Termination of spiral wave breakup in a Fitzhugh-Nagumo model via short and long duration stimuli

Rotating spiral waves have been observed in a variety of nonlinear biological and physical systems. Spiral waves are found in excitable and oscillatory systems and can be stationary, meander, or even degenerate into multiple unstable rotating waves (a process called "spiral wave breakup"). In the heart, spiral wave breakup is thought to be the underlying mechanism of cardiac fibrillation. The spatiotemporal complexity of multiple unstable spiral waves is difficult to control or terminate. Here, the mechanisms of the termination of spiral wave breakup in response to global stimulation are investigated. A modified Fitzhugh-Nagumo model was used to represent cellular kinetics to study the role of the fast (activation) and slow (recovery) variables. This simplified model allows a theoretical analysis of the termination of spiral wave breakup via both short and long duration pulses. Simulations were carried out in both two-dimensional sheets and in a three-dimensional geometry of the heart ventricles. The short duration pulses affected only the fast variable and acted to reset wave propagation. Monophasic pulses excited tissue ahead of the wave front thus reducing the amount of excitable tissue. Biphasic shocks did the same, but they also acted to generate new wave fronts from the pre-existing wave tails by making some active regions excitable. Thus, if the short duration stimuli were strong enough, they acted to fill in excitable tissue via propagating wave fronts and terminated all activity. The long duration wave forms were selected such that they had a frequency spectrum similar to that of the pseudoelectrocardiograms recorded during fibrillation. These long duration wave forms affected both the recovery and activation variables, and the mechanism of unstable multiple spiral wave termination was different compared to the short duration wave forms. If the long duration stimuli were strong enough, they acted to alter the "state" (i.e., combination of fast and slow variables) of the tissue throughout 1.5 cycles, thus "conditioning" the tissue such that by the end of the stimuli almost no excitable tissue remained. The peak current, total energy, and average power of stimuli required to terminate spiral wave breakup were less for the long duration wave forms compared to the short duration wave forms. In addition, closed loop feedback via stimulation with a wave form that was the difference of the pseudoelectrocardiogram and a strongly periodic chaotic signal was successful at terminating spiral wave breakup. These results suggest that it may be possible to improve cardiac defibrillation efficacy by using long duration wave forms to affect recovery variables in the heart as opposed to the traditional brief duration wave forms that act only on the fast variables. (c) 2002 American Institute of Physics.     

The simplified FitzHugh-Nagumo model with action potential duration restitution: Effects on 2D wave propagation

A modification of the simplified FitzHugh-Nagumo (FN) equations is proposed for introducing a residual component of the slow variable, which determines the restitution of action potential duration (APD) also known as the interval-excitation duration relationship. The three-step-wise approximation of ε(E) which is widely used in current publications is replaced in a new model by a four-step approximation. This change is used for studying by computer simulation the effects of APD restitution properties independently of the APD and refractory period on 2D wave propagation in an isotropic matrix (made by 128 × 128 nodes). The method for fitting the model to the given experimental restitution data (obtained from myocardial cells) is presented. The computer simulations implemented on a massively parallel computer (Connection Machine) showed at least three important qualitative distinctions in behavior which demonstrate the effect of APD restitution: changes in the speed and wavelength of propagated waves with the period of stimulation, non-stationary propagation of spiral waves, and site-specific induction of spiral waves with premature stimulation not on the tail of the previous wave. Quantitative effects of differing restitution properties are expressed in the size and location of a window of vulnerability in 2D excitable media. These windows are characterized by the appearance of single and double spiral waves in response to premature stimulation applied inside the window. Thus the APD restitution incorporated in the FN model produces a significant effect on the formation and propagation of spiral waves.     

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.     

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.     

Complex spiral wave dynamics in a spatially distributed ionic model of cardiac electrical activity

This study presents computations and analysis of the dynamics of reentrant spiral waves in a realistic model of cardiac electrical activity, incorporating the Beeler-Reuter equations into a two-dimensional cable model. In this medium, spiral waves spontaneously break up, but may be stabilized by shortening the excitation pulse duration through an acceleration of the dynamics of the calcium current. We describe the breakup of reentrant waves based on the presence of slow recovery fronts within the medium. This concept is introduced using examples from pulse circulation around a ring and extended to two-dimensional propagation. We define properties of the restitution and dispersion relations that are associated with slow recovery fronts and promote spiral breakup. The role of slow recovery fronts is illustrated with concrete examples from numerical simulations. (c) 1996 American Institute of Physics.     

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.     

Conduction block in one-dimensional heart fibers

We present a nonlinear dynamical systems analysis of the transition to conduction block in one-dimensional cardiac fibers. We study a simple model of wave propagation in heart tissue that depends only on the recovery of action potential duration and conduction velocity. If the recovery function has slope >or=1 and the velocity recovery function is nonconstant, rapid activation causes dynamical heterogeneity and finally conduction block away from the activation site. This dynamical mechanism may play a role in the initiation and breakup of spiral waves in excitable media.     

