Dynamical mean-field theory of spiking neuron ensembles: response to a single spike with independent noises

TL;DR

This paper develops a semi-analytical dynamical mean-field theory for large ensembles of FitzHugh-Nagumo neurons, revealing how noise, coupling, and ensemble size influence spike timing and synchronization.

Contribution

The paper introduces a novel DMF approach that simplifies stochastic neuron dynamics into deterministic equations, extending analysis to large clusters and sub-clusters with applications to spike timing and synchronization.

Findings

Spike timing improves with larger ensembles even without coupling.

Neurons exhibit partial synchronization in response to input spikes with coupling.

Moderate noise allows for precise spike propagation in sub-clusters with feed-forward connections.

Abstract

Dynamics of an ensemble of $N$-unit FitzHugh-Nagumo (FN) neurons subject to white noises has been studied by using a semi-analytical dynamical mean-field (DMF) theory in which the original $2 N$-dimensional {\it stochastic} differential equations are replaced by 8-dimensional {\it deterministic} differential equations expressed in terms of moments of local and global variables. Our DMF theory, which assumes weak noises and the Gaussian distribution of state variables, goes beyond weak couplings among constituent neurons. By using the expression for the firing probability due to an applied single spike, we have discussed effects of noises, synaptic couplings and the size of the ensemble on the spike timing precision, which is shown to be improved by increasing the size of the neuron ensemble, even when there are no couplings among neurons. When the coupling is introduced, neurons in ensembles respond to an input spike with a partial synchronization. DMF theory is extended to a large cluster which can be divided into multiple sub-clusters according to their functions. A model calculation has shown that when the noise intensity is moderate, the spike propagation with a fairly precise timing is possible among noisy sub-clusters with feed-forward couplings, as in the synfire chain. Results calculated by our DMF theory are nicely compared to those obtained by direct simulations. A comparison of DMF theory with the conventional moment method is also discussed.