44).

9.3.5 Response-Stimulus Correlation

The second step in retrieving the response-stimulus correlation is to calculate the ex-pected conditional conductances. The expected conductance at some time before orafter an observed response cannot be estimated without knowledge of the relations ofthe various synapses. Simple arguments shed some light on these relations, and sim-ulations sustain the resulting ideas: The expected conditional conductance is the sumof a static expectation which is independent of τ , and a dependent part eq. (9.47).For a leak conductance this dependent part vanishes. This dependent part is propor-tional to the covariances among different conductances, and to the reversal potential ofa conductance eq. (9.48).

Mutual Relations of Synaptic Noise Processes

A synapse feeds the neuron with a spike train. The conditional stimulus density (dPsi(τ))is the density of stimulus spikes si

τ at conductance i and time-distance τ to an observedresponse spike. If a simple conductance model is used (Poisson process or Wiener pro-cess), the number of stimulus spikes is just the expected conditional (on the fact that aresponse spike has been observed at τ = 0) conductance divided by the weight of thesynapse, and the (normalized) RSC is the expected conditional conductance divided bythe un-conditional expectation of that conductance. Obtaining the expected conditionalconductance is the next step which leads to the response-stimulus correlation.

As with the voltage, following conductance values will be conditional on the factthat the voltage has hit threshold at some t, and the same shorter notation - introducingthe distance τ to the voltage hit - shall be used:

⟨Gi⟩

τ = E

Giτ

:= E

Git+τ |Vt = vθ

. (9.45)

116

In addition the distance operator h of a conductance from its unconditional mean isused:

hGit = Gi

t −⟨Gi

∞⟩,

⟨hGi⟩

τ = E

hGiτ

:= E

Git+τ −

⟨Gi

∞⟩|Vt = vθ

.

The symmetry of the voltage eqns. (9.32) and (9.33) can be used to estimate the con-ductance. The equation for the membrane potential does not change for the conditionalcase;

d 〈V 〉τ = ∑i

(vi−〈V 〉τ)d⟨Gi⟩

τ , (9.46)

where the d⟨Gi⟩

τ are the expected conditional conductances. To find the conditionalexpected conductances , the conditional membrane potential eq. (9.46) has to be rear-ranged to extract the d

⟨Gi⟩

τ . Here we encounter a serious problem: when rearrangingfor d

⟨Gi⟩

τ , the scalar product - ∑i(vi−〈V 〉τ)d⟨Gi⟩

τ - cannot be decomposed withoutfurther knowledge of the relations between one synaptic noise process and another.

First step: Obviously there are two distinct parts of the expected conditional con-ductance: a slow gain in probability just before the response spike is emitted - up tothen the function is monotonically increasing - followed by a sudden drop at τ = 0and a static probability from then on. Just before a response spike the probabilityof a stimulus spike is highest, and stimulus and response are highly correlated. Af-ter a response, as well as at very long time distances before a response, stimulus andresponse are uncorrelated. As a start the following hypothesis can be made: the synap-tic stimulus d 〈G〉it can be split into a static (uncorrelated) and a dynamic (correlated)part

d⟨Gi

τ⟩

= d⟨Gi

∞⟩+d

⟨hGi⟩

τ , (9.47)

where the static part 〈G〉i∞ equals the expected synaptic noise and the dynamic partdhGi

t is in some way related to the variance of the synaptic noise. Figures 9.5 (a) and(d) demonstrate the additive influence of different static expectations of a stimulus. It isassumed that the dynamic parts dhGi

t of different channel types i can be expressed byscaling a total conductance d 〈hG〉τ with a synapse-dependent scaling factor αi whichindicates its effective strength in driving the spike mechanism

d⟨hGi⟩

τ = αid 〈hG〉τ .

⇒ 1αi

d⟨hGi⟩

τ =1

α jd⟨hG j⟩

τ = d 〈hG〉τ .

The expected conditional conductance then is the product of the “effective strength” αi

and the “total conductance” 〈G〉τ which is the same for all conductances i.How can “effective synaptic strength” as well as “total conductance” be estimated?

Effective Synaptic Strength

The effective synaptic strength could be split further into a potential part or “influence”,and a real part or “energy”. The “influence” expresses how important the “opinion” of

117

−5 00

0.05

0.1

conditional conductance (RSC)

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S]

−5 00

conditional conductance (RSC)

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S]

−10 −5 0

0

conditional conductance (RSC)

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S]

0 0.02 0.04 0.06 0.080.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09relation of conductances

mean of conductance [µS]

valu

e at

τ=−0

.3m

s

−60 −40 −20 0 20 400.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04relation of conductances

rev. potential of conductance [mV]

valu

e at

τ=

−0.

3ms

0 2 4 6 8

x 10−3

0

0.05

0.1

0.15

0.2

0.25relation of conductances

variance of conductance [(µS)2]

valu

e at

τ=

−0.

3ms

(a) (b)

(d)(c)

(e) (f)

Figure 9.5: Expected conditional conductance of a neuron with four different stimulusconductances. (a)-(c) show the expected conditional conductances, (d)-(f) show therelation of the expected conditional conductance at τ = 0. (a),(d) conductances differonly by their expectation

⟨Gi

t

⟩. (b),(f) conductances differ only by their reversal poten-

tials vi. (c),(f) conductances which differ only by their variance[Gi

t

]. All plots show

the results of a simulation; eq. (9.10) was run and the dGt averaged over the responsespike times according to eq. (9.1) (i.e. the times when Vt hit the threshold and wasreset). For more Details of simulation parameters see appendix 11.11.3.

118

the synapse is to the membrane potential or how influential the synapse is, the “en-ergy” says something about the extent to which the synapse uses its “influence”. Theconjecture regarding the effective synaptic strength,

αi = influence · energy,

means that the success in driving the spike mechanism is the product of how influentiala synapse is, and the energy it invests.

1. Influence of a conductance. Remember the equation for the membrane poten-tial -

dVt = ∑i(vi−Vt)dGi

t .

Obviously a change in the membrane potential is related to a change in the synapticconductance. This synaptic conductance is scaled by the term (vi−Vt). If a suddenconductance change occurs in a synapse of which the reversal potential lies very nearthe membrane potential, the membrane potential will hardly take any notice of thischange. If, on the other hand, the reversal potential lies far away from the membranepotential, even a small quiver in the conductance will have a large effect. This suggeststhat the impact a conductance can have depends on its influence on Vt , and this influenceequals the distance of vi to the current potential Vt . Figures 9.5((b) and (e)) illustratethe linear effect of different reversal potentials.

2. Energy. To what extent does a conductance use this influence? A totally quies-cent synapse doesn’t make any use of its influence. Even if its influence is big, as longas it does not move, the membrane potential does not react to it. The leak potential is anexample for such a conductance. An array of synapses which produces highly variantnoise makes a lot of use of its influence. This can be expressed by the idea of ’energy’which is being invested into changing the membrane potential, and this energy equalsthe variance Var

dGi

t

of the conductance. Figures 9.5((c) and (f)) show the linear

influence of different conductance variances.3. Cooperation among synapses. In case there are cross-correlations, a conduc-

tance doesn’t have to do all the “convincing” on its own, but gets help from others. Itseffective strength is then affected by the cooperation with other synapses. The moreenergetic the helping conductances are, the greater their effect will be. This cooper-ation is expressed in the covariance processes. If there are no cross-correlations, thecooperation is 0 for any two different synapses, and 1 for the same synapse. If thereare cross-correlations, the above conjecture must be re-written to be

αi = ∑j

influence j · cooperationi, j · energy j

= ∑j

(v j−〈V 〉τ) ·d[Gi,G j]τ/dτ (9.48)

This is one of the central theorems of this chapter. The simulations in the results sec-tion will confirm its validity.

4. Other synaptic characteristics. Other characteristics influence the synapticstrength, or the expected conditional conductance in general. Examples are the exactshape of the synaptic input probability distribution, or a synaptic time constant. Theapproach shown here is an approximation which only takes mean and covariation ofconductance processes into account. An example for an asymmetric probability dis-tribution is the Poisson process, and the effect of this asymmetry is demonstrated in

119

the results section. Processes with time correlations are the Gamma process and theOrnstein-Uhlenbeck process. Effects of using an Ornstein-Uhlenbeck process are sub-ject to current research.

Total Conductance

The conditional expected conductance d⟨Gi⟩

t is the sum of the static conductance

〈G〉i∞ and the dynamic conductance d⟨hGi⟩

τ . The total conductance d 〈hG〉τ is equalfor all synapses, and if multiplied by the effective strength αi gives the dynamic con-ductance. Total conductance can be calculated by replacing

⟨Gi⟩

τ in the membrane eq.

(9.46) with d 〈G〉i∞ +α iτd 〈hG〉τ and rearranging for d 〈hG〉τ . This yields: (→ 11.12)

d 〈hG〉τ =d 〈Vτ〉−∑i (vi−〈Vτ〉)d 〈G〉i∞

∑i (vi−〈Vτ〉)α iτ

=d 〈Vτ〉−∑i (vi−〈Vτ〉)d 〈G〉i∞

∑i, j (vi−〈Vτ〉) (v j−〈Vτ〉)d[Gi,G j]τ. (9.49)

Response-Stimulus Correlation

After all contributing measures have been taken into consideration, assembling theresponse-stimulus correlation for synapse type i from all preceding equations is leftas the final step. If time-independent processes are used for the conductances, theresponse-stimulus correlation will be

Riτ =

d⟨Gi⟩

τd 〈Gi〉∞

=d⟨Gi⟩

∞ +α iτ d 〈hG〉τ

d 〈Gi〉∞. (9.50)

This is the main result from all previous theoretical considerations. Figure 9.6 shows anexample. The previous measures which lead to this result shall be summed up briefly.The above equation for the RSC uses the relations of the “effective synaptic strengths”

αi = ∑j

influence j · cooperationi, j · energy j

= ∑j

(v j−〈V 〉τ) ·d[Gi,G j]τ/dτ ,

it uses the notion of a “total conductance” from eqn. (9.49)

d 〈hG〉τ =d 〈Vτ〉−∑i (vi−〈Vτ〉)d 〈G〉i∞

∑i (vi−〈Vτ〉)α iτ

=d 〈Vτ〉−∑i (vi−〈Vτ〉)d 〈G〉i∞

∑i, j (vi−〈Vτ〉) (v j−〈Vτ〉)d[Gi,G j]τ,

and it incorporates an analytical “rectangular” voltage approximation eqns. (9.43) and(9.44)

〈V 〉τ =

23 (〈V 〉∗τ −

√

3 [V ]∗τ )+ 13 vθ ,

〈V 〉∗τ −[V ]∗τ√

3(vθ−〈V〉∗τ ),

(∗)(∗∗)

120

−10 −5 0 5 10

0

RSC, rectangular approximation

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S]

Figure 9.6: This plot shows a match of the analytical response-stimulus correlationand the value measured from a simulation. The neuron receives three conductances- leak, excitatory and inhibitory. Excitatory and inhibitory conductances are modeledusing a Wiener processes. Both conductances are cross-correlated to some extent. Seeappendix 11.11.2 for all parameter values. The plot shows the results of a numericalsimulation; eq. (9.10) was run and the dGt averaged over the response spike times (i.e.whenever Vt hit the threshold and was reset) according to eq. (9.2).

121

and

d 〈V 〉τ =

23

(

d 〈V 〉∗τ −d√

3 [V ]∗τ

)

,(

1− [V ]∗τ√3(vθ−〈V〉∗τ )2

)

d 〈V 〉∗τ −d[V ]∗τ√

3(vθ−〈V〉∗τ ),

(∗)(∗∗)

with

(∗) = 〈V 〉∗τ −√

3 [V ]∗τ < vθ < 〈V 〉∗τ +√

3 [V ]∗τ

(∗∗) = vθ > 〈V 〉∗τ +√

3 [V ]∗τ ,

which depends on the mean of the voltage without reset from eqn. (9.12),

〈V 〉t = V0 +∑i

vi

t∫

0

〈Φ〉t〈Φ〉s

d 〈G〉is ;

and the variance of the voltage without reset eq . (9.18)

[Vt ] = ∑i, j

t∫

0

[Φ]t[Φ]s

(vi−〈Vs〉)(v j−〈Vs〉)d[Gi,G j ]s,

[Φ]t = exp

(

∑i, j

[Gi,G j]t −2∑i

⟨Gi⟩

t

)

.

The values⟨Gi⟩

t and [Gi,G j] depend on the processes used to model the conductancesand are usually so simple that the above integrals can be calculated easily. Examplesto demonstrate this are shown in the following section.

