Repetitive activity and Hopf bifurcation under point-stimulation for a simple FitzHugh-Naguma nerve conduction model

In response to point-stimulation with a constant current, a neuron may propagate a repetitive train of action potentials along its axon. For maintained repetitive activity, the current strength I must be, typically, neither too small nor too large. For I outside some range, time independent steady behavior is observed following a transient phase just after the current is applied. We present analytical results for a piecewise linear FitzHugh-Nagumo model for a point-stimulated (non-space-clamped) nerve which are consistent with this qualitative experimental picture. For each value of I there is a unique, spatially nonuniform, steady state solution. We show that this solution is stable except for an interval (I*, I(*)) of I values. Stability for I too small or too large corresponds to experiments with sub-threshold I or the excessive I which leads to 'nerve block'. For I = I*, I(*) we find Hopf bifurcation of spatially nonuniform, time periodic solutions. We conclude that (I*, I(*)) lies interior to the range of I values for repetitive activity. The values of I* and I(*) and their dependence on the model parameters are determined. Qualitative differences between results for the point-stimulated configuration and the space-clamped case are discussed.