Stability of scroll ring orientation in an advective field

Resonance of scroll rings with periodic external fields in excitable media

By direct numerical simulations of a chemical reaction-diffusion system coupled to a periodic external AC electric field with frequency equal to double frequency of the scroll wave rotation, we find that scroll rings resonate with the electric field and exhibit various dynamical behaviors, for example, their reversals, collapses, or growths, depending both on the initial phase of AC electric fields and on the initial phase of scroll rings. A kinematical model characterizing the drift velocity of the scroll rings along their radial directions as well as that of the scroll rings along their symmetry axes is proposed, which can effectively account for the numerical observations and predict the behaviors of the scroll rings. Besides, the existence of the equilibrium state of a scroll ring under the AC electric fields is predicted by the kinematical model and the predictions agree well with the simulations.

Dynamics of scroll waves in a cylinder jacket geometry

The dynamics of scroll waves in a narrow cylinder jacket-shaped reactor is investigated experimentally by optical tomography. The fate of the scroll waves of excitation in the Belousov-Zhabotinsky reaction depends on the thickness of the cylinder jacket. While at sufficiently wide cylinder jackets vertically oriented scroll waves remain stable, the probability that the filament of a scroll hits a lateral wall increases as the cylinder jacket narrows. This may lead to the rupture of the initial filament and pinning of the filament ends at the lateral walls. Filaments that pin to opposite lateral walls shrink and reorient to a horizontal orientation; such a reorientation corresponds to a transition from an intramural to a transmural scroll wave. The kinetics of the reorientation and shrinkage of the scrolls were studied. Furthermore, we find that no new filaments were generated upon collision of excitation waves at the side of the cylinder jacket opposite to the scroll wave. Thus, under the studied conditions, we do not observe any new generation of filaments due to a phenomenon like reentry.

Dynamics of Scroll Wave in a Three-Dimensional System with Changing Gradient

The dynamics of a scroll wave in an excitable medium with gradient excitability is studied in detail. Three parameter regimes can be distinguished by the degree of gradient. For a small gradient, the system reaches a simple rotating synchronization. In this regime, the rigid rotating velocity of spiral waves is maximal in the layers with the highest filament twist. As the excitability gradient increases, the scroll wave evolutes into a meandering synchronous state. This transition is accompanied by a variation in twisting rate. Filament twisting may prevent the breakup of spiral waves in the bottom layers with a low excitability with which a spiral breaks in a 2D medium. When the gradient is large enough, the twisted filament breaks up, which results in a semi-turbulent state where the lower part is turbulent while the upper part contains a scroll wave with a low twisting filament.

Phase-locked scroll waves defy turbulence induced by negative filament tension

Scroll waves in a three-dimensional media may develop into turbulence due to negative tension of the filament. Such negative tension-induced instability of scroll waves has been observed in the Belousov-Zhabotinsky reaction systems. Here we propose a method to restabilize scroll wave turbulence caused by negative tension in three-dimensional chemical excitable media using a circularly polarized (rotating) external field. The stabilization mechanism is analyzed in terms of phase-locking caused by the external field, which makes the effective filament tension positive. The phase-locked scroll waves that have positive tension and higher frequency defy the turbulence and finally restore order. A linear theory for the change of filament tension caused by a generic rotating external field is presented and its predictions closely agree with numerical simulations.

Dependence of scroll-wave dynamics on the orientation of a gradient of excitability

The dynamics of scroll waves with a variable orientation to a vertically oriented gradient of excitability is studied by optical tomography in the ferroin-catalyzed Belousov-Zhabotinsky reaction. An almost perpendicular orientation between the scroll wave and gradient induces a pair of twists of opposite handedness on the scroll wave. The position of the nodal plane formed between the twists is governed by the time delay of the twist formation and therefore leads to a symmetric or asymmetric twisted scroll wave. Larger inclinations between scroll wave and gradient cause a drift of the filament along the reactor wall until it reaches the bottom of the reaction container. In this case, the scroll wave does not twist, suggesting that a drift acts as an alternative mechanism of responding to the gradient.