9.4 Results

9.4.1 General Performance of the Model Equations

To test the general performance of the equations, a neuron with excitatory and in-hibitory synapses is set up. Expected conditional conductances are measured (thinlines in figure 9.7) and all three approximations are tested (thick lines in figure 9.7).In the case of the Gaussian approximation the theoretical curves lie exactly on top ofthe simulated curves. The curves of the equal distribution approximation are hardlydistinguishable from the Gaussian curves. Obviously the harsh approximation usingan equal distribution has the desired effect. In the case where no voltage correction isapplied, the conductance is underestimated just before the response spike.

9.4.2 Single Excitatory Conductance

To demonstrate the theory presented in the previous part, the response-stimulus cor-relation is derived for a neuron in the simplest setting possible. The neuron is drivenby one leak conductance and one excitatory conductance, which is modeled by whitenoise. The presence of only one noisy conductance reduces the equation for the RSC,since mutual conductance relations don’t have to be taken into account:

d⟨GE⟩

τ =d 〈V 〉τ − (vL−〈V 〉τ)dτ

vE−〈V 〉τ, (9.51)

122

−10 −5 0 5 10

0

RSC, no correction

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S]

−10 −5 0 5 10

0

RSC, rectangular approximation

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S]

−10 −5 0 5 10

0

RSC, gaussian approximation

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S](a) (b) (c)

time to threshold [ms]-10 100

time to threshold [ms]-10 100

time to threshold [ms]-10 100

0 0 0

mea

nco

nduc

tanc

e[µ

S]

mea

nco

nduc

tanc

e[µ

S]

mea

nco

nduc

tanc

e[µ

S]

Figure 9.7: (a) Measured (thin line, numerical conditional expectations d⟨Gi⟩

τ fromeq. (9.10) and estimated (thick line, eq. (9.34)). RSC of a neuron with an excitatory(top curve) and an inhibitory (bottom curve) conductance

⟨Gi

t

⟩. (b) Same as (a), using

the Gaussian approximation (thick line, eq. (9.35)) (c) Same as (a) and (b), using theequal distribution approximation (thick line, eq. (9.43). Details on the simulations andparameters are given in 11.11.5.

In the simulations the equal distribution approximation will be used:

〈V 〉τ =

23(µ−a)+ 1

3 θ ,

µ− a2

3(θ−µ) ,

(θ < µ +a)

θ > µ +a

with

θ = vθ ,

a =√

3 [V ]∗τ ,

µ = 〈V 〉∗τ .

The leak conductance will have the value of µL, the excitatory conductance will havea mean of µE and a variance of σ 2

E . The reversal potentials vi will be called vL andvE , the conductances will be called dGL

t = µLdt (leak) and dGEt = µEdt (excitatory).

Inserting these values into (9.12) and (9.18) results in (→ 11.12):

〈V 〉∗t = (V0−V∞)e−τmt +V∞, (9.52)

with

〈V 〉∞ =vLµL + vE µE

µL + µE,

τm = µL + µE .

And the variance equations reduce to (→ 11.12):

[V ]∗t = e−c1t

t∫

0

ec1t (c22−2c2c3e−τmt + c2

3e−2τmt)ds,

with (9.53)

c1 = σ 2E −2(µL + µE),

c2 = vE −V∞,

c3 = V0−V∞.

In several experiments the parameters of the above equations where changed, whichresulted in a change of the shape of the RSC. This shape change was recorded by

123

0.05 0.1 0.15 0.2 0.25

0

0.5

changing stim. variance

std.dev.

<g>

[µS

]

−10 −5 0 5 10−0.5

0

0.5

1

1.5RSC, rect. apprx.

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [[

µS]

0 0.02 0.04 0.06 0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.5changing stim. mean

exc. mean [µS]

d<g>

t [µS

]

0 0.02 0.04 0.06 0.08 0.1 0.120

0.2

changing stim. mean

exc. mean

<g>

[µS

]

0 20 40 60 80

0

0.2

changing response rate

response rate [Hz]

<g>

t [µS

]

0.05 0.1 0.15 0.2 0.25

0

0.5

changing stim. variance

variance [(µS)2]

<g>

t [µS

]

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25changing step size

step size [ms]

<g> t [µ

S]

0 50 100 150 200 2500

10

20

30

40changing stim. rate

stim. rate [Hz]

stim

. spi

ke d

ensi

ty /

rate

(b)(a)

(c)

(h)(g)

(d)

(f)(e)

Figure 9.8: The plots compare numerical experiments with theory. Thin, rough linesare results of a numerical simulation eq. (9.1), where rt was generated by threshold hitsof eq. (9.10); thick, smooth lines are the theory eq. (9.50). Details are given in the text.(a) Results of one run with the settings described in the text. Points mark the valuesof τ where the following plots are recorded. (b) The effect of increasing the stimulusmean. (c) Same as (b), but at a response rate of 120Hz. (d) The effect of increasingthe response rate on the expected conditional conductance. The correlation is reduced.(e) The effect of increasing the stimulus variance. The response rate is 5Hz. (f) Sameas (e), but at a response rate of 120Hz. (g) The effect of reducing the simulation timestep illustrates the delta peak at τ = 0. In contrast to the other plots, traces are recordedat τ = −[0,0.1,0.2,0.3]mS (h) Influence of an increase of the stimulus rate on thenormalized RSC. Details on the simulations and parameters are given in 11.11.4.124

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

0102030405060708090

100RSC, rectangular approximation

time in relation to threshold crossing [ms]

resp

onse

−st

imul

us c

orre

latio

n

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

0102030405060708090

100RSC, rectangular approximation

time in relation to threshold crossing [ms]

resp

onse

−st

imul

us c

orre

latio

n

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

0102030405060708090

100RSC, rectangular approximation

time in relation to threshold crossing [ms]

resp

onse

−st

imul

us c

orre

latio

n

RSC

0

100

RSC

0

100

RSC

0

100

time to threshold [ms]-10 100

time to threshold [ms]-10 100

time to threshold [ms]-10 100

(a) (b) (c)

Figure 9.9: (a) High leak conductance (1µS): quick neuron (b) Medium leak conduc-tance (0.2µS). (c) Low leak conductance (0.04µS): slow neuron. (a)-(c) It can be seenthat the change of the leak conductance, and therefore the change of the membranetime constant, leads to a change in the speed of integration of the neuron. As before,thin, rough lines are results of a numerical simulation eq. (9.1), where rt was generatedby threshold hits of eq. (9.10); thick, smooth lines are the theory eq. (9.50). Detailsare given in the text. Details on the simulations and parameters are given in 11.11.4.

tracking the RSC values at various time distances. The resulting plots (figure 9.8) showhow the shape of the response-stimulus correlation changes in response to parameterchanges. Theses effects - and whether the theory matches the simulations - shall bediscussed now.

Singularity at τ = 0

Of special interest is the height of the jump at τ = 0, because it states how many spikeshave actually triggered the response spike. This can be calculated by letting V ∗τ → vθin eq. (9.51). The equation says for this point:

d⟨GE⟩

0 =d 〈V 〉0− (vL− vθ )dτ

vE− vθ,

d 〈V 〉0 →(

1− 0√3(02

)

dµ− 10

da2 (θ −µ→ 0)

→ ∞ (θ −µ→ 0).

At a time distance of zero before the observed response, the noisy conductance has adelta peak. A delta peak at τ = 0 seems to be a strange result, but a simple thought canclarify this issue. Obviously there will always be one last spike which lifts the voltageover threshold. This triggering spike appears just before the response. Therefore theprobability P(T ) =

∫

T dP(spike) of having a spike in the interval T = [τ ,0] does notvanish, even if τ → 0. This results in a delta peak which can be seen in figure 9.8(a).To illustrate the behavior at τ = 0, a simulation is run where the RSC is recorded atτ = 0 while the timestep dt is reduced. The result is shown in figure 9.8(g). Thepoints for the traces are taken a lot smaller here. The branching shows how two points,which previously come to lie in the same time bin, are being distributed into differenttime bins, as the width of the time bins (steps) gets smaller. The value at τ = 0 risesexponentially, while the bin width drops linearly. This clearly confirms the singularityat zero time delay.

125

Stimulus Mean, Variance, Rate and Weight

Figures 9.8 (b), (c), (e), (f) demonstrate the effect of changing stimulus mean andvariance. The theory matches the simulations well. If the stimulus rate is increased,this results in a dramatic decrease of the correlation. This change is shown in figure9.8 (h), where the response-stimulus correlation Rτ is plotted against the stimulus rate.Again, theory and simulation match well.

Response Rate

The settings where run in two different versions - with the threshold fixed to yield aresponse rate of 5Hz (figures 9.8 (a,b,e,g,h)), or as a comparison with the thresholdfixed to yield a response rate of 120Hz (figures 9.8 (c), (f)). Figure 9.8 (d) illustrateshow the response-stimulus correlation decreases while the neuron travels from the lowresponse rate regime to a high response rate regime. In the regime 0−20Hz the effectis a nonlinear drop, which reduces to less steep linear decrease at ranges above 20Hz.Changes of mean and variance of the stimulus have a much larger effect in the lowresponse rate regime (figures 9.8 (b,e)) than in the high rate regime (figures 9.8 (c,f)).At higher rates the neuron becomes less sensitive. The analytic expressions (thick lines)for the RSC reproduce this effect (simulated: thin lines) well.

Leak conductance and Integration Time Constant

In the previous example the influence of parameters external to the neuron, such asstimulus mean and variance, was examined. Since in the course of turning a stimulusinto a response the neuron may change its internal parameters as well, they shall beexamined now. Using the same settings, the influence of the leak conductance is in-vestigated, results are shown in figure 9.9. The main effect of a change in the valueof the leak conductance is a change of the membrane time constant. Another effect isthat when the leak conductance is low, fewer ions leak out of the membrane, and there-fore the average membrane potential is higher than when the leak conductance is high.Figure 9.9 demonstrates the change in shape of the RSC (Ri

τ ). Settings are the sameas in the previous simulation, apart from the leak conductance, which is set to 1µS ,see fig. 9.9 (a), 0.2µS fig. 9.9 (b) and 0.04µS, fig. 9.9 (c). This results in an effectivemembrane time constant of 0.9901ms, see fig. 9.9 (a), 4.7619ms fig. 9.9 (b) and 20msfig. 9.9 (c). The figures suggest that the change of the leak conductance results not onlyin a change of the time scale, but that the shape of the RSC is changed as well. Whilethe RSC of a quick neuron is reminiscent of an exponential function, the RSC of a slowneuron seems to include a large linear part before the sharp exponential increase nearτ = 0. This indicates a longer integration phase before the neuron eventually decidesto respond with a spike.

9.4.3 Excitatory versus Inhibitory Synapses

The usual situation for a neuron in the mammalian cortex means that the stimulus itreceives will be cross-correlated. Not only different excitatory stimuli will be cross-correlated. Inhibition is believed to be triggered by feedback and lateral connections,and therefore strongly related to local neural activity. In this example a neuron isdriven by excitatory Wiener and inhibitory Wiener input. The cross-correlation coeffi-cient is scaled from 0 to 1, while the neuron’s spiking is kept at a rate of 5Hz, whichis maintained by adjusting the neuron’s threshold. Figure 9.10 shows the RSC at the

126

−10 −5 0 5 10−0.01

−0.005

0

0.005

0.01covariance = 0

tau

RS

C

−10 −5 0 5 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014covariance = 1

tau

RS

C

0 0.2 0.4 0.6 0.8 13

4

5

6

7x 10

−3 excitatory synapses

covariance

RS

C

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

6

8x 10

−3 inhibitory synapses

covariance

RS

C

(a) (b)

(c) (d)

Figure 9.10: Behavior of excitatory and inhibitory response-stimulus correlation withchanging cross-correlation. Cross-correlation goes from 0 (uncorrelated) to 1 (com-pletely correlated). See text for details. Thin, rough lines are results of a numericalsimulation eq. (9.1), threshold hits from eq. (9.10) formed the response rt ; thick,smooth lines are the theory eq. (9.50). (a) Response-stimulus correlation with uncor-related stimulus. (b) Response-stimulus correlation with correlated stimulus. The fourmarked points indicate at wich time-difference τ the RSC in (c) and (d) was recorded.(c) Shape change of excitatory conductance during increase of cross-correlation. (d)Shape change of inhibitory conductance during increase of cross-correlation. Traceswhere recorded at the points indicated in (a) and (b). Details on the simulations andparameters are given in 11.11.5.

