0244529 A coupled cell system can be described by a collection of individual interacting differential equations. Cell systems are used as models in a variety of applications such as neuronal networks, speciation, arrays of Josephson junctions, and gene dynamics. In this Focused Research Group (FRG) project, the investigators develop a mathematical theory for coupled cell systems and with colleagues explore implications of that theory in applications. The network architecture is a graph that shows the couplings between cells and which cells and couplings are identical. Symmetries in the network architecture have been used previously to explore certain properties of solutions to cell systems, such as synchrony or traveling waves. Symmetry, however, applies directly only to the most regular of networks. For a larger class of coupled cell networks, local symmetries defined on part of the network can replace symmetries as a predictor of interesting and important dynamics. These local symmetries form a groupoid and it is this groupoid structure that is used to analyze properties of solutions and transitions between solutions in coupled cell systems. The investigators and students develop a theory of robust synchrony and investigate synchrony-breaking bifurcations in coupled cell systems (for example, network architecture often forces Takens-Bogdanov singularities to occur in codimension one at such bifurcations). They also investigate the consequences for coupled cell systems of particular features of the internal equations of single cells, such as symmetries and fast/slow variables. Symmetry and symmetry-breaking have been used widely by scientists and mathematicians to investigate a variety of physically and biologically interesting topics, including important types of fluid flows, crystal lattices, the existence of elementary particles, and the characteristic markings of the skins of tigers and leopards. The crucial feature of this approach is that it is model-independent in the sense that in appropriate situations symmetry permits the development of a menu of possible outcomes and the physics or chemistry or biology chooses from this menu. Indeed, the theory of coupled cell systems with symmetry has been applied succesfully to the analysis of different gaits in animals, pattern formation in the visual cortex, and speciation. Relaxing the requirement that the network under consideration has symmetries extends the applicability of this approach to a wider range of problems, in particular to models of gene networks, as well as other network models from neuroscience. In these applications it is rare that the exact model equations are known. It is therefore important to develop mathematical tools that study model-independent features of solutions, tools that focus on solution properties that are determined by the general structure of the equations (such as network architecture) rather than by the details of the equations. This approach will benefit researchers outside of mathematics by describing a menu of possible dynamics that a network can be expected to exhibit.
