Introduction to dynamical mean-field theory of randomly connected neural networks with bidirectionally correlated couplings

TL;DR

This paper provides a pedagogical introduction to dynamical mean-field theory for randomly connected neural networks with correlated couplings, explaining the method, its physical insights, and numerical implementation, including an alternative cavity approach.

Contribution

It offers a clear, accessible explanation of dynamical mean-field theory and the dynamical cavity method applied to complex neural networks with correlated synapses, including detailed numerical procedures.

Findings

Derivation of mean-field equations for correlated neural networks

Introduction of the dynamical cavity method as an alternative derivation

Numerical exploration of fluctuation dissipation theorem in these systems

Abstract

Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.