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 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(More)
We have presented a new generation mechanism of slow spiking or repetitive discharges with extraordinarily long inter-spike intervals using the modified Hodgkin-Huxley equations (Doi and Kumagai, 2001). This generation process of slow firing is completely different from that of the well-known potassium A-current in that the steady-state current-voltage(More)
 The Hodgkin–Huxley equations with a slight modification are investigated, in which the inactivation process (h) of sodium channels or the activation process of potassium channels (n) is slowed down. We show that the equations produce a variety of action potential waveforms ranging from a plateau potential, such as in heart muscle cells, to chaotic bursting(More)
The Hodgkin-Huxley (HH) equations with a modification in which the inactivation process (h variable) of sodium channels is slightly slowed down are investigated. It is shown that this slight modification changes the HH dynamics to a completely different one, with chaotic spiking and very long interspike intervals appearing in a generic manner, although the(More)
A simple mathematical model of living pacemaker neurons is proposed. The model has a unit circle limit cycle and radial isochrons, and the state point moves slowly in one region and fast in the remaining region; regions can correspond to the subthreshold activity and to the action potentials of pacemaker neurons, respectively. The global bifurcation(More)
The Bonhoeffer-van der Pol (BVP) oscillator is a valuable dynamical system model of pacemaker neurons. Isochrons, phase transition curves (PTC), and two dimensional bifurcation diagrams served to analyze the neuron's response to periodic pulse stimuli. Responses are described and explained in terms of the nonlinear dynamical system theory. An important(More)
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(More)
A recent investigation of the influence of periodic inhibitory trains on a crayfish pacemaker neuron showed not only well-known locked periodic responses but also intermittent, messy, and hopping responses. This communication studies the responses of the Bonhoeffer-van der Pol (BVP) model with self-sustained oscillation when exposed to periodic pulse trains(More)
The eight-variable model for the giant neuron localized in the esophageal ganglia of the marine pulmonate mollusk Onchidium verruculatum is reduced to four-and-three-dimensional systems by regrouping variables with similar time scales. These reduced models replicate the complex behavior including beating, periodic bursting and aperiodic bursting displayed(More)