Communications in Mathematical Sciences

Volume 15 (2017)

Number 3

A dynamical state underlying the second order maximum entropy principle in neuronal networks

Pages: 665 – 692

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n3.a5

Authors

Zhi-Qin John Xu (School of Mathematical Sciences, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China; and NYUAD Institute, New York University Abu Dhabi, United Arab Emirates)

Guoqiang Bi (CAS Key Laboratory of Brain Function and Disease, and School of Life Sciences, University of Science and Technology of China, Anhui, China)

Douglas Zhou (School of Mathematical Sciences, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)

David Cai (School of Math. Sciences, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong Univ.; Courant Institute and Center for Neural Sciences, New York Univ., New York, U.S.A.; and NYUAD Institute, New York Univ. Abu Dhabi, United Arab Emirates)

Abstract

The maximum entropy principle is widely used in diverse fields. We address the issue of why the second order maximum entropy model, by using only firing rates and second order correlations of neurons as constraints, can well capture the observed distribution of neuronal firing patterns in many neuronal networks, thus, conferring its great advantage in that the degree of complexity in the analysis of neuronal activity data reduces drastically from $\mathcal{O}(2^n)$ to $\mathcal{O}(n^2)$, where $n$ is the number of neurons under consideration. We first derive an expression for the effective interactions of the $n$th order maximum entropy model using all orders of correlations of neurons as constraints and show that there exists a recursive relation among the effective interactions in the model. Then, via a perturbative analysis, we explore a possible dynamical state in which this recursive relation gives rise to the strengths of higher order interactions always smaller than the lower orders. Finally, we invoke this hierarchy of effective interactions to provide a possible mechanism underlying the success of the second order maximum entropy model and to predict whether such a model can successfully capture the observed distribution of neuronal firing patterns.

Keywords

maximum entropy model, high dimensional data, neuronal networks, mechanism

2010 Mathematics Subject Classification

62B10, 82B05, 92C20

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Published 24 February 2017