resonances and noise in a stochastic hindmarsh–rose...

doi:10.1016/S0092-8240(03)00026-0Bulletin of Mathematical Biology (2003)65, 641–663

Resonances and Noise in a Stochastic Hindmarsh–RoseModel of Thalamic Neurons

STEFAN REINKERDepartment of Mathematics,Institute of Applied Mathematics,University of British Columbia,Vancouver,Canada BC V6T 1Z2E-mail: [email protected]

ERNEST PUILDepartment of Pharmacology & Therapeutics,University of British Columbia,Vancouver,Canada BC V6T 1Z3E-mail: [email protected]

ROBERT M. MIURA∗

Departments of Mathematical Sciences and Biomedical Engineering,New Jersey Institute of Technology,Newark, NJ 07102,U.S.A.Departments of Mathematics and Pharmacology & Therapeutics,Institute of Applied Mathematics,University of British Columbia,Vancouver,Canada BC V6T 1Z2E-mail: [email protected]

Thalamic neurons exhibit subthreshold resonance when stimulated with small sinewave signals of varying frequency and stochastic resonance when noise is addedto these signals. We study a stochastic Hindmarsh–Rose model using Monte-Carlosimulations to investigate how noise, in conjunction with subthreshold resonance,leads to a preferred frequency in the firing pattern. The resulting stochastic reso-nance (SR) exhibits a preferred firing frequency that is approximately exponentialin its dependence on the noise amplitude. In similar experiments, frequencydependent SR is found in the reliability of detection of alpha-function inputs undernoise, which are more realistic inputs for neurons. A mathematical analysis ofthe equations reveals that the frequency preference arises from the dynamics of

∗Address for correspondence: Department of Mathematical Sciences, New Jersey Institute ofTechnology, University Heights, Newark, NJ 07102, U.S.A.

0092-8240/03/040641 + 23 $30.00/0 c© 2003 Society for Mathematical Biology. Published byElsevier Science Ltd. All rights reserved.

642 S. Reinker et al.

the slow variable. Noise can then transfer the resonance over the firing thresholdbecause of the proximity of the fast subsystem to a Hopf bifurcation point. Ourresults may have implications for the behavior of thalamic neurons in a network,with noise switching the membrane potential between different resonance modes.

c© 2003 Society for Mathematical Biology. Published by Elsevier Science Ltd. Allrights reserved.

1. INTRODUCTION

In the nervous system, neurons process information by converting inputs intotrains of action potentials. In response to identical input stimuli, the neurons mayfire different action potential patterns. This is attributable to random electricalevents in the membrane that change a neuron’s probability of firing actionpotentials. The random events are not necessarily detrimental to the quality ofsignal transduction in neurons. This may seem counterintuitive because noise inan electronic circuit, for example, impairs signal transmission. However, noise atoptimal strength can enhance the detection of weak signals. This phenomenon,termed stochastic resonance (SR), is present in many physical, chemical, andbiological systems (Mosset al., 1994; Gammaitoniet al., 1998).

In biological systems, SR was first found experimentally in crayfish tail fins(Douglasset al., 1993). Here, SR in the mechanoreceptors may improve thedetection of water currents, evoked by a predator in a turbulent or ‘noisy’environment. SR also was found in shark mechanoreceptors (Braun et al.,1994), auditory hair cells (Jaramillo and Wiesenfeld, 1998), and mammaliancold receptors (Longtin and Hinzer, 1996). In humans, a behavioral type of SRparticipates in the perception of ambivalent pictures (Nagaoet al., 2000). Morseand Evans(1996) have proposed a clinical application of SR for a cochlear implantin humans with impaired hearing.

SR may be an important mechanism that neurons use to amplify weak, subthre-shold signals. When enhanced by the addition of noise, the subthreshold dynamicshave a major role in the coding of input signals in a spike train (Braunet al., 1994).The main sources of cellular noise are the random firing of presynaptic neurons,synaptic noise due to released neurotransmitter, and membrane noise arising fromstochastic channel openings and closings (Tuckwell, 1989). For detection, thestandard measure of signal quality is the signal-to-noise ratio (SNR), which givesa correlation between signal inputs and noisy outputs. Statistical measures forthe SNR can be quantified from whole-cell patch-clamp recordings from thalamiccells. However, the short life spans of patch-clamped neurons limit the amountof data that can be collected from such studies. Hence, a mathematical modelingapproach is useful to produce simulated statistical SR data.

In this paper, we study the interactions between the nonlinearities in a neuronmodel and input signals with noise. We analyze the subthreshold membraneresonance, a phenomenon resulting in preferred oscillatory activity in response to

Resonances in a Stochastic Neuron 643

small sine wave stimulation (Hutcheonet al., 1994). Our studies focus on the effectof this subthreshold resonance on the transduction of excitatory inputs into a trainof action potentials. Employing both computer simulations and analysis methodson the Hindmarsh–Rose system of equations, a simple qualitative model of burstingthalamic neurons, we demonstrate the occurrence of SR in this model under sinewave stimulation [cf. Longtin (1993), Longtin and Chialvo(1998), Osipov andPonizovskaya(2000), Wu et al. (2001)]. Under stimulation with sine waves atdifferent frequencies and noise, the model also shows a preferred frequency, orfrequency dependent SR, i.e., for a certain noise level, one frequency is detectedbest. The dependence of this stochastic preferred frequency on the noise level isapproximately an exponential function.

