Isolated periodic solutions of the Hodgkin-Huxley equations

Abstract

Periodic solutions of the current clamped Hodgkin-Huxley equations (Hodgkin & Huxley, 1952 J. Physiol. 117, 500) that arise by degenerate Hopf bifurcation were studied recently by Labouriau (1985 SIAM J. Math. Anal. 16, 1121, 1987 Degenerate Hopf Bifurcation and Nerve Impulse (Part II), in press). Two parameters, temperature T and sodium conductance gNa were varied from the original values obtained by Hodgkin & Huxley. Labouriau's work proved the existence of small amplitude periodic solution branches that do not connect locally to the stationary solution branch, and had not been previously computed. In this paper we compute these solution branches globally. We find families of isolas of periodic solutions (i.e. branches not connected to the stationary branch). For values of gNa in the range measured by Hodgkin & Huxley, and for physically reasonable temperatures, there are isolas containing orbitally asymptotically stable solutions. The presence of isolas of periodic solutions suggests that in certain current space clamped membrane experiments, action potentials could be observed even though the stationary state is stable for all current stimuli. Once produced, such action potentials will disappear suddenly if the current stimulus is either increased or decreased past certain values. Under some conditions, "jumping" between action potentials of different amplitudes might be observed.