Breakup of spiral wave under different boundary conditions

In this paper, we investigate the breakup of spiral wave under no-flux, periodic and Dirichlet boundary conditions respectively. When the parameter \varepsilon is close to a critical value for Doppler-induced wave breakup, the instability of the system caused by the boundary effect occurs in the last two cases, resulting in the breakup of spiral wave near the boundary. With our defined average order measure of spiral wave (AOMSW), we quantify the degree of order of the system when the boundary-induced breakup of spiral wave happens. By analysing the AOMSW and outer diameter R of the spiral tip orbit, it is easy to find that this boundary effect is correlated with large values of R, especially under the Dirichlet boundary condition. This correlation is nonlinear, so the AOMSW sometimes oscillates with the variation of \varepsilon.     

Motion of spiral waves induced by local pacing

The motion of spiral waves in excitable media driven by a weak pacing around the spiral tip is investigated numerically as well as theoretically. We presented a Bifurcations diagram containing four types of the spiral motion induced by different frequencies of pacing: rigidly rotating, inward-petal meandering, resonant drift, and outward-petal meandering spiral. Simulation shows that the spiral resonantly drifts when the frequency of pacing is close to that of the spiral rotation. We also find that the speed and direction of the drift can be efficiently controlled by means of the strength and phase of the local pacing, which is consistent with analytical results based on the framework of the weak deformation approximation.      

From local to global spatiotemporal chaos in a cardiac tissue model

Two kinds of chaos can occur in cardiac tissue, chaotic meander of a single intact spiral wave and chaotic spiral wave breakup. We studied these behaviors in a model of two-dimensional cardiac tissue based on the Luo-Rudy I action potential model. In the chaotic meander regime, chaos is spatially localized to the core of the spiral wave. When persistent spiral wave breakup occurs, there is a transition from local to global spatiotemporal chaos.     

时空调制对可激发介质螺旋波波头动力学行为影响及控制研究

本文首先研究了时空调制对可激发介质中周期螺旋波波头动力学行为的影响. 随着时空调制的增大, 螺旋波经历了周期螺旋波、外滚螺旋波、旅行螺旋波和内滚螺旋波的显著变化. 通过定义序参量来定量的描述由时空调制引起的螺旋波在不同态之间非平衡跃迁的临界条件, 及漫游螺旋波波头圆滚圆半径随调制参数的变化情况. 当时空调制增大到某个临界值时, 螺旋波发生了破碎; 再增加时空调制, 螺旋波则发生了衰减, 系统最终演化为空间均匀静息态. 在文中给出了螺旋波发生破碎和衰减的机理和原因. 最后将时空调制方法运用于漫游螺旋波, 实现了将漫游螺旋波控制成周期螺旋波, 或将其控制为空间均匀静息态.     

Dynamics of Spiral Wave Tip in Excitable Media with Gradient Parameter

Using a Barkley model as an example, we study spiral waves and spiral tips in a gradient excitable medium. The gradient distribution of parameters is introduced to depict the inhomogeneous medium. It is found that the parameter fluctuations play an important role in the morphology of spiral pattern and the movements of spiral tips. For varied gradient parameters, it is observed that there exist three kinds of spiral behaviors, stable rotation, rebound of spiral tip from the boundary, and spiral breakup.     

Breakup of Spiral Waves in Coupled Hindmarsh--Rose Neurons

Breakup of spiral wave in the Hindmarsh-Rose neurons with nearest-neighbour couplings is reported. Appropriate initial values and parameter regions are selected to develop a stable spiral wave and then the Gaussian coloured noise with different intensities and correlation times is imposed on all neurons to study the breakup of spiral wave, respectively. Based on the mean field theory, the statistical factor of synchronization is defined to analyse the evolution of spiral wave. It is found that the stable rotating spiral wave encounters breakup with increasing intensity of Gaussian coloured noise or decreasing correlation time to certain threshold.     

