Neural Field Models with Transmission Delays and Diffusion

TL;DR

This paper extends neural field models by incorporating diffusion to represent electrical connections, analyzes spectral properties, and demonstrates how diffusion influences neural synchronization and oscillations.

Contribution

It introduces a diffusion term into neural field models, extends mathematical analysis for delay equations, and characterizes spectral effects on neural dynamics.

Findings

Diffusion suppresses non-synchronized steady states.

Diffusion promotes synchronized oscillatory modes.

Spectral properties are explicitly computed for certain connectivity functions.

Abstract

A neural field models the large scale behaviour of large groups of neurons. We extend results of van Gils et al. [2013] and Dijkstra et al. [2015] by including a diffusion term into the neural field, which models direct, electrical connections. We extend known and prove new sun-star calculus results for delay equations to be able to include diffusion and explicitly characterise the essential spectrum. For a certain class of connectivity functions in the neural field model, we are able to compute its spectral properties and the first Lyapunov coefficient of a Hopf bifurcation. By examining a numerical example, we find that the addition of diffusion suppresses non-synchronised steady-states, while favouring synchronised oscillatory modes.