Mathematical Analysis of Systems with Multiple Components and Multiple Time Scales

研究者研究的系统有多个相互作用的成分，共同创造节奏。这些系统来自神经、化学和基因调控环境。主要有两个主题。第一个是降维，说明在什么情况下，大规模的模型类似于低维的微分方程或地图。其中一些工作使用几何奇异摄动理论来研究多自由度系统如何“凝聚”成基本上更低维的系统，至少在相空间中是局部的。另一组与降维有关的问题涉及快速部分同步。第二个主题涉及系统中许多组件的相互作用如何导致某些组件的活动抑制。化学模式形成和神经系统中有两个需要研究的例子，每一个都有某种类型的全局抑制反馈。具有许多独立但相互作用的组分的系统出现在各种各样的应用中，包括生物技术、化学工程和神经生物学。一般来说，这类系统具有大量的自由度，通常通过数值模拟来研究。然而，模拟本身并不能深刻理解为什么系统会以这样的方式运行，或者如何操纵它们。将方程式简化为更小的系统可能非常有用。但是，如果没有一个原则性的方法来做的减少，一个人不知道有多少的大系统的行为是丢失。这个项目关注的方法来找到这样的原则性的减少，使用数学工具，使人们能够调查的情况下，部分的动态行为像系统的自由度少得多。其他的工具可以帮助我们理解在给定的时间内，即使所有的组件都是耦合的，也只有组件的一部分是可以参与的。该方法适用于从神经，化学和基因调控系统的问题。该项目还为学生和博士后提供了跨学科的培训机会。

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.