April 2009 EPL, 86 (2009) 18005 www.epljournal.org doi: 10.1209/0295-5075/86/18005 Spiral wave annihilation by low-frequency planar fronts in a model of excitable media Miguel A. de la Casa 1 , F. Javier de la Rubia 1 and Plamen Ch. Ivanov 2,3,4(a) 1 Departamento de F´ ısica Fundamental, Universidad Nacional de Educaci´ on a Distancia c/Senda del Rey 9, 28080 Madrid, Spain, EU 2 Department of Sleep Medicine, Harvard Medical School and Brigham and Women’s Hospital Boston, MA 02115, USA 3 Center for Polymer Studies and Department of Physics, Boston University - Boston, MA 02215, USA 4 Institute of Solid State Physics, Bulgarian Academy of Sciences - 1784 Sofia, Bulgaria, EU received 25 November 2008; accepted in final form 12 March 2009 published online 20 April 2009 PACS 87.19.Hh – Cardiac dynamics PACS 87.85.Tu – Modeling biomedical systems PACS 89.75.Kd – Patterns Abstract – We perform numerical lattice simulations of an excitable medium. We show that the interaction of a spiral wave with a periodic train of planar fronts leads to annihilation of the spiral wave even when i) the period of the fronts is longer than the period of the spiral and ii) the annihilating fronts are released at a significant distance from the spiral. The observed annihilation is not due to spiral drift, and occurs well inside the lattice. Copyright c  EPLA, 2009 The dynamics of wave propagation in excitable media has been extensively studied in the last years [1–15] and the existence of spiral waves in this kind of system has been reported in many cases, including aggregating slime-mould cells [16], retinae [17], the Belousov-Zhabotinsky chemical oscillator [18], CO oxidation [19], and heart muscle [20,21]. Of special interest, in areas like cardiology, is the study of wavefront stability, as wave breaks generate spatio- temporal patterns that are associated with potentially fatal arrhythmias [1–3,22–32]. Therefore, it is of great interest to find out how to attenuate and annihilate spiral waves. Here, we investigate the possibility of annihilating spiral waves by a train of periodic plain waves, which is of relevance to the dynamics in the myocardium, where fronts descending from the heart pacemaker (the sinoatrial node) interact with spiral waves. It is widely assumed that stable spiral waves cannot be eliminated by wavefronts of similar or lower frequency, since the domains of faster spiral waves grow at the expense of the slower wavefronts [31,33–37]. In this study, we consider the interaction of a stable spiral wave and planar wavefronts with frequency lower than the frequency of the spiral rotation and we show that this interaction leads to annihilation of the spiral wave over a range of physiologically meaningful parameter values. (a) E-mail: plamen@buphy.bu.edu It has been reported recently [38,39] that the interaction of a single spiral wave with a train of periodic planar fronts can lead, under certain circumstances, i.e., when the fronts have long excitation duration, and are delivered at a specific phase relative to the rotational phase of the spiral, to the formation of complex spatio-temporal patterns characterized by the presence of periodic attenuation of the spiral wave. Since the results reported in [38,39] are based on a cellular automaton model, it is natural to raise the question of the validity of those findings for continuous models based on partial differential equations. To perform our analysis we choose the Aliev-Panfilov model of the cardiac tissue [40]. This is a modification of the classical Fitzhugh-Nagumo model aimed at introduc- ing the experimentally observed restitution curve, which the original model lacks [41–45]. As we will explain later, this feature is crucial for the annihilation of the spiral wave. The model equations are ∂u ∂t = ∂ ∂x i d ij ∂u ∂x j - ku(u - a)(u - 1) - uv, (1) ∂v ∂t = ǫ(u, v)[-v - ku(u - a - 1)] , (2) where u is the transmembrane potential, v is the recovery variable (related to the conductance of the cell membrane), ǫ(u, v)= ǫ 0 + μ 1 v/(u + μ 2 ) and the typical parameter 18005-p1