COMPUTER-AIDED EXPERIMENTS ON THE HOPF-BIFURCATION OF THE FITZHUGH-NAGUMO NERVE MODEL

A space-clamped FitzHugh-Nagumo (FHN) nerve model subjected to a stimu lating electrical current, I, is investigated by a combination of pert urbation and numerical methods. Our goal is to trace out the path of p eriodic solutions initiated by a Hopf bifurcation, especially when the FHN model presents a slow recovery mechanism denoted here by the smal l control parameter P. It is shown in the computed period diagram, tha t 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 s teady state, while satisfying I-M < I- < I+ < I-N and forming the turn ing points for models with multiple steady states. If beta is sufficie ntly small, the results are accompanied with cusp formation at I-M and I-N This fact indicates a discontinuous transition between oscillatio ns of different characters. Further evidences are given by other bifur cation diagrams. For FHN models with multiple steady states, a similar hysteresis phenomenon is also observed for periodic solutions.