University of Nice Sophia-antipolis

INRIA - I3S

Midterm Report

Parameter Tuning in virtual retinausing synaptic plasticity

Author:Hassan Nasser

Supervisor:Dr. Bruno CessacDr. Thierry Vieville

Dr. Pierre Kornprobst

tutor:Dr. Marc Antonini

August 13, 2010

Contents

1 The Virtual Retina 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The Vertebrate Retina . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 The underlying retina model . . . . . . . . . . . . . . . . . . . . . . . . 91.4 The VirtualRetina software . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Simulation via VirtualRetina . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Performing statistics with real and VirtualRetina data 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 What is a spike train? . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Performing statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 The statistical models . . . . . . . . . . . . . . . . . . . . . . . 202.4 The EnaS Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Performing statistics with the EnaS library and results . . . . . . . . . 22

2.5.1 Synthetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.2 Real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.3 VirtualRetina data . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Infering connectivity between Retinal Ganglion Cells 313.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 The connectivity between ganglion cells from real data acquisition . . . 31

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List of Figures

1.1 The vertebrate Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The visual pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Up: the action potential. Down: The corresponding spike train . . . . . 61.4 The different cell types in the vertebrate retina . . . . . . . . . . . . . 71.5 The different layers in the retina . . . . . . . . . . . . . . . . . . . . . . 81.6 The different layers in the retina . . . . . . . . . . . . . . . . . . . . . . 81.7 The different stages of the retina model . . . . . . . . . . . . . . . . . . 101.8 The VirtualRetina software logo . . . . . . . . . . . . . . . . . . . . . . 111.9 An example about a gray scale input image . . . . . . . . . . . . . . . 111.10 The structure of the .spk file . . . . . . . . . . . . . . . . . . . . . . . . 121.11 the log scheme distribution of the ganglion cells . . . . . . . . . . . . . 121.12 The beginning of the retina.xml file . . . . . . . . . . . . . . . . . . . . 131.13 The Signal at each layer of the retina model . . . . . . . . . . . . . . . 141.14 Spike train for different cell types . . . . . . . . . . . . . . . . . . . . . 151.15 The spike train in a large scale simulation . . . . . . . . . . . . . . . . 16

2.1 A typical spike train (produced with VirtualRetina) . . . . . . . . . . . 182.2 The technical details of a raster plot . . . . . . . . . . . . . . . . . . . 182.3 The Enas Logo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 The correlation between non-correlated data . . . . . . . . . . . . . . . 232.5 Evolving of the correlation for different probability distributions (Bernouilli

distribution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Evolving of the correlation for different time coincidence. The delay is

the term we use to express that the two neurons fire between a τ timeinterval. We studied the correlation for different daly values, going from1 to 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 The MEA chip for ganglion recordings . . . . . . . . . . . . . . . . . . 252.8 The evoked potential recording . . . . . . . . . . . . . . . . . . . . . . 252.9 Correlation between ganglion cells (at rest) in real acquisition . . . . . 262.10 Correlation between ganglion cells (evoked potential) in real acquisition 272.11 Correlation between X-ON ganglion cells in VirtualRetina . . . . . . . 282.12 Correlation between Y-ON ganglion cells in VirtualRetina . . . . . . . 282.13 Correlation between X-OFF ganglion cells in VirtualRetina . . . . . . . 292.14 Correlation between Y-OFF ganglion cells in VirtualRetina . . . . . . . 29

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3.1 The MEA chip for ganglion recordings . . . . . . . . . . . . . . . . . . 323.2 The λij in term of distance between the retinal ganglion cells . . . . . . 33

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Motivation

The virtual Retina, recently developed in the Odyssee Team at INRIA Sophia-Antipolisallows large scale simulation of the retina with a reasonable computational cost. Thissoftware simulates the Vertebrate Retina by implementing the steps of light-to-spikestransformations. The light is the input, a spike train is the output. The output dependson the retina parameters. The actual implementation doesn’t take into account thecorrelation between the ganglion cells at the ouput level.Beginning with the understanding of the VirtualRetina software and the statisticalframework that helps to study the ganglion cell, the aim of this work is to introduce animportant retina characteristic in the simulator to improve the biological plausibility.The motivation behind is twofold:

1. To introduce a tool for visual neuroscientists that allows large scale input forcortical simulation.

2. The bio inspired image compression. Due to our collaboration with the I3S whoare working on image compression and reconstruction from spiking data, theparallel work aims to see what is the effect of introducing connections betweenganglion cells on the image compression efficiency.

