Solution of a large nonlinear recurrent neural network at fixed connectivity

TL;DR

This paper analytically characterizes the behavior of large nonlinear recurrent neural networks, linking connectivity, spontaneous activity correlations, and responses to perturbations without weight averaging.

Contribution

It introduces a novel analytical method to compute moments and response functions in large nonlinear networks, confirming a recent conjecture.

Findings

Derived the first nontrivial $1/ ext{sqrt}(N)$ term for correlation functions.

Established an analytical connection between connectivity and activity correlations.

Proved a conjecture by Shen and Hu in the context of nonlinear neural networks.

Abstract

We calculate the moments and response functions of a nonlinear random recurrent neural network in the large $N$ limit. Our approach does not require averaging over synaptic weights and gives the first nontrivial term in a $1/\sqrt{N}$ expansion of general intensive-order correlation functions, proving a recent conjecture by Shen and Hu as a special case. Our results provide an analytical link between synaptic connectivity, correlations in spontaneous activity, and the response of a network to small perturbations.