Hopf bifurcations on invariant manifolds of a modified Fitzhugh–Nagumo model

Abstract

The dynamical features of a modified Fitzhugh–Nagumo (FHN) nerve model are addressed. The model considered accounts for a relaxation time, induced by diffusion and finite propagation velocities, resulting in a hyperbolic system. Bifurcation analysis of the local kinetic system with the relaxation constant as the principal bifurcation parameter reveals a threshold of the relaxation constant beyond which a supercritical Hopf bifurcation occurs. It is shown that the frequency of the ensuing cycles depends on the relaxation time. The existence of Hopf bifurcations on invariant center manifolds is established using the projection method. Analytical formulas for the critical value of the relaxation constant and the first Lyapunov coefficient are derived; results are confirmed via numerical simulations. The addition of external current, at small values of the relaxation constant, produces excitable behavior consistent with the classical FHN model such as periodic firing, bistability, bursting and canard explosions. At higher values of this constant, chaotic motion and other new dynamical objects such as period-two, -three and -four orbits are observed numerically.