Correlation structure of stochastic neural networks with generic connectivity matrices

TL;DR

This paper develops a perturbative method to analytically compute the correlation structure of stochastic neural networks with arbitrary connectivity, including random topologies, using Wilson and Cowan rate neuron models.

Contribution

It introduces a generic perturbative expansion approach for analyzing correlation structures in neural networks with arbitrary connectivity matrices, including random topologies.

Findings

Analytic expressions for correlation structures in weakly connected neural networks.

Method applicable to networks with random and biologically relevant topologies.

Perturbative expansion can be extended to any order for detailed analysis.

Abstract

Using a perturbative expansion for weak synaptic weights and weak sources of randomness, we calculate the correlation structure of neural networks with generic connectivity matrices. In detail, the perturbative parameters are the mean and the standard deviation of the synaptic weights, together with the standard deviations of the background noise of the membrane potentials and of their initial conditions. We also show how to determine the correlation structure of the system when the synaptic connections have a random topology. This analysis is performed on rate neurons described by Wilson and Cowan equations, since this allows us to find analytic results. Moreover, the perturbative expansion can be developed at any order and for a generic connectivity matrix. We finally show an example of application of this technique for a particular case of biologically relevant topology of the synaptic connections.