Instability and Death of Spiral Wave in a Two-Dimensional Array of Hindmarsh-Rose Neurons

Spiral wave could be observed in the excitable media, the neurons are often excitable within appropriate parameters. The appearance and formation of spiral wave in the cardiac tissue is linked to monomorphic ventricular tachycardia that can denervate into polymorphic tachycardia and ventricular fibrillation. The neuronal system often consists of a large number of neurons with complex connections. In this paper, we theoretically study the transition from spiral wave to spiral turbulence and homogeneous state （death of spiral wave） in two-dimensional array of the Hindmarsh-Rose neuron with completely nearest-neighbor connections. In our numerical studies, a stable rotating spiral wave is developed and selected as the initial state, then the bifurcation parameters are changed to different values to observe the transition from spiral wave to homogeneous state, breakup of spiral wave and weak change of spiral wave, respectively. A statistical factor of synchronization is defined with the mean field theory to analyze the transition from spiral wave to other spatial states, and the snapshots of the membrane potentials of all neurons and time series of mean membrane potentials of all neurons are also plotted to discuss the change of spiral wave. It is found that the sharp changing points in the curve for factor of synchronization vs. bifurcation parameter indicate sudden transition from spiral wave to other states. And the results are independent of the number of neurons we used.     

Spontaneous spiral wave breakup caused by pinning to the tissue defect

The work presents a mechanism of spiral wave initiation due to the specific boundary conditions on a border of cardiac tissue defect. There are known scenarios when anatomical or functional defects in cardiac tissue may provoke the spiral wave origination, including unidirectional blockage while passing through the narrow gates, bent over critical curvature wave fronts, inhomogeneous recovery of the tissue, etc. We show a new scenario of spiral wave breakup on a small defect, which is unexcitable but permeable for ionic currents supporting the excitation wave. It was believed that such defects stabilize the rotating wave; however, as shown, instead of stabilizing it leads to the spiral breakup and subsequent multiplication of the rotating waves.     

用同步复极化终止心脏中的螺旋波和时空混沌

为了模拟电击除颤导致动作电位持续时间缩短, 在Luo-Rudy相I心脏模型中引入了同步复极化. 研究了同步复极化对螺旋波和时空混沌动力学的影响. 数值结果表明: 在控制周期比较小的情况下, 同步复极化可以有效消除螺旋波和时空混沌, 在有一些控制参数下, 同步复极化只能消除螺旋波, 或者只能消除时空混沌. 当螺旋波不被控制时, 观察到螺旋波转变为长周期和长波长的螺旋波或破碎成时空混沌的现象. 并对控制机制进行了分析. 关键词： 螺旋波 时空混沌 同步复极化 控制     

Resetting wave forms in dictyostelium territories

The mechanism by which spiral wave patterns appear in populations of Dictyostelium was probed experimentally by external chemical perturbation. Spiral waves, which often arise from the breakup of circular waves driven by pacemakers, typically entrain those pacemakers. We studied these processes by resetting the waves with a spatially uniform pulse of extrinsic cyclic AMP. A pattern of spirals reappeared if resetting was early in the signaling stage, but only targets emerged following late resetting, in a manner analogous to cardiac defibrillation. This supports recent hypotheses that wave pattern selection naturally occurs by slow temporal variation of the excitability of the cells.     

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.     

Controlling Spiral Dynamics in Excitable Media by a Weakly Localized Pacing

Spiral dynamics controlled by a weakly localized pacing around the spiral tip is investigated. Numerical simulations show two distinct characteristics when the pacing is applied with the weak amplitude for suitable frequencies： for a rigidly .rotating spiral, a transition from rigid rotation to meandering motion is observed, and for unstable spiral waves, spiral breakup can be prevented. Successfully preventing spiral breakup is relevant to the modulation of the tip trajectory induced by a localized pacing.     

时变反应扩散系统中螺旋波和湍流的控制

研究了一类参数时变的反应扩散系统中螺旋波和湍流对外电场的响应问题.在数值模拟中,以一类改进的Fitzhugh-Nagumo模型为研究对象(在恰当参数值下可分别描述激发介质和振荡介质),考虑随机和不确定因素(如内外噪声、气压、温度梯度分布和介质形变等)所引起的系统参数涨落对斑图演化的影响,在模拟中选取的参数涨落范围确保系统可以观测到稳定旋转的螺旋波、漫游的螺旋波和湍流,经历一定的暂态过程后,对介质施加极化电场,研究螺旋波和湍流在外电场中的演化.数值计算结果表明：在系统参数发生涨落和外电场强度比较小情况下,主 关键词： 螺旋波 湍流 时变系统 Fitzhugh-Nagumo模型     

