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

Abstract

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 I M and I N satisfying I − < I M < I N < I + and forming the points of maximum period for FHN models with single steady state, while satisfying I M < I − < I + < I N and forming the turning points for models with multiple steady states. If β is sufficiently small, the results are accompanied with cusp formation at I M and I N . 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.