Meander pattern of spiral wave and the spatial distribution of its cycle length

One of the most interesting dynamics of rotating spiral waves in an excitable medium is meandering. The tip of a meandering spiral wave moves along a complex trajectory, which often takes the shape of an epitrochoid or hypotrochoid with inward or outward petals. The cycle lengths (CLs) of a meandering spiral wave are not constant; rather, they depend on the meandering dynamics. In this paper, we show that the CLs take two mean values, outside T^{out} and inside T^{in} the meandering trajectory with a ratio of T^{in}/T^{out}=(n+1)/n for the inward and (n-1)/n for the outward petals, where n is the number of petals in the tip trajectory. We illustrate this using four models of excitable media and prove this result. These formulas are shown to be suitable for a meandering spiral wave in an anatomical model of the heart. We also show that the effective periods of overdrive pacing of meandering spiral waves depend on the electrode location relative to the tip trajectory. Overall, our approach can be used to study the meandering pattern from the CL data; it should work for any type of dynamics that produces complex tip trajectories of the spiral wave, for example, for a drift due to heterogeneity.

Response function framework for the dynamics of meandering or large-core spiral waves and modulated traveling waves

In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a unified theoretical framework for the drift of such patterns under small external perturbations, in terms of overlap integrals between the perturbation and the adjoint critical eigenfunctions of the linearized operator (i.e., response functions). For spiral waves, the finite radius of the spiral tip trajectory and spiral wave meander are taken into account. Different coordinate systems can be chosen, depending on whether one wants to predict the motion of the spiral-wave tip, the time-averaged tip path, or the center of the meander flower. The framework is applied to analyze the drift of a meandering spiral wave in a constant external field in different regimes.

A Huygens principle for diffusion and anomalous diffusion in spatially extended systems

We present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media. For anisotropic systems, strong chaos leads to diffusive behavior (Brownian motion with drift) and weak chaos leads to superdiffusive behavior (Lévy processes with drift). For isotropic systems, the drift term vanishes and strong chaos again leads to Brownian motion. We establish the existence of a nonlinear Huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behavior in odd space dimensions.

Wave-particle dualism of spiral waves dynamics

We demonstrate and explain a wave-particle dualism of such classical macroscopic phenomena as spiral waves in active media. That means although spiral waves appear as nonlocal processes involving the whole medium, they respond to small perturbations as effectively localized entities. The dualism appears as an emergent property of a nonlinear field and is mathematically expressed in terms of the spiral waves response functions, which are essentially nonzero only in the vicinity of the spiral wave core. Knowledge of the response functions allows quantitatively accurate prediction of the spiral wave drift due to small perturbations of any nature, which makes them as fundamental characteristics for spiral waves as mass is for the condensed matter.

Chaos in physiology

A bird's eye view of the role of nontrivial dynamical phenomena in physiological systems is presented. Many levels in the physiological hierarchy are affected. It appears that physiology has a particularly high affinity to chaos. The old central place of physiology in the circle of sciences can perhaps be regained if the two disciplines join resources. Even physics may turn out to be dependent on the chaotic micro dynamics of the brain.

Spatiotemporal irregularities of spiral wave activity in isolated ventricular muscle

Voltage-sensitive dyes and high resolution optical mapping were used to analyze the characteristics of spiral waves of excitation in isolated ventricular myocardium. In addition, analytical techniques, which have been previously used in the study of the characteristics of spiral waves in chemical reactions, were applied to determine the voltage structure of the center of the rotating activity (ie, the core). During stable spiral wave activity local activation occurs in a periodic fashion (ie, 1:1 stimulus: response activation ratio) throughout the preparation, except at the core, which is a small elongated area where the activity is of low voltage and the activation ratio is 1:0. The voltage amplitude increases gradually from the center of the core to the periphery. In some cases, however, regular activation patterns at the periphery may coexist with irregular local activation patterns near the core. Such a spatiotemporal irregularity is attended by variations in the core size and shape and results from changes in the core position. The authors conclude that functionally determined reentrant activity in the heart may be the result of spiral waves of propagation and that local spatiotemporal irregularities in the activation pattern are the result of changes in the core position.