FOLD–HOPF BIFURCATIONS OF THE ROSE–HINDMARSH MODEL WITH TIME DELAY

Abstract

In neural processing information, when we consider the finite propagation speed of signals in synapses, a time delay signal self-feedback mechanism is introduced into the well-known Rose–Hindmarsh Model. The Fold–Hopf bifurcation of this model is analyzed near equilibria. To discuss the singularity of this Codim-2 bifurcation, we discuss the normal form near the Fold–Hopf points on the center manifold. Bifurcations are studied and the stability is investigated by classifying neighborhood regimes near the Fold–Hopf points. Finally, as one of the most important electrical behaviors in neurons, the complex bursting-spiking firing modes associated with bifurcations are illustrated by numerical simulations.