The Hodgkin-Huxley equations (HH) are parameterized by a number of parameters and shows 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(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.

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.

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The analyses of recordings of magnetoencephalography (MEG) and other imaging techniques may require the separation of periodic signals from the observed signals. Blind source separation (BSS) is widely used for the separation of specific signals these days. Though several algorithms based on the BSS scheme for the separation of periodic signals have been proposed, they usually assume the system to be well-posed, satisfactory results often cannot be obtained for practical recordings. In this study, we show that a method based on the joint approximate diagonalization of correlation matrices with several time delays (JADCM) is robust and good results can be obtained by choosing the time delays carefully, especially in practical illposed situations such as signal separation from MEG recordings. The performance of the proposed method is compared with that of Periodic BSS and JADCM using the conventional parameter set.