Response functions for drift of spiral and scroll waves

旋转螺旋波(二维)和涡旋波(三维)是在许多空间扩展的物理、化学和生物自然系统中观察到的一种自组织形式。其中最重要的是心肌，旋转波是导致折返性心律失常的原因，包括最致命的室颤。在理想条件下，螺旋/涡旋波通常围绕静止的中心/细丝稳定地旋转。然而，现实中总是存在的任何对称破坏扰动都会导致中心/细丝的旋转频率和空间位置的逐渐变化，即漂移。理解这种漂移对于应用程序至关重要。虽然在直接的数值模拟中可以观察到漂移，但这些计算通常是昂贵的并且缺乏通用性。存在由小扰动引起的漂移的普遍渐近理论。它的适用性取决于对所谓的响应函数(RF)的了解。在一些已知的情况下，RF基本上只在核心附近为非零值。这种本地化的结果是，螺旋波/滚动波的行为类似于点/弦对象，尽管它显然是非局部的。这种独特的波粒二象性与螺旋波/涡旋波的显著稳定性直接相关。渐近理论利用了这一特性，并在原则上允许比直接数值模拟更简单和更有效地预测它们的漂移。一旦发现，特定模型的RFs允许人们预测螺旋线和涡卷的漂移，以响应任意扰动。目前的建议旨在发展规则和通用的方法来获得RFS，然后使渐近理论成为理解和控制真实系统中旋转波的实际工作工具。

Rotating spiral waves (in two dimensions) and scroll waves (in three dimensions) are a form of self-organization observed in numerous spatially extended systems of physical, chemical and biological nature. The most important of these is heart muscle where rotating waves are responsible for re-entrant arrhythmias, including the most lethal one, the ventricular fibrillation. Under ideal conditions, a spiral/scroll wave commonly rotates steadily around a nonmoving center/filament. However, any symmetry-breaking perturbation, always present in reality, causes a gradual change in rotation frequency and in spatial location of the centre/filament, i.e. a drift. Understanding this drift is vitally important for applications. While drift may be observed in direct numerical simulations, these computations are often expensive and lack generality. There exists a universal asymptotic theory of drift caused by small perturbations. Its applicability is contingent on knowledge of so called response functions (RFs). In a few known cases, the RFs are essentially nonzero only near the core. As a result of this localization, spiral/scroll waves behave like point/string objects, despite being apparently nonlocal regimes. This unique kind of wave-particle duality is directly related to the remarkable stability of spiral/scroll waves. The asymptotic theory exploits this property and allows, in principle, a much simpler and orders of magnitude more efficient prediction of their drift than direct numerical simulations. Once found, RFs of a particular model allow one to predict the drift of spirals and scrolls in response to arbitrary perturbations. The current proposal aims to develop regular and generic methods of obtaining the RFs and then to make the asymptotic theory into an actually working tool for understanding and controlling rotating waves in real systems.