Scroll Rings Research Articles

Pinned Scroll Rings in an Excitable System

Three-dimensional spiral waves in the Belousov-Zhabotinsky reaction are pinned to unexcitable heterogeneities. This pinning can prevent the collapse of scroll rings even if the heterogeneity does not extend along the entire wave filament. In the latter case, frequency differences create stationary gradients in the rotation phase. These twist patterns and their frequencies agree with algebraic solutions of the forced Burgers equation revealing insights into the phase coupling of scroll waves.

Dynamics of Vortex-Wave under a Travelling-Wave Modulation

The mechanism of destabilization is studied for the rotating vortices (scroll waves and spiral waves) in excitable media induced by a parameter modulation in the form of a travelling-wave. It is found that a rigid rotating spiral in the two-dimensional (2D) system undergoes a synchronized drift along a straight line, and a 3D scroll ring with its filament closed into a circle can be reoriented only if the direction of wave number of a travelling-wave perturbation is parallel to the ring plane. Then, in order to describe the behaviour of the synchronized drift of spiral wave and the reorientation of scroll ring, the approximate formulas are given to exhibit qualitative agreements with the observed results.

Three-dimensional spiral waves in an excitable reaction system: Initiation and dynamics of scroll rings and scroll ring pairs

We report experimental results on spiral and scroll waves in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagating concentration waves are detected by two-dimensional photometry and optical tomography. Wave pulses can disappear in front-to-front and front-to-back collisions. This anomaly causes the nucleation of vortices from collisions of three nonrotating waves. In three-dimensional systems, these vortices are scroll rings that rotate around initially circular filaments. Depending on reactant concentrations, the filaments shrink or expand indicating positive and negative filament tensions, respectively. Shrinkage results in vortex annihilation. Expansion is accompanied by filament buckling and bending, which is interpreted as developing Winfree turbulence. We also describe the initiation of scroll ring pairs in four-wave collisions. The two filaments are stacked on top of each other and their motion suggests filament repulsion.

Negative Filament Tension at High Excitability in a Model of Cardiac Tissue

One of the fundamental mechanisms for the onset of turbulence in 3D excitable media is negative filament tension. Thus far, negative tension has always been obtained in media under low excitability. For this reason, its application to normal (nonischemic) cardiac tissue has been questionable, as such cardiac turbulence typically occurs at high excitability. Here, we report expansion of scroll rings (low curvature negative filament tension) in a medium with high excitability by numerical integration of the Luo-Rudy model of cardiac tissue. We discuss the relation between negative tension and the meandering of 2D spiral waves and the possible applications to cardiac modeling.

Stability of scroll ring orientation in an advective field

The stability of the orientation of scroll rings in the excitable Belousov-Zhabotinsky reaction under an applied electrical current was investigated in experiments and simulations. The parallel and antiparallel orientations of the scroll ring unit vector with respect to the current are two stationary states, the first one unstable, the latter stable. For any other orientation, the scroll rings were forced to rotate by the current. At the stable stationary orientation, the scroll rings may contract or expand under the same applied current depending on the radius of the scroll rings. In simulations, delicate adjustments caused a scroll ring to propagate with a constant radius in an advective field.

Reorientation of scroll rings in an advective field

When scroll rings in the excitable Belousov-Zhabotinsky reaction are subjected to an applied electrical current, a reorientation of the scroll ring is induced which is accompanied by a linear drift towards the cathode. The findings can be explained using a modified theory of local filament dynamics under parameter gradients. Numerical simulations using the Oregonator model with an additional advective term accounting for the applied electric field reproduce the experimental results and provide insights into the deformation of the structure of the filament during the reorientation.

Negative filament tension of scroll rings in an excitable system

Scroll rings are three-dimensional spiral waves of excitation that rotate around circular filaments. In a modified Belousov-Zhabotinsky reaction, these filaments expand, buckle, and build up gradients in rotation phase. The instability is caused by negative filament tension (-4.3x10;{-4}cm;{2}s) . Initial deformations are strongest in the direction normal to the filament's osculating plane, and their growth rates decrease rapidly with increasing wave number.