Additionally, we study the influence of noise on more realistic forms of inputsignals. With the probability of detection measure, we can demonstrate both SRand frequency dependent SR whenα-functions are used as inputs. The preferredfrequency in the spike train also manifests itself in the absence of any input signals,with a similar dependence between preferred frequency and noise. We show thatthis autonomous stochastic resonance-like phenomenon (Longtin, 1997) resultsfrom the subthreshold resonance in the HR model.

2. THE MATHEMATICAL MODEL

The well-known FitzHugh–Nagumo equations (FitzHugh, 1961; Nagumoet al.,1962) are a polynomial model derived from the Hodgkin–Huxley (HH) equations.Using a similar approach,Hindmarsh and Rose(1984) derived a polynomial modelof a bursting neuron from a thalamocortical neuron model with detailed ioniccurrents, namely,

dx

dt= y − ax3 + bx2 − z + I0 + Isignal(t), (1)

dy

dt= c − f x2 − y, (2)

dz

dt= ε

(x − 1

4(z − z0)

)(3)

wherex is a voltage-like variable,y controls the voltage recovery after an actionpotential, andz describes the slow dynamics of an adapting current. The parametervalues are given bya = 1, b = 3, I0 = −0.5, c = 1, f = 5, ε = 0.005,and z0 = 5.1. The Hindmarsh–Rose equations (HR model) form an excitablesystem that may fire action potentials or bursts in response to input signal currents,Isignal(t). The magnitude and frequency content ofIsignal determine the evokedfiring pattern (Wanget al., 1998). The polynomial form of the HR model simplifiesboth the mathematical analysis and numerical simulation in comparison to the morephysiological HH-type models.


In this paper, we present results of simulations, using two types of input signals:(a) a sine wave of amplitudeA and periodλ = 2π/ω,

Isignal(t) = A sin(ωt),

in order to study SR in the usual context of periodic inputs, and (b) pairs ofα-function inputs that simulate the shape of postsynaptic potentials (PSPs) withfixed arrival timesti :

Isignal(t) =∑i=1,2

αi(t)

where

αi(t) = αmaxH (t − ti )t − ti

τexp

(− t − ti

τ

).

Here,αmax is a constant,τ denotes the decay time constant of the PSPs,H is theHeaviside function, andti is the arrival time of thei th PSP.

In order to model a stochastic background current, a stochastic termη(t) is addedto a constant background current,i0,

I0 = i0 + η(t).

Assuming noise autocorrelation times that are short compared to the characteristictimes for the dynamics of the equations, we chooseη(t) as Gaussian white noisewith strengthσ , i.e., 〈η(t)〉 = 0, and〈η(t),η(s)〉 = σ 2δ(t − s). Thus, the systemof differential equations is transformed into a system of stochastic differentialequations, and the solution is a stochastic process with realizations that dependon the particular noise realization.

The stochastic equations are simulated using Euler’s method modified to incor-porate the stochastic termη(t) (Kloeden and Platen, 1992). Gaussian white noiseis generated with the Box–Mueller algorithm (Press, 1992). A step size of 0.01proves small enough to give statistically stable results.

Action potentials in the simulation are identified with a Poincar´e map when thex variable crosses thex = 0 plane with a positive slope. Trains of these actionpotentials (also called spikes) are analyzed using interspike-interval histograms bycomputing the distribution of interspike times from long spike trains.

3. RESULTS

3.1. Noisy sine wave stimulation. Classical studies of SR in dynamical systemshave used statistical analysis of the responses to stimulation with sine waves. Whenpresented with a subthreshold signal and noise, the HR model may fire actionpotentials or bursts of action potentials (Fig. 1). These usually occur near the peaksof the signal [sine wave inFig. 1(a) andα-functions inFig. 1(b)] where the model


Figure 1. Voltage (x-variable) trace under sine wave [(a),λ = 325] andα-function [(b),t1 = 50, t2 = 475] stimulation with noise (i0 = −0.7, σ = 0.5 in both traces).

is closest to firing threshold and has been pushed over this threshold by noise.Using different noise amplitudes(σ ) and sine wave inputs of different lengthperiods(λ), we simulate the equations for long times (1000 times the period of


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Figure 2. Interspike interval histograms from sine wave stimulation as inFig. 1 (i0 =−0.5). Spikes during a burst produce a large peak near zero interspike times, and thefirst peak away from zero interspike times is associated with the period of the sine wave.The multipeak (harmonic) structure occurs because one or more subsequent peaks of thesine wave may fail to produce action potentials. (a) At low noise levels, spikes are mostlikely to occur on the peak of the sine wave. Thus, the ISIH has peaks at the input period(hereλ = 1000), at multiples ofλ (skipping peaks), and near 0 (multiple spikes). (b) Atintermediate noise levels, skipping of consecutive peaks of the signal is less likely, andthe peak atλ is large. This is the essential feature of SR, namely, the input frequency ismaximally present in the output. (c) At high noise levels, the system can cross the firingthreshold during all phases of the input signal, and hence, there is not a distinct peak at theinput period. (d) For different sine wave frequencies, the resonant peaks are at the inputperiod (hereλ = 400) and its multiples. The basic shape of the ISIH is preserved.

the input). The times of action potential occurrence are identified by a Poincar´emap and plotted in interspike interval histograms (ISIH;Fig. 2).

