Response functions for drift of spiral and scroll waves

旋转螺旋波（二维）和涡旋波（三维）是一种自组织形式，在物理、化学和生物性质的许多空间扩展系统中观察到。其中最重要的是心肌，那里的旋转波负责再入性心律失常，包括最致命的心室颤动。在理想条件下，螺旋/涡旋波通常围绕不移动的中心/细丝稳定旋转。然而，现实中总是存在的任何对称性破坏扰动都会导致旋转频率和中心/灯丝空间位置的逐渐变化，即漂移。理解这种漂移对应用程序至关重要。虽然在直接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。存在一个普遍的小扰动引起漂移的渐近理论。它的适用性取决于所谓的响应函数（RFs）的知识。在一些已知的情况下，只有在核心附近，rf本质上是非零的。由于这种局域化，螺旋/涡旋波的行为类似于点/弦对象，尽管它们显然是非局域的。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，并且在原则上允许比直接数值模拟更简单和更有效的预测它们的漂移。一旦发现，特定模型的RFs允许人们预测螺旋和涡旋在任意扰动下的漂移。目前的建议旨在发展正则和通用的方法来获得rf，然后使渐近理论成为一个实际工作的工具，用于理解和控制实际系统中的旋转波。

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.