Nucleation and Collapse of Scroll Rings in Excitable Media

We describe a novel nucleation mechanism of scroll rings in three-dimensional reaction-diffusion systems with anomalous dispersion. The vortices form after the collision of two spherical wave fronts from a third, trailing wave that only partially annihilates in the wake of its predecessor. Depending on the relative positions of the three relevant wave sources, one obtains untwisted or twisted scroll rings. The formation of both vortex structures is demonstrated for a modified Belousov-Zhabotinsky reaction.

Periodic forcing of scroll rings and control of Winfree turbulence in excitable media

By simulations of the Barkley model, action of uniform periodic nonresonant forcing on scroll rings and wave turbulence in three-dimensional excitable media is investigated. Sufficiently strong rapid forcing converts expanding scroll rings into the collapsing ones and suppresses the Winfree turbulence caused by the negative tension of wave filaments. Slow strong forcing has an opposite effect, leading to expansion of scroll rings and induction of the turbulence. These effects are explained in the framework of the phenomenological kinematic theory of scroll waves.

Spontaneous scroll ring creation and scroll instability in oscillatory medium with gradients

Scroll waves in a quasi-three-dimensional reaction-diffusion medium with a gradient in the third dimension are studied by numerical simulations using the Fitzhugh-Nagumo model. Under a simple initial condition with only one straight filament, we show a spontaneous creation of twisted scroll rings surrounding the initial filament when the control parameter is increased across a threshold. We find that due to the presence of the gradient, the difference of oscillation frequencies in the third dimension is the underlying cause of the phenomenon. Further increase of the control parameter will lead to the spiral breakups, as a result of the interaction of filaments. Observations in the experiments conducted in the same type of medium with the Belousov-Zhabotinsiky reaction qualitatively agree with our simulation results.

Expanding scroll rings and negative tension turbulence in a model of excitable media

Scroll waves in excitable media, described by the Barkley model, are studied. In the parameter region of weak excitability, negative tension of wave filaments is found. It leads to expansion of scroll rings and instability of wave filaments. A circular filament tends to stretch, bend, loop, and produce an expanding tangle that fills up the volume. The filament does not undergo fragmentation before it touches the boundaries. Statistical properties of such Winfree turbulence of scroll waves are numerically investigated.

Interaction dynamics of a pair of vortex filament rings.

Vortex filament-filament interactions are believed to underlie lethal arrhythmias in cardiac tissue but their dynamics remain poorly understood. We numerically replicate an experimentally postulated reentrant filament configuration as a pair of adjacent circular filaments (scroll rings) with common symmetry axes and varying initial radii and separation distances. The interaction properties are quantified in terms of the scroll-ring lifetime T(L) and direction of initial velocity V0. Two cases were examined, differing only in the direction of the wave around the filament, and observed drastic differences in T(L) between the cases as the separation distance between the rings was decreased. We conclude that ring interactions present unexpected behaviors associated with competing interaction and decay mechanisms.

Parametric forcing of scroll-wave patterns in three-dimensional excitable media

The dynamics of axisymmetric and twisted scroll rings under homogeneous periodic forcing is studied numerically. The collapse of axisymmetric rings can be enhanced or retarded significantly with resonant forcing. Twisted scroll rings exhibit the phenomenon of resonant drift in a direction normal to the axis of the central filament. The scaling of mean and fluctuating axial drift due to forcing is determined. Evidence is provided showing that the appropriate symmetry group for the dynamics of the unforced twisted scroll ring is SE(2)× R . Ordinary differential equations based on symmetry considerations explain the dynamics without resorting to laws of filament motion and the local geometry hypothesis.

Lifetime enhancement of scroll rings by spatiotemporal fluctuations

The dynamics of three-dimensional scroll rings with spatiotemporal random excitability is investigated numerically using the FitzHugh-Nagumo model. Depending on the correlation time and length scales of the fluctuations, the lifetime of the ring filament is enlarged and a resonance effect between the time scale of the scroll ring and the time correlation of the noise is observed. Numerical results are interpreted in terms of a simplified stochastic model derived from the kinematical equations for three-dimensional excitable waves.

Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications.