In the absence of periodic stimulation, the ISIH of a model exhibiting neuronaltonic firing, such as the FitzHugh–Nagumo model or the HH model, is approxi-mately exponentially decaying (Tuckwell, 1989). This implies that the neuron firesrandomly with no correlations between subsequent spikes. A multi-peak structurefor the ISIH is found on application of a periodic input, e.g., a sine wave [Fig. 2(a)].


The first peak near zero interspike interval times in the ISIH represents actionpotentials that form multiple spikes near the maximum of a sine wave (burst). Thenext ISIH peak occurs at an interspike interval time that is roughly equal to theperiod of the input signal. At integer multiples of this time, the harmonic peakscorrespond to skipping of maxima of the sine wave signal.

The locations of the peaks in the ISIH generally do not change with increasingnoise levels, in contrast to their shapes and relative heights [seeFig. 2(a)–(c)]. Thearea under the ISIH peak at the stimulus period is a measure of how much of theinput sine wave frequency is present in the spike train. Hence, varying the noiselevel changes the area under the histogram peak at the period of the stimulus, whichcan be used to define the SNR for this period and noise level,

SNR= 10 log10

(# of spikes near input period

total # of spikes

),

as a measure of the correlation between the input signal and the output spike train.This approach, similar to that ofLongtin et al. (1991), ignores the contributionof the higher order peaks in order to isolate the response at the input frequency.The model shows SR, demonstrated by the SNR going through a maximum atan intermediate noise level where the spike train has a strong component at thestimulus frequency. For low noise levels, the probability of crossing the spikethreshold and the number of action potentials are too small to correlate well withthe signal, leading to decreased SNR. For high noise levels, extraneous actionpotentials appear in the spike train, implying that the neuron fires due to the noiselevel and not to the periodic sine wave stimulation. Again, the SNR decreases.This gives rise to the typical SR shape of SNR going through a maximum at anintermediate (optimal) noise level.

The surface plot of the SNR of HR inFig. 3(a) shows typical SR curves for fixedinput periodλ with one maximum as the noise levelσ varies. The plot also showsthat for fixed noise levelσ , the SNR is dependent on the input periodλ of the sinewave. For high noise levels, the SNR goes through a maximum for an input periodof λ between 250 and 400, near the resonant period of the deterministic system forsubthreshold signals (see below). At lower noise levels, however, the SNR peaks atperiods up toλ = 1200 and reaches an absolute maximum atλ = 750 forσ = 1.0.The relationship between optimal noise level and input period is an exponential-like function shown inFig. 3(b). This frequency dependent SR is a maximal firingresponse (SNR) to small periodic inputs at a resonant frequency.

A similar exponential-like relationship between mean firing frequency and noiselevel has been reported byLindner and Schimansky-Geier(2000) and byLongtin(1997) in the context of autonomous SR in the absence of an input signal. A similarphenomenon also appears in the HH models of firing (Tanabe and Pakdaman,2001b) and the integrate-and-fire neuron (Shimokawaet al., 1999). Stochasticphase locking in the HR model does not depend on the input noise level, as shown


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Figure 3. Signal-to-noise dependence onσ andλ. (a) SR is demonstrated by the maximumin SNR for varyingσ . SR is also frequency dependent. (b) Contour plot of SNR. Thedependence of the optimal noise level [maximum in (a)] for each input wavelength isdenoted by the solid black line.

by Baltanas and Casado(2002). They found two different SR-like phenomena inthe HR system, one at a fixed high frequency near the eigenfrequency of the Hopfbifurcation, and a variable low frequency SR. This latter SR, where the preferredfrequency changes with noise, is the same phenomenon that we have demonstratedin Fig. 3, but they did not study it extensively and only calculated the frequency ofmaximal SR for two noise levels.

3.2. Alpha-function stimulation. In many cases, living neurons receive discreteinputs such as excitatory postsynaptic potentials (EPSPs) that result from mem-brane depolarizing actions of synaptic neurotransmitter, in contrast to the peri-odic inputs discussed inSection 3.1. We useα-function signals to mimic EPSPs[Fig. 1(b)]. In vivo, neurons often experience single or short sequences of EPSPsas carriers of information. In order to define SR in this context, we require a newmeasure for quantifying the congruence of the input and output. We can define the


probability of making a detection error for a single EPSP as

PE = q ∗ PF + (1 − q) ∗ PM

whereq is the probability of an EPSP,PF is the probability of a spike without anEPSP, andPM is the probability of no spike with an EPSP (Robinsonet al., 1998).Therefore,PE describes the probability of making an error, including false firingsin response to no EPSP or not detecting an EPSP. Then,PE should be minimizedfor optimal signal detection.

