Statistical physics of pairwise probability models.

Yasser Roudi

doi:10.3389/neuro.10.022.2009

Abstract

Statistical models for describing the probability distribution over the states of biological systems are commonly used for dimensional reduction. Among these models, pairwise models are very attractive in part because they can be fit using a reasonable amount of data: knowledge of the mean values and correlations between pairs of elements in the system is sufficient. Not surprisingly, then, using pairwise models for studying neural data has been the focus of many studies in recent years. In this paper, we describe how tools from statistical physics can be employed for studying and using pairwise models. We build on our previous work on the subject and study the relation between different methods for fitting these models and evaluating their quality. In particular, using data from simulated cortical networks we study how the quality of various approximate methods for inferring the parameters in a pairwise model depends on the time bin chosen for binning the data. We also study the effect of the size of the time bin on the model quality itself, again using simulated data. We show that using finer time bins increases the quality of the pairwise model. We offer new ways of deriving the expressions reported in our previous work for assessing the quality of pairwise models.

Highlights

In biological networks the collective dynamics of thousands to millions of interacting elements, generating complicated spatiotemporal structures, is fundamental for the function
In this paper we have revisited some classical and some more recent approximations of this kind that take their inspiration from statistical physics
In the case of neural data analysed here, the parameter δ depends on the size of the time bin, δt

Summary

INTRODUCTION

In biological networks the collective dynamics of thousands to millions of interacting elements, generating complicated spatiotemporal structures, is fundamental for the function. Given the strong dependence of the quality of these simple approximations on population size and the size of the time bin, we describe below how one can extend these approximations to obtain more accurate expressions for finding the external fields and couplings of the pairwise model for large N and δt. The IT approximation yields a relation between the couplings and the mean values and pairwise correlations that coincides with leading term and the corrections found by low-rate expansion, as shown in Section “Independent-triplet Approximation” in Appendix. EXTENDING THE NAIVE MEAN-FIELD: TAP EQUATIONS The nMF, IP and IT approximations are good for fitting the model parameters when the typical number of spikes generated by the whole population in a time bin is small compared to 1, i.e. when Nδ 1 (Roudi et al, 2009a,b), in which. The TAP approximations does well in all cases, and this is quantified in

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DISCUSSION

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