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. (C) 2001 Elsevier Science B.V. All rights reserved.