Complex Structures in Spatially Extended Dynamical Systems

建议：DMS-0309657PI：Hermann Riecke[h-Riecke@northwestern.edu]机构：西北大学标题：空间扩展动力系统中的复杂结构摘要由这笔赠款资助的研究将有助于理解两种类型的多自由度动力系统的复杂时空结构。1)用解析和计算方法研究了多频率近共振强迫对空间扩展的连续振荡介质的影响。多频率强迫实质上提供了对系统的更大的控制，并且可以获得动态迷宫图案，并且可以允许螺旋波的湮灭；通过允许调节空间周期模式之间的相互作用，它可以稳定和选择表现出多个长度尺度的空间结构，例如超晶格或准图案。与垂直振动流体中已被充分研究的法拉第波相比，振子可能会失去相对于作用力的锁相，这可能会增加超晶格和准图案的时间复杂性。在目前实验研究的光敏化学反应中，可以实现振荡介质的多频强迫。2)由离散元件组成并呈现复杂拓扑结构的网络的重要性日益受到重视。人们对它们的几何性质给予了极大的关注。第二个项目将阐明由局部耦合的离散可激发元件组成的网络的动力学如何受到随机远程连接的影响，这些连接将网络转换为小世界网络。这项研究的灵感来自于对大脑皮层组织的研究，在这些组织中，具有本地和非本地连接的神经网络在没有外部输入的情况下显示出持续的活动。在没有外部输入的情况下，长距离连接能否诱导持续的活动？它如何取决于网络拓扑？相对于噪声，活动状态和静止状态之间的双稳态的健壮性如何？动力系统的数学理论为科学和工程的许多领域理解和预测系统的动力学行为提供了强有力的工具。PI和他的合作者的工作将集中在由大量相互作用的动力元素组成的两类不同的系统上。1)自发振荡存在于许多空间扩展的自然系统中，例如化学系统。它们可以导致具有重要生物功能的波的传播。例如，cAMP波在网柄菌细胞聚集形成多细胞有机体时提供它们之间的信号，而钙波提供各种细胞内的通讯。重要的是要了解这种振荡如何对环境的变化做出反应。当其频率接近振荡固有频率的倍数时，这种变化的影响最大。第一个项目将确定这种近乎共振的变化造成的各种后果。预计螺旋动力学的结果也将与心肌等兴奋性介质相关。鉴于螺旋波在危及生命的室颤中的重要性，多频强迫是否能消灭螺旋波的问题尤其令人感兴趣。2)神经元网络的自我维持活动对于大脑的各种任务至关重要，例如在执行基于信息的任务时(例如，当拨打电话号码时)，大脑能够在短时间内快速存储信息，然后删除该信息。第二个项目将阐明神经元之间的哪种连接对这样的任务有利。-教授研究生分析和计算方法及其应用和沟通技能是这两个研究项目不可分割的一部分。

Proposal: DMS-0309657PI: Hermann Riecke [h-riecke@northwestern.edu]Institution: Northwestern UniversityTitle: Complex Structures in Spatially Extended Dynamical SystemsABSTRACTThe research funded by this grant will contribute to the understanding of complex spatio-temporal structures in two types of dynamical systems with many degrees of freedom. 1) The impact of near-resonant forcing with multiple frequencies on spatially extended continuous oscillatory media will be investigated using analytical and computational methods. Multi-frequency forcing affords substantially greater control of the system and may give access to dynamical labyrinthine patterns and may allow the annihilation of spiral waves; by allowing to tune the interaction between spatially periodic modes it may stabilize and select spatial structures exhibiting multiple length scales such as superlattices or quasipatterns. In contrast to the well-studied Faraday waves in vertically vibrated fluids, the oscillators may loose their phase-locking relative to the forcing, which may add temporal complexity to the superlattices and quasipatterns. Multi-frequency forcing of oscillatory media can be implemented in the light-sensitive chemical reactions currently investigated experimentally. 2) The importance of networks comprised of discrete elements and exhibiting complex topology has been increasingly appreciated. Most attention has been given to their geometrical properties. The second project will elucidate how the dynamics of a network of locally coupled discrete excitable elements is impacted by the addition of random long-range connections, which transform the network into a small-world network. The research is inspired by studies of cortical brain tissue in which neural networks with local and non-local connectivity exhibit persistent activity without external input. Can the long-range connections induce the persistent activity in the absence of external input? How does it depend on the network topology? How robust is the bistability between the active and the quiescent state with respect to noise? The mathematical theory of dynamical systems provides powerful tools to understand and predict the dynamical behavior of systems in many areas of science and engineering. The work of the PI and his collaborators will focus on two distinct classes of systems that are comprised of a large number of interacting dynamical elements. 1) Spontaneous oscillations occur in many spatially extended natural systems, e.g. chemical systems. They can lead to the propagation of waves that have important biological functions. For instance, cAMP-waves provide the signaling between Dictyostelium cells when they aggregate to form a multi-cellular organism, and calcium waves provide communication inside a wide variety of cells. It is important to understand how such oscillations respond to a variability of their environment. The variation has strongest impact when its frequency is close to a multiple of the natural frequency of the oscillation. The first project will identify various consequences resulting from such near-resonant variations. It is expected that the results on spiral dynamics will also be relevant for excitable media like heart muscle. In view of the significance of spirals during life-threatening ventricular fibrillation the question whether multi-frequency forcing can annihilate spirals is of particular interest. 2) Self-sustained activity of networks of neurons is essential for various tasks of the brain such as its ability to quickly store information for a brief duration while performing a task based on that information (e.g. when dialing a phone number) and then deleting it. The second project will shed light on what kind of connectivity between the neurons is favorable for tasks like that. - Teaching graduate students analytical and computational methods and their application as well as communication skills is an integral part of both research projects.