127

−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5conditional conductance

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S]

−10 −5 0 5 10

0

RSC, rectangular approximation

time in relation to threshold crossing [ms]

mea

n co

nduc

tanc

es [u

S](a) (b)

Figure 9.11: (a) Expected conditional conductances of a neuron driven by an excitatoryWiener process and an inhibitory Poisson process. (b) Same as (a), with the theory(equal distribution approximation: thick lines) added. Explanations in the text. Thin,rough lines are results of a numerical simulation eq. (9.1), threshold hits from eq.(9.10) formed the response rt ; thick, smooth lines according to eq. (9.50). Details onthe simulations and parameters are given in 11.11.6.

cross-correlation of 0 and 1. Of course a complete correlation of inhibition and excita-tion is biologically unlikely, and the results of the scenario are meant to underline theimportance of cross-correlation as such.

Since the response spike is produced by a membrane potential which is below athreshold, and then rises to cross this threshold, a response indicates an excitation.Independent on other parameters, the RSC will always show an excitatory action atτ = 0. With an increasing cross-correlation both conductances will behave in a moreand more similar manner. If the cross-correlation equals one, they are driven by thesame process altogether. Since the curves will exhibit an excitatory action, and sinceboth curves are generated by the same process, the inhibitory conductance eventuallyhas an excitatory effect. This may occur even if the reversal potential of the inhibitorysynapse lies below the membrane voltage. See figure 9.10 (b) for this effect.

9.4.4 Negative Conductances

In this test a neuron was driven by an excitatory conductance which was modeled us-ing the Wiener process, and an inhibitory Poisson process conductance (see 11.11.6for exact values). The effect of the asymmetric density of the Poisson process can beseen clearly. Here a Poisson process drives an inhibitory conductance, and it has alow rate and a high weight. In contrast to white noise which very well can becomenegative (compare appendix 11.10 for a discussion of the impossibility of strictly pos-itive white noise), the Poisson process always remains positive, a fact which is notexpressed in the mean or variance of the process. Because of the inhibitory character-istics of the synapse, the expected conditional conductance will decrease in the vicinityof a response spike. The expected conditional conductance is assumed in the part ofthe probability density which lies below the mean. Since the mean of the process liesvery near to zero, the expectation becomes negative. In the figure 9.11(a) the resultsof the simulation are displayed; in figure 9.11(b) the prediction of the theory - whichshows negative values - is compared to the simulation. In this case it serves to cut offthe prediction at v = 0 to provide a good fit.

128

9.5 Discussion

9.5.1 Ito or Stratonovitch?

Lindner (Lindner et al., 2003) has shown in a neural context that it really does makea difference whether Ito or Stratonovitch integrals are used. Both integrals serve todescribe observable physical and biological entities in a stochastic manner. Since theresulting difference may become quite substantial, the values and drawbacks of bothapproaches shall be discussed briefly.

A real physical system can be expected to have continuous trajectories in statespace, in this case usually Stratonovitch integrals are chosen (it is traditional). Stratonovitchintegrals have the drawback that expectation and moments are quite hard to obtain.Therefore a Stratonovitch-interpreted equation is (often) transformed into its Ito coun-terpart. The resulting equation will include an additional drift term and become morecomplex.

Some observed entities are countable and discrete - such as population sizes oramounts of money. When modeling these quantities the emphasis usually lies on thefact that underlying processes cannot look into the future. This results in Ito differentialequations. If a standard Euler algorithm is used, this results in the SDE being Ito. Itointegrals are easier to deal with mathematically, and the mean can be obtained easily.Their drawback is that the usual chain rule is tranformed into a more complex rule,which provides an additional term (see appendix).

The integrate-and-fire neuron is a little machine might be considered to handlediscrete amounts of stimulus spikes to produce response spikes. Although the conduc-tances are slowly varying processes, which make a Stratonovitch interpretation consid-erable, they are driven by discrete quantities - spike vs. no spike. Hence the Ito SDE isa valid tool.

As a last remark it can be said that mathematically both methods are absolutelycorrect (Kloeden & Platen, 1992) and the choice of which interpretation is used is asubstantial part of the model. The possibility to interprete a sufficiently well posedproblem like the leaky integrate-and-fire neuron in two different ways is a result of thedifficulty to handle noise.

9.5.2 RSC Analysis of Neural Behavior

Many effects and aspects of neural behavior are reflected in changes of the stimulus-response correlation. In most cases the equations from this chapter can reproduceknown effects of neural information processing. Analysis of these effects using theframework of response-stimulus correlation suggested here could provide some fruitfulinsights in mechanisms already known. Some influential measures which conduct theshape and its change of RSC curves shall be discussed briefly.

Membrane time constant and leak conductance The main effect of changing themembrane time constant is the reaction speed of the neuron. A change of the effectivemembrane time constant can result either from a change of the leak conductance orfrom a change of the stimulus rate. In the first case the neuron would be able to fine-tune its detection speed, provided the leak conductance can be changed internally bythe neuron itself. In the latter case the resulting change in detection speed can be seenas an automatic adaptation. If the stimulus rate is increased, the neural detection speedis increased as well.

129

Rate of stimulus and response. Many synaptic learning mechanisms (like synap-tic scaling or synaptic redistribution) are rate based (Abbott & Nelson, 2000), thereforethe influence of rates on neural behavior is very important. Other learning mechanismslike STDP are correlation based, and scale a synapse up if its RSC is high, or scale itdown if its RSC is low. The results of this work confirm that an increase in the totalstimulus rate decreases neural sensitivity, while a rate increase for a single synapseincreases neural sensitivity to this synapse. This change is reflected in the simulatedRSC and its analytic solutions. Rate-based and correlation-based learning mechanisminteract, and more insight into the way they do this could be achieved from RSC anal-ysis. The RSC is also influenced by the response rate. A higher response rate reducesthe correlation; in a high rate regime the presence of a response spike alters the prob-ability of a stimulus spike to have occurred in a less dramatic manner than it does in alow rate regime. This may shed some light on the type of code used - rate code (lowsingle-spike correlations) vs. spike code (high single-spike correlations), and underwhich conditions they are more typical.

9.6 Summary

The investigation presented here is based on the idea of stochastic processes. Thedifferential equation which describes the voltage of the neural membrane was reducedso that synaptic conductances and leak conductances are treated in the same way eq.(9.6). This equation was solved using Ito calculus, subsequently mean and variance arededucted eqns. (9.12) and (9.18). Various noise models where introduced - the Poissoncounting process, the Wiener process and the Ornstein-Uhlenbeck process.

The phenomenon of time-symmetry was explained eqns. (9.33) and (9.32), whichresults in a simple expression of the expected free (no reset) membrane potential beforeand after the crossing of a threshold. Introducing a reset puts the membrane voltage inan absorbing boundary scenario, the effect of which was approximated using the freevoltage as it is eq. (9.34), a (numeric) Gaussian approximation introduced by Bruneland Hakim (Gerstner & Kistler, 2002; Brunel & Hakim, 1999) eq. (9.35), and ananalytically motivated approximation eq. (9.43).

The mutual relations of synaptic conductances where described, and simulationspresented which sustain the resulting ideas: The RSC is the sum of a static expectationwhich is independent of τ , and a dependent part. For a leak conductance this dependentpart vanishes. This dependent part is proportional to the variance of a conductance,and to the reversal potential of a conductance. The above assumptions where taken asa starting point to assemble the response-stimulus correlation.

In various tests the validity of the assumptions was demonstrated. It could be shownthat in all cases investigated, the hypothesis of mutual synaptic relations which resultsin an expression for the effective synaptic strength holds true. It was also demonstratedthat the equations which describe the response-stimulus correlations are exact in caseof Wiener process conductances, and a good approximation for Poisson process con-ductances if the weight is not too high.

The influence of various contributing parameters like stimulus mean, variance, rate,weight, cross-correlation and response rate was discussed, and it was demonstrated thateffects which result from changes in these variables can be investigated by the analyticframework presented in this text.

130

Chapter 10

Summary and Discussion

In this thesis investigations on different aspects of noisy neural information processingare presented. All studies are closely related to one or several of the following fourmajor themes:

• The relation between neural information transmission, processing and modelproperties,

• adaptation to optimal noise levels,

• consequences of metabolic cost of neural activity for neural coding,

• and the interplay between the statistics of the input and neural spiking activity.

Chapters 4-9 present original research. In chapter 4 information transmission ofweak inputs in single model neurons is investigated. The phenomenon of stochasticresonance occurs. The optimal noise level, i.e. the variance of the membrane potentialfluctuations, depends on the mean input intensity. A biologically plausible adaptationprocedure is suggested, which allows the neuron to adjust its membrane potential fluc-tuations, such that information transmission becomes optimal. The procedure dependson the input/output properties, i.e. the transfer function of the neuron. Measuring suchtransfer functions could be a test of whether real neurons are - in principle - capable ofemploying the above procedure.

Information transmission is an important aspect of neural information processing.Information transmission is related to neural activity, neural activity is metabolicallyexpensive. In chapter 5 optimal neural input distributions are calculated which max-imize information transmission. If metabolic cost is taken into account low intensity(unreliable) signals are preferred compared to high intensity (reliable, but costly) sig-nals. Thus, there is an optimal trade-off between metabolic cost and reliable signaltransduction. The phenomenon of stochastic resonance is particularly important incase of weak signals.

It is plausible to assume that neurons perform their computations in groups, calledpopulations. The optimal noise level in a stochastic resonance and information trans-mission setting does not only depend on the input signals, the noise statistics and themodel properties. It does also depend on the properties of such a population. In chap-ter 6 information transmission is investigated in an abstract model of a population ofsingle neurons. In general the optimal noise level in a single neuron scenario and in a

131

population framework need not be the same. It is discussed under what circumstancesa local learning rule, as suggested in chapter 4, makes sense in a population of neurons.Measuring the information rate in neural populations constitutes a considerable techni-cal problem. The performance measure used to quantify the information transmissionand the procedure to measure it do have a significant influence. Relating these differ-ent methods to each other and evaluating an optimal procedure is a subject of currentresearch.

In chapter 7 it is investigated how the presence and properties of membrane po-tential fluctuations influence the ability of a neuron to detect transient inputs. Thismight be considered a basic neural computation. In an abstract as well as in a biophys-ically more plausible model neuron it is demonstrated that temporal correlations in themembrane potential fluctuations yield a more robust performance in the detection oftransient inputs. An approximate theory describes the results. The most fundamentaleffect - a decrease in noise induced neural spiking as a result of increased temporalcorrelations - might be a good candidate for an experimental test.

Higher moments of the background synaptic activity have a significant influence onneural activity. In chapter 8 the neural response to coincident and temporally correlatedinputs is investigated. Both aspects of the input statistics have a significant influenceon the membrane potential variance. An abstract model neuron is employed to studycoincident and temporally correlated inputs. In the presented framework these can becompared on a common ground. From studying the dynamics of the neuronal responseto coincidences and temporal correlations it can be deduced, that temporal correlationshave an important modulating effect on the neuronal response. Transient coincidencesseem to be much more appropriate for neural coding than transient changes in thetemporal correlations. The main results are verified in a biophysically more realisticmodel neuron. Continuing these studies could yield a better understanding of neuralintegration.

The response-stimulus correlation (RSC) is a function of the time-distance to aneuron’s observed response spike. It expresses the correlation of response and stimulus,as well as the expected number of stimulus spikes which have led to a response. TheRSC thus provides an important characterization of neural behavior in a stochasticsetting. In chapter 9 an abstract model neuron is investigated, and the response-stimuluscorrelation is obtained analytically. Fits of the results of this analysis are compared tosimulations, resulting consequences are discussed. As a next step, it is interesting toinvestigate in what aspects the model does not describe a biophysically more realisticneuron model, as a Hodgkin-Huxley type point neuron with fluctuating conductanceinputs.

132

Chapter 11

Appendix

11.1 HH Framework, Slope of the fI-Curve

The slope of the firing rate in a LIF model (as introduced in chapter 2) does notdisplay resonant behavior for all sub-threshold signal intensities, see also (Stemmler,1996). For intensities which induce voltage jumps of less then 0.5 times the voltagedistance from reset to threshold no maximum appears. This is different for the HH typemodel with fluctuating conductance inputs (see chapter 2), which is employed in thisthesis. Figure 11.1 displays the slope of the firing rate of the HH type model introducedin chapter 2 versus noise level. Three different constant currents are injected. Theweakest constant current has about 20 % of the intensity of the rheobase current. Adistinct maximum exist.

11.2 Information Measures, Small Signals

In (Stemmler, 1996) several information measures have been related to each otherin context of constant additive input signals µ and small deviations ∆µ from µ .

