Degenerate Hopf Bifurcation and Nerve Impulse. Part II

Abstract

The bifurcation from equilibrium of periodic solutions of the Hodgkin and Huxley equations for the nerve impulse is studied. In earlier work singularity theory techniques were used to establish that these equations have a branch of periodic solutions undergoing two Hopf bifurcations, and the equations were conjectured to be equivalent to a member of a one-parameter family of generalized Hopf bifurcation problems. Here the invariants for equivalence to this family and the value of the modal parameter are computed (see [W. W. Farr et al., “Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem,” SIAM J. Math. Anal., 20 (1989), pp. 13–30]). The value of this parameter determines the type of bifurcation, and in this way it is decided which of the proposed bifurcation diagrams are actually to be found. Thus a topological description of periodic orbits of the Hodgkin and Huxley equations near the equilibrium solution is obtained. In this way, a periodic solution branch is found that does not arise thro...