Finite size effects for spiking neural networks with spatially dependent coupling

TL;DR

This paper extends finite-size effect theory to spatially dependent spiking neural networks, deriving a local mean field approach and analyzing fluctuations due to finite network size.

Contribution

It generalizes previous uniform coupling models to non-uniform coupling, establishing a local mean field framework and deriving a perturbation expansion for fluctuations.

Findings

Local mean field theory is well-defined for spatially dependent coupling.

A perturbation expansion for input covariance and firing rate fluctuations is derived.

Finite-size fluctuations are characterized in non-uniform neural networks.

Abstract

We study finite-size fluctuations in a network of spiking deterministic neurons coupled with non-uniform synaptic coupling. We generalize a previously developed theory of finite size effects for uniform globally coupled neurons. In the uniform case, mean field theory is well defined by averaging over the network as the number of neurons in the network goes to infinity. However, for nonuniform coupling it is no longer possible to average over the entire network if we are interested in fluctuations at a particular location within the network. We show that if the coupling function approaches a continuous function in the infinite system size limit then an average over a local neighborhood can be defined such that mean field theory is well defined for a spatially dependent field. We then derive a perturbation expansion in the inverse system size around the mean field limit for the covariance of the input to a neuron (synaptic drive) and firing rate fluctuations due to dynamical deterministic finite-size effects.