Relationships between J, as in eq. (4.9), and other information measures:

Discriminability d, as in eq. (4.14):

d = ∆µ√

J(µ). (11.1)

Mutual information I, see chapter 3:

I =(∆µ)2

8ln2J(µ). (11.2)

11.3 LIF in the Population Coding Chapter

Dynamics of the membrane potential:

CmdV (t)

dt=−gL(V (t)−EL)+ Istim(t)+σ

dW (t)dt

. (11.3)

133

0 5 10 15 20 25 30 35 40−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2

0.2

1.6

1.0

α

diff

eren

ce in

spi

ke c

ount

Figure 11.1: Sensitivity of the spike count in a HH (section 2.11.2) model w.r.t. to de-viations from constant current inputs. The spike count is measured for constant currentinjections µ and µ +∆µ within T = 1000ms. The difference is plotted versus the noiselevel α . Three baseline currents are injected. µ = 0.35,0.25,0.1 nA, ∆µ = 0.02nA.Different noise conditions are modeled as a change of the conductances by a (gain)factor α , α × (ge0,gi0), and by a corresponding change in the diffusion coefficientsα × (

√De0,√

Di0). The values of ge0,gi0,De0,Di0 correspond to the standard valuesgiven in the appendix (section 11.5).

Model parameters: τm = 20ms, gL = 25nS,Cm = τmgL = 0.5nF, EL =−74mV , Vreset =−60mV , Tre f rac = 1.72ms, threshold is −54mV . Parameters are taken from(Salinas & Seijnowki, 2000). Istim(t) is generated by a Fourier transform of a whitenoise power spectrum with a cut-off frequency of 20Hz. All simulations were doneusing the Euler integration scheme with an exact update equation (Gillespie, 1996) anda fixed time step of dt = 0.2ms.

11.4 Hodgkin-Huxley Type Neuron

Dynamics of the membrane potential:

Cm∂V∂ t

=−gL(V −EL)− INa− IK− IM− Isyn + Istim(t). (11.4)

Model parameters: Cm = 1 µFcm2 , gL = 0.045mS and EL =−70mV .

134

Voltage-dependent sodium current, INa:

INa = gNa m3h (V −ENa)

dmdt

= αm(V ) (1−m) − βm(V ) m

dhdt

= αh(V ) (1−h) − βh(V ) h

αm(V ) =−0.32(V −VT −13)

exp[−(V −VT −13)/4]−1

βm(V ) =0.28(V −VT −40)

exp[−(V −VT −40)/5]−1

αh(V ) = 0.128exp[−(V −VT −VS−17)/18]

βh(V ) =4

1+ exp[−(V −VT −VS−40)/5].

Model parameters: gNa = 3mS/cm2, VT =−58mV , VS =−10mV and ENa = 0mV.

Delayed-rectifier potassium current, IK:

IK = gK n4 (V −EK)

dndt

= αn(V ) (1−n) − βn(V ) n

αn(V ) =−0.032(V−VT −15)

exp[−(V −VT −15)/5]−1

βn(V ) = 0.5exp[−(V −VT −10)/40].

Model parameters: gK = 5mS/cm2, VT =−58mV and EK =−80mV.

Noninactivating potassium current, IM:

IM = gM p (V −EK)

dpdt

= αp(V ) (1− p) − βp(V ) p

αp(V ) =0.0001(V +30)

1− exp[−(V +30)/9]

βp(V ) =−0.0001(V +30)

1− exp[(V +30)/9].

Model parameters: gM = 1 µS/cm2. The spike threshold is approximately −57mV .The simulations of the population of Hodgkin-Huxley neurons were done with theNEURON simulation environment (Hines & Carnevale, 1997) using the model fromDestexhe et al. (Destexhe et al., 2001).

11.5 Synaptic Background Activity

Synaptic current:

Isyn = ge(t)(V −Ee) + gi(t)(V −Ei). (11.5)

135

Model parameters: Ee = 0mV, Ei =−80mV.

Model of synaptic conductances:

dge(t)dt

= − 1τe

[ge(t)−αge0] + α√

DedW (t)

dt, (11.6)

dgi(t)dt

= − 1τi

[gi(t)−αgi0] + α√

DidW (t)

dt. (11.7)

Model parameters: ge0 = 0.01 µS, gi0 = 0.032 µS, σe0 = 0.003 µS, σi0 = 0.00825 µS, τe =2.7ms, τi = 10.5ms. For stationary membrane potential distributions the relation σ 2

e,i =De,iτe,i

2 holds.

The code for the original point conductance model is available fromhttp://senselab.med.yale.edu/senselab/.

11.6 Optimization of IF with respect to the Noise Level

The Fisher information IF is given by

IF = H(X)−∫ ∞

−∞dxPX(x)

12

log2

(2πeF(x)

)

. (11.8)

Because the entropy of the input distribution H(X) does not depend on the noise level,it is sufficient to minimize

∫ ∞

−∞dxPX(x) ln

(1

F(x)

)

= min . (11.9)

Inserting eq. (6.8) for F(x) one gets:

∫ ∞

−∞dxPX(x)

(

−2ln∂P1|x

∂x− lnN

)

+

∫ ∞

−∞dxPX(x)

(ln(P1|x(1−P1|x)

))= min . (11.10)

Under the assumption of independent Gaussian noise one obtains for∂ P1|x

∂ x (see eq. (6.3))

∂P1|x∂x

=1√

2πσηexp

(

− (Θ− x)2

2σ 2η

)

. (11.11)

Replace ln(P1|x(1−P1|x)

)by its second order Taylor expansion around x = Θ,

ln(P1|x(1−P1|x)

)=− ln4 − 2

π(x−Θ)2

σ2η

+O(x4), (11.12)

and obtain:∫ ∞

−∞dxPX(x)

(

2lnση +

(

1− 2π

)(x−Θ)2

σ2η

)

= min, (11.13)

136

where all terms independent of ση have been omitted. Define Θ = 0 as the point oforigin and solve the integral which yields

2lnση +

(

1− 2π

)(µ2

x +σ 2x )

σ2η

= min . (11.14)

Setting the derivative with respect to ση to zero, one finally obtains

σFopt =

√(

1− 2π

)

(µ2x +σ 2

x ). (11.15)

If the Taylor expansion (see eq. (11.12)) is extended to higher order, then the evaluationof the integral in eq. (11.13) leads to higher order moments of the input distribution.For example, taking the Taylor expansion to 4th order, the evaluation of the integral

∫ ∞

−∞dxPX(x)

(

2lnση +

(

1− 2π

)(x−Θ)2

σ2η

)

(11.16)

+2

3π

(

1− 3π

)∫ ∞

−∞dxPX(x)

(

(x−Θ)4

σ4η

)

= min,

yields the following optimal noise level:

(σ Fopt)

2 =

(12− 1

π

)(µ2

x +σ 2x

)(11.17)

±√

36π

√

a1µ4x +6a2µ2

x σ2x +3a2σ4

x .

Parameters a1 and a2 are: a1 = 3π2 +4π−36, a2 = π2 +12π−44.

11.7 Discriminability in Populations of Neurons

Discrimanility, eq. (4.14) from chapter 4, is investigated in an array of independentmodel neurons receiving constant current signals and additive white noise as inputs.It is assumed that the resulting spike count distribution of each neuron is Gaussianwith mean µ1 and standard deviation σ1. Summing N of these spike count distribu-tions yields a Gaussian with mean µN = Nµ1 and standard deviation σN =

√Nσ1. It is

shown , that the discriminability for the whole array, dN , is proportional to the discrim-inability for a single neuron d1 and scales with the square root of the array size

√N,

see following calculation:

dN =2µN(s)−µN(s+∆s)σN(s)+σN(s+∆s)

(11.18)

= 2N(µ1(s)−µ1(s+∆s))√N(σ1(s)+σ1(s+∆s))

=√

Nd1.

(11.19)

137

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Noise level

Dis

crim

inab

ility

N = 1N = 8N = 32

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Noise level

Dis

crim

inab

ility

N = 1N = 8N = 32N = 100

1 8 32 100 2000.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

Number of neurons

Sta

ndar

d de

viat

ion

* sq

rt(N

)

Noise level = 0.5Noise level = 1Noise level = 1.5

1 8 32 100 200 5000.9

0.95

1

1.05

1.1

1.15

Number of neurons

Sta

ndar

d de

viat

ion

* sq

rt(N

)

Noise level = 0.5Noise level = 1Noise level = 1.5

(a) (b)

(c) (d)

stan

dart

devi

atio

n *

sqrt

(N)

0.85

1.35

1 8 32 100 200

disc

rem

inab

ilit

y

1.0

0α0 1.0 2.0

stan

dart

devi

atio

n *

sqrt

(N)

0.9

1.15

0.2

1.6

disc

rem

inab

ilit

y

1 8 32 100 200 500 0.2 1.20.7 α

N

N

Figure 11.2: Discriminability, eq. (4.14), for a summing array of N HH neurons withconstant current inputs and conductance noise, as introduced in chapter 4. For (a) and(b) Ibias = 0.2nA. (a) Standard deviation of the spike count distribution versus thenumber of neurons. (b) Discriminability versus noise level α , see eq. (4.14) and eq.(11.18), for N = 1,8,32. (c) and (d): Same as in (a) and (b) but Ibias = 0.3nA andN = 1,8,32,100. Discriminabilites in (b) and (d) are scaled to lie on top of each other,according to eq. (11.18). For all simulations: deviation from bias current ∆I = 0.01nA.Results are averaged over 1000 repetitions, each trial has a length of 1s. The errorbarsin (a) and (c) denote the standard deviation of the empirical estimate of the standarddeviation of the spike count distribution. Calculations are done by Thomas Hoch.

Thus, in the above scenario the optimal noise level does not depend on the number ofneurons in the population. The above observation is verified in a summing array ofHodgkin-Huxley neurons with conductance noise, as presented in chapter 2. Constantcurrents are injected to measure the discriminability eq. (4.14) and eq. (11.18).Figures 11.2 (b) and (d) show the discriminability for two different bias currents Ibias

as a function of the array size. Several population sizes are tested. It can be observed,that the optimal noise level does not depend on the population size. Subfigures 11.2 (a)and (c) demonstrate that the standard deviation of the summed spike count distributionscales with the square root of N for two different bias currents.

138

11.8 Moments of the Membrane Potential, LIF

One can yield the differential equation for < X 2t >, eq. (11.20), by multiplying eq.

(8.1) with itself, average over an ensemble (<>) and neglect all terms of order > 1 indt. Note that < Xt dWt > is zero. The differential equations for < XtVt >, eq. (8.6), and< V 2

t >, eq. (8.7), are derived in an analog way.

d < X2t >

dt+

2τX

< X2t >= D (11.20)

< XtVt >

dt+

1τ

< XtVt >=< X2t >, τ =

τX τV

τX + τV(11.21)

< V 2t >

dt+

2τV

< V 2t >= 2 < XtVt > (11.22)

Note that < V 2t >, eq. (11.22), depends on τX , D, and τV plus a new time constant τ

which is introduced by the covariance between Xt and Vt and depends on τX and τV .For differential equations of the form

dyt

dt+gxyt = hx, (11.23)

the solution is

yx = e−Gxη +∫ x

θhte

Gt dt, (11.24)

where Gx =∫ x

θ gtdt and yθ = η .

Using the above expressions the analytic solution is easy to obtain. Constant values forτX and D are assumed. At time t0 = 0 a step like change in τX and/or D occurs. Fort < t0 the parameters are denoted as τX1 and D1, for t > t0 as τX2 and D2.The solution to eq. (11.20) is

< X2t >=

D1τX1

2e−2(t−t0)

τX2 +D2τX2

2(1− e

−2(t−t0)τX2 ), (11.25)

where τi = τXiτVτXi+τV

, i = 1,2. The solution to eq. (11.21) is

< XtVt > =D1τX1τ1

2e−(t−t0)

τ2 (11.26)

+D2τX2τ2

2(1− e

−(t−t0)τ2 )

+ D(e−2(t−t0)

τX2 − e−(t−t0)

τ2 ),

where D =D1τX1τ p

22 − D2τX2τ p

22 , τ p

2 = τ2τX2τX2−2τ2

.The solution to eq. (11.22) is

139

P false positive P false positive

P co

rrec

t det

ectio

n

P co

rrec

t det

ectio

n

0 0.4 1.00 0.41.0

0.3

1.0

0.3

1.0

N

τX

(a) (b)

Figure 11.3: (a) ROC curves for different populations sizes, N = 4,8,15,80 (alongthe arrow). Pulse height is 2mV, membrane potential variance is 100 mV 2. Reset isat 0mV, threshold at 20mV. Pulses are applied every 50ms, τV = 5ms. See text forthe procedure which yields the ROC curve. (b) ROC curves with different temporalcorrelations of the additive noise. Temporal correlation τX = 0.1,0.5,1.0,10ms (alongthe arrow). Pulse height is 1mV, N = 40, other parameters as in (a).