In order to computePF , we perform 10 000 simulations on the equations withnoise but no input signal, where each run is 100 time steps long. ThenPF forthis noise levelσ is the fraction of trials where an action potential was evokedwithout an EPSP. SimilarlyPM can be obtained from trials with an EPSP. ThenPM is the fraction of trials where no action potential is evoked in response to theEPSP within 100 time steps after the onset of the EPSP. We use 10 000 simulationseach with and without an EPSP and henceq = 0.5. All probabilities PM , PF ,and PE are dependent on the noise level,σ , because an increased noise amplitudeproduces more spike discharges. Thus, an increase in the noise amplitude resultsin increasedPF and decreasedPM [Fig. 4(a)]; hence, 1− PE goes through amaximum for intermediate noise levels [Fig. 4(b)], exhibiting another occurrenceof SR.

In experiments withα-function stimulation and noise, thePE measurementhas an advantage over SNR since it obtains relevant information about the firingprobabilities of the system at every trial. In contrast, long experimental runs arerequired to obtain enough ISIH data for evaluation of SNR. By using mathematicalmodels, PE can be estimated from repeated simulations of the equations withα-function inputs and determining the frequency of spikes, with and withoutEPSPs.

As with sine wave inputs, our simulations exhibit a preferred frequency; thedetection probability for an EPSP depends on the time since a previous EPSP.Again, 10 000 simulations, each for varying time delays�t = t2 − t1, give PE forthe detection of the second EPSP.Figure 5shows 1− PE for the second EPSP asa function of�t . The error is minimized (1− PE maximized) for delays between400 and 700 time steps, depending on the noise level. These delays are in thesame range as the preferred stochastic frequency, observed with periodic sine wavestimulation and noise. The preferred frequency again shows a dependence on thenoise level where an increase in noise decreases the preferred frequency.

3.3. Noise stimulation with no signal. Even when only noise (with no signal) isadded to a neuron model, SR can occur in the spike output at an intrinsic frequencyof the system. In order to investigate whether this SR is present in the HR model,we simulate the equation withIsignal = 0 and noise levelσ for 107 time steps andcompute the interspike-interval histogram for each noise level.Figure 6showsthe three-dimensional plot of the ISIH againstσ . Most of the spikes occur close


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Figure 4. Firing probabilities underα-function stimulation with dependence onσ . (a) PF ,PM for detecting a single EPSP. (b) 1− PE . The typical SR shape with maximum SNR atintermediate noise levels appears.

together at small interspike intervals. However, there is also a local maximum inthe interspike time at intermediate times for each noise level. This is a preferredfrequency of the system because it fires at this rate without any external forcing. Inthe figure, there is no single preferred firing period, but rather a range of periodsbetween approximately 100 and 600. The dependence of the preferred frequencyon the noise level is again an exponential-like function [cf.Fig. 3(b)] as with sinewave stimulation.

This demonstrates that the preferred frequency SR is not simply caused by theinput signal but is a manifestation of the internal dynamics of the system. This is in


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Figure 5. Preferred frequency SR withα-function stimulation and noise. The probabilityof reliably detecting the second EPSP(1 − PE ) depends on the delay(�t) followingthe first one. Traces shown are forσ = 0.75, 1.0, 1.25, and 1.5, with higher noiselevel corresponding to higher probabilities. For all noise levels, the curve goes througha maximum for�t between 400 and 700. Similar to sine wave stimulation (cf.Fig. 3), theresonance period decreases as the level of noise increases.

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Figure 6. Three-dimensional interspike interval histogram with dependence on the noiselevel σ without any input signal. Each line is an ISIH for fixedσ . There is a noise leveldependent local maximum in the ISIH distribution. Similar toFig. 3(a), this indicates apreferred firing frequency even when no signal is present.

analogy to autonomous or coherent SR (Longtin, 1997) found in bursting neuronmodels, where an optimal noise level maximizes firing at the burst frequency.

3.4. Stochastic resonance in the fast subsystem. The different time scales of thenoise, thez and thex , y dynamics, and the complicated bifurcation dynamics of thefull HR model (discussed below) motivate us to simplify the system of equations.In order to separate the different time scales, we observe thatequation(3), which


governsz, contains the small parameterε. Therefore,z is a slow variable, and wecan formally simplify the model by settingε = 0 andz to its resting valuez0. Thuswe obtain the reduced two-dimensional stochastic HR model

dx = (y − ax3 + bx2 + I ) dt + dη(t), (4)

dy = (c − f x2 − y) dt, (5)

so thatI = I0 − z0. Then, z0 plays the role of an additional current bias. Thereduced model is capable of firing action potentials, but for the given parametervalues, it does not exhibit bursting and is similar to the FitzHugh–Nagumomodel of tonic firing neurons. We study SR in this model under sine wave,α-function, and no signal stimulation with noise to understand the effect of theslow variable.

