Asymptotic dynamics of reflecting spiral waves

Abstract

Resonantly forced spiral waves in excitable media drift in straight-line paths, their rotation centers behaving as pointlike objects moving along trajectories with a constant velocity. Interaction with medium boundaries alters this velocity and may often result in a reflection of the drift trajectory. Such reflections have diverse characteristics and are known to be highly nonspecular in general. In this context we apply the theory of response functions, which via numerically computable integrals, reduces the reaction-diffusion equations governing the whole excitable medium to the dynamics of just the rotation center and rotation phase of a spiral wave. Spiral reflection trajectories are computed by this method for both small- and large-core spiral waves in the Barkley model. Such calculations provide insight into the process of reflection as well as explanations for differences in trajectories across parameters, including the effects of incidence angle and forcing amplitude. Qualitative aspects of these results are preserved far beyond the asymptotic limit of weak boundary effects and slow resonant drift.

Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation

“…Moreover, the spacing ∆ζ between the roots of the shift function corresponds well to the distance ∆r between the nodal lines of the response function. This confirms our conjecture 32 that the interaction of spiral waves with a physical no-flux boundary is controlled by the response functions, just as in the case of resonantly driven spirals interacting with effective boundaries formed by a step-wise change in the excitability of the medium 19,20 . The saddle point approximation noticeably overestimates the magnitude of the shift due to interaction of the spiral wave with the boundary, while integrating (33) directly produces an estimate that is in reasonable quantitative agreement with the result of direct numerical simulations.…”

“…The existence of stable equilibria suggests the presence of bound states, where a spiral would drift along a planar no-flux boundary forever. Similar bound states were found for resonantly driven spirals next to effective boundaries 19,20 .…”

“…As a result, despite the dissipative nature of excitable media, one finds a wave-particle duality that is more akin to that found in Hamiltonian systems. For example, Langham and Barkley 19,20 used the response function formalism to show that core of a resonantly driven spiral in a bounded domain moves almost like a classical particle, although reflections from the "walls" are characterized by a strongly nonlinear relation between the incident and reflected angle.…”

Robust approach for rotor mapping in cardiac tissue

“…These variables are associated with the Euclidean symmetry of the problem and can be interpreted in terms of the lowdimensional dynamics (translation and rotation) of the core of the spiral wave, which serves as its source and anchor. Notable examples include meander and drift [3][4][5][6] of spiral waves and their interaction with boundaries [7][8][9] . In fact, even for complex patterns of excitation which involve multiple spiral waves, many features of the dynamics can be understood and described reasonably well in terms of the wave core interaction 9,10 .…”