Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I. Global organization of bistable periodic solutions

Abstract

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.

Complex nonlinear dynamics of the Hodgkin–Huxley equations induced by time scale changes

“…In fact, the equations could produce various kinds of membrane waveforms that are observed in diverse neurons. If a parameter such as the potassium equilibrium potential V K of the original HH equations in addition to the I ext value is changed, various complicated phenomena appear (Guckenheimer and Labouriau 1993;Fukai et al 2000a). The V K value changes the number and stability of equilibrium points, and this change makes the bifurcation analysis very complicated.…”

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations II. Singularity theoretic approach and highly degenerate bifurcations

“…3 of Fukai et al (1999) for corresponding waveforms. For the numerical procedure, see Fukai et al (1999).…”

“…In the previous paper (Fukai et al 1999), dH2 was simply referred to as the``degenerate Hopf bifurcation'' (the points marked as dH in the 2BDs), and the points labeled by an asterisk in the 2BDs (Figs. 2 and 4 in Fukai et al 1999) correspond to dH1 when I ext is taken as the bifurcation parameter.…”

“…In our previous paper (Fukai et al 1999), we looked at the bistabilities of periodic solutions in the HH and identi®ed the parameter regions in which either two stable periodic solutions with dierent amplitudes and periods and an equilibrium point or two stable periodic solutions coexist. The analysis of the global organization of the bifurcation structure in the multiple-parameter space of the HH suggested that the existence of highly degenerate Hopf bifurcation points in I ext -V K -V Na parameter space is associated with the emergence of bistability and tristability.…”

A GUI (Graphical User Interface) framework to Introduce H-H Model in Comparison with Kv3.3 Model

“…2000 Fukai, H., et al [11] Described an artificial neural network to identify and model the physiological behavior with proper threshold value.…”