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

TL;DR

This paper studies the large deviations and asymptotic behavior of spatially extended stochastic neural networks with random interactions, revealing convergence properties and complex non-Markovian dynamics.

Contribution

It introduces a large-deviation principle for neural networks with spatial structure and delays, characterizing the limit through a non-local Gaussian process.

Findings

Empirical measure satisfies a large-deviation principle.

Convergence of empirical measure and propagation of chaos.

Limit characterized by a non-Markovian implicit equation.

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 (i) the presence of interaction delays that depend on the distance between cells and (ii) Gaussian random interaction amplitude whose mean and variance depend on the neurons positions and scale as the inverse of the network size. We show that the empirical measure satisfies a large-deviation principle with good rate function reaching its minimum at a unique spatially extended probability measure. This result implies averaged convergence of the empirical measure and propagation of chaos. The limit is characterized through complex non-Markovian implicit equation in which the network interaction term is replaced by a non-local Gaussian process whose statistics depend on the solution over the whole neural field.