Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation Parameters

Abstract

Abstract:In this paper, we study the dynamics of a certain Hodgkin-Huxley model describing the action potential (AP) of a cardiac muscle cell for a better understanding of the occurrence of a special type of cardiac arrhythmia, the so-called early afterdepolarisations (EADs). EADs are pathological voltage oscillations during the repolarisation or plateau phase of cardiac APs. They are considered as potential precursors to cardiac arrhythmia and are often associated with deficiencies in potassium currents or enhancements in the calcium or sodium currents, e.g., induced by ion channel diseases, drugs or stress. Our study is focused on the enhancement in the calcium current to identify regions, where EADs related to enhanced calcium current appear. To this aim, we study the dynamics of the model using bifurcation theory and numerical bifurcation analysis. Furthermore, we investigate the interaction of the potassium and calcium current. It turns out that a suitable increasing of the potassium current adjusted the EADs related to an enhanced calcium current. Thus, one can use our result to balance the EADs in the sense that an enhancement in the potassium currents may compensate the effect of enhanced calcium currents.

Early Afterdepolarisations Induced by an Enhancement in the Calcium Current

“…Then, these ion channels break open and/or up such that this interaction allows an ion current flow, which changes the membrane potential. A normal AP is always uniform and the cardiac muscle cell AP is typically divided into four phases, i.e., the resting phase, the upstroke phase, the (long) plateau phase and the repolarisation phase, see for more details [15]. The resting phase is designated by high potassium (K + ) currents.…”

“…with V T y ∈ R, k y ∈ R\ {0} denotes the equilibrium of the corresponding gating variable and τ y is the corresponding relaxation time constant for each of d, f and x. The gating variables d, f and x ∈ [0, 1] are important for the activation (opening) and inactivation (closing) of the ion channels and therefore for the ion current interaction, see [15]. Moreover, the Nernst potentials of these ion currents are denoted by E Ca 2+ and E K + , while the corresponding conductance are represented by G Ca 2+ = 0.025 mS cm 2 and G K + = 0.05 mS cm 2 , respectively.…”

“…In Figure 1 some examples of EADs are presented with τ d = 0.1 ms and G Ca 2+ ∈ 0.029 mS cm 2 ; 0.03 mS cm 2 ; 0.031 mS cm 2 ; 0.035 mS cm 2 (from left to right). Please compare Figure 6a in [15] with the second trajectory in Figure 1. Here, we see that the four dimensional system behaves very similar to the three dimensional system, provided τ d is small enough and the other system parameters are the same.…”

“…In this paper, we will use the geometric singular perturbation theory [11,12] and bifurcation analysis [13] to investigate reasons for the appearing of EADs. Here, we are focused on EADs related to an enhancement in the calcium currents, see [14,15]. The main novelty is the combination of these theories to study EADs and mainly the use of the needed time scale separation argument to derive the parameter sensitivity of the considered system.…”

Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

“…Later on, many mathematical models had been proposed for the transmission dynamics of infectious diseases [2][3][4][5][6][7][8][9]. In recent years, some works have been studied for mathematical analysis of human diseases and epidemic models also utilising dynamical system approaches as stability analysis, LaSalle's invariance principle, Routh-Hurwitz criterion, or Lyapunov function in combination with numerical studies [10][11][12][13][14]. These models provided theoretical and quantitative bases for the prevention and control of infectious diseases.…”