A large deviation principle for networks of rate neurons with correlated synaptic weights

TL;DR

This paper establishes a large deviation principle for the asymptotic behavior of large networks of correlated rate neurons, characterizing the limiting distribution and its properties, with implications for neuroscience modeling.

Contribution

It introduces a novel large deviation framework for networks with correlated Gaussian synaptic weights, providing a detailed characterization of the limit law.

Findings

The empirical measure satisfies a large deviation principle with a unique global minimum.

The limit measure is characterized as the image of a stationary Gaussian measure.

The approach facilitates analysis of finite-size effects in neural networks.

Abstract

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of firing rate neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. We introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of neurons. The main result of this article is that the image law through the empirical measure satisfies a large deviation principle with a good rate function which is shown to have a unique global minimum. Our analysis of the rate function allows us also to characterize the limit measure as the image of a stationary Gaussian measure defined on a transformed set of trajectories. This is potentially very useful for applications in neuroscience since the Gaussian measure can be completely characterized by its mean and spectral density. It also facilitates the assessment of the probability of finite-size effects.