Turbulence control with local pacing and its implication in cardiac defibrillation

In this review article, we describe turbulence control in excitable systems by using a local periodic pacing method. The controllability conditions of turbulence suppression and the mechanisms underlying these conditions are analyzed. The local pacing method is applied to control Winfree turbulence (WT) and defect turbulence (DT) induced by spiral-wave breakup. It is shown that WT can always be suppressed by local pacing if the pacing amplitude and frequency are properly chosen. On the other hand, the pacing method can achieve suppression of DT induced by instabilities associated with the motions of spiral tips while failing to suppress DT induced by the instabilities of wave propagation far from tips. In the latter case, an auxiliary method of applying gradient field is suggested to improve the control effects. The implication of this local pacing method to realistic cardiac defibrillation is addressed.     

Electrical alternans and spiral wave breakup in cardiac tissue

This paper reports the results of a theoretical investigation of spiral wave breakup in model equations of action potential propagation in cardiac tissue. A general formulation of these equations is described in which arbitrary experimentally determined restitution and dispersion curves can in principle be fitted. Spiral wave behavior is studied in two-dimension as a function of a parameter Re which controls the steepness of the restitution curve at short diastolic intervals. Spiral breakup is found to occur when the minimum period T(min), below which a periodically stimulated tissue exhibits alternans in action potential duration, exceeds by a finite amount the spiral rotation period T(S). At this point, oscillations in action potential duration are of sufficiently large amplitude to cause a spontaneous conduction block to form along the wavefront. The latter occurs closer to the initiation point of reentry (spiral tip) with increasing steepness and, hence, in smaller tissue sizes. Spiral breakup leads to a spatially disorganized wave activity which is always transient, except for tissues larger than some minimum size and within a very narrow range of Re which increases with dispersion.     

Interaction of Wave Trains with Defects

The evolution and transition of planar wave trains propagating through defects(obstacles) in an excitable medium are studied. When the frequency of the planar wave trains is increased, three different dynamical regimes,namely fusion, "V" waves, and spiral waves, are observed in turn and the underlying mechanism is discussed. The dynamics is concerned with the shapes of the defects. Circle, triangle, and rectangle defects with different sizes are considered. The increase of pacing frequency broadens the fan-shaped broken region in the behind of a rectangle defect.The increase of width of a triangle defect leads to breakup of wave trains easier while the change of height shows opposite effect, which is presented in a phase diagram. Dynamical comparison on defects with different shapes indicates that the decrease of the defect width along the propagation of wave trains makes the fan-shaped region and the minimal frequency for breakup of spiral both increased.     

Filament instability and rotational tissue anisotropy: A numerical study using detailed cardiac models

The role of cardiac tissue anisotropy in the breakup of vortex filaments is studied using two detailed cardiac models. In the Beeler-Reuter model, modified to produce stable spiral waves in two dimensions, we find that anisotropy can destabilize a vortex filament in a parallelepipedal slab of tissue. The mechanisms of the instability are similar to the ones reported in previous work on a simplified cardiac model by Fenton and Karma [Chaos 8, 20 (1998)]. In the Luo-Rudy model, also modified to produce stable spiral waves in two dimensions, we find that anisotropy does not destabilize filaments. A possible explanation for this model-dependent behavior based on spiral tip trajectories is offered. (c) 2001 American Institute of Physics.     

多可激性障碍下的螺旋波动力学

在许多实际可激系统中局部不均匀是广泛存在的, 它们是螺旋波形成以及动力学行为改变的重要因素. 本文研究了可激性障碍对螺旋波动力学行为的影响. 研究表明, 在障碍区域内可激性参数大于区域外情况下障碍会对其附近的螺旋波波头有吸引作用, 多局部障碍共存时吸引行为不仅依赖障碍分布, 而且依赖障碍的大小以及区域内可激性参数的具体取值. 通过抑制变量小值区域的变化分析了这些行为发生的原因. 在障碍区域内可激性参数小于区域外情况下障碍对其近邻的螺旋波波头有排斥作用, 排斥后波头的运动依赖初始螺旋波是刚性旋转的还是漫游的. 多局部障碍共存时排斥作用对螺旋波动力学行为的改变依赖障碍的分布、大小与区域内可激性参数的具体取值以及初始螺旋波的类型. 关键词： 螺旋波 时空混沌 可激性障碍     

