Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains

Abstract

Abstract:We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. To find the maximum entropy Markov chain, we use the thermodynamic formalism, which provides insightful connections with statistical physics and thermodynamics from which large deviations properties arise naturally. We provide an accessible introduction to the maximum entropy Markov chain inference problem and large deviations theory to the community of computational neuroscience, avoiding some technicalities while preserving the core ideas and intuitions. We review large deviations techniques useful in spike train statistics to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability, and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.

Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics

“…To obtain the unique Markov transition matrix of maximum entropy we follow the procedure explained in section 2.3.2. Thus, for a given choice of monomials, we associate a potential of the form (21) where the λ l s that do not correspond to a chosen monomial is set to 0. Then, one computes the empirical average of the chosen monomials from data.…”

“…This potential can also be written in terms of monomials using the Hammersley-Clifford decomposition (3.3), through a series expansion of the function f . This procedure generate a series of monomials with coefficients that can be explicitly computed (using the fact that, from the monomials definition (21) x…”

“…There have been a few attempts to use the Thermodynamic Formalism in a different way other than as a natural mathematical foundation of statistical mechanics, for example, studying population dynamics [13,14], self-organised criticality [15], and statistics of spike trains in neural systems [16][17][18][19][20][21][22]. Neuronal networks are biological systems, whose components such as neurons, synapses, ionic channels, ..., are ruled by the laws of physics and written in terms of differential equations, hence, are dynamical systems.…”

An Introduction to the Non-Equilibrium Steady States of Maximum Entropy Spike Trains

“…Here, we build from a previous article [52] where it is shown that the SCGF (7) can be obtained directly from the inferred Markov transition matrix P through the Gärtner-Ellis theorem (8). Consider a MEMC with transition matrix P. Let f be an observable of finite range and k ∈ R. We introduce the tilted transition matrix by f of P, parametrized by k and denoted by P ( f ) (k) [53] as follows:…”

An Introduction to the Non-Equilibrium Steady States of Maximum Entropy Spike Trains

“…Here, we build from a previous article [42] where it is shown that the SCGF (7) can be obtained directly from the inferred Markov transition matrix P through the Gärtner-Ellis theorem (8). Consider a MEMC with transition matrix P. Let f be an observable of finite range and k ∈ R. We introduce the tilted transition matrix by f of P, parametrized by k and denoted by P ( f ) (k) [29] as follows:…”