Dynamic solutions of next generation neural field models with delays

TL;DR

This paper analyzes neural field models with delays, revealing how delay variations induce bifurcations leading to traveling waves and breathing bump solutions in neural networks.

Contribution

It introduces methods to efficiently solve self-consistency equations for dynamic solutions in delayed neural field models, expanding understanding of pattern formation.

Findings

Delays cause Hopf bifurcations in neural states.

Traveling waves and breathing bumps emerge from bifurcations.

Delay parameters significantly influence pattern dynamics.

Abstract

We study networks of theta neurons arranged on a ring with delayed interactions. In the continuum limit the systems are described by next generation neural field models with delays. We consider distributed delays with both finite and infinite support, and conduction delays. The stability of spatially uniform and localized bump states is determined, and we find that they undergo Hopf bifurcations as parameters related to the delays are varied. These bifurcations create traveling waves and ``breathing'' bump solutions. These dynamic solutions satisfy self-consistency equations and we show how to efficiently solve these equations. Following traveling waves and periodic solutions as parameters are varied provides a global picture of the influence of different delays on pattern formation processes in spatially extended networks of theta neurons.