Mathematical Analysis of Systems with Multiple Components and Multiple Time Scales

研究人员研究了具有多个相互作用的组件的系统，这些组件共同创造了节奏。这些系统来自神经、化学和基因调控环境。有两个主要主题。第一种是降维，研究高维模型在何种情况下表现为低维的微分方程式或地图。其中一些工作使用几何奇异摄动理论来研究具有多个自由度系统如何至少在相空间中局部地“凝聚”到本质上较低维的系统。另一组与降维相关的问题涉及快速部分同步。第二个主题涉及系统中多个组件的交互如何导致某些组件中的活动被抑制。要研究的两个例子来自非对称模式形成和神经系统，每个系统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件的系统存在于大量的应用中，包括生物技术、化学工程和神经生物学。一般情况下，这类系统具有较多的自由度，通常采用数值模拟的方法进行研究。然而，仅靠模拟并不能深入理解系统为什么会这样运行，或者它们是如何被操纵的。将方程简化为较小的系统是非常有用的。但是，如果没有适当的方法进行简化，人们就不知道大系统的行为损失了多少。这个项目涉及找到这样的原则性约化的方法，使用数学工具使人们能够研究在哪些情况下动力学部分的行为类似于具有更少自由度的系统的情况。可以使用其他工具来理解如何在给定时间只涉及组件的某个子集，即使所有组件都是耦合的。这些方法被应用于神经、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.