Orbital Motion of Spiral Waves in Excitable Media

Abstract

Spiral waves in active media react to small perturbations as particlelike objects. Here we apply asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation center along circular orbits around the inhomogeneity. The stationary orbits have fixed radii and alternating stability, determined by the properties of the bulk medium and the type of inhomogeneity, while the drift speed along an orbit depends on the strength of the inhomogeneity. Direct numerical simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment. [7] biology). In a perfectly uniform medium the core of a spiral wave may be anywhere, depending on initial conditions. However, real systems are always heterogeneous, and therefore spiral drift due to inhomogeneity is of great practical interest to applications. Understandably, such drift has been mostly studied in excitable chemical reactions and the heart, where drift due to a gradient of medium properties [8,9] and pinning [10] (anchoring, trapping) to a localized inhomogeneity [11][12][13] have been observed in experiments and simulations. Interaction with localized inhomogeneity can be considered to be a particular case of the general phenomenon of vortex pinning to material defects [14].Here we identify a new type of spiral wave dynamics: precession around a localized inhomogeneity along a stable circular orbit. We predict this novel phenomenon theoretically, describe its key features, and confirm it by numerical simulations. We argue that this orbital motion of spiral waves is robust and prevalent, has nontrivial and surprising consequences for applications, and should be directly observable in experiments.We consider reaction-diffusion equations, which is the most popular class of models describing spiral waves:where u, f 2 R ' , D 2 R 'Â' , p 2 R m , uðr; tÞ is the dynamic vector field,r 2 R 2 , pðrÞ ¼ p 0 þ p 1 ðrÞ, jp 1 j ( 1, is the vector of parameters, D is diffusion matrix. For p ¼ (1) is assumed to have spiral wave solutions rotating with angular velocity ! (taken here to be clockwise for ! > 0),where ( , #) are polar coordinates defined with respect to the center of rotationR ¼ ðX; YÞ T , and È is the initial rotation phase.In the presence of a small perturbation p 1 ðrÞ Þ 0, the spiral's center of rotation R ¼ X þ iY is not constant but slowly evolves with the equation of motionwhere ¼ ðr ÀRÞ and ¼ #ðr ÀRÞ þ ! À È are polar coordinates in the corotating frame of reference, and h is the perturbation to the right-hand side of Eq. (1

Nonlinear physics of electrical wave propagation in the heart: a review

“…Hartmann et al report that the spiral frequency increases substantially if the size of the domains is smaller than half the spiral wavelength in a large domain and that spirals cannot be sustained for domains smaller than one tenth of the spiral wavelength in a large domain [209]. Other interesting aspects recently found include boundary-induced meander [210] as well as the existence of discrete orbits for particular distances to the boundary [211] as well as changes in the stability of scroll waves near boundaries [212,213].…”

Drift of Scroll Waves in Thin Layers Caused by Thickness Features: Asymptotic Theory and Numerical Simulations

“…[6] and is analogous to the "orbital motion" described in Ref. [13] for localized parametric heterogeneities.…”

Effective dynamics of twisted and curved scroll waves using virtual filaments

“…Thereafter, he applied the Fredholm alternative theorem to obtain necessary conditions on the filament motion, which are the desired equations of motion. The introduction of critical adjoint eigenmodes of the system, which are also known as "response functions" (RFs) [26], formed the basis of many subsequent analytical results on pattern evolution in excitable systems [8,22,25,[27][28][29][30][31][32][33][34][35]. Strikingly, the RFs were observed to be strongly localized near the spiral wave's rotation center, which can be rightfully called "particle-wave dualism of spiral waves" [31].…”