Mathematical Analysis of Systems with Multiple Components and Multiple Time Scales

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The investigator studies systems with multiple interactingcomponents that together create rhythms. The systems come fromneural, chemical, and gene regulatory contexts. There are twomain themes. The first is reduction of dimensions, investigatingthe circumstances under which large-dimensional models behavelike much lower-dimensional differential equations or maps. Someof that work uses geometric singular perturbation theory toinvestigate how systems with multiple degrees of freedom"condense" to essentially lower-dimensional systems, at leastlocally in phase space. Another set of issues related toreduction of dimension concerns fast partial synchronization.The second theme concerns how interaction of many components in asystem can lead to the suppression of activity in some of thecomponents. Two examples of this to be studied come from achemical pattern formation and neural systems, each with sometype of global inhibitory feedback. Systems with many separate but interacting components arisein a large variety of applications, including biotechnology,chemical engineering, and neurobiology. In general, such systemshave a large number of degrees of freedom, and are usuallyinvestigated by numerical simulation. However, simulations alonedo not provide a deep understanding of why the systems behave theway they do, or how they can be manipulated. Reductions ofequations to smaller systems can be very useful. But without aprincipled way to do the reductions, one does not know how muchof the behavior of the large system is lost. This projectconcerns methods to find such principled reductions, usingmathematical tools that enable one to investigate circumstancesunder which parts of the dynamics behave like that of systemswith a much smaller number of degrees of freedom. Other tools tobe used allow one to understand how only some subset of thecomponents can be involved at a given time, even though allcomponents are coupled. The methods are applied to problems fromneural, chemical and gene regulation systems. The project alsoprovides interdisciplinary training opportunities for studentsand postdocs.