Mathematical Sciences: Mathematical Problems Concerning Oscillations and Singular Perturbations

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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.