Mathematical Sciences: Mathematical Problems Concerning Oscillations and Singular Perturbations

研究人员开发了数学方法来分析 振荡神经元网络 这种网络在世界上很常见， 中枢神经系统，但到目前为止， 它们的结构和功能在很大程度上尚未探索。 一些 具体项目与分析紧密相连， 特定的神经生理网络，如脊椎动物， 波动运动的中枢模式发生器 这项工作 是与一组 神经生理学家另一些则旨在开发新的方法， 分析这些网络。 特别是，神经元和神经网络 它们在许多不同的时间尺度上运行，新的数学 需要一些技术来理解 可能导致的后果。 数学是用来帮助 确定神经元中的哪些特征或它们的连通性是 负责生理上的紧急行为 准备或计算机模拟它们。 数学 这些技术在这项工作的早期阶段是有用的 包括产生更简单方程的简化过程 有类似的行为。有待开发的新技术包括 解析多时间方程的几何方法 鳞片 一些项目主题涉及发展 这些技术与生物学问题无关。 中枢模式发生器是神经网络--集群 连接在一起的神经元--被认为控制着 有节奏的运动活动，如步行，跑步，游泳， 呼吸了 这些生成器是振荡网络的例子 的神经元。 该项目旨在更好地了解 振荡网络的结构之间的关系 以及网络的行为。 这种理解是 是各种神经生理学问题的基础。

The investigator develops mathematical methods for analyzing oscillatory networks of neurons. Such networks are common in the central nervous system, but so far the relationship between structure and function in them is largely unexplored. Some of the specific projects are tightly connected to the analysis of particular neurophysiological networks, such as the vertebrate central pattern generator for undulatory locomotion. This work is undertaken in collaboration with a group of neurophysiologists. Others are designed to develop new methods to analyze such networks. In particular, neurons and networks of them operate on many different time scales, and new mathematical techniques are needed to understand the range of behaviors that can result as a consequence. The mathematics is used to help determine which features in the neurons or their connectivity is responsible for the emergent behavior seen in physiological preparations or in computer simulations of them. Mathematical techniques that have been useful for earlier stages of this work include reduction procedures that produce simpler equations having similar behavior. New techniques to be developed include geometric methods for the analysis of equations with many time scales. Some of the project topics concern the development of such techniques independent of biological questions. Central pattern generators are neural networks -- clusters of neurons connected together -- that are thought to govern rhythmic motor activity such as walking, running, swimming, and breathing. These generators are examples of oscillatory networks of neurons. The project aims to develop a better understanding of the relations between the structure of oscillatory networks and the behaviors of the networks. Such understanding is fundamental for a variety of neurophysiological questions.