The determining role of covariances in large networks of stochastic neurons

TL;DR

This paper develops a low-dimensional dynamical model incorporating covariances to better understand stochastic neural networks, revealing that second-order moments significantly influence network behavior beyond mean-field predictions.

Contribution

The authors derive a novel dynamical system that includes covariances, improving the modeling of stochastic neural networks beyond traditional mean-field approaches.

Findings

Second-order moments alter fixed points and limit cycles.

Including covariances captures stochastic network behavior more accurately.

Mean-field approximation may miss oscillatory dynamics.

Abstract

Biological neural networks are notoriously hard to model due to their stochastic behavior and high dimensionality. We tackle this problem by constructing a dynamical model of both the expectations and covariances of the fractions of active and refractory neurons in the network's populations. We do so by describing the evolution of the states of individual neurons with a continuous-time Markov chain, from which we formally derive a low-dimensional dynamical system. This is done by solving a moment closure problem in a way that is compatible with the nonlinearity and boundedness of the activation function. Our dynamical system captures the behavior of the high-dimensional stochastic model even in cases where the mean-field approximation fails to do so. Taking into account the second-order moments modifies the solutions that would be obtained with the mean-field approximation, and can lead to the appearance or disappearance of fixed points and limit cycles. We moreover perform numerical experiments where the mean-field approximation leads to periodically oscillating solutions, while the solutions of the second-order model can be interpreted as an average taken over many realizations of the stochastic model. Altogether, our results highlight the importance of including higher moments when studying stochastic networks and deepen our understanding of correlated neuronal activity.