Liénard-type models for the simulation of the action potential of cardiac nodal cells

Abstract

Existing models of cardiac cells which include multi-variable cardiac transmembrane current are too complex to simulate the long time dynamical properties of the heart rhythm. The large number of parameters that need to be defined and set for such models make them not only cumbersome to use but also require a large computing power. Consequently, the application of such models for the bedside analysis of heart rate of a specific patient may be difficult. Other ways of modelling need to be investigated.We consider the general problem of developing a model of cardiac pacemaker tissue that allows to combine the investigation of phenomena at a time scale of thousands of heart beats with the ability to reproduce realistic tissue-level characteristics of cell dynamics. We propose a modified van der Pol–Duffing equation–a Liénard-type oscillator–as a phenomenological model for cardiac nodal tissue, with certain important physiological similarities to ion-channel models of cardiac pacemaker cells.The model presented here is specifically designed to qualitatively reproduce mesoscopic characteristics of cell dynamics, including action potential duration (APD) restitution properties, phase response characteristics, and phase space structure.We show that these characteristics agree qualitatively with the extensive ionic models and experimental results in the literature [Anumonwo et al., 1991, [33], Cao et al., 1999, [49], Coster and Celler, 2003, [31], Qu, 2004, [45], Tsalikakis et al., 2007, [32], Inada et al., 2009, [14], Qu et al., 2010, [50]]. Furthermore, we discuss reasons for the memory effect [Cherry and Fenton, 2007, [42]] obtained for the pacemaker cells.The usefulness of the general concepts presented here is illustrated by a simulation of the atrio-ventricular re-entry tachycardia (AVNRT) in a pseudo one dimensional strip of tissue. We compare the results with other kinds of simulation in the literature.