## Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations. I: Global organization of bistable periodic solutions.(English)Zbl 0962.92005

Summary: The Hodgkin-Huxley equations (HH) are parameterized by a number of parameters and show a variety of qualitatively different behaviors depending on the parameter values. We explored the dynamics of the HH for a wide range of parameter values in the multiple-parameter space, that is, we examined the global structure of bifurcations of the HH. Results are summarized in various two-parameter bifurcation diagrams with $$I_{\text{ext}}$$ (externally applied DC current) as the abscissa and one of the other parameters as the ordinate.
In each diagram, the parameter plane was divided into several regions according to the qualitative behavior of the equations. In particular, we focused on periodic solutions emerging via Hopf bifurcations and identified parameter regions in which either two stable periodic solutions with different amplitudes and periods and a stable equilibrium point or two stable periodic solutions coexist. Global analysis of the bifurcation structure suggested that generation of these regions is associated with degenerate Hopf bifurcations.

### MSC:

 92C20 Neural biology 34C23 Bifurcation theory for ordinary differential equations

Zbl 0962.92006
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