Large Deviations of a Spatially-Stationary Network of Interacting Neurons

TL;DR

This paper establishes a large deviation principle for a spatially-stationary neural network model with correlated noise, enabling quantification of rare deviations and extending traditional mean-field approaches.

Contribution

It introduces a process-level LDP for a lattice-based neural network with correlated noise, generalizing mean-field models and applicable to various learning rules.

Findings

LDP for stationary empirical measures of neural networks.

Quantification of the likelihood of deviations from the system's limit.

Extension of mean-field models to correlated, spatially-structured neurons.

Abstract

In this work we determine a process-level Large Deviation Principle (LDP) for a model of interacting neurons indexed by a lattice $\mathbb{Z}^d$. The neurons are subject to noise, which is modelled as a correlated martingale. The probability law governing the noise is strictly stationary, and we are therefore able to find a LDP for the probability laws $\Pi^n$ governing the stationary empirical measure $\hat{\mu}^n$ generated by the neurons in a cube of length $(2n+1)$. We use this LDP to determine an LDP for the neural network model. The connection weights between the neurons evolve according to a learning rule / neuronal plasticity, and these results are adaptable to a large variety of neural network models. This LDP is of great use in the mathematical modelling of neural networks, because it allows a quantification of the likelihood of the system deviating from its limit, and also a determination of which direction the system is likely to deviate. The work is also of interest because there are nontrivial correlations between the neurons even in the asymptotic limit, thereby presenting itself as a generalisation of traditional mean-field models.