Previous works have shown that there are connections between the retina cells at dif-ferent levels. At different levels in the VirtualRetina, the connections between cells aretaken into account, but not at the level of the ganglion cells. Theoretically, this relationbetween cells (at previous levels) doesn’t propagate within the different layers of Virtu-alRetina there is no correlation between the ganglion cells. This hypothesis is verifiedby statistics that are presented in the second chapter. This work demands multidisci-plinary knowledge. Thanks to the team and the collaboration and people from differentdomain are working to accomplish the task. The work is divided into two part. Thefirst part, which is described in this preliminary report, is to study of previous workabout VirtualRetina and then performing statistics from the VirtualRetina output toverify if that is no correlation between ganglion cells in Virtual Retina.

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Chapter 1

The Virtual Retina

1.1 Introduction

The goal of this chapter is to introduce the VirtualRetina software. This chapter isdivided it into three parts:

1. In the first part we briefly present vertebrate retina.

2. The second part explains the model which is behind the VirtualRetina Software.

3. The third part explains the software itself.

The main reference of this chapter is the thesis of Adrien Wohrer [1].

1.2 The Vertebrate Retina

The vertebrate retina (situated at the back of the eye, Figure 1.1 ) is the interfacebetween the Incoming light and the neural pathways. Its main role is the transductionof light rays into electrical currents. These electrical currents are the action potentialsthat go to the cortex. The right eye current go to the left cortex and vice versa (Figure1.2). The light transduction is processed by consecutive complex phenomena thathappen at different layer in the retina.

The complexity of the retinal structure makes it a wide world to be studied byscientists. To traduce light into action potentials, several procedure take place throughdifferent stages, going from the photo transducers to the ganglion cells. Note that weuse the term spike instead of action potential. In general, a spike denotes that theneuron fires, which is equivalent to the fact that this neuron achieves the threshold ofthe action potential and fires a potential at its output (Figure 1.3).

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Figure 1.1: At left, the eye from the cornea to the optic nerve. At the right, a magni-fication shows the retina with some details

Figure 1.2: The visual pathways: From the optics nerves to cortex

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Figure 1.3: Up: the action potential. Down: The corresponding spike train

Back to the retina structure. The following layers define the retina:

• Light receptors.

• Horizontal cells.

• Bipolar cells.

• Amacrine Cells.

• Ganglion cells

These different types of cells are represented in the Figure 1.4. Each of these celltypes has a role in the ’external work image processing’. At last count, the retinalnetwork is composed of at least 50 clearly distinct cell types.

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Figure 1.4: The different cell types in the vertebrate retina

The Figure 1.5 shows the trajectory of the light-then-spikes process through thedifferent layers. The light comes from outside, being optically processed by the eyecompartments (Pupil, lens, ...), it then goes through the neural layers and then hits theextreme back of the retina where the photo receptors are. A chemical process happensin the photo receptors to transform light into electrical current. There are two typesof receptors: the cones and rods. The rods are very sensitive to light and motion. Thecones get activated at high illumination level and they are sensitive to the colors andshapes.

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Figure 1.5: The different layers in the retina

Figure 1.6: The different layers in the retina

The electrical current produced by these receptors goes then to the OPL (OuterPlexiform Layer) where there are two kinds of cells: the horizontal and bipolar cells. Asthe Figure 1.6 shows, there are direct connections between the consecutive components

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as well as between the non consecutive components. The light receptors excite thehorizontal and bipolar cells. The bipolar and ganglion cells excite the amacrine cells.There are also feedback actions: the horizontal cells give inhibitory feedback to thebipolar cells. The amacrine cells give also inhibitory feedback to the ganglion andbipolar cells.

1.3 The underlying retina model

This part explains the virtual retina model. The creator of the VirtualRetina softwarehas chosen to use the contrast gain control model to implement the retinal function-ality because he wanted to allow large scale simulation keeping in mind the biologicalplausibility of the underlying model. The term ’contrast gain control’ refers to theability of the retinal system to control the transfer function of the contrast information.The Figure 1.7 represents a retina model. From incoming light to spike generation,the process is represented by a processed image and the corresponding mathematicalequation.

