FRG: Synchrony and Structure in Coupled Cell Systems

Golubitsky0244529 耦合单元系统可以通过一组单独的相互作用的微分方程来描述。 细胞系统在各种应用中用作模型，例如神经元网络、物种形成、约瑟夫森连接阵列和基因动力学。 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。 网络架构是一个图表，显示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以前曾被用来探索细胞系统解决方案的某些属性，例如同步或行波。 然而，对称性仅直接适用于最规则的网络。 对于更大类的耦合单元网络，在网络的一部分上定义的局部对称性可以取代对称性作为有趣且重要的动态的预测因子。 这些局部对称性形成群形，正是这种群形结构用于分析耦合单元系统中解的性质以及解之间的转换。 研究人员和学生发展了鲁棒同步理论，并研究耦合细胞系统中的同步破坏分岔（例如，网络体系结构通常迫使 Takens-Bogdanov 奇点出现在此类分岔处的余维一中）。 他们还研究了单细胞内部方程特定特征（例如对称性和快/慢变量）对耦合细胞系统的影响。 对称性和对称性破缺已被科学家和数学家广泛用于研究各种物理和生物学方面有趣的主题，包括流体流动的重要类型、晶格、基本粒子的存在以及老虎和豹子皮肤的特征标记。 这种方法的关键特征是，它是模型无关的，因为不适当的情况对称性允许开发可能结果的菜单，并且物理或化学或生物学可以从此菜单中进行选择。 事实上，对称耦合细胞系统理论已成功应用于动物不同步态、视觉皮层模式形成和物种形成的分析。 放宽对所考虑的网络具有对称性的要求，将这种方法的适用性扩展到更广泛的问题，特别是基因网络模型以及神经科学的其他网络模型。 在这些应用中，很少有人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非常重要，这些工具关注由方程的一般结构（例如网络体系结构）而不是方程的细节决定的解属性。 这种方法通过描述网络可能表现出的可能动态的菜单，将使数学以外的研究人员受益。

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.