Filter Based Encoding Model of Mammalian Retina Ganglion Cells

Using uniform field input, we model mouse retina ganglion cells as linear filters with a stochastic rate code to encode the filtered value. For each cell, the genetic algorithm is used to find the linear filter of the input, which has the optimal mutual information with the output of the cell. Based on the shape of the filters (the encoding time course of the neurons), we cluster the cells into 3 broad classes: FAST-ON, SLOW-ON, and OFF (as well as smaller sub-clusters). Finally, we show that the optimal filters and the clustering structure can be used to improve the performance of predicting the neurons’ firing rate. I. I NTRODUCTION The quality and quantity of experimental data from biological neuronal systems is dramatically rising [1]. With more data, there is a need for algorithms which are guided by the data, rather than guided by human analysis of that data. Information theory [2], [3] then becomes a natural framework to study how neurons communicate information in the nervous system. It introduces themutual informationbetween two random variablesS andR, denoted asI(S;R), which is a measure of the statistical dependency. It is also the upper bound on how many bits aboutS (or R) we know on average. In spite of many advances, it is still not known precisely how information is represented by neural code [4], [5]. In this work, we model the retina ganglion cells of a mouse as linear filters using a stochastic rate code to encode the filter values. With the genetic algorithm, we find the filter which maximizes mutual information between the filtered stimulus and the number of spikes in a time frame. In section II we describe our data and analysis techniques in detail. In section III, we use computational optimization techniques to find optimal filters for each cell. With those filters in hand, in section IV we cluster the cells into several classes. Finally in section V we use the optimal filters as well as the clustering structure to improve the performance of predicting neurons’ firing rate. II. M ETHODOLOGY AND TECHNIQUES The data set we analyze consists of 17 retina ganglion cells from a mouse, recorded from a multi-electrode array. The firing rate ranges from4.2 to 77.4 Hz. The visual stimulus is a 1.5 hour long spatially uniform flicker, in which intensity values (ranging from0 to 255) are drawn randomly from a Gaussian distribution with mean μ ≈ 127 and standard deviation σ ≈ 54 every 100 ms. There areN ≈ 54, 000 frames. [More about the stimulus. ] The stimulus ranges over 256 possible values (8 bits). We bin the stimulus into 16 approximately equal sized bins to reduce the number of bits to describe the stimulus to at most 4. The stimulus is represented as a vector S = (s1, s2, . . . , sN ) of intensity values. We say that si is the intensity of framei. For each cell, we bin the spike train in 100 ms bins and count the number of spikes in every bin. We call the response of the j cell the sequence of counts for each bin: R = (r 1, r j 2, . . . , r j N ) wherer k is the number of spikes in the k th bin for the j cell. Ignoring any temporal dependencies, we may consider the random variable(S,R) created by pairing each (si, r j i ) for all i. We can calculate the mutual information: I(S;R) = H(S) + H(Rj)−H(S,Rj). (1) It may be that the response at frame i is not most informative about the stimulus at frame i, but instead about an earlier stimulus. Thus we introduce the R l = (r j 1+l, r j 2+l, . . . , r j N ), which has the lengthN − l. Of course,R 0 = R . If l ¿ N , thenH(R) ≈ H(R l ), since both are empirical estimations of the entropy with onlyl fewer samples in the latter. For each delayl, we can compute the mutual information I(S;R l ). The encoding efficiency [6] with delayl is defined as ηl = I(S;R j l )/H(R j l ). (2) Fig. 1 shows the encoding efficiency with different delays for every neuron. Note that no cell has a maximum of ηl at l = 0. Instead, all of them reach a maximum at l = 1 or l = 2. Thus all the cells in this study are most informative about the stimulus100 − 300 ms in the past. Remember that our stimulus is drawn independently, thus the mutual information being spread over a wide time window is due only to neural processing and not due to the statistics of the stimulus. In our setting, a general model for the cell as a rate-encoder is to suppose there is a probability pj(ri|si, si−1, . . . si−k) for each cellj for some value ofk that captures how much past dependency there is in each cell. In Fig. 1 we see that many c lls have non-zero information about stimuli 500 ms or more into the past. This indicates that several frames of stimulus are processed jointly to generate its subsequent response. Thus we would need to use at least 5 − 10 frames of past data to predict the response at a given time. If we suppose that each stimulus and response can be represented by only 4 bits, then we need a table with ( 2 )1+k elements, fork = 7, that comes 0 5 10 15 0 0.1 0.2 1 0 5 10 15 0 0.1 0.2 2 0 5 10 15 0 0.1 0.2 3 0 5 10 15 0 0.1 0.2 4 0 5 10 15 0 0.1 0.2 5 0 5 10 15 0 0.1 0.2 6 0 5 10 15 0 0.1 0.2 0.3