Zhenxing Pan

Bifurcation analysis and effects of changing ionic conductances on pacemaker rhythm in a sinoatrial node cell model.

The electrical excitation (action potential generation) of sinoatrial node (cardiac pacemaker) cells is directly related to various ion channels (pore-forming proteins) in cell membranes. In order to analyze the relation between action potential generation and ion channels, we use the Yanagihara-Noma-Irisawa (YNI) model of sinoatrial node cells, which is described by the Hodgkin-Huxley-type equations with seven variables. In this paper, we analyze the global bifurcation structure of the YNI model by varying various conductances of ion channels, and examine the effects of these conductance changes on pacemaker rhythm (frequency of action potential generation). The coupling effect on pacemaker rhythm is also examined approximately by applying external current to the YNI model.

Variability of pacemaker rhythm in a detailed model of cardiac sinoatrial node cells

The sinoatrial node (cardiac pacemaker) cell spontaneously generates periodic electrical signals (action potentials), and the action potential generation is deeply related to various ion channels in its cell membrane. The abnormalities of ion channels cause serious sinus arrhythmia, and such diseases are usually treated by applying drugs which have effects on ion channels. This paper uses the Zhang model of sinoatrial node cells to investigate the relation between pacemaker rhythm (frequency of action potential generation) and ion channels. The Zhang model is described by the Hodgkin-Huxley-type nonlinear ordinary differential equations, and its parameter values depend on the region (periphery or center) of the sinoatrial node. We analyze the bifurcation structure of the Zhang model, and examine the variabilities of pacemaker rhythm and its sensitivities on conductance changes of ion channels for both periphery and center cells. Moreover, these results are compared to the previous results of another sinoatrial node cell model (Yanagihara-Noma-Irisawa model). The above results are expected to be useful in the treatment of sinus arrhythmia.

Computational and Mathematical Models of Neurons

What are models? The HH equations (2.3) are often called a physiological model, whereas the models appeared in the following sections are simplified models or abstract models. However, there is no model in which any simplifications or abstractions have not been made. Of course, many features of real neurons are ignored even in the HH equations. All models have their applicability and limits to describe natural phenomena. Therefore, all types of models whatever simplified or physiological, have their own values to model real phenomena. Starting with the BVP or FHN model which is a simplification of the HH equations, this chapter proceeds to several neuronal models with higher abstractions which are useful to tract some essential features of neurons.

Whole System Analysis of Hodgkin–Huxley Systems

In Chap. 2, we have explored the dynamics of the original Hodgkin–Huxley equations of a squid giant axon where only the parameter Iext was changed. The HH equations, however, include various constants or parameters whose values were determined based on physiological experiments, and thus the values inherently possess a certain ambiguity. Also, the “constants” are not really constant but change temporally. Thus, in this chapter, we study the effects of the change of the constants or parameters on the dynamics of the HH equations and consider the robustness and sensitivity of the equations; we study the bifurcation structure of the HH equations by changing their various parameters. To do so, in this chapter, we consider a slight modification of the original HH equations since the modification is mathematically more tractable.

The Hodgkin–Huxley Theory of Neuronal Excitation

Hodgkin and Huxley (1952) proposed the famous Hodgkin–Huxley (hereinafter referred to as HH) equations which quantitatively describe the generation of action potential of squid giant axon, although there are still arguments against it (Connor et al. 1977; Strassberg and DeFelice 1993; Rush and Rinzel 1995; Clay 1998). The HH equations are important not only in that it is one of the most successful mathematical model in quantitatively describing biological phenomena but also in that the method (the HH formalism or the HH theory) used in deriving the model of a squid is directly applicable to many kinds of neurons and other excitable cells. The equations derived following this HH formalism are called the HH-type equations.

Hodgkin–Huxley-Type Models of Cardiac Muscle Cells

Following the HH formalism introduced in Chap. 2, various kinds of HH-type models of neurons and other excitable cells are proposed (Canavier et al. 1991; Chay and Keizer 1983; Cronin 1987; Gerber and Jakobsson 1993; Hayashi and Ishizuka 1992; Keener and Sneyd 1998; Noble 1995; Rinzel 1990; Traub et al. 1991), and are analyzed (Alexander and Cai 1991; Av-Ron 1994; Bertram 1994; Bertram et al. 1995; Butera 1998; Canavier et al. 1993; Chay and Rinzel 1985; Doi and Kumagai 2005; Guckenheimer et al. 1993; Maeda et al. 1998; Rush and Rinzel 1994; Schweighofer et al. 1999; Terman 1991; Tsumoto et al. 2003, 2006; Yoshinaga et al. 1999). The HH-type equations include many variables depending on the number of different ionic currents and their gating variables considered in the equations, whereas the original HH equations possess only four variables (a membrane voltage, activation and inactivation variables of Na+ current and an activation variable of K+ current). Among the diverse family of HH-type equations, this chapter explores the dynamics and the bifurcation structure of the HH-type equations of heart muscle cells (cardiac myocytes).