The incoming light is represented by L(x, y, t), the illumination quantity at eachpixel of the image, at the time t. This light, being convoluted in time and spaceat the receptor layer, gives a positive action for the bipolar cells. On the opposite,the horizontal cells do another time-space convolution for this signal and acts witha negative feedback. From both actions, comes the notation difference of Gaussian(DOG); the output of the OPL for a Light input. The DOF is hence the input of theIPL; the loge of the contrast gain control process. Our main interest is the ganglionlayer which receive an input current Igang (x,y,t). This current goes to the ganglion cellswhich fire spikes when the threshold is reached. The ganglion cells axons, altogether,are the optic nerve that goes to the cortex. It’s the output from the ganglion cells whatneuroscientists need to do simulation and research in the cortex.

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Figure 1.7: The different stages of the retina model

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1.4 The VirtualRetina software

This software (Figure 1.8 represents its logo) has been developed by Adrien Wohrer, aformer PhD student at INRIA and it exists in two releases:

• February 2009 - May 2010

Figure 1.8: The VirtualRetina software logo

The main aim of this software is to produce spike trains knowing an image sequenceand a retina configuration. An image sequence is a set of successive images appearingwith a fixed speed (24 frame/sec.). The Retina configuration is the set of model pa-rameters. This configuration could be done thanks to an xml file. To summarize, theinput files are:

• An xml file (the retina configuration).

• An image sequence (The Figure 1.9 is one of the images in the input sequence).

Figure 1.9: An example about a gray scale input image

The software shows how the image appears at the different layers and, at the end, itgives 4 output files summarizing the simulation, the spike train, the neuron positions.The spike train file, with a .spk extension, contains two columns. The first columncontains the unit of the ganglion cell and the second contains the corresponding timeof firing.

As the distribution of the ganglion cells is already configured in the input retina.xmlfile, the output retina.xml file contains the data of the input.xml file and the metadata

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Figure 1.10: The structure of the .spk file

Figure 1.11: The log polar scheme distribution of the ganglion cells in VirtualRetina.The distribution is mainly concentrated at the center of the retina (fovea loge). Theconcentration of this distribution follows an exponential decay along the distance fromthe fovea. The background color shows the pixels of the image and the cell(s) for eachpixel. The VirtualRetina allows the simulation through another scheme, the rectangularone, where the distribution of the ganglion cells in uniform overall the retina. Wecan herein simulate a rectangular scheme with 1 cell/pixel density where each pixelcorresponds to one ganglion cell.

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concerning the units configuration; the identifier (unit) of each cell and it’s spatialposition in degree (Figure 1.11). The degree is a space unit measure in VirtualRetinaand it refers to how much the cell is far from the center, in degree.

The input retina.xml file looks like the Figure 1.12

Figure 1.12: The xml file is the ’written virtual retina’. It carries all the properties ofthe different layers in the model.

1.5 Simulation via VirtualRetina

In this section we show some of the results that VirtualRetina produces. The Figure1.13 shows the signal at the different layers of the retina model. This is the first outputof the software, showing in real time the variation of the signal following the variationof the input images overall the sequence.

Mainly, we used retina configurations for X, Y, ON and OFF cells 1 for the ganglioncells layer. The input sequence is the default one (Figure 1.9). To change the configu-ration of the ganglion layer it’s sufficient to change 4 parameters in the retina.xml file.

1The X-cells are A specific retinal ganglion cells (neurons) involved in visual information processing(Troy and Shou, 2002; Hughes, 1979). These cells differ from related Y cells (called also alpha-cells)and W cells (called also gamma-cells) by their morphology, response properties, and their projectioninto cell layers of the lateral geniculate nucleus of the thalamus that transmits information to the visualcortex (Lennie, 1980; Bowling and Michael, 1984; Sur et al, 1987; Tamamaki et al, 1995; Stanford etal, 1983; Boykott and Wassle, 1991; Wassle and Boycott, 1991) and represent a class of horizontal cells.The ON cells have the property to get activated in response to a positive stimilus. In the opposite,the OFF cells get activated in response to and inhibition.

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(a) Receptor Layer (b) Horizontal Layer (c) OPL Current

(d) Bipolar Current (e) Fast adaptation layer (f) Ganglion inputs cur-rent

Figure 1.13: The Signal at each layer of the retina model

We used 24 Frame/sec to simulate natural retina ability in caption images. Figure 1.14shows several spike trains for different cells. The Difference between ON and OFF cellsis that the ON are sensitive to illumination while, on the opposite, the OFF cells aresensitive to obscurity).

