Large deviations for randomly connected neural networks: II. State-dependent interactions

TL;DR

This paper extends large deviation analysis to state-dependent neural network models with Gaussian interactions, demonstrating convergence, uniqueness, and propagation of chaos, including a stochastic Kuramoto model variant.

Contribution

It introduces a novel large deviations framework for neural networks with state-dependent interactions and random Gaussian weights, establishing convergence and chaos propagation results.

Findings

Empirical measures satisfy a large deviation principle.

Unique probability measure minimizes the rate function.

Convergence in both averaged and quenched cases, with propagation of chaos in the averaged case.

Abstract

This work continues the analysis of large deviations for randomly connected neural networks models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and (ii) have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case, we show that under sufficient regularity assumptions, the empirical measure satisfies a large-deviation principle with good rate function achieving its minimum at a unique probability measure, implying in particular its convergence in both averaged and quenched cases, as well as a propagation of chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of Kuramoto model with random connections.