< V 2t > =

D1τX1τV τ1

2e−2(t−t0)

τV (11.27)

+ D1τX1τ1τa2 (e

t−t0τa2 −1)

+ D2τX2τ2τV

2(e−2(t−t0)

τV −1)

+ D2τX2τ2τa2 (1− e

t−t0τa2 )

+ Dτb2 (e

2(t−t0)

τb2 −1)

+ 2Dτa2 (1− e

t−t0τa2 ),

where τa2 = τ2τV

2τ2−τV, τb

2 = τX2τVτX2−τV

.The stationary moments are given by

< X2t >=

DτX

2, < XtVt >=

DτX τ2

, < V 2t >=

DτX τV τ2

. (11.28)

11.9 Pulse Detection in Populations of Neurons

As could be seen in chapter 7, pulse detection based on single neurons performs poorly.To study pulse detection in a population of neurons a setting similar to the one inchapter 6 is chosen. A population of N LIF neurons (see chapter 2), receives colored,additive and independent noise via Ornstein-Uhlenbeck processes. The pooled outputis considered as the activity, A, of the population. A sub-threshold current pulse isinjected every 50ms. If A exceeds a threshold value CA within a short time (0.2ms)

140

after the application of the pulse a correct detection is presumed. PC (for a given CA)is the average of correct detections over all trials and varies between zero and one asa function of CA. Denote Ai as the pooled activity in time bin i. If Ai exceeds CA

between two pulses (outside of the critical time window for correct detections), this isconsidered as a false positive. Pf , quantifying the false positives is calculated as

Pf = 〈∑i

1(Ai>CA)/∑i

1(Ai>0)〉trials. (11.29)

1(x) is the indicator function and is equal to 1 if x > 0 and zero otherwise. Pf variesbetween zero and one as a function of CA. Figure 11.3 displays two ROC curves. PC isplotted versus Pf , Pf is varied by changing CA. Low CA corresponds to high Pf and viceversa. Figure 11.3 (a) displays the ROC curve for different numbers of neurons in thepopulation. The pulse is very weak, it induces voltage jumps of 2mV, threshold is at20mV, reset at 0mV. One can observe that even small populations perform well in thisdetection task, with increasing population size, performance gets better. Figure 11.3(b) displays the ROC curve for different temporal correlations of the additive noise.Increasing the temporal correlation yields a better performance.

11.10 Strictly positive Conductances with White Noise

Unfortunately using the increments of the Wiener process (white noise) to describeconductances has a major disadvantage. With low expectations µ and high variancesσ2 negative conductances may occur. While being mathematically absolutely correct,this is not really biologically feasible, and one may be tempted to use a process whichremains strictly positive instead. Although the equations below could handle such aprocess, there is a mathematical problem using a process like

dGit = µidt +σi|dWt |.

To see why, one must bear in mind that dt is actually short for ∆t,(∆t → 0), whereasdWt means

√∆tξt ,(∆t → 0) with the ξt being a random Gaussian variable with mean

zero and variance one. Using strictly positive noise has the following consequences:

dGit = µidt +σi|dWt |

= lim∆t→0

µi∆t +σi

√∆t|ξt |

E

dGit

= lim

∆t→0µi∆t +σi

√∆tE|ξt | ,

= lim∆t→0

µi∆t +σi

√∆t

√

2π

E

∫

dGit

= lim∆t→0

∑t

µi∆t +σi

√∆t

√

2π

,

= ∞

- the integrals do not converge any longer. To overcome this problem, the absolute ofwhite noise could be used and its expectation subtracted, to yield noise with expectation0. The parameter µ could then be scaled to find the point where no negative values

141

occur:

dGit = µidt +σi(|dWt |−EdWt)

dGit > 0

µidt > σiE|dWt |

µi > σiE|dWt |

dt

µi > lim∆t→0

σi

√2∆t√π∆t

µi > ∞

On the other hand, using (dWt)2, or any noise with increments proportional to dt -

dGit = µidt +σi(dWt)

2

= lim∆t→0

µi∆t +σi(√

∆tξt)2

= lim∆t→0

µi∆t +σi∆t(ξt)2

∫

dGit = lim

∆t→0∑

tµi∆t +σi∆t(ξt)

2

Var

∫

dGit

= lim∆t→0

(

∑t

µi∆t +σi∆t(ξt)2)2

= lim∆t→0

∑t

µi(∆t)2 +2µi(ξt)2(∆t)2

+(σi∆t(ξt)2)2

= 0

- makes the noise disappear. The conclusion is: There is no possible way of findingnoisy increments which are strictly positive and still have converging integrals.

11.11 Notes On The Simulations

11.11.1 General Notes

Simulations where run to illustrate the influence of certain single parameters on theRSC. Usually while neural parameters are changed, the spiking behavior of the neuronis affected. The neuron can operate in two distinctly different regimes - sub-threshold(where spikes are generated by small potential fluctuations, and influence of synapticvariance and cross-correlation is dominant) and super-threshold (where spikes are gen-erated mainly by the potential mean being driven across the threshold, and influenceof synaptic mean is dominant). Since these regimes are different with regard to thespike-generation mechanism, the influence of a possible regime-change would blur theinfluence of the change of the parameter in question (although distinguishing betweenthe two might be a bit artificial). To ensure that the neuron remains in a scenario whereit spikes normally, parameters have to be adjusted very carefully. The usual way is toeither tweak the stimulus parameters, or to adjust the threshold. Since the aim here is todemonstrate the analytic solution of the RSC in a wide range of different stimulus sce-narios, adjusting the stimulus would be difficult, therefore it was decided on the latter.

142

Most simulations (excluding the rate dependency test, see figure 9.8(h)) are run withthe neuron’s rate fixed to NHz, with N = 5, and the threshold adjusted accordingly. Thefollowing technique is used to achieve this: starting from v0 the membrane potential istaken for 1000

N ms without reset, and the voltage maximum is recorded. This is averagedover 500 times. The resulting voltage is used as the threshold.

The conditional values for the conductances are obtained as follows: The mem-brane voltage according to eq. (9.10) is evaluated numerically until the voltage reachesthe threshold. This is done many times. Each time the voltage reaches the threshold,the value of the conductances from a certain time t− (which is the hitting time minusa time difference of τ = 10ms) up to a certain time t+ (hitting time plus τ = 10ms) isadded to the conductance average. This average is then divided by the amount of hits.

11.11.2 Simulations in Part One

Throughout the first parts of the text (Introduction; Models, Materials, Methods) fig-ures from a simulation are used to illustrate the text. The simulation was run using thefollowing parameters:

• Leak process with G0t =

⟨G0

t

⟩= t ·0.2µS and reversal potential at v0 = 70mV

• Excitatory Wiener process with mean 〈Gt〉= t0.08µS, standard deviation√

[Gt ] =√t ·0.02µS and reversal potential at v1 = 20mV

• Inhibitory Wiener process with mean 〈Gt〉= t0.04µS, standard deviation√

[Gt ]=√t ·0.04µS and reversal potential at v1 =−75mV

• Cross-correlations between excitatory and inhibitory conductances where set to0.2

• Rate was fixed to 5Hz

This results in an effective membrane time constant of µm = 10.2µS+0.08µS+0.04µS =

3.1250µS−1

11.11.3 Simulation illustrating mutual synaptic relations

This simulation was run to illustrate mutual synaptic relations of a neuron receivinginput from multiple conductance processes. The setting was as follows:

• The neuron was fed by five conductances, the first of which was a leak conduc-tance with an expectation of

⟨G0⟩

t = t · 0.2µS and a variance of 0 at a reversalpotential of v0 =−70mV .

• Conductances G1t to G4

t where modeled by a Wiener process with mean t ·0.01µSand standard deviation of 0.01µS. In three different settings one of the aboveparameters was changed.

• In the setting illustrating the effect of different means, conductances G1t to G4

thad a mean of 0.01, 0.03, 0.05 and 0.07µS.

• In the setting illustrating the effect of different reversal potentials, conductancesG1

t to G4t had reversal potential of −60,−30, 0 and 30mV

• In the setting illustrating the effect of different variances, conductances G1t to G4

thad a variance of 0.001, 0.003, 0.005 and 0.007(µS)2

143

11.11.4 Single Excitatory Conductance

• The simulation was run using a leak conductance with vL =−70mV and dGLt =

0.2µS and one excitatory conductance modeled by Gaussian white noise withvE = 20mV ,

⟨dGE

⟩= 0.01µS ·dt and

[dGE

]= 0.005(µS)2.

• In various settings different parameters where explored. The values of theseparameters are shown in the figures which show the results.

11.11.5 Simulation showing Excitation and Inhibition

• Leak conductance with reversal potential vL =−70mV , mean⟨GL⟩

t = t ·0.2µS.

• Excitatory conductance modeled by a Wiener process, reversal potential vE =20mV , mean

⟨GE⟩

t = t ·0.08µS and standard deviation√

[GE ]t = 0.05µS.

• Inhibitory conductance modeled by a Poisson process, reversal potential vI =−75mV , mean

⟨GI⟩

t = t ·0.02µS and standard deviation√

[GI ]t = 0.05µS.

Threshold is fixed to yield a response rate of 5Hz. The mixing matrix for the correla-tions was

a =

1 0 00 1-c/2 c/20 c/2 1-c/2

and c was run from 0 to 1.

11.11.6 Poisson Inhibition Simulation

• Leak conductance with reversal potential vL =−70mV , mean⟨GL⟩

t = t ·0.2µS.

• Excitatory conductance modeled by a Wiener process, reversal potential vE =20mV , mean

⟨GE⟩

t = t ·0.08µS and standard deviation√

[GE ]t = 0.05µS.

• Inhibitory conductance modeled by a Poisson process, reversal potential vI =−75mV , mean

⟨GI⟩

t = t ·0.02µS and standard deviation√

[GI ]t = 0.05µS.

Threshold is fixed to yield a response rate of 5Hz. The mixing matrix for the correla-tions was

a =

1 0 00 0.8 0.20 0.2 0.8

144

11.12 Details of the calculations in the RSC Chapter

Equation 9.7:

d lnΦt = Φ−1t dΦt −

12

Φ−2t d[Φ,Φ]t

= Φ−1t

(

−Φt ∑i

dGit

)

− 12

Φ−2t

(

Φ2t ∑

i, j

d[Gi,G j]t

)

= −∑i

dGit −

12 ∑

i, jd[Gi,G j]t

lnΦt = −∑i

Git −

12 ∑

i, j[Gi,G j ]t

Φt = exp

(

−∑i

Git −

12 ∑

i, j[Gi,G j]t

)

Equation 9.8:

dΦ−1t = −Φ−2

t dΦt +Φ−3t d[Φ,Φ]t

= −Φ−2t

(

−Φt ∑i

dGit

)

+Φ−3t

(

Φ2t ∑

i, j

d[Gi,G j ]t

)

= Φ−1t ∑

i

dGit +Φ−1

t ∑i, j

d[Gi,G j]t

= Φ−1t

(

∑i

dGit +∑

i, j

d[Gi,G j]t

)

Equation 9.9:

d[Φ−1,V ]t = dΦ−1t ·dVt

= −Φ−1t

(

∑i

dGit −∑

i, jd[Gi,G j]t

)

·∑i(vi−Vt)dGi

t

= −Φ−1t ∑

i, j(vi−Vt)d[Gi,G j]t

145

Equation 9.10:

d(Φ−1

t Vt)

= Φ−1t dVt +VtdΦ−1

t +d[Φ−1,V ]t

= Φ−1t ∑

i(vi−Vt)dGi

t

+Φ−1t Vt

(

∑i

dGit +∑

i, jd[Gi,G j]t

)

+Φ−1t ∑

i, j(vi−Vt)d[Gi,G j]t

= Φ−1t ∑

ividGi

t −VtΦ−1t ∑

idGi

t

︸ ︷︷ ︸

−c1

+VtΦ−1t ∑

idGi

t

︸ ︷︷ ︸

c1

+VtΦ−1t ∑

i, jd[Gi,G j]t

︸ ︷︷ ︸

c2

+Φ−1t ∑

i, jvid[Gi,G j]t −VtΦ−1

t ∑i, j

d[Gi,G j]t

︸ ︷︷ ︸

−c2

= Φ−1t ∑

ividGi

t +Φ−1t ∑

i, jvid[Gi,G j]t

The terms marked with c1 and c2 cancel out and voltage dependency on the right sidevanishes.