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(a) X-ON cells (b) Y-ON cells

(c) X-OFF cells (d) Y-OFF cells

Figure 1.14: Spike train for different cell types. The retina was configured with 16 cellsin the fovea and with a log polar distribution around. The input image is the defaultsequence (Figure 1.9).

The Figure 1.15 shows the spike trains for different colonies of cells but there aremore larger simulation, thousands of cells.

1.6 Conclusion

We saw in this chapter a fast view about the retina from biological and simulation pointof view. As it is supposed to offer, we can use it to produce spike trains for more thanthousands of ganglion cells. It also offers access to other layer of the simulator suchas the input current to the ganglion cells. The time that the simulator take with anordinary personal machine (Ubunto 9.10, 2 GB RAM, 1.66 GHZ Core2Duo Processor)is somehow reasonable: seconds to simulate a retina with hundred of fovea cells and anda 50 images as an input sequences. The simulation time arises to minutes for simulatinglong time sequences,

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(a) XY-cat cells

(b) Magno cells

Figure 1.15: The spike train for a large scale simulation; thousands of cells. The twofigures show the response for ganglion cells of two types (Up: XY cat cells; two layerof cells. Down: Magno ON and OFF cells). We see the time that the cells take at thebeginning to achieve the asymptotic behavior. This response corresponds to a movingbar stimulus.

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Chapter 2

Performing statistics with real andVirtualRetina data

2.1 Introduction

In this chapter we explain the basics and tools with which we are performing statisticsof spike trains. We begin with an explanation about the theoretical basis. We thenintroduce the EnaS library; the main tool we use to perform our statistics. Finally,results from synthetic, real and VirtualRetina data are presented.

2.2 What is a spike train?

As previously presented, the spike train is a structure that contains boolean data (0 and1) about a neuron activity. To create a spike train we need to know at which time theneuron fires. A raster is the spike train of several neurons or a neural network (Figure2.1). We tag the neuron as a ’unit ’.By definition, a spike train can be written as:

S(t) =N∑i=1

δ(t− ti) (2.1)

Where ti is the list of times where the neuron i fires, N is the total number of units orneurons and δ is the Dirac function.

The Figure 2.1 shows a typical spike train for a network of 600 ganglion cells during3 secondes (From VirtualRetina simulation).

The length of the raster is the number of the time sample for this raster, N is thenumber of neurons in the network.

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Figure 2.1: A typical spike train (produced with VirtualRetina)

Figure 2.2: The raster of 5 neurons is presented here with an explanation about itslength, size and a spike block of Range R

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2.3 Performing statistics

The data of a spike train are boolean, so that, if we want to perform statistics we will bedealing with ’ones ’ and ’zeros ’ that represent the activity and non-activity of neurons.Denote that i = 0, 1...N − 1; the neuron index. We consider that the activity of theneuron i is represented by wi(t) where:

ωi(t) =

{1 if the neuron i fires0 elsewhere

(2.2)

We define here also another three terms:

1. A spike pattern: represents the activity of the N neurons at a specific time t.

ω(t) = [ωi(t)]N−1i=0 (2.3)

A spike pattern is -for numerical purpose- encoded as follow:

ω(t) =N−1∑i=0

2iωi(t) (2.4)

Where i is an integer.

2. A spike block: represents the activity of N neurons between the time t1 and t2,in another way, it’s several consecutive spiking patterns.

ωt1t2 = ω(t)t1≤t≤t2 (2.5)

3. A raster plot: the activity of the N neurons overall the time.

ω(t) = [ωi(t)]0i=−∞ (2.6)

We consider that the neuron fires in a time interval with a precision δ, i.e., betweent and t + δ. Hence, we consider that, typically, δ = 1ms, the smallest time unit in theraster time scale. A simple characteristics about a spike train is the firing rate:

ri(t) =ni(t)

tfire/sec. (2.7)

Where n the number of times the neuron i fired.

