FRG: Synchrony and Structure in Coupled Cell Systems

FRG：耦合单元系统中的同步和结构

• 批准号: 0244529
• 负责人: Kresimir Josic
• 金额: $ 88.6万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244529&HistoricalAwards=false
• 关键词: FRG; Synchrony; Structure; Coupled; Cell

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.

戈卢比茨基0244529 耦合细胞系统可以用一组相互作用的微分方程来描述。 细胞系统在神经网络、物种形成、约瑟夫森结阵列和基因组学等许多应用中被用作模型。 在这个聚焦研究小组（FRG）项目中，研究人员开发了一个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意义。 网络结构是一个图表，显示了细胞之间的耦合，以及哪些细胞和耦合是相同的。 网络结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性质，如同步性或行波。 然而，对称性只直接适用于最规则的网络。 对于一类较大的耦合胞元网络，局部对称性可以代替对称性，作为重要动力学的预测因子。 这些局部对称性形成了一个群胚，正是这种群胚结构被用来分析耦合胞腔系统中解的性质和解之间的转换。 研究人员和学生发展了一个强大的同步理论，并研究了耦合细胞系统中的同步断裂分叉（例如，网络结构通常迫使Takens-Bogdanov奇点在这种分叉处发生余维1）。 他们还研究了耦合细胞系统的特殊功能的内部方程的单细胞，如对称性和快/慢变量的后果。 对称性和对称破缺已被科学家和数学家广泛用于研究各种物理学和生物学上有趣的课题，包括重要类型的流体流动、晶格、基本粒子的存在以及虎豹皮的特征标记。 这种方法的关键特征是，它是独立于模型的，在这个意义上，不适当的情况下，对称性允许一个菜单的可能结果的发展和物理或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，以及神经科学中的其他网络模型。 在这些应用中，精确的模型方程是已知的。 因此，重要的是要开发数学工具，研究模型无关的解决方案，工具，侧重于解决方案的属性，由一般结构的方程（如网络架构），而不是由细节的方程。 这种方法通过描述一系列可能的网络动态，将使数学之外的研究人员受益。