Under sine wave stimulation with noise [Fig. 7(a)], the typical SR curve with onemaximum for varying noise levels appears as expected. However, in the reducedmodel, SNR decays for increased signal periods, and there is no frequency thatis detected optimally. A comparison withFig. 3(a) shows that thez dynamicsboost the SR for longer periods and a preferred frequency arises from the interplaybetween the firing and the slowz variable.

A preferred firing frequency is also absent underα-function stimulation with theprobability of detection measure [Fig. 7(b)]. Only for a short delay between twoEPSPs is 1−PE increased when the second EPSP catches the tail of the first one foran increased maximum; this is the well-known temporal summation phenomenon.There is no maximum for intermediate delays like in the full model (cf.Fig. 5).

Similarly, the ISIH distribution of the fast subsystem without an input signal[Fig. 7(c), cf. Fig. 6] is decaying for eachσ and no inherent frequency of themodel is revealed by the noise input.

In summary, all three forms of inputs that we investigated in the reduced modelshow no preferred firing frequency. This demonstrates that thez dynamics areresponsible for the preferred frequency SR.

4. MATHEMATICAL ANALYSIS

4.1. Subthreshold dynamics. The deterministic HR equations exhibit a classicalfrequency preference or resonance for small, subthreshold inputs, similar tosubthreshold resonance found experimentally in thalamic and cortical neurons(Puil et al., 1994; Hutcheonet al., 1994, 1996; Hutcheon and Yarom, 2000).Subthreshold resonance is a maximal deterministic voltage response to smallperiodic current inputs at the resonant frequency.

The responses of dynamical systems to small periodic variations can be obtainedin an analytical form [seeHutcheonet al. (1994), for an analysis of the HH-equations]. Assuming thatIsignal(t) = δeiωt , the system (1)–(3) can be linearized


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Figure 7. SR in the reduced HR model. (a) SNR dependence onσ andλ under sine wavestimulation [cf. Fig. 3(a)]. Again SR is apparent in the maximum for each frequencyinput. There is no frequency dependence of the maximum for varying noise levelsσ .(b) Detection probability of the EPSP following an earlier EPSP (cf.Fig. 5). For all noiselevels, the detection probability decays or is approximately constant. (c) ISIH distributionof the fast subsystem (cf.Fig. 6). There is no preferred frequency, and the ISIH distributionis decaying for each noise levelσ .


around itsI0-dependent steady state(x0, y0, z0) by introducing the perturbations,X = x − x0, Y = y − y0, Z = z − z0:

d X

dt= Y − 3ax2

0 X + 2bx0X − Z + δeiωt ,

dY

dt= −2 f x0X − Y,

d Z

dt= εX − ε

4Z .

Letting X = Aeiωt , Y = Beiωt , Z = Ceiωt , andδ = 1 and then dividing byeiωt ,the system becomes

Aiω = B − 3ax20 A + 2bx0 A − C + 1,

Biω = −2 f x0A − B,

Ciω = εA − ε

4C.

Solving for A in the first equation after eliminatingB andC yields the result:

A =[

iω + 2 f x0

1 + iω+ 4ε

ε + 4iω+ 3ax2

0 − 2bx0 + 1

]−1

. (6)

This expression corresponds to the complex impedance for a biophysical neuronmodel and describes the magnitude and phase of the voltage responses to currentinputs. Figure 8(a) shows the dependence of the impedance magnitude|A| on x0

and periodλ = 2π/ω. For short periods (high frequencies), thex response to aperiodic signal is small whereas longer periods give a nearly constant response.There is a singularity atλ ∼ 336 andx0 ∼ −1.33 [see parameter values after(3) above], as seen by setting the denominator ofA (the quantity in brackets)to 0. The infinite impedance of the linearized system does not correspond to aninfinitely increasing solution of the sine wave stimulated system, because awayfrom equilibrium the nonlinear terms dominate the dynamics. The impedanceis large for a wide frequency range aroundλ = 336 for x0 just below−1.33,signifying an amplification of the impedance over this range of frequencies. Awayfrom thisx0 value, the impedance magnitude quickly diminishes.

There also exist three linearly independent solutions of the homogeneous linearequations. However, as shown inFig. 8(b), these solutions are unstable forx0 >

−1.33 because at least one eigenvalue associated with these solutions has a positivereal part. Atx0 = −1.33, the real part of the pair of complex-conjugate eigenvaluescrosses 0, and the corresponding homogeneous solution becomes dominant. Thisis the point of a Hopf bifurcation, which gives rise to burst firing behavior (seebelow), the solutionAeiωt (6) loses stability, and the nonlinear terms in (1)–(3)dominate the dynamics with action potential firing.


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Figure 8. (a) Impedance diagram of the Hindmarsh–Rose model. Log10(|impedance|) vs.x0 and stimulus period,λ, obtained from (6). The peak nearx0 = −1.33,λ = 335 signifiesa maximum response to inputs of this wavelength. (b) Eigenvalues of the linearized matrixof the HR model with dependence onx0. At x0 = −1.33, a pair of complex-conjugateeigenvalues crosses thex0-axis for increasingx0, resulting in a loss of stability of thesolution (4). The third eigenvalue is negative and does not influence the stability.