Observables

We call observable, the function that associate a real number to a raster plot. Forexample, we can say that observing if the neuron i fires at the time t = 0 is an observable.We can also observe if the neurons i and j fired at time t = 0 and we attribute thefollowing function to this event:

φ(ω) = ωi(0)ωj(0) (2.8)

Idem, if the neuron j fired a τ time interval after the neuron i:

φ(ω) = ωi(0)ωj(τ) (2.9)

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2.3.1 The statistical models

We want to characterize spike train statistics. Our main assumptions in this scope arethat the spike train is stationary and non deterministic. We also assume to study theasymptotic behavior of this spike train.The simplest statistical model is the Bernoulli distribution.The probability that a block ωts corresponds to a known word 1 ats is:

µ(ωts = ats) = µ(ωi(k) = ai(k), i = 1...N, k = s...t)

= ΠNi=1Π

tk=sµ(ωi(k) = ai(k))

= ΠNi=1r

ni(t,s)i (1− ri)(t−s−ni(t−s))

(2.10)

µ refers to the probability that the events (between parenthesis) happen, ni and ri arerespectively the number of spike and firing rate of the neuron i in the time interval[s, t] This probability (In the Bernoulli Distribution) corresponds to the firing rate ofthe neuron i. The correlation between two neuron is then:

Ci,j = |µ(ωi(t)ωj(t))− µ(ωi(t))µ(ωj(t))| (2.11)

Thus, Ci,j is equal to how many time the two neurons i and j fired together minus theproduct of their firing rate.More generally, given a spike train, we are looking to find the statistical model of thedata representing this spike train. For example, in a Bernoulli model, we search theprobability µ that corresponds to the firing rate ri (µ(ωi) = ri).The Entropy measures how disorganized the system is. For a spike block w having aprobability µ we define the entropy as:

h(µ) = −∑

µ($)logµ($) (2.12)

Where $(t) =∑N−1

i=0 2iωi(t). $(t) is the sum overall possible spike blocks. We takeinto account some constraint when maximizing the entropy:

1. The first constraint means to approach the experimental and measured probabil-ity: µ(ϕl) = ϕexpl , the experimental average of a funtion ϕl.

2. The second constraint is a classic assumption in probability:∑

w µ(w) = 1.

Gibbs Potential

A Gibbs distribution is a probability that maximizes the statistical entropy under theconstraint that the experimental average ϕl the same prescribed functions ϕl is equalto the average with respect to µ.

1We note by the term word, the structure that represents the neuron activity through a spike block

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This distribution is such that the probability of a block $, µ($), behaves like eψ($).ψ($) is the Gibbs Potential, defined by:

ψ =L∑l=1

λlφl (2.13)

Where λl are the set of Lagrange multipliers. The equation 3.1 represent the statisticalmodel. Additionally, the entropy os such a potential obeys:

P [ψ] = h[µ] + µ[ψ] (2.14)

Where µ is the probability distribution we are locking for (µ(φl) = Cl), and P is thetopological pressure.

The process of Entropy Maximization is equivalent then to find the parameters λl.For example, to see the correlation between two neurons, we suppose that the statisticalmodel is given by the following Gibbs Potential:

ψ(ω) = λ1ω0(0) + λ2ω1(0) + λ3ω0(0)ω1(0) (2.15)

In this equation, we have three monomials, each of then is multiplied by a coefficientλi. The developed EnaS library allow to find these parameters given a spike train andby the mean of the Entropy Maximization. To find out the correlation, we read andinterpret the coefficient λ3 which means that if this coefficient is important in the modelthen, the number of time the neurons 0 and 1 fired together is high, which means thatthey are strongly correlated.

2.4 The EnaS Library

EnaS (Figure 2.3), the Event neural assembly Simulation is a dedicated C++ library.It has been developed in collaboration between the NeuroMathComp and Cortex teamwithin the INRIA Sophia-Antipolis. This library is dedicated for neural simulation anddoing statistics for real or synthetic data.

Figure 2.3: The Enas Logo

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The code source is available on the EnaS website (http://enas.gforge.inria.fr/) with some tutorial documentations and examples about the uses if its classes.The source doesn’t need to be installed or configured. It’s sufficient to put it in thesame work director and include ‘‘EnaS.h’’; and include namespace enas;.The compilation of the program that holds ‘‘EnaS.h’’; is a bit different because thelibrary uses another external libraries such that gsl, gpl and glpk which have to beinstalled and configured before using this library, and hence, the compilation commandhas to be the following:

g++ -Wall -lgsl -lgslcblas -lglpk -lm MyFile.cpp -o MyOutputFile.o

The main classes we used from this library are the following:

• FileTimeSequence: to load a ’time-unit’or a .spk file containing the boolean dataof a spike train.

• PrefixTree: That defines a tree from a Gibbs potential which allows to estimatesome statistical parameters.

• GibbsPotential: the class that allows the modeling of a set of data and the ex-traction of the model parameters such as the λ coefficients.