Vulnerability to re-entry in simulated two-dimensional cardiac tissue: effects of electrical restitution and stimulation sequence

Ventricular fibrillation is a lethal arrhythmia characterized by multiple wavelets usually starting from a single or figure-of-eight re-entrant circuit. Understanding the factors regulating vulnerability to the re-entry is essential for developing effective therapeutic strategies to prevent ventricular fibrillation. In this study, we investigated how pre-existing tissue heterogeneities and electrical restitution properties affect the initiation of re-entry by premature extrastimuli in two-dimensional cardiac tissue models. We studied two pacing protocols for inducing re-entry following the "sinus" rhythm (S1) beat: (1) a single premature (S2) extrastimulus in heterogeneous tissue; (2) two premature extrastimuli (S2 and S3) in homogeneous tissue. In the first case, the vulnerable window of re-entry is determined by the spatial dimension and extent of the heterogeneity, and is also affected by electrical restitution properties and the location of the premature stimulus. The vulnerable window first increases as the action potential duration (APD) difference between the inside and outside of the heterogeneous region increases, but then decreases as this difference increases further. Steeper APD restitution reduces the vulnerable window of re-entry. In the second case, electrical restitution plays an essential role. When APD restitution is flat, no re-entry can be induced. When APD restitution is steep, re-entry can be induced by an S3 over a range of S1S2 intervals, which is also affected by conduction velocity restitution. When APD restitution is even steeper, the vulnerable window is reduced due to collision of the spiral tips.     

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.     

Development and transition of spiral wave in the coupled Hindmarsh--Rose neurons in two-dimensional space

The dynamics and the transition of spiral waves in the coupled Hindmarsh--Rose (H--R) neurons in two-dimensional space are investigated in the paper. It is found that the spiral wave can be induced and developed in the coupled HR neurons in two-dimensional space, with appropriate initial values and a parameter region given. However, the spiral wave could encounter instability when the intensity of the external current reaches a threshold value of 1.945. The transition of spiral wave is found to be affected by coupling intensity D and bifurcation parameter r. The spiral wave becomes sparse as the coupling intensity increases, while the spiral wave is eliminated and the whole neuronal system becomes homogeneous as the bifurcation parameter increases to a certain threshold value. Then the coupling action of the four sub-adjacent neurons, which is described by coupling coefficient D’, is also considered, and it is found that the spiral wave begins to breakup due to the introduced coupling action from the sub-adjacent neurons (or sites) and together with the coupling action of the nearest-neighbour neurons, which is described by the coupling intensity D.     

Fibroblasts alter spiral wave stability

We consider a three-domain model of cardiac tissue consisting of fibroblasts, myocytes, and extracellular space. We show in the one dimensional case that the fibroblasts with different resting potentials may alter restitution properties of tissue. On this basis we demonstrated that in two dimensional slice of cardiac tissue, a spiral wave break up can be caused purely by the influence of fibroblasts and, vice-versa, initially unstable spiral can be stabilized by fibroblasts depending on the value of their resting potential.     

激发介质相对不应态对螺旋波动力学行为的影响

采用元胞自动机模型研究激发介质相对不应态对螺旋波动力学行为的影响。数值模拟表明：元胞激发阈值存在一临界区间,该区间的螺旋波周期会突然增加,并存在一最大周期,在合适的系统尺寸和状态数下,螺旋波周期不再受相对不应态的影响而只取决于系统的激发阈值;相对不应态导致螺旋波“Z”型漫游、小范围无规律漫游、花瓣状漫游、锯齿状漫游、风车状漫游等复杂的波头运动。观察到稳定螺旋波、漫游螺旋波和螺旋波消失,并对产生这些现象的机制作简要的解释。     

Stability conditions for the traveling pulse: Modifying the restitution hypothesis

As a simple model of reentry, we use a general FitzHugh-Nagumo model on a ring (in the singular limit) to build an understanding of the scope of the restitution hypothesis. It has already been shown that for a traveling pulse solution with a phase wave back, the restitution hypothesis gives the correct stability condition. We generalize this analysis to include the possibility of a pulse with a triggered wave back. Calculating the linear stability condition for such a system, we find that the restitution hypothesis, which depends only on action potential duration restitution, can be extended to a more general condition that includes dependence on conduction velocity restitution as well as two other parameters. This extension amounts to unfolding the original bifurcation described in the phase wave back case which was originally understood to be a degenerate bifurcation. In addition, we demonstrate that dependence of stability on the slope of the restitution curve can be significantly modified by the sensitivity to other parameters (including conduction velocity restitution). We provide an example in which the traveling pulse is stable despite a steep restitution curve. (c) 2002 American Institute of Physics.