Computer-aided experiments on the Hopf bifurcation of the FitzHugh-Nagumo nerve model

A space-clamped FitzHugh-Nagumo (FHN) nerve model subjected to a stimulating electrical current, I, is investigated by a combination of perturbation and numerical methods. Our goal is to trace out the path of periodic solutions initiated by a Hopf bifurcation, especially when the FHN model presents a slow recovery mechanism denoted here by the small control parameter β. It is shown in the computed period diagram, that in addition to the two Hopf bifurcation points I− and I+, there are another two critical points IM and IN satisfying I− < IM < IN < I+ and forming the points of maximum period for FHN models with single steady state, while satisfying IM < I− < I+ < IN and forming the turning points for models with multiple steady states. If β is sufficiently small, the results are accompanied with cusp formation at IM and IN. This fact indicates a discontinuous transition between oscillations of different characters. Further evidences are given by other bifurcation diagrams. For FHN models with multiple steady states, a similar hysteresis phenomenon is also observed for periodic solutions.