This library is helpul to our project because it contains classes that can help usto perform some statistics. For example, EnaS can estimate the parameters of thestatistical model of a spike train (The λs in the Eq. 2.15).

2.5 Performing statistics with the EnaS library and

results

The main idea behind using EnaS with spike trains data is to estimate the parametersλl between neurons. As a special case, we are interested in correlation. This correlationdepends on the statistical model that the spike train follows. In an Bernoulli model,the correlation tends to zero along a raster.

Data from Bernoulli distribution could be generated at different probability value.We note that, in Bernoulli distribution, the probability of the two independent possibleevents is complementary, i.e., if p is the probability that the event happens, so, theprobability that the event doesn’t happen is equal to 1− p.The purpose to measure the correlation between two neurons doesn’t mean that weestimate this correlation and we see if its value is very low or not, but, the matter isto see how does this correlation evolve in term of the Raster length. Actually, we canshow theoretically and experimentally that this correlation decreases with the Rasterlength. The evolving function in a log scale is:

Ci,j(t) =k√(T )

(2.16)

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where k is a real number and T is the raster length.

2.5.1 Synthetic data

Figure 2.4 shows the property of the exponential decay of correlation in term of theraster length. The data were generated randomly with a Bernoulli distribution andp = 0.5. We can also show the same idea for different Bernoulli probability distribution(From p = 0.1 to p = 0.9), Figure 2.5.A fitting technique (Lavenberg-Marquadt) allowed us to fit the results with the k

Ta . Ifwe saw the results for the whole time scale, we can observe that they really follow theline of equation 0.6√

T(Where k = 0.6 and a = 0.5).

Figure 2.4: The correlation between two neurons whose raster plots follow a Bernouillidistribution. The data are synthetic with a probability distribution p = 0.4, and theplot is in the logarithmic scale. The evolving of the calculated correlation with empiricalalgorithms decreases in a closed manner to the k√

(T )line.

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Figure 2.5: Evolving of the correlation for different probability distributions (Bernouillidistribution)

Figure 2.6: Evolving of the correlation for different time coincidence. The delay is theterm we use to express that the two neurons fire between a τ time interval. We studiedthe correlation for different daly values, going from 1 to 6.

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2.5.2 Real data

One of our collaborator (Kolo Bodgan, INSERM)supplied us with some real data ac-quisition for ganglion cells activity. The signals were acquired with an “MEA Chip”that contains 58 electrodes (Figure 3.1).

Figure 2.7: The MEA chip for ganglion recordings

Data characteristics:

• The total duration of the acquisition is 93 s.

• 30 sec. of spontaneous activity followed by 63 sec. of evoked potential activity(Figure 2.8). The 63 sec. of evoked potential were done as 1 sec. of light stimulusfollowed by 5 sec. inter-interval whit non stimulus.

Figure 2.8: The evoked potential recording. A light stimulus is given in face to theeye and, back, at the ganglion cells layer, the recording of electric potential takes placewith the MEA acquisition grid.

Statistics about the correlation between two random neuron within the real acquireddata have shown that the correlation doesn’t tend to 0 when the raster length tend to

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∞. This fact emphasizes the hypothesis that the neurons in the vertebrate retina arecorrelated, from where our motivation to add the correlation property to the ganglioncells in VirtualRetina.

We applied the Gibbs potential model on this data (Equation 2.15) in order toestimate the correlation between several couples of neurons, in term of the raster length.

Figure 2.9: Correlation between ganglion cells in real acquisition. The figure showsthe correlation for three couples of neurons, at severals distances. The time scalecorresponds to the first 30 sec. of the acquisition 2.5.2; neurons at rest.

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Figure 2.10: Correlation between ganglion cells in real acquisition. The figure showsthe correlation for three couples of neurons, at severals distances. The time scalecorresponds to the last 60 sec. of the acquisition 2.5.2; in evoked potential

2.5.3 VirtualRetina data

Data with virtual retina could be generated through the installed software or the web-service implementation. It’s hence prefered to use the software after installation becauseit’s more controllable and all the parameter you put in the model are also accessible.The below figures (2.11, 2.12, 2.13, 2.14) show that the correlation between differentcouple of neurons at at several distances. For the 4 cell types, the correlation tends tozero for the highe raster length.These figures show the increasing of the correlation value in term of the raster length.There are some perturbation at the extermities of the curves. The perturbation at thebeginning comes from the non asymptotic behavior of the spike train.

