9303706 Ermentrout The proposer continues his work on the modelling of central pattern generators with particular application to the lamprey spinal cord. He collaborates with several physiologists on modelling of the propagation and genesis of electrical activity in muscular and nervous tissues. Techniques for reducing complex membrane models to ``connectionist'' models and the requisite numerical routines will be developed. Interactive computational tools for solving of parabolic PDES and integral equations that arise in biological models will also be developed. The formation of regular patterns of muscular contraction activity is critical in many living systems. This proposal aims to study the common features of this organization. The rhythmic waves of contraction as food moves down the small intestine, the small regular changes in diameter of blood vessels, and the regular beating of the heart are all controlled by oscillating electrical activity of cells. Understanding such patterned oscillators is critical in developing a fundamental understanding of physiological structures and functions. Moreover, irregularities in these activities can lead to diseased states and in some cases even death. Mathematical models of interacting rhythmical units can shed light on how the units normally behave and interact, how irregularities in the units arise, and sometimes on how to reset them to their normal mode. The proposer continues his work on such models as well as the development of computational tools to aid in this type of modeling. ***