The subthreshold resonance is dependent on thez variable, which can be seen bysettingε = 0. In this two-dimensional reduced model, the impedance plot is flatwithout a resonant frequency (not shown). In general, the location of the resonancemaximum depends onε, the time scale of the slow equation.

This analysis demonstrates that the HR system has one single subthresholdpreferred frequency atλ = 336 that is dependent on thez variable.

4.2. Bifurcation and phase plane analysis. In deterministic dynamical systems,classical bifurcation theory describes qualitative changes of behavior. The changefrom a steady state to burst firing in the HR model occurs as the input current,I0, is increased. AtI0 = −0.03, a subcritical Hopf bifurcation occurs and thesystem jumps to a burst firing cycle that arises from a supracritical Hopf bifucation


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Figure 9. Phase plane and bifurcation diagrams of the deterministic Hindmarsh–Rosemodels. (a) Bifurcation diagram of the deterministic HR model. AsI0 increases, themodel goes through a subcritical Hopf bifurcation atI0 = −0.03 that gives rise to burstfiring. There are also two additional Hopf bifurcations atI0 = 4.01 and 4.90 that donot contribute to the dynamics of tonic or burst firing. (b) Phase plane of the reduceddeterministic HR model forI0 = −0.5. The broken lines denote the nullclines and thediamonds are the three fixed points. Overlaid is the trace of a trajectory of the noisyreduced system model. Most of the time, the trajectory stays near the stable fixed point,(x, y) = (−1.72, −14.1). However, when threshold is crossed, a large excursion takesplace that encircles the right-most fixed point and corresponds to an action potential.(c) Bifurcation diagram of the reduced deterministic HR model. AsI0 increases, thestable fixed point becomes unstable through a saddle-node bifurcation at the right lowerknee(I0 = 0.23). A homoclinic bifurcation gives rise to tonic action potential firing atI0 = −0.92. Thus there is hysteresis forI0 in (−0.92, 0.23) because there are both astable fixed point and a stable cycle present. Another Hopf bifurcation near the left kneedoes not contribute to firing of action potentials.

at I0 = 23.96 [Fig. 9(a)]. Thus the stable fixed point loses stability atx0 = −1.33.The valueI0 = −0.5 used in our stochastic simulation corresponds to a restinglevel of x0 = −1.72. The bifurcation dynamics are very complicated, includingchaotic regimes [seeTerman(1991, 1992)], not shown in the bifurcation diagram.In our case, however, the chaotic dynamics can be ignored because the stochasticterm will remove any dependency on initial conditions.

In stochastic systems, bifurcation analysis is replaced by the study of invariantmeasures that describe the probability distribution of the variables. These can


be obtained from the Fokker–Planck equation associated with the model or bycomputing the probability distribution of the variables from long simulations ofthe equations (Gardiner, 1983).

To understand the separate mechanisms of the fast and the slow subsystems, it isuseful to study the dynamics of the two-dimensional HR model by visualization ina phase plane plot [Fig. 9(b)]. The nullclines and the fixed point are plotted toge-ther with an exemplary trajectory of the noisy reduced HR model. Most of thetime, the trajectory stays in a neighborhood of the stable fixed point,(x, y) =(−1.72,−14.1), driven by the stochastic term. Only when the firing threshold isexceeded will there be a large excursion around an unstable fixed point, resultingin the action potential. With a sufficiently large positive input current,I , thecubic nullcline is shifted upwards so that the stable fixed point collides with theunstable fixed point in a saddle node bifurcation, eliminating both steady states.Then, solutions can move towards a periodic limit cycle solution around theremaining fixed point, resulting in continuous action potential firing, called tonicfiring. Figure 9(c) summarizes the bifurcation dynamics of the reduced HR model.With the presence of three fixed points for every value ofI between−0.92 and0.23, corresponding to the resting level, threshold, and center of the oscillation,respectively, small changes of total input currentI can switch the model betweenthe fixed point and tonic firing. Such small changes can arise from noise or changesin thez variable of the full system.

4.3. Stochastic bifurcation analysis. In the reduced HR model, (4) and (5), thestochastic termdη perturbs the trajectories away from the stable deterministic tra-jectories, and the solution of the model can be represented as a probability distribu-tion p(x, y), representing a stochastic process. Then

∫pdxdy over a setD ⊂ R2

denotes the probability of finding a realization of the stochastic differential equa-tion in the setD. Figure 10shows the development of the probability distributionfor increasingI when there is noise withσ = 0.5 and no additional input sig-nal. The plots are two-dimensional histograms of residence probability forx andyobtained from long (106 time steps) simulations of (4) and (5).

