Three-dimensional Excitable Media Research Articles

Influence of Acidity on Spiral Waves in a Bubble-Free Belousov-Zhabotinsky Reaction with Pyrogallol

Spiral waves are ubiquitously observed in a variety of physical and biological systems, including superconductors, superfluids, CO-oxidation on platinum surfaces, cell aggregation of slime mold and arrhythmia in cardiac tissues. Such spiral waves are uniquely explained by a reaction-diffusion mechanism. Due to easy preparation and convenient detection, the excitable chemical Belousov-Zhabotinsky (BZ) reaction is employed to study spiral waves in experiments. We studied the influence of initial concentration of H 2 SO 4 ((H 2 SO 4 )) on the dynamics of spiral waves in a thin layer of the BZ reaction with pyrogallol. This reaction has an advantage over the classical BZ reaction with malonic acid, as it is bubble-free. We found that the spiral tip, i.e., the organizing center, moved along so-called meandering trajectories with three or four outward petals. In addition, the area occupied by the spiral tip decreased when (H 2 SO 4 ) was increased. We further investigated the dynamics far from the organizing center by measuring properties of propagating fronts. An increase of (H 2 SO 4 ) resulted in a simultaneous decrease of the wavelength and wave period. In contrast, the wave speed grew with (H 2 SO 4 ). Since disturbances by the byproduct CO 2 bubbles are avoided and the wave velocity is sufficiently low, the results present a suitable guideline for further investigations on propagating excitation waves in two- and three-dimensional excitable media, especially observations of wave instabilities in three-dimensional systems using optical tomography.

Interaction of a Pair of Parallel Scroll Waves

Interactions of pairs of scroll waves in three-dimensional excitable media were studied experimentally in the Belousov-Zhabotinsky reaction by optical tomography. The behavior of two scroll waves depended on the distance d between their filaments. When the interfilament distance was shorter than the wavelength λ of the scroll waves (but larger than the diameter of the spiral core), the filaments repelled each other. Once d ≈ λ, the two scroll waves synchronized, rotating around their filaments with both a common rotation frequency and a common pitch. The interfilament distance of synchronized scroll waves did not change. When fluctuations broke the symmetry of the rotation periods, the scroll with higher rotation frequency displaced the slower rotating one, until the latter was ousted or even annihilated. These behaviors were independent of the sense of rotation (co- or counter-rotating), the filament dynamics (rigidly rotating or meandering tip motion in two-dimensional media), and the presence or absence of a gradient of excitability parallel to the filaments.

Critical role of pinning defects in scroll-wave breakup in active media

The breakup of rotating scroll waves in three-dimensional excitable media has been linked to important biological processes. The known mechanisms for this transition almost exclusively involve the dynamics of the scroll filament, i.e., the line connecting the phase singularities. In this paper, we describe a novel defect-induced route to breakup of a scroll wave pinned by an inexcitable obstacle partially extending through the bulk of the medium. The wave is helically wound around the defect inducing sudden changes in velocity components of the wavefront at the obstacle boundary. This results in breakup far from the filament, eventually giving rise to spatiotemporal chaos. Our results suggest a potentially critical role of pinning obstacles in the onset of life-threatening disturbances of cardiac activity.

Scroll wave meandering induced by phase difference in a three-dimensional excitable medium

We investigated scroll waves in an inhomogeneous excitable 3D system with gradient of excitability. The gradient promotes twisting of the scroll waves. Sufficiently large excitability gradient enhances the twisting and causes simple scroll waves to transition to meandering scroll waves. For the twist-induced instability of scroll waves, we analyzed the stability of 2D spiral waves sliced from the twisted scroll in the vertical direction. The 3D problem is simplified by taking into account the diffusive coupling in the third direction as a time-delayed perturbation to the 2D spiral wave. An additional "negative mass" term measuring the twist thus arises in the 2D system and induces the transition from simple rotation to meandering. A further increase in the gradient ruins partially the unity of the meandering scrolls and generates semiturbulence, the analogs of which were observed in the Belousov-Zhabotinski reaction. We also generated the phase diagram in the parameter space by adjusting the threshold for excitation of the media.

Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics

Scroll waves are vortices that occur in three-dimensional excitable media. Scroll waves have been observed in a variety of systems including cardiac tissue, where they are associated with cardiac arrhythmias. The disorganization of scroll waves into chaotic behavior is thought to be the mechanism of ventricular fibrillation, whose lethality is widely known. One possible mechanism for this process of scroll wave instability is negative filament tension. It was discovered in 1987 in a simple two variables model of an excitable medium. Since that time, negative filament tension of scroll waves and the resulting complex, often turbulent dynamics was studied in many generic models of excitable media as well as in physiologically realistic models of cardiac tissue. In this article, we review the work in this area from the first simulations in FitzHugh-Nagumo type models to recent studies involving detailed ionic models of cardiac tissue. We discuss the relation of negative filament tension and tissue excitability and the effects of discreteness in the tissue on the filament tension. Finally, we consider the application of the negative tension mechanism to computational cardiology, where it may be regarded as a fundamental mechanism that explains differences in the onset of arrhythmias in thin and thick tissue.

