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

Abstract

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 firings. When h is slowed down--differently from the case of n variable being slow --chaotic bursting oscillations are observed for a wide range of parameter values although both variables cause a decrease in the membrane potential. The underlying nonlinear dynamics of various action potentials are analyzed using bifurcation theory and a so-called slow-fast decomposition analysis. It is shown that a simple topological property of the equilibrium curves of slow and fast subsystems is essential to the production of chaotic oscillations, and this is the cause of the large difference in global firing characteristics between the h-slow and n-slow cases.

Roles of sarcoplasmic reticulum Ca2+ cycling and Na+/Ca2+ exchanger in sinoatrial node pacemaking: Insights from bifurcation analysis of mathematical models

“…Bifurcation diagrams can be constructed by the ordinary numerical integration of differential equations if the concepts of bifurcation theory are applied, which is the brute-force method (7,39). However, strict bifurcation (stability) analysis, called the "continuation" method (5,39,47), can determine the stability and bifurcation of EPs (and LCs) more rapidly and accurately even when a system exhibits multistability (coexistence of multiple EPs and LCs) with hysteresis, as suggested for nonlinear systems, including cardiac myocytes (10,21,22).…”

The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales

“…was derived in [6] from a version of the classical HH equations obtained by introducing independent time constants τ m , τ h , τ n in the conductance equations, as suggested in [2,3,4,5]. The value of ǫ that results from this derivation is 1/120 ≈ .0083, which represents the reciprocal of the largest maximal ionic conductance, that of the sodium current, in the original model.…”

“…Past work has demonstrated that a step increase in τ h or τ n can lead to the existence of mixed-mode oscillations (MMOs), featuring alternating sets of large excursions and of small (or subthreshold) oscillations (STOs) in v, as solutions of the dimensional form of (1.1) [2,3,4,5]; see Figure 2. In [6], we used recently developed mathematical theory [8,10,11] to explain the onset and offset of MMOs in system (1.4) under the systematic, independent variation of τ h , τ n , and I.…”

“…Recent studies [2,3,4,5,6] have shown that a modified version of the HH equations, where only the speeds of the gating dynamics of the corresponding ion channels are changed, can lead to very slow firing rates of action potentials. In [6], we explained this observation and the corresponding complex oscillatory patterns, known as mixed-mode oscillations (MMOs), within the framework of multiple time scales analysis via the generalized canard phenomenon [8].…”

Spontaneous high-frequency action potential

“…Since Hodgkin-Huxley model was described in 1952 [93], the chaotic property of spontaneous high-frequency action potential in excitable cells had been regarded as a vital characteristic of fibrillation action potential and kept on being studied in non-linear system by the intensive research [94,95,[97][98][99], the chaotic property exhibited system instability with the unstable action potential amplitude and duration. The Hodgkin-Huxley model per se is a set of nonlinear ordinary differential equations that approximate the electrical characteristics of excitable cells such as neurons and cardiac myocytes (non-linear system refers to the system whose variations of results are non-consistent).…”