We have presented numerical results on the resonant drift of rigidly rotating spiral waves within a circular domain, and interaction of resonantly drifting spiral waves with medium obstacles and boundaries. We have considered media with a radius the same order of magnitude as the spiral wavelength, and the simplest possible spiral wave motions, a periodic rigidly rotating wave and a biperiodic one, drifting along the boundary. In the context of applications to cardiac arrhythmia, the rigidly rotating wave corresponds to a re-entrant propagation of the leading circle type, around a functional block (the core, in spiral wave terminology), while the biperiodic motion corresponds to re-entrant wave around an anatomical obstacle, such as the inferior vena cava in a right atrial flutter. Both types of re-entry were subjected to both a constant periodic and feedback controlled resonant external time forcing.
Although a circular domain provides a more natural model for cardiac tissue than a rectangular domain, special care should be taken in carrying out numerical experiments in polar coordinates to study the drift phenomenon. As mentioned in Sect.2.8, integration in polar coordinates shows a very slow radial drift of spiral waves even in the absence of external forcing. Such a slow drift has been described in experiments with the Belousov-Zhabotinsky reaction, Gómez-Gesteira et al [15], however, in the numerical experiments it is a numerical artefact, as its velocity varies with the step size, and is due to grid anisotropy caused by nonuniform errors of the finite difference approximation of the original equations.
Earlier work on the effects of periodic forcing on spiral waves has been by a kinematic approach Davydov et al [13] and Mikhailov et al [19], by numerical, phenomenological and asymptotical studies by Biktashev and Holden [4]-[9], and by the forcing of a model ODE system by Mantel and Barkley [18]. These studies provide insight into the directed motion or resonant drift of spiral waves, and in the context of cardiac defibrillation deal with moving the reentrant wave towards a boundary. Interactions with a boundary have not been so intensively studied - however see Biktashev [3], Ermakova et al [14], Aranson et al [2] and Biktashev and Holden [7], and the implicit assumption is that the spiral will either be reflected or extinguished at the boundary. For the purposes of efficient extinction of spiral wave at the medium boundary it is of great importance to understand possible types of spiral wave behavior near the boundary.
Biktashev and Holden [4]-[9] used resonant drift under feedback control to overcome boundary interactions, with the feedback triggered by a recording electrode located in a corner of the medium. If the recording electrode is located within the interior of the circular domain then, independently of however far from the wavetip of a rigidly rotating wave it is located, synchronization of the tip motion occurs and after a transient the wavetip approaches a closed circular trajectory centered at the recording electrode point, Figure 4. An analogous synchronization phenomenon has been observed earlier in the case of meandering spiral waves in Grill et al. [16]. Thus the recording electrode feedback stimulation produces a synchronized dynamical behavior for the cases of rigidly rotating or meandering spiral waves, and the case of biperiodic motion near the boundary within a circular medium. The recording electrode imposes a centre of symmetry for the system. Motion is a circle around the recording electrode. Since simple re-entry in the wall of a heart chamber occurs in an anatomically defined, complicated three-dimensional structure some locations for a recording electrode used to control re-entrant activity by resonant drift will be more effective than others in driving re-entrant activity to a boundary with inexcitable tissue.
When interaction between the drifting spiral wave and inexcitable obstacles in the medium traps the spiral wave, as in the pinning described by Davidenko et al [12], recording electrode feedback-controlled forcing can, in some (see Fig. 7(a)), but not all cases (see Fig. 7(b)) , detach the spiral from the obstacle. Even in the cases when the wavetip could not be detached from the obstacle by recording electrode feedback-controlled forcing a properly chosen delay allowed one to manage the drift in such a way to avoid the obstacle and finally successfully extinguish the wave at the boundary, Fig. 7c,d. If the obstacle is an anatomical obstacle in the heart, then choice of recording site and time delay could, in principle, produce resonantly drifting trajectories that avoid the obstacle.