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.

Filament-Induced Surface Spiral Turbulence in Three-Dimensional Excitable Media

Filament-induced surface defect-mediated turbulence in bounded three-dimensional (3D) excitable media is investigated in the regime of negative line tension. In this regime turbulence arises due to unstable filaments associated with scroll waves and is purely a 3D phenomenon. It is shown that the statistical properties of the turbulent defect dynamics can be used to distinguish surface defect-mediated turbulence from its 2D analog. Mechanisms for the creation and annihilation of surface defects are discussed and Markov rate equations are employed to model the results.

Control of scroll-wave turbulence using resonant perturbations

Turbulence of scroll waves is a sort of spatiotemporal chaos that exists in three-dimensional excitable media. Cardiac tissue and the Belousov-Zhabotinsky reaction are examples of such media. In cardiac tissue, chaotic behavior is believed to underlie fibrillation which, without intervention, precedes cardiac death. In this study we investigate suppression of the turbulence using stimulation of two different types, "modulation of excitability" and "extra transmembrane current." With cardiac defibrillation in mind, we used a single pulse as well as repetitive extra current with both constant and feedback controlled frequency. We show that turbulence can be terminated using either a resonant modulation of excitability or a resonant extra current. The turbulence is terminated with much higher probability using a resonant frequency perturbation than a nonresonant one. Suppression of the turbulence using a resonant frequency is up to fifty times faster than using a nonresonant frequency, in both the modulation of excitability and the extra current modes. We also demonstrate that resonant perturbation requires strength one order of magnitude lower than that of a single pulse, which is currently used in clinical practice to terminate cardiac fibrillation. Our results provide a robust method of controlling complex chaotic spatiotemporal processes. Resonant drift of spiral waves has been studied extensively in two dimensions, however, these results show for the first time that it also works in three dimensions, despite the complex nature of the scroll wave turbulence.

Topological Aspect of Knotted Vortex Filaments in Excitable Media

Scroll waves exist ubiquitously in three-dimensional excitable media. The rotation centre can be regarded as a topological object called the vortex filament. In three-dimensional space, the vortex filaments usually form closed loops, and can be even linked and knotted. We give a rigorous topological description of knotted vortex filaments. By using the φ-mapping topological current theory, we rewrite the topological current form of the charge density of vortex filaments, and using this topological current we reveal that the Hopf invariant of vortex filaments is just the sum of the linking and self-linking numbers of the knotted vortex filaments. We think that the precise expression of the Hopf invariant may imply a new topological constraint on knotted vortex filaments.

Negative filament tension in the Luo-Rudy model of cardiac tissue

Scroll waves are vortices that occur in three-dimensional excitable media. Scroll waves have been observed in a variety of systems including cardiac tissue, where they are associated with cardiac arrhythmias. The disorganization of scroll waves into chaotic behavior is thought to be the mechanism of ventricular fibrillation, which lethality is widely known. One of the possible mechanisms of scroll wave instability is negative filament tension, which was studied theoretically using low-dimensional models of excitable medium. In this article we perform a numerical study of negative filament tension using the Luo-Rudy phase 1 model, which is widely used in cardiac electrophysiology. We show that this instability exists in this model, study its manifestation and discuss its relation to cardiac arrhythmogenesis.

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.

An elegant method to study an isolated spiral wave in a thin layer of a batch Belousov–Zhabotinsky reaction under oxygen-free conditions

A method to prepare a uniform thin layer of a batch Belousov-Zhabotinsky (BZ) reaction under oxygen-free conditions for the study of an isolated spiral wave is presented. After a first layer of gel soaked with the BZ solution has been delivered into the reactor, a single spiral wave was initiated, and finally the remaining reactor volume was filled with gel and BZ medium. The completely filled reactor is sealed gas-tightly, yielding oxygen-free, and thus more controlled, reaction conditions. A systematic study of the behaviour of an isolated spiral wave in a ferroin-catalyzed BZ reaction under batch conditions has been performed. Recipes for BZ media that support a slowly rotating meandering spiral were developed. In cases of extremely low excitability (i.e., relative large stimuli are required to induce a propagating wave), the number of petals in the trajectory of a spiral tip decreased due to aging of the reaction system. Since oxygen-free conditions are necessary for the study of the dynamics in three-dimensional excitable media, and the wave velocities of a spiral are sufficiently low, the developed chemical recipes are suitable for studies of the behaviour of scroll waves in three-dimensional systems by optical tomography.

Scroll waves meandering in a model of an excitable medium

We study numerically the dynamics of a scroll wave in a three-dimensional (3D) excitable medium in the presence of substantial meandering of the corresponding 2D spiral wave in the Aliev-Panfilov model. We identify three types of dynamics of the scroll wave filament--quasi-2D, periodic, and aperiodic meandering--and we study their dependence on parameter settings and thickness of the medium.

