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The traditional Wilson-Cowan model of excitatory and inhibitory mean fieldinteractions in neuronal populations considers a weak Gamma distribution oftime delays when processing inputs, and is obtained via a time-coarse grainingtechnique that averages the population response. Previous analyses of thestability of the Wilson-Cowan model focused on more simplified cases, where thedelays were either not present, constant or were of a specific type. Sincethese simplifications may significantly alter the behavior of the model, wefocus on understanding the behavior of the system before time-course graining,and for a wider range of delay distributions.  For these generalized delay equations, we perform stability and bifurcationanalyses with respect to parameters that capture both the coupling profile, andthe time delay. The investigation is done through the examination of thesystem's associated characteristic equation. Under mild assumptions, we givecomplete mathematical proofs of our theoretical results, for the model withgeneral delay distributions and prove the transversality condition for thepossible Hopf bifurcations, in a generalized context. The stability region inthis parameter space is described theoretically for several types of delaykernels, and numerical simulations are presented to substantiate thetheoretical results.  We illustrate these theoretical principles in an application to a basalganglia circuit, in which $\beta$-band oscillations have been associated withParkinson's Disease.

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