WhenI0 is far from the bifurcation point (I0 = −1.5, corresponding toI = 0.13;cf. Fig. 9(a) and9(c)), the real parts of the eigenvalues for the fixed point are largeand negative (not shown). Thus the noise perturbs the trajectories in a smallneighborhood of the fixed point [Fig. 10(a)] and almost never evokes an actionpotential such that the probability distribution approximates a Gaussian centered atthe fixed point. ForI closer to the bifurcation point, the noise can boost the systemabove threshold, producing an action potential. During such an action potential,the trajectory roughly follows the nullclines, giving rise to a positive probabilitynear the nullclines [compareFig. 10(b) with Fig. 9(b)]. Action potentials are morelikely to occur close to the bifurcation point where the probability distribution hasa bimodal shape with the probability maxima corresponding to the fixed point andthe lower limit of the oscillation. This change from a unimodal to a bimodal


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -16-14

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-4-2

02

4

00.0020.0040.0060.008

0.010.0120.0140.016

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02

46

00.0020.0040.0060.008

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0.02

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00.010.020.030.040.050.060.07

p

xy

p

xy

p

xy

p

xy

(a) I0=–1.5 (b) I0=–0.7

(c) I0=–0.5 (d) I0=+0.1

Figure 10. Probability distribution of the reduced Hindmarsh–Rose model forσ = 0.5and varyingI0. (a) I0 = −1.5, the distribution is close to a delta function. (b)I0 =−0.7, decreasing the distance from threshold broadens the distribution, and occasionalaction potential firing gives positive residence probability near the nullclines, cf.Fig. 9(c).(c) I0 = −0.5, the double peaked distribution indicates that a stochastic bifurcation hastaken place. (d)I0 = 0.1, after the bifurcation, there is tonic firing with positive residencyprobability along the nullclines, cf.Fig. 9(b). Because action potentials are narrow in time,the probability is highest near the baseline of the action potential where the orbit spendsmost of its time.

distribution during stochastic bifurcations is also called a phenomenological orP-bifurcation, as described byArnold (1998).

Beyond the bifurcation, tonic firing occurs, and the noise only influences thetiming of the action potentials. Hence, the probability distribution is concentratedaround the trajectory of the deterministic action potential. Residence probability ispositive all along the firing orbit and highest near the minimum value of the actionpotential (x-variable) because that is where the dynamics are slowest. For details,seeMeunier and Verga(1988), who performed the first investigations of stochasticbifurcations. Arnold (1998) gives an overview of modern stochastic bifurcationtheory, andJansons and Lythe(1998) provide an analytical treatment of stochasticbifurcations.

The stochastic bifurcation in the reduced HR system corresponds to the saddle-node and homoclinic bifurcations of the deterministic two-dimensional system.The appearance of the stochastic homoclinic bifurcation is similar to the stochasticHopf-bifurcation in the Fitzhugh–Nagumo model described byTanabe and Pak-daman(2001a), with the difference that the stochastic homoclinic bifurcationimmediately shows a large excursion from the rest state. The destruction of an


unstable and a stable fixed point through a saddle-node bifurcation manifests itselfin the stochastic case as the disappearance of the lower maximum inFig. 9(d).

This stochastic bifurcation is the mechanism through which action potentialfiring occurs under noise. When an excitatory(Isignal > 0) subthreshold signalarrives, the probability distribution goes through the bifurcation inFig. 10and anaction potential is likely to occur. Because thez dynamics are slower, thebifurcation of the fast subsystem is a good approximation of the firing in the fullmodel, wherez acts as a modulating parameter. Hence, a resonance from the slowdynamics can influence firing through thez term in (3).

5. DISCUSSION

In this paper, we used two different measures of SNR and demonstrated thepresence of SR in the Hindmarsh–Rose model for burst firing. Firstly, a classicalanalysis of interspike interval histograms revealed SR in the model, as expected forthreshold systems. The frequency of maximal SNR in response to the underlyingsubthreshold signal is not fixed but depends on the noise level. This preferredfrequency SR depends on the subthreshold resonance which occurs in responseto small oscillatory inputs. Hence, thalamic neurons that exhibit subthresholdresonance may display preferred frequency stochastic firing when presented withsmall periodic signals and noise.

We observed that the noise level required for most faithful transduction of theinput signal frequencies to the spike train, i.e., the point of SR, decreased in anexponential-like manner with the input period [Figs3(b), 5]. The range of thepreferred frequency SR period depended on the noise level and was betweenλ =250 and 1200, compared to subthreshold resonance atλ = 336. In the standarddouble-well model for SR,Berdichevsky and Gitterman(1996) analytically founda similar exponential-like dependence of optimal noise level on input frequencyunder sine wave stimulation.

We also investigated SR in a more realistic context ofα-function stimulation andreliability of detection. The SR curves were similar to those of the sine wave case,including a preferred firing frequency that depended on the noise level. Thus, thestochastic and subthreshold resonance phenomena may interact in living neurons toimprove the detection of pairs or short sequences of EPSPs. Even in the absence ofan input signal, a preferred firing frequency appeared in the ISIH. Our observationsof analogous results with ISIH and probability of detection measures imply that thedependence of the optimal noise level on the input frequency is a feature of manysystems under noise stimulation.

