Hopf Bifurcation to Repetitive Activity in Nerve

Abstract

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. We consider these stimulus-response properties 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. We associate this with the onset of repetitive activity. We derive analytic bifurcation formulae and evaluate these for the case of a cubic nonlinearity to determine regions in parameter space where the steady solution is unstable. We interpret physiologically the effect of stimulus form; either voltage or current input, and either spatially localized or spatially uniform stimulus.