非线性耦合可激发介质中螺旋波动力学行为的研究
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摘要
近几十年来,时空斑图动力学一直是非线性科学研究的热点,通过对他们的研究,有助于了解发生在自然界和人类社会中的现象。螺旋波是一种重要的时空斑图,在一些物理、化学和生物系统中都观察到螺旋波,例如在心脏组织、蛙类卵细胞中的钙离子波、Belousov-Zhabotinsky化学反应系统等。在某些情况下,螺旋波和时空混沌对人类是有害的,例如,心肌组织中出现螺旋波会导致心动过速,心脏中的螺旋波自发破碎成时空混沌会导致纤维性颤动,如果不及时有效地治疗,甚至会危及患者生命。因此,对螺旋波和时空混沌的控制研究具有很大应用价值,但要彻底解决抑制心脏中螺旋波和时空混沌问题,依赖我们对时空斑图的动力学的彻底了解。出现螺旋波的系统一般都是反应扩散系统,例如可激发系统,生物中的神经元、心肌细胞都是典型的可激发系统。目前对单层介质中的螺旋波和其它时空斑图的动力学已经进行了广泛地研究,取得了大量的研究结果,然而对耦合可激发介质中的螺旋波等时空斑图动力学的研究仍局限在线性耦合情况,对非线性耦合研究较少。在真实的心脏系统,心室壁由心内膜、心外膜和其中的心肌三层组成,这三层细胞的电生理特性各不相同,只有研究耦合系统中斑图动力学才能更好理解在心脏系统中出现的现象,在这样的动机下,本文以Bar模型为基础,研究了两层耦合可激发介质中的螺旋波动力学,完成了两个研究工作,分别介绍如下：
第一章是本文的综述部分。简要的介绍了时空混沌、反应扩散系统、Bar模型的一些基本性质以及在激发介质中螺旋波的产生和波头的运动,此外还简单的介绍了时空斑图的同步和控制。
第二章基于Bar模型研究了两层耦合可激发介质系统中时空混沌对螺旋波的影响。提出无条件和有条件两种驱动响应耦合,这两种耦合都是驱动响应线性耦合。耦合前,驱动和响应子系统分别处于时空混沌态和螺旋波态。在不同的参数下,我们发现螺旋波表现出不同的动力学行为。在很小的耦合强度下,时空混沌对螺旋波动力学行为几乎无影响。当耦合强度很大时,无条件耦合总是导致螺旋波的破碎。当相关参数适当选取时,时空混沌既能提高受其作用介质的激发性,也能降低它的激发性。此外,它还能使稳定螺旋波和漫游螺旋波作无规漫游或无规漂移,甚至导致螺旋波漂移出系统；对于不稳定螺旋波,时空混沌能极大延迟螺旋波出现破碎。特别是,在有条件耦合下,可以使不稳定螺旋波成为稳定或漫游螺旋波。
第三章研究了两个非线性相互耦合可激发系统中螺旋波的演化,我们发现耦合螺旋波的演化依赖相位差、耦合强度和螺旋波波头之间的距离。在两子系统参数相同的情况下,在适当选取相位和耦合强度下,可观察到两螺旋波同步、反同步、子系统出现从螺旋波态到不同的定态或静息态的相变、螺旋波漫游、螺旋波波头相向运动等；特别是我们首次观察到两系统螺旋波的激发同步现象,不能实现完全同步的原因是：在波峰期间子系统出现高频振荡,而且在相互耦合的空间点上的振荡是反相的,这种局域介质动力学行为与神经元的峰放电行为相似。上述大部分现象在两子系统参数不同时也会出现,除了两系统不能达到完全同步外。但在适当选取相位和耦合强度下,可观察到两螺旋波相同步、反相同步。这些结果表明,我们提出的耦合方法,有助于了解神经元系统的动力学行为,也有助于了解心室结构对波传播的影响。
In recent decades, the investigation of spatiotemporal pattern dynamics has become a hot topic of nonlinear science. The investigation can help us to understand the phenomena that occur in nature and human society. Spiral wave is the important spatiotemporal pattern, which has been observed in some physical, chemical and biological systems, such as cardiac tissues, the calcium wave in the frog egg cell, the Belousov-Zhabotinsky chemical reaction system, and so on. In some cases, spiral waves and spatiotemporal chaos is harmful to humans. For example, spiral wave in myocardial tissue would lead to tachycardia and the breakup of it into spatiotemporal chaos will lead to fibrillation, endangering the lives of patients if the fibrillation does not timely and effectively treated. Therefore, study on the control of spiral wave and spatiotemporal chaos is of great value of application. However, it relies on our thorough understanding of the dynamics of the spatiotemporal pattern to solve completely the suppression of spiral wave and spatiotemporal in cardiac tissues. The systems, in which spiral waves can be generated, are the reaction diffusion systems, such as the excitable system. The biological neurons, cardiac cells are the typical excitable systems. The dynamics of spiral wave and other spatiotemporal pattern in single-layer medium has been widely studied, and a lot of results have been obtained so far. However, the related investigations in the coupled excitable media still focus on linearly coupled media, and nonlinearly coupled media are less applied. The ventricular wall in the real cardiac system is composed of myocardium, epicardium and endocardium, and the cells in three layers media have different electrophysiological properties. In order to better understand the phenomenon occurring in cardiac system, the pattern dynamics in the coupled systems need to be studied. So we investigate dynamics of spiral waves in two-layer coupled excitable media based on Bar model. The two results that we obtain are respectively introduced as follows:
The first chapter is the overview section of this article. The some basic properties of spatiotemporal chaos, reaction diffusion system Bar model, as well as the generation of spiral wave in excitable medium, the movement of spiral wave's tip are briefly introduced. Synchronization and control of spatiotemporal pattern are briefly introduced too.
