HOPF BIFURCATION TO REPETITIVE ACTIVITY IN NERVE.

Various nerve axons and other excitable systems exhibit repetitive activity (e. g. , trains of propagated impulses) in response to a spatially localized, time-independent stimulus. If the stimulus is too weak, or in some cases too strong, one finds rather a spatially nonuniform, steady response which attenuates with distance from the input site. These stimulus-response properties are considered for a qualitative model of nerve conduction, the FitzHugh-Nagumo (parabolic partial differential) equations. For each stimulus amplitude there is a unique steady state solution. At critical stimulus values this steady solution loses stability and a branch of time periodic (spatially nonuniform) solutions appears via Hopf bifurcation. This is associated with the onset of repetitive activity. Analytic bifurcation formulae are derived and evaluated for the case of a cubic nonlinearity to determine regions in parameter space where the steady solution is unstable. The effect of stimulus forms, either voltage or current input, and either spatially localized or spatially uniform stimulus is interpreted physiologically.