Mathematical Analysis of Systems with Multiple Components and Multiple Time Scales

Trustees Of Boston University, Boston

Investigators

Abstract

The investigator studies systems with multiple interacting components that together create rhythms. The systems come from neural, chemical, and gene regulatory contexts. There are two main themes. The first is reduction of dimensions, investigating the circumstances under which large-dimensional models behave like much lower-dimensional differential equations or maps. Some of that work uses geometric singular perturbation theory to investigate how systems with multiple degrees of freedom "condense" to essentially lower-dimensional systems, at least locally in phase space. Another set of issues related to reduction of dimension concerns fast partial synchronization. The second theme concerns how interaction of many components in a system can lead to the suppression of activity in some of the components. Two examples of this to be studied come from a chemical pattern formation and neural systems, each with some type of global inhibitory feedback. Systems with many separate but interacting components arise in a large variety of applications, including biotechnology, chemical engineering, and neurobiology. In general, such systems have a large number of degrees of freedom, and are usually investigated by numerical simulation. However, simulations alone do not provide a deep understanding of why the systems behave the way they do, or how they can be manipulated. Reductions of equations to smaller systems can be very useful. But without a principled way to do the reductions, one does not know how much of the behavior of the large system is lost. This project concerns methods to find such principled reductions, using mathematical tools that enable one to investigate circumstances under which parts of the dynamics behave like that of systems with a much smaller number of degrees of freedom. Other tools to be used allow one to understand how only some subset of the components can be involved at a given time, even though all components are coupled. The methods are applied to problems from neural, chemical and gene regulation systems. The project also provides interdisciplinary training opportunities for students and postdocs.