FRG: Synchrony and Structure in Coupled Cell Systems

耦合单元系统可以用相互作用的微分方程的集合来描述。细胞系统被用作各种应用的模型，如神经网络、物种形成、约瑟夫森连接阵列和遗传动力学。在这个重点研究小组（FRG）项目中，研究人员开发了耦合细胞系统的数学理论，并与同事一起探索该理论在应用中的含义。网络架构是一个图，它显示了单元之间的耦合以及哪些单元和耦合是相同的。网络架构中的对称性先前已被用于探索蜂窝系统解决方案的某些特性，例如同步或行波。然而，对称性只直接适用于最规则的网络。对于更大的一类耦合细胞网络，局部对称性定义在网络的一部分可以取代对称性作为一个有趣的和重要的动态预测。这些局部对称性形成了类群，正是这种类群结构被用来分析耦合单元系统中溶液的性质和溶液之间的转换。研究人员和学生发展了鲁棒同步理论，并研究了耦合细胞系统中的同步中断分叉（例如，网络结构经常迫使Takens-Bogdanov奇点在这种分叉处出现协维1）。他们还研究了单细胞内部方程的特定特征对耦合细胞系统的影响，如对称性和快/慢变量。对称性和对称性破缺被科学家和数学家广泛用于研究各种物理和生物学上有趣的话题，包括重要类型的流体流动、晶体晶格、基本粒子的存在，以及老虎和豹子皮肤上的特征标记。这种方法的关键特征是它是模型独立的，在某种意义上，不适当的情况下，对称允许发展出一系列可能的结果，物理学、化学或生物学从这个菜单中选择。事实上，具有对称性的耦合细胞系统理论已经成功地应用于分析动物的不同步态、视觉皮层的模式形成和物种形成。放宽所考虑的网络具有对称性的要求扩展了这种方法在更广泛问题中的适用性，特别是基因网络模型，以及来自神经科学的其他网络模型。在这些应用中，很少知道确切的模型方程。因此，开发研究解的模型独立特征的数学工具是很重要的，这些工具关注的是由方程的一般结构（如网络结构）而不是方程的细节决定的解的性质。这种方法将使数学以外的研究人员受益，因为它描述了网络可能表现出的一系列可能的动态。

Golubitsky0244529 A coupled cell system can be described by a collection ofindividual interacting differential equations. Cell systems areused as models in a variety of applications such as neuronalnetworks, speciation, arrays of Josephson junctions, and genedynamics. In this Focused Research Group (FRG) project, theinvestigators develop a mathematical theory for coupled cellsystems and with colleagues explore implications of that theoryin applications. The network architecture is a graph that showsthe couplings between cells and which cells and couplings areidentical. Symmetries in the network architecture have been usedpreviously to explore certain properties of solutions to cellsystems, such as synchrony or traveling waves. Symmetry, however,applies directly only to the most regular of networks. For alarger class of coupled cell networks, local symmetries definedon part of the network can replace symmetries as a predictor ofinteresting and important dynamics. These local symmetries forma groupoid and it is this groupoid structure that is used toanalyze properties of solutions and transitions between solutionsin coupled cell systems. The investigators and students developa theory of robust synchrony and investigate synchrony-breakingbifurcations in coupled cell systems (for example, networkarchitecture often forces Takens-Bogdanov singularities to occurin codimension one at such bifurcations). They also investigatethe consequences for coupled cell systems of particular featuresof the internal equations of single cells, such as symmetries andfast/slow variables. Symmetry and symmetry-breaking have been used widely byscientists and mathematicians to investigate a variety ofphysically and biologically interesting topics, includingimportant types of fluid flows, crystal lattices, the existenceof elementary particles, and the characteristic markings of theskins of tigers and leopards. The crucial feature of thisapproach is that it is model-independent in the sense that inappropriate situations symmetry permits the development of a menuof possible outcomes and the physics or chemistry or biologychooses from this menu. Indeed, the theory of coupled cellsystems with symmetry has been applied succesfully to theanalysis of different gaits in animals, pattern formation in thevisual cortex, and speciation. Relaxing the requirement that thenetwork under consideration has symmetries extends theapplicability of this approach to a wider range of problems, inparticular to models of gene networks, as well as other networkmodels from neuroscience. In these applications it is rare thatthe exact model equations are known. It is therefore importantto develop mathematical tools that study model-independentfeatures of solutions, tools that focus on solution propertiesthat are determined by the general structure of the equations(such as network architecture) rather than by the details of theequations. This approach will benefit researchers outside ofmathematics by describing a menu of possible dynamics that anetwork can be expected to exhibit.