Mathematical Sciences: Traveling Waves in Excitable Media

在多种生物系统中，激发行波为通信和空间组织提供了一种机制，如神经肌肉组织中的膜去极化波、果蝇胚胎中的有丝分裂波、黏菌聚集过程中的环状AMP波以及在易感群体中传播的感染波。这些和其他可兴奋性的例子有许多共同的特征，这些特征可以用可兴奋介质的一般数学模型来表示。研究者和他的同事使用已建立的理论工具和新的计算方法研究这种模型中的波传播，主要关注三维可激介质中的旋转涡旋波。当涡旋波围绕其中心“灯丝”旋转时，灯丝根据某些运动规律在空间中缓慢移动，这些规律至今仍知之甚少。研究人员基于可激发介质的元胞自动机模型，开发了高效准确的计算工具，以模拟长时间内大域中滚动细丝的运动。哺乳动物心脏的厚壁心室是一个三维的可激发介质，传播电化学活动的波。通过调节细胞内钙水平，这些波控制心脏的跳动。在正常的心跳过程中，波诱导左心室的肌肉纤维同步收缩，血液有效地从心室排出。但在病理条件下（颤振和纤颤），心跳退化成这个项目中研究的那种旋转的涡旋波。旋转的涡旋波使肌纤维异步收缩，因此当其他部分收缩时，部分腔体松弛。因此，腔内的血液不能有效地泵送，心脏必须“去纤颤”才能挽救受害者的生命。为了了解心室颤动的根本原因和有效的治疗方法，需要更好地了解三维可激介质中旋转涡旋波的发生、发展和湮灭，这是本项目的基本目标。

9525766 Tyson Traveling waves of excitation provide a mechanism for communication and spatial organization in a variety of biological systems, such as waves of membrane depolarization in neuromuscular tissue, waves of mitosis in fruit fly embryos, waves of cyclic AMP during slime mold aggregation, and waves of infection spreading through susceptible populations. These and other examples of excitability share many common features that can be expressed in generic mathematical models of excitable media. The investigator and his colleague study wave propagation in such models using established theoretical tools and new computational approaches, focusing primarily on rotating scroll waves in three-dimensional excitable media. As a scroll waves rotates around its central "filament," the filament moves slowly through space according to certain laws of motion that are still poorly understood. The investigators develop efficient and accurate computational tools, based on a cellular automaton model of excitable media, in order to simulate the motion of scroll filaments in large domains over long periods of time. The thick-walled ventricle of the mammalian heart is a three-dimensional excitable medium that propagates waves of electrochemical activity. By modulating intracellular calcium levels, these waves control the beating of the heart. During a normal heartbeat the wave induces the muscle fibers of the left ventricle to contract synchronously and blood is efficiently expelled from the chamber. But under pathological conditions (flutter and fibrillation), the heartbeat degenerates into a rotating scroll wave of the sort studied in this project. The rotating scroll wave causes the muscle fibers to contract asynchronously, so that part of the chamber is relaxed when other parts are contracting. As a result the blood within the chamber is not pumped efficiently and the heart must be "defibrillated" to save the victim's life. To understand the fundamental causes of ventricular fibrillation and its effective treatment, one needs a better understanding of the onset, development and annihilation of rotating scroll waves in three-dimensional excitable media, which is the basic goal of this project.