Bifurcations in a planar system of differential delay equations modeling neural activity

A planar system of differential delay equations modeling neural activity is investigated. The stationary points and their saddle-node bifurcations are estimated. By an analysis of the associated characteristic equation, Hopf bifurcations are demonstrated. At the intersection points of the saddle-node and Hopf bifurcation curves in an appropriate parameter plane, the existence of Bogdanov–Takens singularities is shown. The properties of the Bogdanov–Takens singularities are studied by applying the center manifold and normal form theory. A numerical example illustrates the obtained results.