Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations. II: Singularity theoretic approach and highly degenerate bifurcations. (English) Zbl 0962.92006

Fukai, Hidekazu; Nomura, Taishin; Doi, Shinji; Sato, Shunsuke

Biol. Cybern. 82, No. 3, 223-229 (2000).

[For part I of this paper see the preceding entry, Zbl 0962.92005).]
In the Hodgkin-Huxley equations (HH), we have identified the parameter regions in which either two stable periodic solutions with different amplitudes and periods and an equilibrium point or two stable periodic solutions coexist. The global structure of bifurcations in the multiple-parameter space in the HH suggested that the bistabilities of the periodic solutions are associated with the degenerate Hopf bifurcation points by which several qualitatively different behaviors are organized.
In this paper, we clarify this by analyzing the details of the degenerate Hopf bifurcations using the singularity theory approach which deals with local bifurcations near a highly degenerate fixed point.

MSC:

92C20	Neural biology
34C23	Bifurcation theory for ordinary differential equations
37N25	Dynamical systems in biology

Keywords:

Hodgkin-Huxley equations; periodic solutions; Hopf bifurcations

Citations:

Zbl 0962.92005