Short communication Origin of arrhythmias in a heart model Hiba Sheheitli, Richard Rand * Dept. of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA article info Article history: Received 18 March 2009 Accepted 19 March 2009 Available online 26 March 2009 PACS: 05.45.Xt 87.19.Hh Keywords: Nonlinear vibrations Cardiology Relaxation oscillations Alternans Arrhythmias Coupled oscillators abstract An investigation of the nonlinear dynamics of a heart model is presented. The model com- partmentalizes the heart into one part that beats autonomously (the x oscillator), repre- senting the pacemaker or SA node, and a second part that beats only if excited by a signal originating outside itself (the y oscillator), representing typical cardiac tissue. Both oscillators are modeled by piecewise linear differential equations representing relaxation oscillators in which the fast time portion of the cycle is modeled by a jump. The model assumes that the x oscillator drives the y oscillator with coupling constant a. As a decreases, the regular behavior of y oscillator deteriorates, and is found to go through a ser- ies of bifurcations. The irregular behavior is characterized as involving a large amplitude cycle followed by a number n of small amplitude cycles. We compute critical bifurcation values of the coupling constant, a n , using both numerical methods as well as perturbations. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction It is well known that heart arrhythmias are often characterized by an arterial pulse which consists of alternating strong and weak beats called alternans. Electrical alternans of the heart, defined as beat to beat variability in electrocardiogram (ECG) signal, have been associated with ventricular arrhythmias in many clinical settings [2]. In particular, a recent study has showed that alternans affecting the T-wave is common among patients at increased risk for ventricular arrhythmias [2], where the T-wave is the component of ECG associated with the repolarization phase of action potentials of the ventric- ular cells [1]. Ventricular heart cells are of the excitable type that possess an equilibrium membrane potential and will nor- mally only fire upon receiving a strong enough electric signal. This signal is generated by the autonomously firing cells of the sinoatrial (SA) node, known as the pacemaker, and conducted to the ventricals through cardiac tissue. The idea of this work is to model the heart as two oscillators, one for the SA node (call it x) and one for the rest of the heart (call it y), which could represent excitable ventricular cells. The x oscillator is modeled as beating autonomously when uncoupled from the y oscillator, while the y oscillator is modeled as not beating at all, but rather as staying fixed in an equi- librium position, when uncoupled from the x oscillator. Our goal is to describe the bifurcation sequence which occurs as the coupling constants vary. The oscillators are modeled as relaxation oscillators with instantaneous jumps. This model of relaxation oscillators has been used previously in a model of a forced oscillator [3], two coupled limit cycle oscillators [4] and three coupled limit cycle oscillators [5]. 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.03.016 * Corresponding author. Fax: +1 607 255 2011. E-mail address: rhr2@cornell.edu (R. Rand). Commun Nonlinear Sci Numer Simulat 14 (2009) 3707–3714 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns