  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.
