Vortex Turbulence due to the Interplay of Filament Tension and Rotational Anisotropy

The mechanism of scroll wave turbulence is investigated in excitable media with rotational anisotropy. We adopt the Barkley model with heterogeneity in the diffusion constants. Through comparative numerical studies, we demonstrate the vortex turbulence results from the rotational anisotropy＇s cooperation with negative filament tension or competition with positive filament tension. The presence of rotational anisotropy can enlarge the parameter region leading to negative-tension induced wave turbulence in isotropic media.     

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.     

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.     

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.     

Scroll waves in spherical shell geometries

The evolution of scroll waves in excitable media with spherical shell geometries is studied as a function of shell thickness and outer radius. The motion of scroll wave filaments that are the locii of phaseless points in the medium and organize the wave pattern is investigated. When the inner radius is sufficiently large the filaments remain attached to both the inner and outer surfaces. The minimum size of the sphere that supports spiral waves and the maximum number of spiral waves that can be sustained on a sphere of given size are determined for both regular and random initial distributions. When the inner radius is too small to support spiral waves the filaments detach from the inner surface and form a curved filament connecting the two spiral tips in the surface. In certain parameter domains the filament is an arc of a circle that shrinks with constant shape. For parameter values close to the meandering border, the filament grows and collisions with the sphere walls lead to turbulent filament dynamics. (c) 2001 American Institute of Physics.     

Intermittent self-organization of scroll wave turbulence in three-dimensional excitable media

We study the asymptotic behavior of scroll wave turbulence in large three-dimensional excitable media modeled by FitzHugh-Nagumo equations. The focus is on the type of turbulence caused by negative tension of scroll wave filaments, which is considered to be one of the mechanisms of cardiac fibrillation. We discovered that the initial increase in turbulence complexity can be followed by intermittent self-organization, when complex filament tangles are replaced by a small number of relatively stable triple filament strands. The intermittency is the result of a competition between the destabilizing effect of negative tension and mutual attraction of filaments with similar orientation.     

Excitable patterns in active nematics

We analyze a model of mutually propelled filaments suspended in a two-dimensional solvent. The system undergoes a mean-field isotropic-nematic transition for large enough filament concentrations, and the nematic order parameter is allowed to vary in space and time. We show that the interplay between nonuniform nematic order, activity, and flow results in spatially modulated relaxation oscillations, similar to those seen in excitable media. In this regime the dynamics consists of nearly stationary periods separated by "bursts" of activity in which the system is elastically distorted and solvent is pumped throughout. At even higher activity, the dynamics becomes chaotic.     

Cookbook asymptotics for spiral and scroll waves in excitable media

Algebraic formulas predicting the frequencies and shapes of waves in a reaction-diffusion model of excitable media are presented in the form of four recipes. The formulas themselves are based on a detailed asymptotic analysis (published elsewhere) of the model equations at leading order and first order in the asymptotic parameter. The importance of the first order contribution is stressed throughout, beginning with a discussion of the Fife limit, Fife scaling, and Fife regime. Recipes are given for spiral waves and detailed comparisons are presented between the asymptotic predictions and the solutions of the full reaction-diffusion equations. Recipes for twisted scroll waves with straight filaments are given and again comparisons are shown. The connection between the asymptotic results and filament dynamics is discussed, and one of the previously unknown coefficients in the theory of filament dynamics is evaluated in terms of its asymptotic expansion. (c) 2002 American Institute of Physics.     

Effects of reduced discrete coupling on filament tension in excitable media

Wave propagation in the heart has a discrete nature, because it is mediated by discrete intercellular connections via gap junctions. Although effects of discreteness on wave propagation have been studied for planar traveling waves and vortexes (spiral waves) in two dimensions, its possible effects on vortexes (scroll waves) in three dimensions are not yet explored. In this article, we study the effect of discrete cell coupling on the filament dynamics in a generic model of an excitable medium. We find that reduced cell coupling decreases the line tension of scroll wave filaments and may induce negative filament tension instability in three-dimensional excitable lattices.     

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.     

Varieties of spiral wave behavior: An experimentalist's approach to the theory of excitable media

Spiral waves in diverse excitable media exhibit strikingly variegated behavior. Mechanistic interpretations of excitability in laboratory systems are commonly tested by comparing the wavelength, period, and meander patterns of the model's spiral waves with laboratory observations, but models seem seldom to be rejected by such tests. The reason may be that almost any excitable medium behaves in many respects like almost any other, if its parameters are properly adjusted within a reasonable range. What generalizations can be made about "excitable media" in the absence of more specifications? It would be useful to distinguish such generic features from idiosyncrasies of specific models. The range of behavioral flexibility of the FitzHugh-Nagumo excitable medium is explored by varying two of its parameters and comparing the results with other excitable media to suggest a generic pattern of parameter dependence. The results exhibit the remarkable diversity of rotor behavior in a single model and provide a database for quantitative testing of mathematical generalizations.     

Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue

It has been hypothesized that stationary scroll wave filaments in cardiac tissue describe a geodesic in a curved space whose metric is the inverse diffusion tensor. Several numerical studies support this hypothesis, but no analytical proof has been provided yet for general anisotropy. In this Letter, we derive dynamic equations for the filament in the case of general anisotropy. These equations are covariant under general spatial coordinate transformations and describe the motion of a stringlike object in a curved space whose metric tensor is the inverse diffusion tensor. Therefore the behavior of scroll wave filaments in excitable media with anisotropy is similar to the one of cosmic strings in a curved universe. Our dynamic equations are valid for thin filaments and for general anisotropy. We show that stationary filaments obey the geodesic equation.     

Scroll wave instabilities in an excitable chemical medium

Two kinds of scroll wave instabilities were studied experimentally in the excitable Belousov-Zhabotinsky reaction: three-dimensional meandering and negative line tension of the scroll wave filament. The filament displays a flat zigzag shape in the initial stages of the experiment. As the chemical medium ages, the filament assumes a wiggly shape while its length increases substantially. Numerical simulations underpin the experimental findings and their interpretation.     

Chirality dependent component of vortex advection in excitable media

An advective field induces drift of a vortex in excitable media. The component of the drift velocity C( perpendicular ) perpendicular to the field is known to change its sign with the chirality of the vortex. In an experiment with vortices in an electric field in a chemical excitable medium, we have found unexpectedly that C( perpendicular ) changes its sign also independently of chirality with changing composition of the medium. We did not succeed to explain this phenomenon by using existing mathematical models of chemical excitable media. The experiment described calls for more realistic models.(c) 1999 American Institute of Physics.     

A normal form for excitable media

We present a normal form for traveling waves in one-dimensional excitable media in the form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behavior of single pulses in a periodic domain and also the richer behavior of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.     

Excitable media in a chaotic flow

The response to a localized perturbation of an excitable medium under stirring by chaotic advection is investigated. It is found that below a critical stirring rate a localized perturbation produces a coherent global excitation of the system. For very slow stirring, however, the coherence of the global excitation is gradually lost. We propose a simple model to describe the effect of the flow on the excitable dynamics, and explain the observed behavior as a consequence of a steady excited filament state found in the reduced problem.     

Anomalous pulse interaction in dissipative media

We review a number of phenomena occurring in one-dimensional excitable media due to modified decay behind propagating pulses. Those phenomena can be grouped in two categories depending on whether the wake of a solitary pulse is oscillatory or not. Oscillatory decay leads to nonannihilative head-on collision of pulses and oscillatory dispersion relation of periodic pulse trains. Stronger wake oscillations can even result in a bistable dispersion relation. Those effects are illustrated with the help of the Oregonator and FitzHugh-Nagumo models for excitable media. For a monotonic wake, we show that it is possible to induce bound states of solitary pulses and anomalous dispersion of periodic pulse trains by introducing nonlocal spatial coupling to the excitable medium.     

Generation of finite wave trains in excitable media

Spatiotemporal control of excitable media is of paramount importance in the development of new applications, ranging from biology to physics. To this end, we identify and describe a qualitative property of excitable media that enables us to generate a sequence of traveling pulses of any desired length, using a one-time initial stimulus. The wave trains are produced by a transient pacemaker generated by a one-time suitably tailored spatially localized finite amplitude stimulus, and belong to a family of fast pulse trains. A second family, of slow pulse trains, is also present. The latter are created through a clumping instability of a traveling wave state (in an excitable regime) and are inaccessible to single localized stimuli of the type we use. The results indicate that the presence of a large multiplicity of stable, accessible, multi-pulse states is a general property of simple models of excitable media.     

A geometric theory for scroll wave filaments in anisotropic excitable media

Scroll waves are an important example of self-organisation in excitable media. In cardiac tissue, scroll waves of electrical activity underlie lethal ventricular arrhythmias and fibrillation. They rotate around a topological line defect which has been termed the filament. Numerical investigation has shown that anisotropy can substantially affect the dynamics of scroll waves. It has recently been hypothesised that stationary scroll wave filaments in cardiac tissue describe geodesics in a space whose metric is the inverse diffusion tensor. Several computational studies have validated this hypothesis, but until now no quantitative theory has been provided to study the effects of anisotropy on scroll wave filaments. Here, we review in detail the recently developed covariant formalism for scroll wave dynamics in general anisotropy and derive the equations of motion of filaments. These equations are fully covariant under general spatial coordinate transformations and describe the motion of filaments in a curved space whose metric tensor is the inverse diffusion tensor. Our dynamic equations are valid for thin filaments and for general anisotropy and we show that stationary filaments obey the geodesic equation. We extend previous work by allowing spatial variations in the determinant of the diffusion tensor and the reaction parameters, leading to drift of the filament.     

Using weak impulses to suppress traveling waves in excitable media

Here we propose mechanisms for suppressing non-steady-state motions--propagating pulses, spiral waves, spiral-wave chaos--in excitable media. Our approach is based on two points: (1) excitable media are multistable; and (2) traveling waves in excitable media can be separated into fast and slow motions, which can be considered independently. We show that weak impulses can be used to change the values of the slow variable at the front and back of a traveling wave, which leads to wave front and wave back velocities that are different from each other. This effect can destabilize the traveling wave, resulting in a transition to the rest state.