具有早期后除极化现象的可激发系统中螺旋波破碎方式研究

在Greenberg-Hasting元胞自动机模型中引入了正常元胞和老化元胞,并规定只有老化元胞存在早期后除极化现象且早期后除极化可以激发其他元胞.在正常元胞和老化元胞均匀分布的情况下,研究了早期后除极化对螺旋波演化行为的影响,重点探讨了早期后除极化导致的螺旋波破碎方式.数值模拟结果表明:早期后除极化在比率约为26.4%的少数情况下不对螺旋波产生影响,在其他情况下则会对螺旋波产生各种影响,包括使螺旋波漫游、漂移、波臂发生形变以及导致螺旋波破碎和消失等.观察到早期后除极化通过传导障碍消失和通过转变为反靶波消失,早期后除极化导致螺旋波破碎有8种方式,包括非对称破缺导致的破碎、对称破缺导致的破碎、同时激发双波导致的破碎、非对称激发导致的破碎、整体传导障碍导致的破碎、整体快速破碎等.分析发现这些螺旋波破碎现象都与早期后除极化产生回火波有关,得到螺旋波破碎的总比率通常约为13.8%,但是在适当选取老化元胞密度和早期后除极化的激发下,螺旋波破碎比率可达到32.4%,这些结果与心律失常致死的统计结果基本一致,本文对产生这些现象的物理机理做了简要分析.     

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.     

Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity

It has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, i.e., electrical wave(s) that recirculate repeatedly throughout the tissue at a higher frequency than the waves produced by the heart's natural pacemaker (sinoatrial node). However, the complicated structure of cardiac tissue, as well as the complex ionic currents in the cell, have made it extremely difficult to pinpoint the detailed dynamics of these life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. Some, but not all, of these mechanisms have been observed before using other models; therefore, the purpose of this paper is to demonstrate them using just one framework model and to explain the different parameter regimes or physiological properties necessary for each mechanism (such as high or low excitability, corresponding to normal or ischemic tissue, spiral tip trajectory types, and tissue structures such as rotational anisotropy and periodic boundary conditions). Each mechanism is compared with data from other ionic models or experiments to illustrate that they are not model-specific phenomena. Movies showing all the breakup mechanisms are available at http://arrhythmia.hofstra.edu/breakup and at ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-12-039203/ INDEX.html. The fact that many different breakup mechanisms exist has important implications for antiarrhythmic drug design and for comparisons of fibrillation experiments using different species, electromechanical uncoupling drugs, and initiation protocols. (c) 2002 American Institute of Physics.     

心肌组织机械形变的元胞自动机模拟

采用Greenberg-Hastings元胞自动机模型研究机械形变对心肌组织中螺旋波动力学行为的影响.数值模拟表明：对于规则网格下的稳定螺旋波,在生理性机械形变作用下,螺旋波发生漫游但不破碎;在病理性机械形变作用下,螺旋波会发生持续漫游、漫游后消失和破碎进入螺旋波湍流态三种变化.通过对比发现机械形变的振幅变化率对螺旋波的影响较大,而机械形变的角频率对螺旋波的影响较小.结合数值模拟,对心前区受到猛烈撞击会出现心颤致死及耐力运动员在发生心动过速后比一般人员更容易恢复正常进行解释.     

Spiral breakup as a model of ventricular fibrillation

The phenomenon of spiral breakup in a 2D and a 3D excitable medium is described. Differences between breakup in two dimensions and in three dimensions are discussed. Spiral breakup in an anatomical model of the ventricles of the heart is also studied. The patterns of excitation in the heart are presented at different wavelengths together with their electrocardiograms. Finally it is suggested that the phenomenon of spiral breakup is a possible mechanism of the ventricular fibrillation (VF). (c) 1998 American Institute of Physics.     

Mechanism for spiral wave breakup in excitable and oscillatory media

We study spiral wave breakup using a Fitzhugh-Nagumo-type system. We find that spiral wave breakup can occur near the core or far from it in both excitable and oscillatory regimes. There is a faraway breakup scenario in both excitable and oscillatory media that depends on long wavelength modulation modes. We observed three distinct scenarios, including one that involves breakup that does not develop into turbulence. However, we find that the mechanisms behind these three scenarios are the same: they are caused by the interaction between the dispersion relation and the asymptotic behavior of the modulation mode. The difference in phenomenology is due to the asymptotic behavior of the modulation mode.     

