## Delocalized Stimulation Resonant Drift of Spiral Waves

Another approach is based on an alternative idealization of the action of the electric current on cardiac tissue. Suppose, for simplicity and in the first approximation, that a reasonably spatially uniform electric field (say as produced by a transthoracic defibrillator) acts simultaneously and similarly on all cells in the tissue. Mathematically, that is equivalent to introduction into the model of a parameter that explicitly depends on time. Davydov et al.6 considered a simplified "kinematic" description of spiral waves and predicted that if the parameters of the model are changed periodically with a period close to the rotation period of the spiral wave, then the spiral exhibits large-scale wandering, which, in the case of a precise resonance, degenerates into a drift along a straight line (Fig. 2). This theoretical prediction was supported by numerical simulations of a piecewise linear FitzHugh-Nagumo model and then immediately confirmed by experiments in BZ reaction.7 Subsequent studies have demonstrated that this resonant drift phenomenon is not restricted to the two particular cases but can be reproduced in a wide variety of spiral wave models, including cardiac models.9

Following the same logic as with the high-frequency induced drift, if the excursion of the resonantly drifting vortex is large enough to bring it into an inexcitable boundary, this can lead to extermination of the spiral wave, and thus can be thought of as another low-voltage defibrillation strategy. Some difficulties in practical application of this idea are immediately obvious. As with the case of the high-frequency induced drift, one needs to know the appropriate frequency of the stimulation: the further it is from the resonance, the more compact is the trajectory of the drift. The theory proposed in Davydov et al.6 gives the following expression for the radius of the drift trajectory Rd (up to a choice of notations):

where cd is the resonant drift speed depending on the forcing mode and magnitude, wf is the angular frequency of the forcing, and ws is the angular frequency of the spiral. So the lower the stimulation amplitude, the lower the drift speed cd and the more precise the resonance to achieve needed Rd should be.

However, even if the resonant frequency is found, it is still not enough to eliminate the spiral. Figure 3 illustrates a simulation in a variant FitzHugh-Nagumo model in which a spiral wave drifting in a straight line reaches the vicinity of an inexcitable boundary. However, the spiral does not annihilate there, but instead turns around and drifts away from the boundary. The mechanism of such resonant repulsion has been considered in Biktashev and Holden,10,11 where it was shown that the resonant drift can be approximately described by a system of ordinary differential equations of the form d$

Figure 2: Resonant drift of spiral waves. (a-c) Snapshots of a spiral wave in a Belousov-Zhabotinsky (BZ) experiment at a precise resonance; black cross is reference. (Reprinted with permission from Agladze et al.,7 © 1987, IEEE) (d) In a piecewise variant of FitzHugh-Nagumo system at a precise resonance. (Reprinted from Davydov et al.6 with kind permission of Springer Science and Business Media) (e) In a kinematic model of a generic excitable medium without refractoriness, away from a precise resonance. (Reprinted with permission from Mikhailov et al.,8 © 1994, Elsevier) (f) In the reaction-diffusion model with OXSOFT rabbit atrium kinetics, away from a precise resonance. (Reprinted with permission from Biktashev and Holden,9 © 1995, Royal Society)

Figure 2: Resonant drift of spiral waves. (a-c) Snapshots of a spiral wave in a Belousov-Zhabotinsky (BZ) experiment at a precise resonance; black cross is reference. (Reprinted with permission from Agladze et al.,7 © 1987, IEEE) (d) In a piecewise variant of FitzHugh-Nagumo system at a precise resonance. (Reprinted from Davydov et al.6 with kind permission of Springer Science and Business Media) (e) In a kinematic model of a generic excitable medium without refractoriness, away from a precise resonance. (Reprinted with permission from Mikhailov et al.,8 © 1994, Elsevier) (f) In the reaction-diffusion model with OXSOFT rabbit atrium kinetics, away from a precise resonance. (Reprinted with permission from Biktashev and Holden,9 © 1995, Royal Society)

where R = R(t) = X(r) + iY(t) is the complex coordinate of the instant center of rotation of the spiral, $ = $(t) is the phase difference between the spiral rotation and the periodic forcing, and Cd are, as before, respectively the spiral's frequency and the speed of the resonant drift, and (Cx, Cy) is the vector of the spontaneous drift of the spiral that would

Figure 3: Mechanism of repulsion of resonantly drifting vortex from an inexcitable boundary. (Reprinted with permission from Biktashev and Holden,10 @ 1993, Elsevier) Shown are successive positions of the vortex in exactly three periods of stimulation. Black dots at each picture denote the positions of the vortex tip at the instant of stimulation. (ac) The stimuli occur in the same rotation phase, and the trajectory is straight. (c, d) The natural frequency of the vortex increases near the boundary, each successive stimulus occurring at a later phase, and the direction of the drift turns. (d-f) The vortex goes away from the boundary, it resumes its original natural frequency, and the trajectory is again straight

Figure 3: Mechanism of repulsion of resonantly drifting vortex from an inexcitable boundary. (Reprinted with permission from Biktashev and Holden,10 @ 1993, Elsevier) Shown are successive positions of the vortex in exactly three periods of stimulation. Black dots at each picture denote the positions of the vortex tip at the instant of stimulation. (ac) The stimuli occur in the same rotation phase, and the trajectory is straight. (c, d) The natural frequency of the vortex increases near the boundary, each successive stimulus occurring at a later phase, and the direction of the drift turns. (d-f) The vortex goes away from the boundary, it resumes its original natural frequency, and the trajectory is again straight happen without external perturbation, say due to spatial gradients of tissue properties or in proximity to inexcitable obstacles. If Cx = Cy =0 and ws, cd = const, then system (2) is easily solved leading to (1). In terms of system (2), the explanation of the resonant repulsion is in the dependence of its key parameters on the spatial position of the spiral, particularly ws = ws(R). In Fig. 3, the closer the spiral is to the boundary, the higher its frequency will be. That destroys the resonance ws = wf, which by the first equation leads to an increase in $ which means a change of the direction of the resonant drift given by cde1$. Such change continues until the spiral is sufficiently far from the boundary. Then ws = wf again, and the spiral drifts along a different straight line, now away from the boundary.

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