Equation 9.14

dhVt = dVt−d 〈V 〉t= ∑

i(vi−Vt)dGi

t −∑i(vi−〈V 〉t)d

⟨Gi⟩

t

= ∑i

((vi−Vt)dGi

t − (vi−〈V 〉t))

146

Equation 9.15:

dhVt = ∑i

((vi−Vt)dGi

t − (vi−〈V 〉t)d⟨Gi⟩

t

)

= ∑i

(vidGi

t −VtdGit − vid

⟨Gi⟩

t + 〈V 〉t d⟨Gi⟩

t

)

= ∑i

vi (dhGit +d

⟨Gi⟩

t)︸ ︷︷ ︸

dGit

−(hVt + 〈V 〉t)︸ ︷︷ ︸

Vt

(dhGit +d

⟨Gi⟩

t)︸ ︷︷ ︸

dGit

−vid⟨Gi⟩

t + 〈V 〉t d⟨Gi⟩

t))

= ∑i

(vidhGi

t + vid⟨Gi⟩

t

−hVtdhGit −hVtd

⟨Gi⟩

t −〈V 〉t dhGit −〈V 〉t d

⟨Gi⟩

t

−vid⟨Gi⟩

t + 〈V 〉t d⟨Gi⟩

t

)

= ∑i

((vi−hVt−〈V 〉t)dhGi

t −hVtd⟨Gi⟩

t

)

Equation 9.16:

d(hVt)2 = 2hVtdhVt +d[hV,hV ]t

= 2hVt ∑i

((vi−hVt−〈V 〉t)dhGi

t −hVtd⟨Gi⟩

t

)

+d[V,V ]t

= 2∑i(vihVt− (hVt)

2−〈V 〉t hVt)dhGit

−2∑i(hVt)

2d⟨Gi⟩

t +d[V,V ]t

Equation 9.17:

d[V,V ]t = E(dVt)

2 in expectation

= E

(

∑i(vi−Vt)dGi

t

)2

= E

∑i, j

(vi−Vt)(v j−Vt)d[Gi,G j]t

= E

∑i, j

(viv j− viVt − v jVt +V 2

t

)d[Gi,G j]t

= ∑i, j

(viv j− vi 〈V 〉t − v j 〈V 〉t +E

V 2

t

)d[Gi,G j]t

147

Equation 9.19:

λ = PdXt (Θ) =1

σ√

2π

Θ∫

−∞

e−22/sσ 2ds

=1

σ√

2π

Θ√2σ∫

−∞

e−t2σ√

2dt

=1√π

Θ√2σ∫

−∞

e−t2dt

=1√π

0∫

−∞

e−t2dt +

Θ√2σ∫

0

e−t2dt

=1√π

(√π

2+

√π

2erf

(Θ√2σ

))

=12

(

erf

(Θ√2σ

)

+1

)

Equation 9.24:

⟨[Gi,G j]⟩ =

⟨

dGitdG j

t

⟩

=⟨

dG jt |dGi

t

⟩⟨dGi

t

⟩

=⟨

µ jdt +σ jdX jt |dX i

t > λi

√dt⟩⟨

dGit

⟩

= σ j

⟨

dX jt |dX i

t > λi

√dt⟩⟨

dGit

⟩

= σ j

⟨

∑k

a jkdW kt |dX i

t > λi

√dt

⟩

⟨dGi

t

⟩

= σ j ∑k

a jk

⟨

dW kt |dX i

t > λi

√dt⟩⟨

dGit

⟩

= σ j ∑k

a jkaik · c⟨dGi

t

⟩

= σ j ∑k

a jkaik

⟨

X it |X i

t > λ√

dt⟩

[dX i

t

]⟨dGi

t

⟩

= σ jci j

⟨

X it |X i

t > λ√

dt⟩

[dX i

t

] w jλ j

The µ j at the second step can be ommited, since any constant c∆t will vanish in thepresence of another constant c

√∆t as soon as ∆t→ 0.

148

Equation 9.28:

git = τiµi +σi

∫

[0,t]

e− 1

τi(t−s)

dWs

dgit = d

τiµi +σi

∫

[0,t]

e− 1

τi(t−s)

dWs

= −σi

τi

∫

[0,t]

e− 1

τi(t−s)

dWs +σidWt

git dt +dgi

t = τiµidt

+

(

σi−1σi

τi

) ∫

[0,t]

e− 1

τi(t−s)

dWs +dWt

1τi

git dt +dgi

t = µidt +σidWt

= stimulusit dt

Equation 9.38:

P(Ω) = P((−∞,∞)) =

∞∫

−∞

dP(v)

θ ≤ µ +a ⇒∞∫

−∞

dP(v) =

θ∫

µ−a

c(θ − v)dv

= c(θθ∫

µ−a

dv−θ∫

µ−a

vdv

= cθ (θ − (µ−a))− 12

c(θ 2− (µ−a)2) = 1

c =

(θ 2

2+(µ−a)

(µ−a

2−θ))−1

θ > µ +a ⇒∞∫

−∞

dP(v) =

µ−a∫

µ−a

c(θ − v)dv

= c(θµ+a∫

µ−a

dv−µ+a∫

µ−a

vdv

= cθ ((µ +a)− (µ−a))− c2((µ +a)2− (µ−a)2)

= 2ac(θ −µ) = 1

c = (2a(θ −µ))−1

149

Equations 9.39 and 9.40:

θ > µ +a ⇒∞∫

−∞

vdP(v) =

µ+a∫

µ−a

vc(θ − v)dv

= cθµ+a∫

µ−a

vdv− c

µ+a∫

µ−a

v2dv

= c

(θ2

((µ +a)2− (µ−a)2)

−13

((µ +a)3− (µ−a)3)

)

= c

(

2θ µa−2µ2a− 23

a3)

=θ µ−µ2− 1

3 a2

θ −µ

= µ− a2

3(θ −µ)

d

∞∫

−∞

vdP(v) = d

(

µ− a2

3(θ −µ)

)

= dµ− da2

3(θ −µ)−a2d

13(θ −µ)

= dµ− da2

3(θ −µ)− a2dµ

3(θ −µ)2

=

(

1− a2

3(θ −µ)2

)

dµ− 13(θ −µ)

da2

150

Equation 9.41:

θ < µ +a ⇒∞∫

−∞

vdP(v) =

θ∫

µ−a

vc(θ − v)dv

= cθµ+a∫

µ−a

vdv− c

µ+a∫

µ−a

v2dv

= c

(θ2

(θ 2− (µ−a)2)

−13

(θ 3− (µ−a)3)

)

= −23

a+23

µ +13

θ

=23(µ−a)+

13

θ

d

∞∫

−∞

vdP(v) = d

(23(µ−a)+

13

θ)

=23(dµ−da)

Equation 9.49:

d 〈Vτ〉 = ∑i

(vi−〈Vτ〉)d⟨Gi

τ⟩

d 〈Vτ〉 = ∑i

(vi−〈Vτ〉)(

d 〈G〉i∞ +α iτd 〈hG〉τ

)

d 〈Vτ〉 = ∑i

(vi−〈Vτ〉)d 〈G〉i∞ +d 〈hG〉τ ∑i

(vi−〈Vτ〉)α iτ

d 〈hG〉τ =d 〈Vτ〉−∑i (vi−〈Vτ〉)d 〈G〉i∞

∑i (vi−〈Vτ〉)α iτ

=d 〈Vτ〉−∑i (vi−〈Vτ〉)d 〈G〉i∞

∑i, j (vi−〈Vτ〉)(v j−〈Vτ〉)d[Gi,G j]τ

Equation 9.51:

d⟨GE⟩

τ

= d⟨GE⟩

∞

+d 〈V 〉τ −∑ j(v j−〈V 〉τ)d

⟨G j⟩

∞(vE −〈V 〉τ)2d[Gi,Gi]τ

(vE −〈V 〉τ)2d[GE ,GE ]τ

= d⟨GE⟩

∞ +d 〈V 〉τ −∑ j(v j−〈V 〉τ)d

⟨G j⟩

∞vE −〈V 〉τ

= d⟨GE⟩

∞ +d 〈V 〉τ − (vL−〈V 〉τ )dτ− (vE−〈V 〉τ)µdτ

vE−〈V 〉τ=

d 〈V 〉τ − (vL−〈V 〉τ )dτvE−〈V 〉τ

151

Equation 9.52:

〈V 〉t = e−(µL+µE )t

V0 +(vLµL + vE µE)

t∫

0

e(µL+µE )sds

= e−(µL+µE )t(

V0 +vLµL + vE µE

µL + µE

[

e(µL+µE )s]t

0

)

= e−(µL+µE )t(

V0 +vLµL + vE µE

µL + µE

(

e(µL+µE)t −1))

=vLµL + vE µE

µL + µE+

(

V0−vLµL + vE µE

µL + µE

)

e−(µL+µE)t

= (V0−V∞)e−τmt +V∞

Equation 9.53:

[V ]t = e−c1t

t∫

0

ec1t(vE − eVs)2ds

with c1 = σ 2E −2(µL + µE)

= e−c1t

t∫

0

ec1t(vE − (V0−V∞)e−τmt −V∞)2ds

= e−c1t

t∫

0

ec1t(c2− c3e−τmt)2ds

with c2 = vE −V∞,c3 = V0−V∞

= e−c1t

t∫

0

ec1t (c22−2c2c3e−τmt + c2

3e−2τmt)ds

11.13 Ito Calculus

Throughout this work the calculus introduced by Ito (Ito, 1944) and extended by F ollmerand Protter (F ollmer & Protter, 2000; F ollmer et al., 1995) is used. This calculus makesit possible to extend ordinary differential equations (ODE) to incorporate noise and be-come stochastic differential equations (SDE).

11.13.1 Ito Integrals

At the beginning of last century various mathematicians examined Brownian motion.Brownian motion was formulated as the integral over white noise. In search for apowerful formalism which involves stochastic integrators, Ito designed the Ito integral(named after him) ∫

Xt dYt

in the following way:∫

XtdYt = lim∆t→0

∑ti

Xti

(Yti+1 −Yti

).

152

Note that the increments of the process Yt are evaluated for timesteps which lie afterthose of Xt . Therefore if Xt is adapted - that is, it cannot look into the future - Xt

and dYt will be independent. This makes the calculation of expectations more easy.The disadvantage which comes with this type of integral is the fact that the chain ruled f (Xt) = f ′(Xt)dXt becomes distorted. The new rule - called the Ito rule - is the heartof Ito calculus.

11.13.2 The Quadratic Covariation Process

The covariation process of two stochastic processes X and Y (also referred to as thebracket process) is defined as (Protter, 1995)

[X ,Y ]t = XtYt −∫

XtdYt −∫

Yt dXt .

The above is equivalent to the limit

lim∀i.ti+1−ti→0

∑i

(Xti+1 −X ti

)(Yti+1 −Y ti

)

where the ti cover the range of t. If X and Y are semimartingales, then [X ,Y ]t will be asemimartingale as well. Moreover, it satisfies

[X ,Y ]0 = X0Y0

∆[X ,Y ] = ∆X∆Y

where ∆X are the jumps of the process X .

11.13.3 The Ito Rule

The Ito Rule is one of the fundamentals of stochastic calculus. It provides a stochasticequivalent to the chain rule of analysis. For any function f : Rn → R, where fk is thederivative of f with respect to the kth dimension of Xt , it states

f (Xt)− f (X0) = ∑k

∫

fk(Xt)dX kt +

12 ∑

k

[

fk(X),X k]

t

and therefore -

d f (Xt) = ∑k

fk(Xt)dX kt +

12 ∑

k

d[

fk(X),X k]

t

= ∑k

fk(Xt)dX kt +

12 ∑

i,k

fi,kd[

X i,Xk]

t.

See also (F ollmer & Protter, 2000) and (F ollmer et al., 1995).

11.13.4 Stratonovitch Integrals

A stochastic integral which behaves according to the chain rule for ordinary differentialequations is the Stratonovitch integral. The Stratonovitch integral

∫

Xt dYt

153

is defined - in terms of Ito integrals - as

∫

Xt dYt =∫

XtdYt +12

[X ,Y ]t .

