Gibbs distribution analysis of temporal correlations structure in retina ganglion cells
Highlights
► A method to estimate general maximum-entropy models (beyond Ising). ► A method to compare statistical models by minimizing Kullback–Leibler divergence. ► Application to analyze multi-electrode arrays spike-trains for small groups of neurons. ► For spatio-temporal patterns of two/three neurons, higher orders terms, and Markovian interactions of finite memory improve the description of the statistics.
Introduction
Modern advances in neurophysiology techniques, such as two-photons imaging of calcium signals or micro-electrode arrays electro-physiology, have made it possible to observe simultaneously the activity of assemblies of neurons (Stevenson and Kording, 2011). Such experimental recordings provide a great opportunity to unravel the underlying interactions of neural assemblies. The analysis of multi-cells spike-patterns constitutes an alternative to descriptive statistics (e.g. cross-correlograms or joint peri-stimulus time histograms) which become hard to interpret for large groups of cells (Brown et al., 2004, Kass et al., 2005). Earlier multi-cells approaches (e.g. Abeles and Gerstein, 1988), focus on synchronization patterns. Using algorithms detecting the most frequent instantaneous patterns in a data set, and calculating their expected probability, these approaches aim at testing whether those patterns were produced by chance (Grün et al., 2002). This methodology relies however on a largely controversial assumption, namely Poisson-statistics (Pouzat and Chaffiol, 2009, Schneidman et al., 2006).
A second type of approach has become popular in neuroscience after works of Schneidman et al., 2006, Shlens et al., 2006. They used a maximum entropy approach model spike trains statistics as the Gibbs distribution of the Ising model. The parameters of this distribution are determined from the mean firing rate of each neuron and their pairwise synchronizations. These works have shown that for a small group of cells (10–40 retinal ganglion cells) the Ising model describes most (∼80–90%) of the statistics of the instantaneous patterns, and performs much better than a non-homogeneous Poisson model.
However, several papers have pointed out the importance of temporal patterns of activity at the network level (Abeles et al., 1993, Lindsey et al., 1997, Villa et al., 1999, Segev et al., 2004a). Recently, Tang et al., 2008, Ohiorhenuan et al., 2010, have shown the insufficiency of the Ising model to predict the temporal statistics of the neural multi-cells activity. Therefore, some authors, Marre et al., 2009, Amari, 2010, Roudi and Hertz, 2010, have attempted to define time-dependent Gibbs distributions on the basis of a Markovian approach (1-step time pairwise correlations). The application of such extended model in Marre et al. (2009) increased the accuracy of the statistical characterization of data with the estimated distributions.
In this paper we propose an extension of the maximal entropy approach to general spatio-temporal correlations, based on the transfer-matrix method in statistical physics (Georgii, 1988; Section 2). We describe a numerical method to perform the estimation of the Gibbs distribution parameters from empirical data (Section 3). We apply this method to the analysis of spike trains recorded from ganglion cells using multi-electrodes devices in the salamander retina (Section 4). We analyze retinal spike trains taking into account spatial patterns of two and three neurons with triplets and quadruplets terms, and temporal terms up to four time steps. Our analysis emphasizes the role of higher order spatio-temporal interactions. Section 5 contains the discussion and conclusions.
Section snippets
Spike trains and raster plots
Let N be the number of neurons and denote i = 1, … , N the neuron index. Assume that we have discretised time in steps of size Δ. Without loss of generality (change of time units) we may set Δ = 1. This provides a time discretization labelled with an integer index n. We define a binary variable ωi(n) ∈ {0, 1}, which is ‘1’ if neuron i has emitted a spike in the nth time interval and is zero otherwise. We use the notation ω to differentiate our binary variables ∈ {0, 1} to the notation σ or S traditionally
Estimation of Gibbs distributions
Let us now show how P(ψ) and μ, the main objects of our approach, can be computed.
Methods
Retinae from the larval tiger salamander (Ambystoma tigrinum) were isolated from the eye, placed over a multi-electrode array and perfused with oxygenated Ringer’s medium at room temperature (22 °C). Extracellular voltages were recorded by a microelectrode array and streamed to disk for offline analysis. Spike sorting was performed as described earlier in Segev et al. (2004b) to extract 40 cells. The stimulus was a natural movie clip showing a woodland scene. The 20–30 s movie segment was
Discussion and conclusion
In this paper, we have developed a Gibbs distribution analysis for general spatio-temporal spike patterns. Our method allows one to handle Markovian models with memory up to the limits imposed by the finite size of the data (see also supplementary material). Our analysis on retina data suggests that higher order interaction terms, as well as interaction between non-consecutive time bins, are necessary to model the statistics of the spatio-temporal spiking patterns, at least for small
Acknowledgments
This work has highly benefited from the collaboration with the INRIA team Cortex, and we warmly acknowledge T. Viéville. It was supported by the INRIA, ERC-NERVI number 227747, KEOPS ANR-CONICYT and European Union Project #FP7-269921 (BrainScales) to B.C and J.C.V; grants EY 014196 and EY 017934 to M.J.B; and FONDECYT 1110292, ICM-IC09-022-P to A.P. J.C Vasquez has been funded by French ministry of Research and University of Nice (ED-STIC).
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