Mathematical Sciences: Nonlinear Differential and Integral Equations in Physiology

[303706] Ermentrout提议者继续他的中央模式发生器的建模工作，特别应用于七鳃鳗的脊髓。他与几位生理学家合作，对肌肉和神经组织中电活动的传播和发生进行建模。将开发将复杂膜模型简化为“连接主义”模型的技术和必要的数值程序。交互式计算工具求解抛物型偏微分方程和积分方程，出现在生物模型也将开发。肌肉收缩活动规律的形成在许多生命系统中是至关重要的。本提案旨在研究该组织的共同特征。当食物沿着小肠向下移动时，有节奏的收缩波，血管直径的小规律变化，心脏的正常跳动都是由细胞的振荡电活动控制的。理解这种模式振荡对于发展对生理结构和功能的基本理解至关重要。此外，这些活动的不规则性可能导致疾病状态，在某些情况下甚至导致死亡。相互作用的节奏单位的数学模型可以揭示单位的正常行为和相互作用，单位中的不规则性是如何产生的，有时还可以说明如何将它们重置为正常模式。提议者继续他在这些模型上的工作，以及开发计算工具来帮助这种类型的建模。* * *

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. ***