Influence of spatially modified tissue on atrial fibrillation patterns: Insight from solutions of the FitzHugh-Nagumo equations

arXiv:1004.0883v1 [physics.bio-ph] 6 Apr 2010 Influence of spatially modified tissue on atrial fibrillation patterns: Insight from solutions of the FitzHugh-Nagumo equations Claudia Lenk1,∗Mario Einax1, and Philipp Maass2† 1Institut f¨ur Physik, Technische Universit¨at Ilmenau, 98684 Ilmenau, Germany 2Fachbereich Physik, Universit¨at Osnabr¨uck, Barbarastraße 7, 49069 Osnabr¨uck, Germany (Dated: 6 April 2010) Abstract We study the interplay between traveling action potentials and spatial inhomogeneities in the FitzHugh-Nagumo model to investigate possible mechanisms for the occurrence of fibrillatory states in the atria of the heart. Different dynamical patterns such as ectopic foci, localized and meandering spiral waves are found depending on the characteristics of the inhomogeneities. Their appearance in dependence of the size and strength of the inhomogeneities is quantified by phase diagrams. Furthermore it is shown that regularly paced waves in a region R, that is connected by a small bridge connection to another region L with perturbing waves emanating from an additional pacemaker, can be strongly disturbed, so that a fibrillatory state emerges in region R after a transient time interval. This finding supports conjectures that fibrillatory states in the right atrium can be induced by self-excitatory pacemakers in the left atrium. PACS numbers: 87.10.Ed, 87.18.Hf ∗Electronic address: claudia.lenk@tu-ilmenau.de †Electronic address: pmaass@uos.de; URL: http://www.statphys.uni-osnabruck.de 1 I. INTRODUCTION Atrial fibrillation (AF) is the most frequently appearing heart arrhythmia since it accounts for one third of all hospitalizations caused by heart arrhythmia in the industrialized countries [1]. During AF the electric conduction system of the heart is disturbed and an increased rate of activation by a factor of 3-12 compared to normal sinus rhythm occurs. Special spatio-temporal patterns of the electric potential like spiral waves, mother waves or ectopic foci are thought to be underlying generating mechanisms of AF [2–5]. These patterns are often located near physiologically modified regions of the heart tissue in the left atrium [5–8]. The question hence arises, how these physiologically modified regions can be responsible for the generation of spiral waves or ectopic foci and how they influence the properties of these patterns. To tackle these questions, we study generating mechanism for AF on the basis of the FitzHugh-Nagumo model [9], which is a simple model for action potential generation and propagation. By modeling physiologically modified regions using a spatial variation of the parameters characterizing cell properties like excitability or resting state stability, we calcu- late phase diagrams, which specify the type of spatio-temporal excitation pattern in depen- dence of the extent of the modified region and the strength of the modification. Thereupon we investigate how self-excitatory sources as spiral waves or ectopic foci with rather regular dynamics in one region can induce irregular, fibrillatory excitation patterns in some other region. Irregular, fibrillatory states are often observed in the right atrium [7, 8, 10] and it was conjectured that these are caused by the perturbation of regular waves generated by the sinus node by waves emanating from an additional pacemaker like a spiral wave or ectopic foci in the left atrium. II. MODEL The FitzHugh-Nagumo (FHN) equations [9] are a set of two coupled nonlinear ordinary differential equations, which describe excitable media via an inhibitor-activator mechanism. They were originally developed by searching for a simplified version of the Hodgkin-Huxley equations for electric pulse propagation along nerves [11]. When combined with a spatial 2 diffusion term, the equations are ∂u ∂t = D ∂2u ∂x2 + ∂2u ∂y2  + c(v + u −u3 3 + z) ∂v ∂t = −1 c (u −a + bv) . (1) This set of partial differential equations serves as a prototype for a large variety of reaction- diffusion systems, which occur, for example, in chemical reactions as the Bhelousov- Zhabotinsky reaction [12, 13] or the catalysis of carbon monoxide [14, 15], in population dynamics [16], in biology in connection with aggregation processes [17] or plancton dynam- ics [19], as well as in the spreading of forest fires [20]. Here we will use Eqs. (1) in their original context as a model to investigate the spatio- temporal evolution of electric excitations in the heart. In this approach the variable u is roughly associated with the membrane potential and the variable v with the ion currents through the cell membrane. The resting state is given by the pair of values u = u0 = 1.2 and v = v0 = −0.6. The diffusion coefficient D describes the coupling between the cells, and z is an applied external current (stimulus). The influence of the parameters a, b and c can be inferred by numerical solutions of Eqs. (1) without the diffusive term. The parameter values have to be limited to some range in order to generate excitability,

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