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Figure 2.11: Correlation between X-ON ganglion cells in VirtualRetina

Figure 2.12: Correlation between Y-ON ganglion cells in VirtualRetina

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Figure 2.13: Correlation between X-OFF ganglion cells in VirtualRetina

Figure 2.14: Correlation between Y-OFF ganglion cells in VirtualRetina

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2.6 Conclusion

We have shown in this chapter the statistical tools we are based on to perform statisticswith spike trains. We also explained the mathematical, numerical and algorithmicframeworks that are behind. With these tools we have shown the statistics for realacquisitions, simulated and synthetic data. We have shown the difference between thebetween the Bernoulli model data, the VirtualRetina data and real acquisitions bystudying the correlation in term of the raster length in each of these cases.

In synthetic data, the correlation increases as the line a√T

increases. In VirtualRetinasimulations, the correlation follows also this line. On the opposite, the correlationbetween different couples of neurons in real acquisitions doesn’t verify this property,which means that the ganglions cells in VirtualRetina are not correlated.

However, there exist connections between cells in the VirtualRetina model but atthe lower layers, not at the ganglion cells layer. The results have shown also that theconnections between the retina cells at the lower layer doesn’t imply automatically thatthe ganglion cells are correlated. We need also that the ganglion cells have connectionsbetween themselves.

In the scope of the second part of the project, we would like to add connections be-tween the ganglion cells in VirtualRetina in order to enhance the biological plausibility,i.e., we will translate the values of the statistical model parameters (for real ganglioncells) in connections and add them to the ganglion cells in the VirtualRetina.

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Chapter 3

Infering connectivity betweenRetinal Ganglion Cells

3.1 introduction

In the first section of this chapter we will show the results that give an idea about theconnectivity between retinal ganglion cells from real data acquisition. The second sec-tion will be a bibliographical study. This chapter is in fact complementary to the secondone; both explain about how to know about connectivity between retinal ganglion cellsbut in two different ways: previousely we measured the evolving of the correlation interm of raster length, and now, we measure quatitatively the connectivity using theparameters of Gibbs Models.

3.2 The connectivity between ganglion cells from

real data acquisition

Thanks to some acquired data by one of our collaborator (Kolomiets Bodgan, Instituede vision de Paris), we could use the power of EnaS library to make some statisticsabout connectivity. For technical details, we can take an idea about the connectivitybetween two neurons from the λl in the Gibbs Potential equation:

ψ =L∑l=1

λlφl (3.1)

This equation is described previousely in the second chapter (Section 2.3.1).In fact, the bigger the λl, the more the occurence of the observable in the train spike.By consequence, if we measure different λl for several couples of neuron, we will havean idea about their connectivity.

Recalling the acquisition electrode MEA (fig. 3.1):In the following we will:

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Figure 3.1: The MEA chip for ganglion recordings

1. Take a collection of six neurons.

2. Apply the Gibbs model.

3. Use Enas to estimate the parameters λl.

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Figure 3.2: The 6 graphes show the value of λ0j in term of distance for different net-work arrangement ( Vertical (Ex: The cells 12,13,14,15,16,17) ,Horizontal , Random).The idea here is to see how does the distance affect the connectivity factor.The x-axisrepresents the distance between the first cell (called Cell 0) and the j-th cell (CalledCell j). The y-axis represents the connectivity factor λij between the cell 0 and the cellj (j=1,2,...5). In all the graphs we can observe that: A part or the whole graph behaveslike a increasing at the beginning then decreasing after some distance (commonly herebetween 400 and 1000 µm).

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Bibliography

[1] Adrien Wohrer, Model and large-scale simulator of a biological retina, with contrastgain control. University of Nice Sophia-Antipolis, INRIA, 2009.

[2] J.C. Vasquez1, T. Viville, B. Cessac, Entropy-based parametric estimation of spiketrain statistics. INRIA, 2009

[3] S. Coccoa, S. Leiblerb and R. Monassond Neuronal couplings between retinal gan-glion cells inferred by efficient inverse statistical physics methods PNAS, 2009

[4] C. Shalizi, K. Shalizi Blind Construction of Optimal Nonlinear Recursive Predictorsfor Discrete Sequences CoRR, 2004

[5] Authors: R. Haslinger, K. Klinkner, C. Shalizi The Computational Structure ofSpike Trains Neural Computation, vol. 22 (2010), pp. 121–157

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