Systematic fluctuation expansion for neural network activity equations

TL;DR

This paper develops a systematic extension of neural population activity equations by incorporating correlations, providing a more comprehensive model that captures phenomena missed by traditional mean field approaches.

Contribution

It introduces a hierarchy of generalized activity equations derived from a stochastic theory, including correlations and coupling terms, improving neural network modeling.

Findings

Captures phenomena missed by mean field equations

Provides closed-form equations at each approximation level

Demonstrates effectiveness in an all-to-all connected network

Abstract

Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate while leaving out higher order statistics like correlations between firing. A stochastic theory of neural networks which includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations, i.e. they depend only upon the mean and specific correlations of interest, without an {\it ad hoc} criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean fieldrate equations alone.