Suppress Winfree Turbulence by Local Forcing Excitable Systems

The occurrence of Winfree turbulence is currently regarded as one of the principal mechanisms underlying cardiac fibrillation. We develop a local stimulation method that suppresses Winfree turbulence in three-dimensional excitable media. We find that Winfree turbulence can be effectively suppressed by locally injecting periodic signals to only a very small subset (around some surface region) of total space sites. Our method for the first time demonstrates the effectiveness of local low-amplitude periodic excitations in suppressing turbulence in 3D excitable media and has fundamental improvements in efficiency, convenience, and turbulence suppression speed compared with previous strategies. Therefore, it has great potential for developing into a practical low-amplitude defibrillation approach.

Influence of nonexcitable cells on spiral breakup in two-dimensional and three-dimensional excitable media.

We study spiral wave dynamics in the presence of nonexcitable cells in two-dimensional (2D) and three-dimensional (3D) excitable media, described by the Aliev-Panfilov model. We find that increasing the percentage of randomly distributed nonexcitable cells can prevent the breaking up of a spiral wave into a complex spatiotemporal pattern. We show that this effect is more pronounced in 2D than 3D excitable media. We explain the observed 2D vs 3D differences by a different dependence of the period and diastolic interval of the spiral wave on the percentage of nonexcitable cells in 2D and 3D excitable media.

Spiral waves in three-dimensional excitable media with light-sensitive reaction

The propagation of spiral waves in three-dimensional excitable media with and without forcing is studied numerically, and it is shown that, when localized forcing is imposed in a cube located in the middle of the computational domain, the spiral wave preserves its integrity, although it exhibits a dent along the cube’s periphery and is unable to penetrate into the cube. However, when forcing is applied in a parallelepiped which extends across the entire domain in the vertical direction and is located in the middle of the three-dimensional excitable medium, the spiral wave is destroyed by the strip forcing and two active regions are created on either side of the strip. These active regions are characterized by high concentration of the activator, move towards the boundaries parallel to the strip, and, upon reaching them, move rather quickly in a direction parallel to the strip. They also form complex spatial patterns, e.g., ear-shaped regions, and convoluted annular regions on both sides of the strip, where the activator’s concentration is small. These spatio-temporal patterns are periodic with a period and a width which are larger than those corresponding to the localized forcing. It is also shown that strip forcing not only destroys the spiral wave, it also increases the period and width of the breathing-type motions that result from such a destruction.

Taming Winfree turbulence of scroll waves in excitable media.

Winfree turbulence of scroll waves is a special kind of spatiotemporal chaos that exists exclusively in three-dimensional excitable media and is currently considered one of the principal mechanisms of cardiac fibrillation. A chaotic wave pattern develops through the negative-tension instability of vortex filaments, which tend to spontaneously stretch, bend, loop, and produce an expanding tangle that fills up the volume. We demonstrate that such turbulence can readily be controlled by weak nonresonant modulation of the medium excitability. Depending on the forcing frequency, either suppression or induction of turbulence can be achieved.

Large-excitability asymptotics for scroll waves in three-dimensional excitable media.

Three-dimensional scroll waves are considered in a reaction-diffusion model of excitable media in the large excitability limit. Coordinates based on the scroll filament are defined and shown to provide a natural extension of the coordinates used for two-dimensional spiral waves. The leading-order free-boundary equations for interface motion in three dimensions are explicitly derived in these coordinates. Three specific examples are considered: straight twisted scroll waves, axisymmetric scroll waves, and helical scroll waves. The equations for the fields at leading and first order in the core region are given.

Three-dimensional organisation of re-entrant propagation during experimental ventricular fibrillation

Ventricular fibrillation is believed to be produced by the breakdown of re-entrant propagation waves of excitation into multiple re-entrant sources. These re-entrant waves may be idealised as spiral waves in two-dimensional, and scroll waves in three-dimensional excitable media. Optically monitored, simultaneously recorded endocardial and epicardial patterns of activation on the ventricular wall do not always show spiral waves. Analysis of optically recorded irregular electrical wave activity on the surface of the heart during experimentally induced fibrillation reveals a strong local temporal periodicity. The spatial distribution of the dominant temporal frequencies of excitation has a domain organisation. The domains are large (≈1 cm 2) and they persist for minutes. We show that numerical simulations, even with a simple homogeneous excitable medium, can reproduce the key features of the simultaneous endo- and epicardial visualisations of propagating activity, and so these recordings may be interpreted in terms of scroll waves within the ventricular wall. The domain structure can be reproduced in a two-dimensional excitable medium governed by the FitzHugh–Nagumo equations with a spatial inhomogeneity. We identified two potential mechanisms that may contribute to the observed experimental dynamics: coexistence of stable spiral waves with non-commensurate frequencies of rotation, and Wenckebach-like frequency division from a single spiral source due to inhomogeneity. Both mechanisms reproduce the uniformity of the dominant frequency within individual domains and sharp boundaries between domains. The possibility of distinguishing between different mechanisms using Lissajous figures is discussed.

Structure and variability of spinning reaction waves in three-dimensional excitable media

A mathematical simulation for a reaction wave that propagates in a cylindrical sample is performed. The propagation modes that have not yet been observed experimentally are predicted. The areas of existence for these modes have been determined.