The main difference to the Ito integral is the fact that the integrator is not evaluated attime steps in the future, but at the mid-point of presence and future. If two measuresXt− and Xt+ are defined as

∫

Xt−dYt = lim∆t→0

∑ti

Xti

(Yti+1 −Yti

)

and ∫

Xt+dYt = lim∆t→0

∑ti

Xti+1

(Yti+1−Yti

),

then the Stratonovitch integral will be half of each:

∫

Xt dYt =12

∫

Xt−dYt +12

∫

Xt+dYt

It can be shown that for this type of integral the additional term from the Ito ruledisappears. In contrast to the Ito integral, expectations are not easy to calculate, whichis why Stratonovitch integrals are often expressed via their Ito counterpart.

11.13.5 Ito Isometry

If Xt is adapted, the Ito integral obeys the following properties - which makes it moreeasy to deal with, as they don’t usually hold for the Stratonovich integral -

E

∫

Xt dWt

= 0

E

(∫

XtdWt

)2

=∫

E

X2t

dt

or more generally for two independent processes X and Y

E

∫

XtdYt

=

∫

EXtEdYt

E

(∫

XtdYt

)2

=

∫

E

X2t

d[Y,Y ]t

Especially the Ito integral over the Wiener process∫

Xt dWt will always have a vanish-ing expectation.

154

11.14 Publications, Gregor Wenning

11.14.1 Publications, Journals and Proceedings

• J. Kanev, G. Wenning and K. Obermayer, Approximating the response stimuluscorrelation for the integrate-and-fire neuron. Neurocomputing, 58-60, 47-52,2004

• T. Hoch, G. Wenning and K. Obermayer, Optimal noise aided signal transmis-sion trough populations of neurons. Physical Review E, 63, 011911/1-11, 2003

• G. Wenning and K. Obermayer, Adaptive stochastic resonance. Physical ReviewLetters, 90, 120602/1-4, 2003

• T. Hoch, G. Wenning and K. Obermayer, Adaptation using local information formaximizing the global cost. Neurocomputing, 52-54, 541-546, 2003

• P. Adorjan, L. Schwabe, G. Wenning and K. Obermayer, Rapid adaptation tointernal states as a coding strategy im visual cortex? NeuroReport, 13(3), 337-342, 2002

• G. Wenning and K. Obermayer, Activity driven adaptive stochastic resonance.Advances in Neural Information Processing systems, 14, 301-308, 2002

• G. Wenning and K. Obermayer, Adjusting stochastic resonance in a leaky integrate-and-fire neuron to sub-threshold stimulus distributions. Neurocomputing, 44-46,225-231, 2002

• I. Dorofeyev, B. Gotsmann, G. Wenning and H. Fuchs, Brownian motion of mi-croscopic solids under the action of fluctuating electromagnetic fields. PhysicalReview Letters, 83, 4906-4909, 1999

• I. Dorofeyev, H. Fuchs, B. Gotsmann and G. Wenning, Fluctuating electromag-netic interactions of solids terminated by non-planar surfaces. Physical ReviewB, 60, 9069-9081, 1999

11.14.2 Manusscripts, Submitted

• G. Wenning and K. Obermayer, Pulse detection in a colored noise setting. Phys-ical Review E, conditionally accepted

• J. Kanev, G. Wenning and K. Obermayer Response stimulus correlation of aleaky integrate-and-fire neuron. Physical Review E, conditionally accepted

• T. Hoch, G. Wenning and K. Obermayer, Correlations in the background activ-ity allow the use of single neuron learning rules in populations. submitted toNeurocomputing

155

References

Abbott, L.F., & Nelson, S.B. 2000. Synaptic plasticity: Taming the beast. NatureNeuroscience, 3, 1178–1183.

Adelsberger-Mangan, D.M., & Levy, W.B. 1929. Information maintenance and statis-tical dependence reduction in simple neural networks. Biological Cybernetics, 67,469–77.

Adorjan, P., Schwabe, L., Wenning, G., & Obermayer, K. 2002. Fast adaptation as acortical coding strategy. NeuroReport, 13(3), 337–342.

Aiello, L.C., & Wheeler, P. 1995. The expensive tissue hypothesis: The brain andthe digestive system in human and primate evolution. Current Anthropology, 36,199–221.

Amemori, K.-I., & Ishii, Shin. 2001. Gaussian process approach to spiking neurons forinhomogeneous Poisson inputs. Neural Computation, 13, 2763–2797.

Ames, A. 2000. CNS energy metabolism as related to function. Brain Research Review,34, 42–68.

Amit, D.J., & Brunel, N. 1997. Model of global spontaneous activity and local struc-tured activity during delay periods in the cerebral cortex. Cerebral Cortex, 7,237–252.

Arimoto, S. 1972. An algorithm for computing the capacity of arbitrary discrete mem-oryless channels. IEEE Transactions on Information Theory, 18, 14–20.

Attwell, D., & Laughlin, S.B. 2001. An energy budget for signaling in the grey matterof the brain. Journal of Cerebral Blood Flow and Metabolism, 21, 1133–1145.

Baddeley, R., Abbott, L.F., Booth, M.C.A., Sengpiel, F., Freeman, T., Wakeman, E.A.,& Rolls, E.T. 1997. Responses of neurons in primary and inferior temporal visualcortices to natural scences. Proceedings of the Royal Society, London, B, 264,1775–1783.

Balasubramanian, V., Kimber, D., & Berry, M. 2001. Metabolically efficient informa-tion processing. Neural Computation, 13, 799–815.

Barlow, H.B. 1969. Information Processing in the Nervous System. Springer, NewYork.

Benzi, R., Sutara, A., & Vulpiani, A. 1981. The mechanism of stochastic resonance.Journal of Physics A, 14, L453–L457.

156

Benzi, R., Parisi, G., Sutera, A., & Vulpiani, A. 1982. Stochastic resonance in climaticchange. Tellus, 34, 10.

Berry, II., Warland, M.J., & Meister, M. 1997. The structure and precision of retinalspike trains. Proceedings of the National Academy of Sciences, 94, 5411–5416.

Bienenstock, E.L., Cooper, L.N., & Munroe, P.W. 1982. Theory of the developmentof neuron selectivity: Orientation specificity and binocular interaction in visualcortex. Journal of Neuroscience, 2, 32–48.

Blahut, R.E. 1972. Computation of channel capacity and rate-distortion function. IEEETransactions on Information Theory, 18(4), 460–473.

Brown, S. P., & Masland, R. H. 2001. Spatial scale and cellular substrate of contrastadaptation by retinal ganglion cells. Nature Neuroscience, 4(1), 44–51.

Brunel, N. 2000. Dynamics of sparsely connected networks of excitatory and inhibitoryspiking neurons. Journal of Computational Neuroscience, 8, 183–208.

Brunel, N., & Hakim, V. 1999. Fast global oscillations in networks of integrate-and-fireneurons with low firing rates. Neural Computation, 11, 1621–1671.

Brunel, N., & Nadal, J. 1998. Mutual information, Fisher information and populationcoding. Neural Computation, 10, 1731–1757.

Brunel, N., Chance, F.S., Fourcaud, N., & Abbott, L.F. 2001. Effects of synapticnoise and filtering on the frequency response of spiking neurons. Physical ReviewLetters, 86(10), 2186–2189.

Bulsara, A., Jacobs, E.W., Zhou, T., Moss, F., & Kiss, L. 1991. Stochastic resonancein a single neuron model: Theory and analog simulation. Journal of TheoreticalBiology, 152, 531–555.

Bulsara, A., Lowen, S. B., & Rees, C.D. 1994. Cooperative behavior in periodicallymodulated Wiener process: Noise-induced complexity in a model neuron. Physi-cal Review E, 53, 3958–3969.

Bulsara, A.R., & Zador, A. 1996. Threshold detection of wideband signals: A noiseinduced maximum in the mutual information. Physical Review E, 54(3), R2185–R2188.

Capurro, A., Pakdaman, K., Nomura, T., & Sato, S. 1998. Aperiodic stochastic reso-nance with correlated noise. Physical Review E, 58(4), 4820–4826.

Carandini, M., & Ferster, D. 1997. A tonic hyperpolarization underlying contrastadaptation in cat visual cortex. Science, 276(5314), 949–52.

Chance, F.S., Abbott, L.F., & Reyes, A.D. 2002. Gain modulation from backgroundsynaptic input. Neuron, 35, 773–782.

Chapeau-Blondeau, F., Godivier, X., & Chambet, N. 1996. Stochastic resonance in aneuron model that transmits spike trains. Physical Review E, 53(1), 1273–1275.

Collins, J.J., Chow, C.C., & Imhoff, T.T. 1995a. Aperiodic stochastic resonance inexcitable systems. Physical Review E, 52, R3321–R3324.

157

Collins, J.J., Chow, C.C., & Imhoff, T.T. 1995b. Stochastic resonance without tuning.Nature, 376(6537), 236–8.

Collins, J.J., Imhoff, T.T., & Grigg, P. 1996. Noise-enhanced information transmissionin rat SA1 cutaneous mechanoreptors via aperiodic stochastic resonance. Journalof Neurophysiology, 76, 642–645.

Cordo, P., Inglis, J.T., Vershueren, S., Collins, J.J., Merfeld, D.M., Rosenblum, S.,Buckley, S., & Moss, F. 1996. Noise in human muscle spindles. Nature, 383,769–770.

Cover, T.M., & Thomas, J.A. 1991. Elements of Information Theory. Wiley Inter-science.

Cox, D.R., & Lewis, P.A. W. 1966. The statistical analysis of series of events. Methuen.

Dayan, P., & Abbott, L.F. 2001. Theoretical Neurosience. MIT Press.

de Polavieja, G.G. 2002. Errors drive the evolution of biological signaling to costlycodes. Journal of Theoretical Biology, 214(4), 657–664.

de Ruyter van Steveninck, R.R., Lewen, G.D., Strong, S.P., Koberle, R., & Bialek,W. 1997. Reproduceability and variability in neural spike trains. Science, 275,1805–1808.

Destexhe, A., & Pare, D. 1999. Impact of network activity on the integrative propertiesof neocortical pyramidal neurons in vivo. Journal of Neurophysiology, 81, 1531–1547.

Destexhe, A., Rudolph, M., Fellous, J.M., & Sejnowski, T.J. 2001. Fluctuating synapticconductances recreate in-vivo-like activity in neocortical neurons. Neuroscience,107, 13–24.

Destexhe, A., Rudolph, M., & Pare, D. 2003. The high-conductance state of neocorticalneurons in vivo. Nature Review Neuroscience, 4, 739–751.

Diesmann, M., Gewaltig, M.O., & Aertsen, A. 1999. Stable propagation of syn-chronous spiking in cortical neural networks. Nature, 402(6761), 529–533.

Douglass, J.K., Wilkens, L., Pantazelou, E., & Moss, F. 1993. Noise enhancementof information transfer in crayfisch mechanoreceptors by stochastic resonance.Nature, 365, 337–340.

Engel, A.K., Fries, P., & Singer, W. 2001. Dynamic predictions: oscillations andsynchrony in top-down processing. Nature Reviews Neuroscience, 2(10), 704–716.

Fauve, S., & Heslot, F. 1983. Stochastic resonance in a bistable system. Physics LettersA, 97, 5–7.

Feldmann, M.L. 1984. Morphology of the neocortical pyramidal neuron, CerebralCortex Vol. 1. Plenum Press.

Foldiak, P. 1990. Forming sparse representations by local anti-Hebbian learning. Bio-logical Cybernetics, 64, 165–70.

158

F ollmer, H., Protter, P., & Shiryaev, A.N. 1995. Quadratic covariation and an extensionof Ito’s formula. Bernoulli, 1, 149–170.

F ollmer, H.F., & Protter, P. 2000. On Ito’s formula for multidimensional Brownianmotion. Probability Theory and Related Fields, 80, 1–20.

Gammaitoni, L., Marchesoni, F., Menichella-Saetta, E., & Santucci, S. 1989. Stochas-tic resonance in bistable systems. Physical Review Letters, 62, 349–352.

Gammaitoni, L., H anggi, P., Jung, P., & Marchesoni, F. 1998. Stochastic resonance.Reviews of Modern Physics, 70(1), 223–287.

Gardiner, C.W. 1985. Handbook of Stochastic Methods for Physics, Chemistry and theNatural Sciences. 2nd ed. Springer.

Gerstner, W. 2000. Coding properties of spiking neurons: Reverse and cross-correlations. Neural Networks, 6-7, 599–610.

Gerstner, W., & Kistler, W. 2002. Spiking Neuron Models. Cambridge UniversityPress.

Gillespie, D. 1996. Exact numerical simulation of the Ornstein-Uhlenbeck process andits integral. Physical review E, 54, 2084–2091.

Gingl, Z., Kiss, B., & Moss, F. 1995. Non-dynamical stochastic resonance: Theoryand experiments with white and arbitrary colored noise. Europhysics Letters, 29,191–196.

