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01-08-2012

Modeling the impact of common noise inputs on the network activity of retinal ganglion cells

Authors: Michael Vidne, Yashar Ahmadian, Jonathon Shlens, Jonathan W. Pillow, Jayant Kulkarni, Alan M. Litke, E. J. Chichilnisky, Eero Simoncelli, Liam Paninski

Published in: Journal of Computational Neuroscience | Issue 1/2012

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Abstract

Synchronized spontaneous firing among retinal ganglion cells (RGCs), on timescales faster than visual responses, has been reported in many studies. Two candidate mechanisms of synchronized firing include direct coupling and shared noisy inputs. In neighboring parasol cells of primate retina, which exhibit rapid synchronized firing that has been studied extensively, recent experimental work indicates that direct electrical or synaptic coupling is weak, but shared synaptic input in the absence of modulated stimuli is strong. However, previous modeling efforts have not accounted for this aspect of firing in the parasol cell population. Here we develop a new model that incorporates the effects of common noise, and apply it to analyze the light responses and synchronized firing of a large, densely-sampled network of over 250 simultaneously recorded parasol cells. We use a generalized linear model in which the spike rate in each cell is determined by the linear combination of the spatio-temporally filtered visual input, the temporally filtered prior spikes of that cell, and unobserved sources representing common noise. The model accurately captures the statistical structure of the spike trains and the encoding of the visual stimulus, without the direct coupling assumption present in previous modeling work. Finally, we examined the problem of decoding the visual stimulus from the spike train given the estimated parameters. The common-noise model produces Bayesian decoding performance as accurate as that of a model with direct coupling, but with significantly more robustness to spike timing perturbations.

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It is also useful to recall the relationship between the spatial covariance C _s and the mixing matrix M here: as noted above, since the common noise terms q _t have unit variance and are independent from the stimulus and from one another, the spatial covariance C _s is given by M ^T M. However, since \(\mathbf{C}_s = (\mathbf{UM})^T(\mathbf{UM})\) for any unitary matrix U, we can not estimate M directly; we can only obtain an estimate of C _s. Therefore, we can proceed with the estimation of the spatial covariance C _s and use any convenient decomposition of this matrix for our calculations below. (We emphasize the non-uniqueness of M because it is tempting to interpret M as a kind of effective connectivity matrix, and this over-interpretation should be avoided.)

Estimating the spatio-temporal filters k _i simultaneously while marginalizing over the common noise q is possible but computationally challenging. Therefore we held the shape of k _i fixed in this second stage but optimized its Euclidean length ||k _i||₂. The resulting model explained the data well, as discussed in Section 3 below.

We make one further simplifying approximation in the case of the population model with no direct coupling terms. Here the dimensionality of the common noise terms is extremely large: dim(q) = n _cells ×T, where T is the number of timebins in the experimental data. As discussed in Appendix A, direct optimization over q can be performed in \(O(n_{\rm cells}^3 T)\) time, i.e., computational time scaling linearly with T but unfortunately cubically in n _cells (see Paninski et al. 2010 for further discussion). This cubic scaling becomes prohibitive for large populations. However, if we make the simplifying approximation that the common noise injected into each RGC is nearly conditionally independent given the observed spikes y when computing the marginal likelihood p(y | Θ, C _s), then the optimizations over the n _cells independent noise terms q _i can be performed independently, with the total computation time scaling linearly in both n _cells and T. We found this approximation to perform reasonably in practice (see Section 3 below), largely because the pairwise correlation in the common-noise terms was always significantly less than one (see Fig. 4 below). No such approximation was necessary in the pairwise case, where the computation always scales as O(T).

In many of the three point correlation functions one can notice persistent diagonal structures. If we consider the three point correlation of neuron 1 with the time shifted firing of neurons 2 and 3, the diagonal structure is the sign of a correlation between the time shifted neurons (2 and 3).

Since x is binary, strictly speaking, u ⁱ is not a Gaussian vector solely described by its covariance. However, because the filters k ⁱ have a relatively large spatiotemporal dimension, the components of u ⁱ are weighted sums of many independent identically distributed binary random variables, and their prior marginal distributions can be well approximated by Gaussian distributions (see Pillow et al. 2011 and Ahmadian et al. 2011 for further discussion of this point). For this reason, we replaced the true (non-Gaussian) joint prior distribution of y _i with a Gaussian distribution with zero mean and covariance Eq. (24).

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Title: Modeling the impact of common noise inputs on the network activity of retinal ganglion cells
Authors: Michael Vidne
Yashar Ahmadian
Jonathon Shlens
Jonathan W. Pillow
Jayant Kulkarni
Alan M. Litke
E. J. Chichilnisky
Eero Simoncelli
Liam Paninski
Publication date: 01-08-2012
Publisher: Springer US
Published in: Journal of Computational Neuroscience / Issue 1/2012
Print ISSN: 0929-5313
Electronic ISSN: 1573-6873
DOI: https://doi.org/10.1007/s10827-011-0376-2

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