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.     

Selection of twisted scroll waves in three-dimensional excitable media

The selection of shape and rotation frequency for scroll waves in reaction-diffusion equations modeling excitable media is investigated. For scrolls with uniform twist about straight filaments, asymptotic methods are used to derive free-boundary equations at leading and first order. Both orders are validated against full solutions of the reaction-diffusion equations. Using these two orders and with no adjustable parameters, the shape and frequency of waves are correctly predicted except possibly near the point of propagation failure where the core becomes large.     

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.     

非均匀可激发介质中的稀密螺旋波

在非均匀可激发介质中,采用Barkley模型数值模拟了稀螺旋波和密螺旋波, 并对二者的动力学行为随参数的变化进行了研究. 结果发现:稀螺旋波的旋转频率随参数b的增加迅速减小,之后趋于饱和, 显示出不同于密螺旋波的行为;两种螺旋波的周期和波长随参数ε 和非均匀区域尺寸R的增加而增加,相对稀螺旋波而言,密螺旋波的性质对R的依赖更为敏感; 稀螺旋波端点的波速随R的增加而减小,与密螺旋波波速变化趋势相反. 另外,由于非均匀区域的影响,当ε 或b 超过某一临界值时,螺旋波臂上出现缺陷点.     

非均匀激发介质中螺旋波的演化

采用简单的离散可激发介质模型研究了不同性质和形状的小异质性颗粒对螺旋波演化的影响,在小异质性颗粒均匀分布的情况下,螺旋波的失稳与小异质性颗粒的形状、分布密度和性质有关.数值研究结果表明:在适当的条件下,可以观察到稳定螺旋波发生漫游和螺旋波漂移出系统边界现象,首次观察到螺旋波破碎成空间无序的周期变化斑图和破碎成时空有序的迷宫斑图现象,简要讨论了产生这些现象的物理机理.     

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.     

Rotating spiral waves in excitable media: The analytical results

In this paper we calculate the rotation frequency of spiral waves as a function of the parameters of the excitable medium. We give the complete analytical solution in the special case of the Rinzel-Keller model and suggest an analytical procedure for general two-component relaxational reaction-diffusion models. This procedure is based on the Greenberg equation and stability analysis; it is applicable when the core of a spiral wave is large as compared with the characteristic diffusion length. Construction of spiral wave solutions for the waves inside ring channels, circular regions, or around holes is discussed. Multi-armed spiral waves are investigated.     

极化电场对可激发介质中螺旋波的控制

螺旋波在不同的物理、化学和生物系统中普遍存在.周期外场,比如极化电场,尤其是具有旋转对称性的圆极化电场可对螺旋波动力学产生重要影响.本文综述了极化电场对可激发介质中螺旋波的控制,包括共振漂移、同步、手征对称性破缺、多臂螺旋波的稳定、次激发介质中的螺旋波、三维回卷波湍流态的控制、心脏组织中螺旋波的去钉扎、心脏组织中螺旋波湍流态的控制等.     

耦合可激发介质中螺旋波的控制研究

采用Br模型研究了三层耦合可激发介质中螺旋波的控制.相邻层之间采用双向耦合.利用加在第二层介质上的局域周期信号产生的平面波来消除螺旋波.数值模拟表明:只有当三层介质的耦合满足一定条件才可能实现螺旋波的控制,可以通过耦合互补方式实现螺旋波的控制;平面波与低频螺旋波的相互作用可以产生高频螺旋波,导致螺旋波不能被消除;存在优化的驱动宽度,过大或过小的驱动宽度需要增加第一、三层介质的耦合强度.观察到控制结果依赖控制时机的现象.研究结果可用于植入式心脏除颤器的设计.     

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.     

随机相位扰动抑制激发介质中漂移的螺旋波

研究了一类二维变量描述的激发系统中漂移螺旋波的抑制问题.通过在整个系统中局部注入带随机相位的电信号，如在系统256×256格点的边界或中心区域中选取4×4或者5×5格点区域施加一个带随机相位的外部激励电信号，在系统内部产生一个持续的靶波信号，实现靶波对螺旋波的动态竞争.数值计算表明：该方法对于Barkley模型中螺旋波有很强的抑制作用，与简单的局部周期信号驱动比较，具有暂态过程比较短的特点，而且对于时空噪声具有一定的抗干扰性.在一定的噪声范围内，即使系统出现不均匀性，也可以观测到靶波，新出现的靶波对螺旋波有抑制作用. 关键词： 螺旋波 靶波 Barkley模型 随机相位     

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.     

Superspiral structures of meandering and drifting spiral waves

The spatiotemporal superstructure of meandering and drifting spiral waves is explained analytically. It is also demonstrated that the Hopf eigenmode that causes the transition to meandering waves is weakly exponentially localized at onset but grows exponentially slightly before onset.     

Dynamical simulations of twisted scroll rings in three-dimensional excitable media

Excitable media such as heart muscle and the Belousov-Zhabotinsky reagent harbor vortex-like sources of periodic wave activity. In three dimensions this vortex has been observed in vivo, in vitro, and in numero. It typically closes in a ring. Using a partial differential equation model of an excitable medium, we present the first computation of a topologically new vortex ring. The computations suggest distinctive properties whereby we may recognize this new organizing center in experiments with excitable media.     

Oscillatory traveling waves in excitable media

A new type of waves in an excitable medium, characterized by oscillatory profile, is described. The excitable medium is modeled by a two-component activator-inhibitor system. Reaction-diffusion systems with diagonal and cross diffusion are examined. As an example, a front (kink) represented by a heteroclinic orbit in the phase space is considered. The wave shape and velocity are analyzed with the use of exact analytical solutions for wave profiles.     

Noise controlled spiral growth in excitable media

We consider a two-dimensional pulse-coupled array of noisy threshold elements with a long-range interaction. For tight coupling, we observe the formation of spatio-temporal excitation waves, such as target and spiral waves. For weak coupling, we find noise sustained spiral growth, where the noise level controls the scale of the spiral. (c) 1995 American Institute of Physics.     

Selection of spiral waves in excitable media with a phase wave at the wave back

Universal relationships between the medium excitability and the angular velocity and the core radius of rigidly rotating spiral waves in excitable media are derived for situations where the wave front is a trigger wave and the wave back is a phase wave. Two universal limits restricting the region of existence of spiral waves in the parameter space are demonstrated. The predictions of the free-boundary approach are in good quantitative agreement with results from numerical reaction-diffusion simulations performed on the Kessler-Levine model.     

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.