Exotic dynamic behavior of the forced FitzHugh-Nagumo equations

Space-clamped FitzHugh-Nagumo nerve model subjected to a stimulating electrical current of form I-o + I cos gamma t is investigated via Poincare map and numerical continuation. If I = 0, it is known that Hopf bifurcation occurs when I-o is neither too small nor too large. Given such an I-o. If gamma is chosen close to the natural frequency of the Hopf bifurcated oscillation, a series of exotic phenomena varying with I are observed numerically. Let 2 pi lambda/gamma denote the generic period we watched. Then the scenario consists of two categories of period-adding bifurcation. The first category consists of a sequence of hysteretic, lambda --> lambda + 2 period-adding starting with lambda = 1 at I = 0+, and ending at some finite I, say I*, as lambda --> infinity. The second category contains multiple levels of period-adding bifurcation. The top level consists of a sequence of lambda --> lambda + 1, period-adding starting with lambda = 2 at I = I*. From this sequence, a hierarchy of m --> m + n --> n, period-adding in between are derived. Such a regular pattern is sometimes interrupted by a series of chaos. This category of bifurcation also terminates at some finite I. Harmonic resonance sets in afterwards. Lyapunov exponents, power spectra, and fractal dimensions are used to assist these observations.