Twists of opposite handedness on a scroll wave

The dynamic interaction of scroll waves in the Belousov-Zhabotinsky reaction with a vertically orientated gradient of excitability is studied by optical tomography. This study focuses on scroll waves, whose filaments were oriented almost perpendicular to the gradient. Whereas scroll waves with filaments exactly perpendicular to the gradient remain unaffected, filaments with a component parallel to the gradient develop a twist. Scroll waves with U-shaped filaments exhibit twists starting from both of its ends, resulting in scroll waves whose filaments display a pair of twists of opposite handedness. These twists are separated by a nodal plane where the filament remains straight and untwisted. The experimental findings were reproduced by numerical simulations using the Oregonator model and a linear gradient of excitability almost perpendicular to the orientation of the filament.

Helical deformation of the filament of a scroll wave

In three-dimensional excitable systems scroll waves may lose their originally straight shape through different instabilities. In experiments, the formation of zigzag-shaped or helical filaments is often observed. Such a deformation may be due to either a twist-induced instability or a 3D meandering instability. We performed a systematic study using a Belousov-Zhabotinsky (BZ) system with a vertical gradient of excitability and determined the necessary twist for the onset of undulation in the filament. Thus, we demonstrate that in the case of a BZ system with a gradient parallel to the filament, the deformation of the filament is induced by a twist-induced instability.

Stationary vortex loops induced by filament interaction and local pinning in a chemical reaction-diffusion system

Scroll rings are three-dimensional excitation waves rotating around one-dimensional filament loops. In experiments with the Belousov-Zhabotinsky reaction we show that the collapse of these loops can be stopped by local pinning to only two unexcitable heterogeneities. The resulting vortices rotate around stationary but curved filaments. The absence of filament motion can be explained by repulsive interaction that counteracts the expected curvature-induced motion. The shape and key dependencies of the stationary filaments are well described by a curvature-flow model with additive interaction velocities that rapidly decrease with filament distance.

Surfactant-induced gradients in the three-dimensional Belousov-Zhabotinsky reaction

Scroll waves are prominent patterns formed in three-dimensional excitable media, and they are frequently considered highly relevant for some types of cardiac arrhythmias. Experimentally, scroll wave dynamics is often studied by optical tomography in the Belousov-Zhabotinsky reaction, which produces CO(2) as an undesired product. Addition of small concentrations of a surfactant to the reaction medium is a popular method to suppress or retard CO(2) bubble formation. We show that in closed reactors even these low concentrations of surfactants are sufficient to generate vertical gradients of excitability which are due to gradients in CO(2) concentration. In reactors open to the atmosphere such gradients can be avoided. The gradients induce a twist on vertically oriented scroll waves, while a twist is absent in scroll waves in a gradient-free medium. The effects of the CO(2) gradients are reproduced by a numerical study, where we extend the Oregonator model to account for the production of CO(2) and for its advection against the direction of gravity. The numerical simulations confirm the role of solubilized CO(2) as the source of the vertical gradient of excitability in reactors closed to the atmosphere.

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.

Alternative stable scroll waves and conversion of autowave turbulence

Rotating spiral and scroll waves (vortices) are investigated in the FitzHugh-Nagumo model of excitable media. The focus is on a parameter region in which there exists bistability between alternative stable vortices with distinct periods. Response functions are used to predict the filament tension of the alternative scrolls and it is shown that the slow-period scroll has negative filament tension, while the filament tension of the fast-period scroll changes sign within a hysteresis loop. The predictions are confirmed by direct simulations. Further investigations show that the slow-period scrolls display features similar to delayed after-depolarization and tend to develop into turbulence similar to ventricular fibrillation (VF). Scrolls with positive filament tension collapse or stabilize, similar to monomorphic ventricular tachycardia (VT). Perturbations, such as boundary interaction or shock stimulus, can convert the vortex with negative filament tension into the vortex with positive filament tension. This may correspond to transition from VF to VT unrelated to pinning.