Influence of Hopf Bifurcation Dynamics on Conduction Failure of Action Potentials Along Nerve C-Fiber

Abstract

Contrary to faithful conduction of every action potential or spike along the axon, some spikes induced by the external stimulation with a high frequency at one end of the unmyelinated nerve fiber (C-fiber) disappear during the conduction process to the other end, which leads to conduction failure. Many physiological functions such as information coding or pathological pain are involved. In the present paper, the dynamic mechanism of the conduction failure is well interpreted by two characteristics of the focus near Hopf bifurcation of the Hodgkin–Huxley (HH) model. One is that the current threshold to evoke a spike from the after-potential corresponding to the focus exhibits damping oscillations, and the other is that the damping oscillations exhibit an internal period. A chain network model composed of HH neurons and stimulated by the external periodic stimulation is used to stimulate C-fiber. In the two-dimensional parameter space of the stimulation period and coupling strength, the conduction failure appears for the coupling strength lower than that of the faithful conduction, which is due to some maximal values of the coupling current for low coupling strength not being strong enough to evoke spikes, and the coupling strength threshold between the faithful conduction and conduction failure exhibiting damping oscillations with respect to the stimulation period, due to the damping oscillations of the current threshold. The damping oscillations of the coupling strength exhibit close correlations to those of the current threshold. The coupling strength for the conduction failure exhibits maximal values as the stimulation period is approximated to 1-, 2-, 3- or 4-times of the internal period and the maximal values decrease with increasing stimulation period. In addition, the correspondence between the simulation results and the previous experimental observations is discussed. The results present deep insights into the dynamics of the conduction failure with Hopf bifurcation and are helpful to investigate the influence of other modulation factors on the conduction failure.