In order to separate stochastic and subthreshold resonance for a mathematicalanalysis of the stochastic firing dynamics, the three-dimensional stochastic dif-ferential equation system can be split into slow and fast subsystems. The timescale of the slow subsystem determines the subthreshold resonance while the two-


dimensional fast subsystem produces action potentials. Addition of noise to thefast variables results in a stochastic bifurcation that gives rise to firing. The evo-lution of the underlying probability density from a unimodal to a bimodal shapedescribes this stochastic bifurcation.

Our simulations of the fast subsystem of the HR equations revealed no preferredstochastic firing frequency. This illustrates that subthreshold resonance, introducedby the slow subsystem, is necessary for preferred frequency SR. The underlyingslow dynamics and input signals push the system through the bifurcation tostochastic firing. Hence, subthreshold resonance may influence firing behaviorthrough this mechanism. From our analysis of the fast and slow subsystems, onemight expect that the SR peaks near the frequency of subthreshold resonance.However, the interplay between subthreshold resonance, firing, input signal, andnoise is complicated and produces the exponential-like dependence of the preferredfrequency on the optimal noise level.

Massanes and Vicente(1999) suggested an explanation for the frequency depen-dence of the optimal noise level, based on a maximal SNR produced by the firingof one action potential per sine wave signal maximum. A shorter period requiresmore noise to have the same likelihood of crossing threshold in the shorter amountof time. Hence, the optimal noise level for SR is higher for higher frequencies(shorter wavelengths), and this mechanism modifies the preferred frequency. Thisrationale can explain the exponential-like dependence of the preferred firing fre-quency under noise in both the sine wave and no signal input cases, where thepreferred frequency arises from the subthreshold resonance caused by the slowzequation.

However, this explanation does not apply to our findings withα-functionstimulation and probability of detection measure with pairs of EPSPs. The shapeand duration of the EPSPs were constant and the noise needed to evoke a spikeshould be independent of the timing of the EPSPs. The maximum probability ofreliably detecting an EPSP at 400 to 700 time steps after the first EPSP arises fromthe subthreshold resonance of the system. The frequency dependence in this caseis more subtle and an additional mechanism might be involved.

In the context of neurons receiving inputs from other neurons, we can interpretthe interdependence between noise and preferred frequency as a cell modulating itsoutput frequency through noise changes, i.e., the internal noise level will changethe preferred output frequency. This may occur through the opening or closingof membrane ion channels which, in small populations, contribute to the noise.In a stochastic neural network, the level of random firing of individual neuronsmay switch between states of network activity when noise changes the preferredfrequency of individual neurons.

In thalamic neurons, as in other resonant neurons, the preferred frequency wouldamplify input spike trains that arrive at the matching frequency, selecting specificinputs out of the synaptic bombardment experienced by neurons. In networksof coupled neurons, resonance would promote synchronization, which is critical


for brain functions, including conscious behavior and sleep. The synchronizationis important in the pathophysiology of epilepsy, in particular, during generalizedabsence seizures, when the cortico-thalamocortical system becomes hypersynchro-nized, disrupting proper brain function. It is interesting to note that in thalamicneurons, the ionic current,IT , that is responsible for subthreshold resonance, ispresent mostly at dendritic sites of EPSP generation. In contrast, the cell bodiesthat receive mostly inhibitory postsynaptic potential (IPSP) input have less reso-nance (Destexheet al., 1998).

In the brain, the spontaneous firing of neurons creates a bombardment of randomsynaptic inputs onto neurons (Stacey and Durand, 2000). This is likely to bethe most important source of noise, in addition to membrane and channel noiseexperienced by neuronsin vivo. A limitation of our study is the use of Gaussianwhite noise, which ignores any autocorrelation in the noise.Godivier and Chapeau-Blondeau(1996) and Chapeau-Blondeauet al. (1996) showed that the morerealistic model of noise from multiple random synaptic inputs also gives rise to SRin neuronal models. When autocorrelation times of the noise are long, it is likelythat there will be interference with the resonance time scale of the model. However,when the noise autocorrelation time is much shorter than the resonance time scale,the influence of the noise time scale on the resonance would be negligible (Hanggiet al., 1993).

In summary, we have demonstrated SR with a noise-dependent preferred fre-quency in the Hindmarsh–Rose model of the thalamic neuron. Frequency pref-erence may be a universal feature of SR. However, the mechanism of interactionbetween the noise and subthreshold resonance which produces the wide range offrequency preference is not clear, warranting more research in this direction. Inneurons, a frequency dependence of the optimal noise level has important implica-tions for the integration of synaptic inputs and generation of oscillations in a noisyenvironment. Thus, the noise level can act as a control parameter rather than beingdetrimental to information processing reliability.

ACKNOWLEDGEMENTS

We gratefully acknowledge financial grant support from the University of BritishColumbia for the University Graduate Fellowship to Mr S Reinker, the NaturalSciences and Engineering Research Council of Canada to Dr RM Miura, and theCanadian Institutes for Health Research to Dr E Puil. We thank a referee for helpfulcomments and additional references.

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Received 8 April 2002 and accepted 20 February 2003

resonances and noise in a stochastic hindmarsh–rose...

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