In a weakly excitable medium, characterized by a large threshold stimulus, the free end of an isolated broken plane wave (wave tip) can either rotate (steadily or unsteadily) around a large excitable core, thereby producing a spiral pattern, or retract, causing the wave to vanish at boundaries. An asymptotic analysis of spiral motion and retraction is carried out in this weakly excitable large core regime starting from the free-boundary limit of the reaction-diffusion models, valid when the excited region is delimited by a thin interface. The wave description is shown to naturally split between the tip region and a far region that are smoothly matched on an intermediate scale. This separation allows us to rigorously derive an equation of motion for the wave tip, with the large scale motion of the spiral wave front slaved to the tip. This kinematic description provides both a physical picture and exact predictions for a wide range of wave behavior, including (i) steady rotation (frequency and core radius), (ii) exact treatment of the meandering instability in the free-boundary limit with the prediction that the frequency of unstable motion is half the primary steady frequency, (iii) drift under external actions (external field with application to axisymmetric scroll ring motion in three dimensions, and spatial- or/and time-dependent variation of excitability), and (iv) the dynamics of multiarmed spiral waves with the prediction that steadily rotating waves with two or more arms are linearly unstable. Numerical simulations of FitzHugh-Nagumo kinetics are used to test several aspects of our results. In addition, we discuss the semiquantitative extension of this theory to finite cores and pinpoint mathematical subtleties related to the thin interface limit of singly diffusive reaction-diffusion models.

Dynamics of scroll rings in a parameter gradient

Experimental observations of scroll rings in excitable media have shown that an externally applied parameter gradient can significantly alter the dynamics of the ring, leading to rotation of the plane of the ring, variations in its rate of collapse, and distortions in its shape. Here we propose a theory to account for these effects by extending the theory of local filament dynamics to include the influence of the parameter gradient. Numerical simulations using the FitzHugh-Nagumo equations show that the proposed theory works well when the differences in rotation periods at different parts of the filament are sufficiently small. When this condition does not hold, the theory must be modified to take into account nonlocal interactions.

Formation and evolution of scroll waves in photosensitive excitable media.

Experimental and computational studies of the formation and evolution of scroll waves in three-dimensional excitable media are presented. Scroll waves are initiated in the photosensitive Belousov-Zhabotinsky reaction by perturbing traveling waves transverse to their direction of propagation. Scroll rings are generated by perturbing circular waves, which expand or contract depending on the strength of an imposed excitability gradient and its direction relative to the rotational direction of the scroll wave. (c) 1998 American Institute of Physics.

The dynamics of scroll wave filaments in the complex Ginzburg-Landau equation

An analytical treatment is presented for scroll waves in the complex Ginzburg-Landau equation in the limit of small filament curvature, torsion, and phase twist. Explicit expressions for the filament velocity and frequency shift are found. The theoretical results are verified numerically in the case of circular untwisted scroll rings and for straight and sinusoidal scroll filaments with phase twist.

Interactions of spiral waves in inhomogeneous excitable media

This paper examines the behavior of pairs of spiral waves in a two-dimensional excitable medium with a parameter gradient, and extends the results to study the effects of self-interactions of scroll waves in three dimensions. When two spirals are placed in regions of different local rotation periods, the interaction between the waves significantly affects their motion. There are two distinct phases of the interaction: initially, the spiral with faster rotation period “unwinds” the slower one. After the unwinding, the wavefronts from the faster spiral interact directly with the center of rotation (core) of the slower one, producing a drift that is significantly faster than that due to the gradient alone. A simple point-source model yields a prediction of the unwinding time, and the motion after unwinding can be understood in terms of planar wavefronts interacting with the core of the slower spiral. The results are then applied to self-interactions of a scroll ring in three dimensions, where they suggest that nonlocal effects are significant when a parameter gradient is present.

Dynamics of Spiral Waves on Unbounded Domains Using Center-Manifold Reductions

An equivariant center-manifold reduction near relative equilibria ofG-equivariant semiflows on Banach spaces is presented. In contrast to previous results, the Lie groupGinduces a strongly continuous group action on the underlying function space. In fact,Gmay act discontinuously. The results are applied to bifurcations of stable patterns arising in reaction–diffusion systems on the plane or in three-space modeling chemical systems such as catalysis on platinum surfaces and Belousov–Zhabotinsky reactions. These systems are equivariant under the Euclidean symmetry group. Hopf bifurcations from rigidly-rotating spiral waves to meandering or drifting waves and from twisted scroll rings are investigated.