Stability of coupled Wilson-Cowan systems with distributed delays

TL;DR

This paper analyzes how coupled Wilson-Cowan systems with distributed delays transition between oscillatory and non-oscillatory states, revealing the influence of delay distributions and coupling on pathological neural rhythms.

Contribution

It introduces a detailed analysis of coupled Wilson-Cowan models with distributed delays, comparing discrete and Gamma delay kernels, and elucidates mechanisms controlling oscillation regimes.

Findings

Weak Gamma delay restricts oscillations to specific coupling and delay ranges.

Different kernels emphasize different critical coupling combinations for oscillation control.

Simulations demonstrate applications to beta and alpha neural rhythms.

Abstract

Building upon our previous work on the Wilson-Cowan equations with distributed delays, we study the dynamic behavior in a system of two coupled Wilson-Cowan pairs. We focus in particular on understanding the mechanisms that govern the transitions in and out of oscillatory regimes associated with pathological behavior. We investigate these mechanisms under multiple coupling scenarios, and we compare the effects of using discrete delays versus a weak Gamma delay distribution. We found that, in order to trigger and stop oscillations, each kernel emphasizes different critical combinations of coupling weights and time delay, with the weak Gamma kernel restricting oscillations to a tighter locus of coupling strengths, and to a limited range of time delays. We finally illustrate the general analytical results with simulations for two particular applications: generation of beta-rhytms in the basal ganglia, and alpha oscillations in the prefrontal-limbic system.