In Chapter2we investigate the evolution of spiral waves in a system of two coupled excitable media by using Bar model. The drive-response coupling schemes with or without constraint condition are proposed. The response and drive subsystems are respectively in the state of spiral wave and spatiotemporal chaos before the coupling turns on. We find that spiral wave exhibits different dynamical behaviors for different parameters. When the coupling strength is weak, the dynamical behavior of spiral wave in response system almost remains unchanged. When the coupling strength is strong, the coupling without constraint condition always leads to the breakup of spiral wave. When the related parameters are properly chosen, spatiotemporal chaos not only can enhance the excitation of the forced medium but also reduce it. In addition, it can induce chaotic meander or drift of a stable or meandering spiral wave, and even causes spiral wave to move out of system. It can greatly delay the breakup of an unstable spiral wave. Specially, it causes an unstable spiral wave become a stable or chaotically meandering spiral wave under the coupling with constraint condition.
In Chapter3we investigate the evolution of spiral waves in nonlinearly coupled excitable media. We find that the evolutions of coupled spiral waves depend on phase difference, coupling strength and the distance between spiral wave's tips. When the two subsystems have same parameters, the synchronization and anti-synchronization of spiral waves, the occurrence of the transition from spiral wave states to the different steady states or the resting state, meandering of spiral wave, and the movements of spiral wave's tips in opposite direction are observed if coupling strength and phase difference are properly chosen. Furthermore, the excitation-synchronization of the spiral waves in subsystems is found for the first time. The reason why two spiral waves do not achieve synchronization is that the high-frequency oscillation arises during the peak stage of the wave. Moreover, oscillations at space points with mutual coupling form anti-phase. The dynamic behavior of local medium is similar to the spiking of neuron. Most of above-mentioned phenomena will also appear when two subsystems have different parameters, in which two subsystems can not achieve synchronization and anti-synchronization. However, phase and anti-phase synchronizations of two subsystems can be observed for the correct choice of phase difference and coupling strength. These results show that our proposed coupling methods can help one to understand the dynamical behavior of neuronal systems, and also help one to understand the effect of ventricular structure on wave propagation.
第一章是本文的综述部分。简要的介绍了时空混沌、反应扩散系统、Bar模型的一些基本性质以及在激发介质中螺旋波的产生和波头的运动,此外还简单的介绍了时空斑图的同步和控制。
第二章基于Bar模型研究了两层耦合可激发介质系统中时空混沌对螺旋波的影响。提出无条件和有条件两种驱动响应耦合,这两种耦合都是驱动响应线性耦合。耦合前,驱动和响应子系统分别处于时空混沌态和螺旋波态。在不同的参数下,我们发现螺旋波表现出不同的动力学行为。在很小的耦合强度下,时空混沌对螺旋波动力学行为几乎无影响。当耦合强度很大时,无条件耦合总是导致螺旋波的破碎。当相关参数适当选取时,时空混沌既能提高受其作用介质的激发性,也能降低它的激发性。此外,它还能使稳定螺旋波和漫游螺旋波作无规漫游或无规漂移,甚至导致螺旋波漂移出系统；对于不稳定螺旋波,时空混沌能极大延迟螺旋波出现破碎。特别是,在有条件耦合下,可以使不稳定螺旋波成为稳定或漫游螺旋波。
第三章研究了两个非线性相互耦合可激发系统中螺旋波的演化,我们发现耦合螺旋波的演化依赖相位差、耦合强度和螺旋波波头之间的距离。在两子系统参数相同的情况下,在适当选取相位和耦合强度下,可观察到两螺旋波同步、反同步、子系统出现从螺旋波态到不同的定态或静息态的相变、螺旋波漫游、螺旋波波头相向运动等；特别是我们首次观察到两系统螺旋波的激发同步现象,不能实现完全同步的原因是：在波峰期间子系统出现高频振荡,而且在相互耦合的空间点上的振荡是反相的,这种局域介质动力学行为与神经元的峰放电行为相似。上述大部分现象在两子系统参数不同时也会出现,除了两系统不能达到完全同步外。但在适当选取相位和耦合强度下,可观察到两螺旋波相同步、反相同步。这些结果表明,我们提出的耦合方法,有助于了解神经元系统的动力学行为,也有助于了解心室结构对波传播的影响。
In recent decades, the investigation of spatiotemporal pattern dynamics has become a hot topic of nonlinear science. The investigation can help us to understand the phenomena that occur in nature and human society. Spiral wave is the important spatiotemporal pattern, which has been observed in some physical, chemical and biological systems, such as cardiac tissues, the calcium wave in the frog egg cell, the Belousov-Zhabotinsky chemical reaction system, and so on. In some cases, spiral waves and spatiotemporal chaos is harmful to humans. For example, spiral wave in myocardial tissue would lead to tachycardia and the breakup of it into spatiotemporal chaos will lead to fibrillation, endangering the lives of patients if the fibrillation does not timely and effectively treated. Therefore, study on the control of spiral wave and spatiotemporal chaos