Giorno, V., Nobile, A.G., & Ricciardi, L. 1992. Instantaneous return processes andneural firings. Cybernetics and Systems Research, 1, 829–236.

Heneghan, C., Chow, C.C., Collins, J.J., Imhoff, T.T., Lowen, S.B., & Teich, M.C.1996. Information measures quantifying aperiodic stochastic resonance. PhysicalReview E, 54(3), R2228–R2231.

Hines, M.L., & Carnevale, N.T. 1997. The NEURON simulation environment. NeuralComputation, 9, 1179–1209.

Hoch, T., Wenning, G., & Obermayer, K. 2003a. Adaptation using local informationfor maximizing the global cost. Neurocomputing, 52-54, 541–546.

Hoch, T., Wenning, G., & Obermayer, K. 2003b. Optimal noise-aided signal transmis-sion through populations of neurons. Physical Review E, 68, 011911/1–11.

Hodgkin, A.L., & Huxley, A.F. 1952. A quantitative description of membrane currentand its application to conduction and excitation in nerve. Journal of Physiology,117, 500–544.

Ito, K. 1944. Stochastic integral. Proceedings of the Imperial Academy of Tokyo, 20,519–524.

Jaramillo, F., & Wiesenfeld, K. 1998. Mechanoelectrical transduction assisted byBrownian motion: A role for noise in the auditory system. Nature Neuroscience,1, 384–388.

159

Jaynes, E.T. 1957. Information theory and statistical mechanics. Physical Review, 106,620–30.

Kandel, E. R., Schwarz, J. H., & Jessel, T.M. 2001. Principles of Neural Science, 4thed. Appleton and Lange.

Kanev, J., Wenning, G., & Obermayer, K. 2004. Approximating the response stimuluscorrelation in the integrate-and-fire neuron model. Neurocomputing, 58-60, 47–52.

Kanev, J., Wenning, G., & Obermayer, K. submitted. Response stimulus correlation ofa leaky integrate-and-fire neuron.

Katz, P.S. 1999. Beyond Neurotransmission. Oxford University Press.

Kay, S. M. 1993. Fundamentals of statistical signal processing, Volume 1: EstimationTheory. Prentice Hall.

Kempter, R., Gerstner, W., van Hemmen, J.L., & Wagner, H. 1998. Extracting oscil-lations: Neuronal coincidence detection with noisy periodic spike input. NeuralComputation, 10, 1987–2017.

Kempter, R., Gerstner, W., & van Hemmen, J.L. 2001. Intrinsic stabilization of outputrates by spike-based Hebbian learning. Neural Computation, 13, 2709–2741.

Kim, K.J., & Rieke, F. 2001. Temporal contrast adaptation in the input and outputsignals of salamander retinal ganglion cells. Journal of Neuroscience, 21(1), 287–299.

Kitajo, K., Nozaki, D., Ward, L.M., & Yamamoto, Y. 2003. Behavioral stochasticresonance within the human brain. Physical Review Letters, 90(21), 218103/1–4.

Kloeden, P.E., & Platen, E. 1992. Numerical Solution of Stochastic Differential Equa-tions. Springer.

Koch, C., & Segev, I. (eds). 1998. Methods in Neural Modeling: From ions to Net-works, 2nd edition. Bradford Book.

Lansky, P., & Sacerdote, L. 2001. The Ornstein-Uhlenbeck neuronal model withsignal-dependent noise. Physics Letters A, 285, 132–140.

Lapicque, L. 1907. Recherches quantitatives sur l’exitation electrique des nerfs traiteecomme une polarization. Journal of Physiology (Paris), 9, 620–635.

Laughlin, S., de Ruyter van Steveninck, R., & Anderson, J. 1998. The metabolic costof neural information. Nature Neuroscience, 1(1), 36–40.

Laughlin, S.B. 1981. A simple coding procedure enhances a neuron’s informationcapacity. Zeitschrift fur Naturforschung, 36(c), 910–912.

Levin, J.E., & Miller, J. P. 1996. Broadband neural encoding in the cricket cercalsensory system enhanced by stochastic resonance. Nature, 380(6570), 165–168.

Levy, W., & Baxter, R. 1996. Energy efficient neural codes. Neural Computation, 8,531–543.

160

Lindner, B., Longtin, A., & Bulsara, A. 2003. Analytic expressions for rate and CV ofa type I neuron driven by white Gaussian noise. Neural Computation, 15, 1761–1788.

Longtin, A. 1993. Stochastic resonance in neuron models. Journal of StatisticalPhysics, 70(1/2), 309–327.

Longtin, A., Bulsara, A., & Moss, F. 1991. Time-interval sequences in bistable systemsand the noise induced transmission of information by sensory neurons. PhysicalReview Letters, 67, 656–659.

L ubke, J., Egger, V., Sackmann, B., & Feldmeyer, D. 1999. Reliable synaptic connec-tions between pairs of excitatory layer 4 neurons within a single barrel of devel-oping rat somatosensory cortex. Journal of Physiology, 521(1), 169–190.

L ucke, H.D. 1995. Signalubertragung. Springer Berlin.

Martin, R.D. 1996. Scaling of the mammalian brain - the maternal energy hypothesis.News in Physiological Sciences, 11, 149–156.

Mato, G. 1998. Stochastic resonance in neural systems: Effect of temporal correlationin the spike trains. Physical Review E, 58(1), 876–880.

McNamara, B., Wiesenfeld, K., & Roy, R. 1988. Observation of stochastic resonancein a ring laser. Physical Review Letters, 60, 2626–2629.

Mitaim, S., & Kosko, B. 1998. Adaptive stochastic resonance. Proceedings of theIEEE, Special Issue on intelligent signal processing, 86(11), 2152–2183.

Nozaki, D., Mar, D.J., Grigg, P., & Collins, J.J. 1999a. Effects of colored noise onstochastic resonance in sensory neurons. Physical Review Letters, 82(11), 2402–2405.

Nozaki, D., Collins, J.J., & Yamamoto, Y. 1999b. Mechanism of stochastic resonanceenhancement in neuronal models driven by 1/f noise. Physical Review E, 60(4),4637–4644.

Olshausen, B.A., & Field, D.J. 2004. What is the other 85% of V1 doing? NeuralComputation, submitted.

Panzeri, S., Petroni, F., Petersen, R.S., & Diamond, M.E. 2003. Decoding neuralpopulation activity in rat somatosensory cortex: Role of columnar organization.Cerebral Cortex, 13, 45–52.

Plesser, H.E., & Gerstner, W. 2000. Noise in integrate-and-fire neurons: From stochas-tic input to escape rates. Neural Computation, 12(2), 367–384.

Plesser, H.E., & Tanaka, S. 1997. Stochastic resonance in a model neuron with reset.Physics Letters A, 225, 228–234.

Pouget, A., Zemel, R.S., & Dayan, P. 2000. Computation and inference with populationcodes. Nature Review Neuroscience, 1, 125–132.

Protter, P. 1995. Stochastic Integration and Differential Equations. Springer.

Redlich, A.N. 1993. Supervised factorial learning. Neural Computation, 5, 750–66.

161

Ricciardi, L.M., & Sacerdote, L. 1979. The Ornstein-Uhlenbeck process as a modelfor neural activity. Biological Cybernetics, 35, 1–9.

Rieke, F., Warland, D., de Ruyter van Steveninck, R., & Bialek, W. 1997. Spikes:Exploring the Neural Code. MIT Press: Cambridge.

Risken, H. 1984. The Fokker Planck Equation: Methods of Solution and Applications.Springer, Berlin.

Rolfe, D.F.S., & Brown, G.C. 1997. Cellular energy utilization and molecular originof standard metabolic rate in mammals. Physiological Review, 77, 731–758.

Rudolph, M., & Destexhe, A. 2001a. Correlation detection and resonance in neuralsystems with distributed noise sources. Physical Review Letters, 86(16), 3662–3665.

Rudolph, M., & Destexhe, A. 2001b. Do neocortical pyramidal neurons displaystochastic resonance? Journal of Computational Neuroscience, 11, 19–42.

Rudolph, M., & Destexhe, A. 2003. Tuning neocortical pyramidal neurons betweenintegrators and coincidence detectors. Journal of Computational Neuroscience,14, 239–251.

Russell, D.F., Wilkens, L.A., & Moss, F. 1999. Use of behavioural stochastic resonanceby paddle fish for feeding. Nature, 402, 291–294.

Sakmann, B., & Neher, E. (eds). 1983. Single channel recording. Plenum Press, NewYork.

Salinas, E., & Seijnowki, T.J. 2000. Impact of correlated synaptic input on outputfiring rate and variability in simple neuron models. Journal of Neuroscience, 20,6193–6209.

Sato, S. 1978. On the moments of the firing interval of the diffusion approximatedmodel neuron. Mathematical Biosciences, 39, 53–70.

Schreiber, S., Machens, C.K., Herz, A.V., & Laughlin, S.B. 2002. Energy-efficientcoding with discrete stochastic events. Neural Computation, 14(6), 1323–46.

Shadlen, M.N., & Newsome, W.T. 1998. The variable discharge of cortical neurons:Implications for connectivity, computation, and information coding. The Journalof Neuroscience, 18(10), 3870–3896.

Shannon. 1948. A mathematical theory of communication. The Bell System TechnicalJournal, 27, 379–423, 623–656.

Silberberg, G., Bethge, M., Markram, H., Pawelzik, K., & Tsodyks, M. 2004. Dy-namics of population rate codes in ensembles of neocortical neurons. Journal ofNeurophysiology, 91(2), 704–9.

Simonotto, E., Riani, M., Seife, C., Roberts, M., Twitty, J., & Moss, F. 1997. Visualperception of stochastic resonance. Physical Review Letters, 78, 1186–1189.

Song, S., Miller, K.D., & Abbott, L.F. 2000. Competitive hebbian learning throughspiketiming-dependent synaptic plasticity. Nature Neuroscience, 3, 919–926.

162

Stemmler, M. 1996. A single spike suffices: the simplest form of stochastic resonancein model neurons. Network: Computation in Neural Systems, 7, 687–716.

Stemmler, M., & Koch, C. 1999. How voltage-dependent conductances can adept tomaximize the information encoded by neuronal firing. Nature Neuroscience, 2(6),521–527.

Stocks, N.G. 2000. Suprathreshold stochastic resonance in multilevel threshold sys-tems. Physical Review Letters, 84, 2310–2314.

Stocks, N.G. 2001. Information transmission in parallel threshold arrays: Suprathresh-old stochastic resonance. Physical Review E, 63, 041114/1–9.

Stopfer, M., Jayaraman, V., & Laurent, G. 2003. Intensity versus identity coding in anoffactory system. Neuron, 39, 991–1004.

Strassberg, A.F., & DeFelice, L.J. 1993. Limitations of the Hodgkin-Huxley formal-ism: Effects of single channel kinetics on transmembrane voltage dynamics. Neu-ral Computation, 5, 843–855.

Tougaard, J. 2000. Stochastic resonance and signal detetection in an energy detector- implications for biological receptor systems. Biological Cybernatics, 83, 471–480.

Tsodyks, M.V., & Sejnowski, T.J. 1995. Rapid state switching in balanced corticalnetwork models. Network: Computation in Neural Systems, 6, 111–124.

Tuckwell, H.C. 1998. Introduction to theoretical neurobiology: volume 2 Nonlinearand Stochastic Theories. Cambridge University Press.

Turrigiano, G.G., Leslie, K.R., Desai, N.S., Rutherford, L.C., & Nelson, S.B. 1998.Activity-dependent scaling on quantal amplitudes in neocortical neurons. Nature,391, 892–896.

Wenning, G., & Obermayer, K. 2002a. Activity driven adaptive stochastic resonance.Pages 301–308 of: Advances in Neural Information Processing Systems, vol. 14.MIT Press.

Wenning, G., & Obermayer, K. 2002b. Adjusting stochastic resonance in a leakyintegrate-and-fire neuron to sub-threshold stimulus distributions. Neurocomput-ing, 44-46, 225–231.

Wenning, G., & Obermayer, K. 2003. Activity driven adaptive stochastic resonance.Physical Review Letters, 90(12), 120602/1–4.

Wenning, G., Hoch, T., & Obermayer, K. submitted. Pulse detection in a colored noisesetting.

Wickens, T.D. 1991. Elementary Signal Detection. Wiley Series in Telecommunica-tions.

Wiesenfeld, K., Pierson, D., Pantazelou, E., Dames, C ., & Moss, F. 1994. Stochasticresonance on a circle. Physical Review Letters, 72, 2125–2129.

163