Stability and bifurcations in Wilson-Cowan systems with distributed delays, and an application to basal ganglia interactions

• We study an extension of the Wilson-Cowan, for general delay distributions. analyze bifurcations in terms connectivity and average time delay. illustrate dynamics differences between different suggest that distributions may underlie functional behaviors. application to Beta oscillations a basal ganglia circuit. The traditional Wilson-Cowan model excitatory inhibitory meanfield interactions neuronal populations considers weak Gamma distribution delays when processing inputs, is obtained via time-coarse graining technique averages population response. Previous analyses stability focused on more simplified cases, where were either not present, constant or specific type. Since these simplifications significantly alter behavior model, we focus understanding system before time-course graining, wider range For generalized equations, perform bifurcation with respect parameters capture both coupling profile, investigation done through examination system’s associated characteristic equation. Under mild assumptions, give complete mathematical proofs our theoretical results, prove transversality condition possible Hopf bifurcations, context. region this parameter space described theoretically several types kernels, numerical simulations are presented substantiate results. found regions differ distributions: promote stable firing rates, while strong regular oscillations, Dirac appear facilitate complex aperiodic patterns. This supports unexplored possibility being used as substrate behaviors, emphasizes importance careful choice kernel model. principles circuit, which ? -band have been Parkinson’s Disease.