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Large deviations for randomly connected neural networks: I. Spatially extended systems

Part of: Probability theory on algebraic and topological structures Mathematical biology in general Limit theorems Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  16 November 2018

Tanguy Cabana  and
Jonathan D. Touboul
Tanguy Cabana
Affiliation:
Collège de France
Jonathan D. Touboul*
Affiliation:
Collège de France and Brandeis University
*
* Postal address: Department of Mathematics and Volen National Center for Complex Systems, Brandeis University, 415 South Street, Waltham, MA 02453, USA. Email address: [email protected]
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Abstract

In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider first the presence of interaction delays that depend on the distance between cells and then the Gaussian random interaction amplitude with a mean and variance that depend on the position of the neurons and scale as the inverse of the network size. We show that the empirical measure satisfies a large deviations principle with a good rate function reaching its minimum at a unique spatially extended probability measure. This result implies an averaged convergence of the empirical measure and a propagation of chaos. The limit is characterized through a complex non-Markovian implicit equation in which the network interaction term is replaced by a nonlocal Gaussian process with a mean and covariance that depend on the statistics of the solution over the whole neural field.

Keywords

Large deviationconvergence of measureneural networkdisordered system

MSC classification

Primary: 60F10: Large deviations
Secondary: 60B10: Convergence of probability measures 92B20: Neural networks, artificial life and related topics 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 944 - 982
Copyright
Copyright © Applied Probability Trust 2018 

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