  The traditional Wilson-Cowan model of excitatory and inhibitory mean field
interactions in neuronal populations considers a weak Gamma distribution of
time delays when processing inputs, and is obtained via a time-coarse graining
technique that averages the population response. Previous analyses of the
stability of the Wilson-Cowan model focused on more simplified cases, where the
delays were either not present, constant or were of a specific type. Since
these simplifications may significantly alter the behavior of the model, we
focus 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 bifurcation
analyses with respect to parameters that capture both the coupling profile, and
the time delay. The investigation is done through the examination of the
system's associated characteristic equation. Under mild assumptions, we give
complete mathematical proofs of our theoretical results, for the model with
general delay distributions and prove the transversality condition for the
possible Hopf bifurcations, in a generalized context. The stability region in
this parameter space is described theoretically for several types of delay
kernels, and numerical simulations are presented to substantiate the
theoretical results.
  We illustrate these theoretical principles in an application to a basal
ganglia circuit, in which $\beta$-band oscillations have been associated with
Parkinson's Disease.