is of great value of application. However, it relies on our thorough understanding of the dynamics of the spatiotemporal pattern to solve completely the suppression of spiral wave and spatiotemporal in cardiac tissues. The systems, in which spiral waves can be generated, are the reaction diffusion systems, such as the excitable system. The biological neurons, cardiac cells are the typical excitable systems. The dynamics of spiral wave and other spatiotemporal pattern in single-layer medium has been widely studied, and a lot of results have been obtained so far. However, the related investigations in the coupled excitable media still focus on linearly coupled media, and nonlinearly coupled media are less applied. The ventricular wall in the real cardiac system is composed of myocardium, epicardium and endocardium, and the cells in three layers media have different electrophysiological properties. In order to better understand the phenomenon occurring in cardiac system, the pattern dynamics in the coupled systems need to be studied. So we investigate dynamics of spiral waves in two-layer coupled excitable media based on Bar model. The two results that we obtain are respectively introduced as follows:
The first chapter is the overview section of this article. The some basic properties of spatiotemporal chaos, reaction diffusion system Bar model, as well as the generation of spiral wave in excitable medium, the movement of spiral wave's tip are briefly introduced. Synchronization and control of spatiotemporal pattern are briefly introduced too.
In Chapter2we investigate the evolution of spiral waves in a system of two coupled excitable media by using Bar model. The drive-response coupling schemes with or without constraint condition are proposed. The response and drive subsystems are respectively in the state of spiral wave and spatiotemporal chaos before the coupling turns on. We find that spiral wave exhibits different dynamical behaviors for different parameters. When the coupling strength is weak, the dynamical behavior of spiral wave in response system almost remains unchanged. When the coupling strength is strong, the coupling without constraint condition always leads to the breakup of spiral wave. When the related parameters are properly chosen, spatiotemporal chaos not only can enhance the excitation of the forced medium but also reduce it. In addition, it can induce chaotic meander or drift of a stable or meandering spiral wave, and even causes spiral wave to move out of system. It can greatly delay the breakup of an unstable spiral wave. Specially, it causes an unstable spiral wave become a stable or chaotically meandering spiral wave under the coupling with constraint condition.
In Chapter3we investigate the evolution of spiral waves in nonlinearly coupled excitable media. We find that the evolutions of coupled spiral waves depend on phase difference, coupling strength and the distance between spiral wave's tips. When the two subsystems have same parameters, the synchronization and anti-synchronization of spiral waves, the occurrence of the transition from spiral wave states to the different steady states or the resting state, meandering of spiral wave, and the movements of spiral wave's tips in opposite direction are observed if coupling strength and phase difference are properly chosen. Furthermore, the excitation-synchronization of the spiral waves in subsystems is found for the first time. The reason why two spiral waves do not achieve synchronization is that the high-frequency oscillation arises during the peak stage of the wave. Moreover, oscillations at space points with mutual coupling form anti-phase. The dynamic behavior of local medium is similar to the spiking of neuron. Most of above-mentioned phenomena will also appear when two subsystems have different parameters, in which two subsystems can not achieve synchronization and anti-synchronization. However, phase and anti-phase synchronizations of two subsystems can be observed for the correct choice of phase difference and coupling strength. These results show that our proposed coupling methods can help one to understand the dynamical behavior of neuronal systems, and also help one to understand the effect of ventricular structure on wave propagation.
引文
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