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Entropy 2018, 20(8), 573; https://doi.org/10.3390/e20080573

Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains

1
Centro de Investigación y Modelamiento de Fenómenos Aleatorios, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso 2340000, Chile
2
IPICYT/División de Matemáticas Aplicadas, Instituto Potosino de Investigación Científica y Tecnológica, San Luis Potosí 78216, Mexico
3
Centre of Complexity Science and Department of Mathematics, Imperial College London, London SW7 2AZ, UK
4
Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK
*
Author to whom correspondence should be addressed.
Received: 6 June 2018 / Revised: 4 July 2018 / Accepted: 11 July 2018 / Published: 3 August 2018
(This article belongs to the Special Issue Maximum Entropy and Bayesian Methods)
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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. View Full-Text
Keywords: computational neuroscience; spike train statistics; maximum entropy principle; large deviation theory; out-of-equilibrium statistical mechanics; thermodynamic formalism; entropy production computational neuroscience; spike train statistics; maximum entropy principle; large deviation theory; out-of-equilibrium statistical mechanics; thermodynamic formalism; entropy production
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Cofré, R.; Maldonado, C.; Rosas, F. Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains. Entropy 2018, 20, 573.

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