Li´ enard-Type Models for the simulation of the action potential of cardiac nodal cells P. Podziemski * , J.J. ˙ Zebrowski Faculty of Physics, Warsaw University of Technology, ul.Koszykowa 75, 00-662 Warszawa, Poland Abstract 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 Li´ enard-type oscillator - as a phenomenological model for cardiac nodal tissue, with certain important similarities to physiological models of cardiac pacemaker cells. The model presented here is specifically designed to qualitatively reproduce meso- scopic characteristics of cell dynamics, including action potential duration (APD) resti- tution properties, phase response characteristics, and phase space structure. We show that these characteristics agree qualitatively with the extensive ionic models and experimental results in literature [1, 2, 3, 4, 5, 6, 7]. We discuss the idea of the memory effect [8] in the pacemaker cells and show that the reason for such effects, at least partially, lies in the non-stationarity of the problem. The usefulness of the general concepts presented here is illustrated by a simulation of the atrio-ventricular reentry tachycardia (AVNRT) in a pseudo-one dimensional strip of tissue. We compare the results with other kinds of simulation in the literature. Keywords: cardiac modeling, cardiac nodal tissue, phase responce, restitution, cardiac electrophysiology 1. Introduction 1 Natural human heart rate is irregular with the time. This feature, named heart 2 rate variability (HRV), can be an important diagnostic factor and is associated with 3 an effective, physiological functioning of the cardiovascular system [9, 10]. Much effort 4 has gone into the creation of diagnostic tools for HRV - including the application of 5 complexity measures, chaos theory and nonlinear signal analysis [11, 12]. Simultaneously, 6 a variety of models from the cell level to whole heart models are used to study the 7 functioning of cardiac tissue. Very rarely such simulations address the problem of the 8 variability of the heart rate, while the results of HRV studies are rarely discussed in 9 * Corresponding author. Tel. +48 22 234 79 58 Fax. +48 226 282 171 Email address: podziemski@if.pw.edu.pl (P. Podziemski) Preprint submitted to Physica D March 15, 2013