Spiral breakup in an array of coupled cells: the role of the intercellular conductance

We study numerically how the intercellular conductance affects the process of spiral breakup in an array of coupled excitable cells. The cell dynamics are described by the Aliev-Panfilov model, and the intercellular connection is made via Ohmic elements. We find that decreasing intercellular conductance can prevent the breaking up of a spiral wave into a complex spatiotemporal pattern. We study the mechanism of this effect and show that the breakup disappears because of increasing the diastolic interval of an initial spiral wave.     

Two forms of spiral-wave reentry in an ionic model of ischemic ventricular myocardium

It is well known that there is considerable spatial inhomogeneity in the electrical properties of heart muscle, and that the many interventions that increase this initial degree of inhomogeneity all make it easier to induce certain cardiac arrhythmias. We consider here the specific example of myocardial ischemia, which greatly increases the electrical heterogeneity of ventricular tissue, and often triggers life-threatening cardiac arrhythmias such as ventricular tachycardia and ventricular fibrillation. There is growing evidence that spiral-wave activity underlies these reentrant arrhythmias. We thus investigate whether spiral waves might be induced in a realistic model of inhomogeneous ventricular myocardium. We first modify the Luo and Rudy [Circ. Res. 68, 1501-1526 (1991)] ionic model of cardiac ventricular muscle so as to obtain maintained spiral-wave activity in a two-dimensional homogeneous sheet of ventricular muscle. Regional ischemia is simulated by raising the external potassium concentration ([K(+)](o)) from its nominal value of 5.4 mM in a subsection of the sheet, thus creating a localized inhomogeneity. Spiral-wave activity is induced using a pacing protocol in which the pacing frequency is gradually increased. When [K(+)](o) is sufficiently high in the abnormal area (e.g., 20 mM), there is complete block of propagation of the action potential into that area, resulting in a free end or wave break as the activation wave front encounters the abnormal area. As pacing continues, the free end of the activation wave front traveling in the normal area increasingly separates or detaches from the border between normal and abnormal tissue, eventually resulting in the formation of a maintained spiral wave, whose core lies entirely within an area of normal tissue lying outside of the abnormal area ("type I" spiral wave). At lower [K(+)](o) (e.g., 10.5 mM) in the abnormal area, there is no longer complete block of propagation into the abnormal area; instead, there is partial entrance block into the abnormal area, as well as exit block out of that area. In this case, a different kind of spiral wave (transient "type II" spiral wave) can be evoked, whose induction involves retrograde propagation of the action potential through the abnormal area. The number of turns made by the type II spiral wave depends on several factors, including the level of [K(+)](o) within the abnormal area and its physical size. If the pacing protocol is changed by adding two additional stimuli, a type I spiral wave is instead produced at [K(+)](o)=10.5 mM. When pacing is continued beyond this point, apparently aperiodic multiple spiral-wave activity is seen during pacing. We discuss the relevance of our results for arrythmogenesis in both the ischemic and nonischemic heart. (c) 1998 American Institute of Physics.     

热敏神经元网络中螺旋波死亡和破裂的数值模拟

实验发现大脑皮层内出现螺旋波且螺旋波对神经元电信号传递有积极作用.利用细胞网络方法从对大脑皮层观察到的螺旋波进行数值模拟.以包含温度因子的热敏神经元模型在二维空间构造规则网络,研究了神经元膜片温度参数对神经元网络中螺旋波演化影响;定义了一类统计同步因子来刻画温度因子引起螺旋波相变(破裂和死亡)的临界条件.发现在规则网络下,当温度超过一定值后螺旋波会死亡和消失而导致整个网络达到均匀同步;在考虑了弱通道噪声情况下,螺旋波温度超越一定临界值则引起螺旋波的破裂.进一步分析了暂时性发烧昏迷的可能机制在于神经系统某些功能区螺旋波传播电信号的中断.