Mathematical Sciences: Mathematical Problems Concerning Populations of Oscillators

8901913科佩尔这个项目是研究与中央图案生成器有关的各种问题(C.P.G.S)，这是一种神经网络，控制着节律运动活动的刻板印象。这些网络通常包括振荡的子网络或细胞与其他振荡器或非振荡的神经元相互作用。研究最多的CPG之一S是一个大致被配置为一系列振荡器的网络。数学的一个目标是直观和测试关于连通性的一般类型约束的猜想，以便网络能够正常运行。目前的努力建立在以前结果的基础上，并集中在三个领域：(1)远程耦合在网络中的作用，(2)网络的构建不是由模块振荡子部分组成，而是表现为离散振荡器的链，(3)C.P.G.的神经活动与它所引导的机械活动之间的相互作用。此外，还提出了关于小无脊椎动物网络C.P.G.S所引起的问题的工作。这项工作的重点是将细胞的性质与网络的涌现性质联系起来。所有工作都要在实验者的配合下完成。最后，提出了一些纯粹的数学问题。

8901913 Kopell This project is to study a variety of questions relating to central pattern generators (C.P.G.'s), which are neural networks that govern the stereotypic aspects of rhymthmic motor activity. These networks often involve oscillating subnetworks or cells interacting with other oscillators or non-oscillating neurons. One of the most studied C.P.G.'s is a network that is roughly configured as a chain of oscillators. One goal of the mathematics is to intuit and test conjectures about general kinds of constraints on the connectivity, in order that the network be able to function appropriately. The current effort builds on former results and focuses on three areas: (1) the role of long-range coupling in the network, (2) the construction of networks not composed of modular oscillatory subpieces, but behaving like chains of discrete oscillators, (3) the interaction of the neural activity of the C.P.G. with the mechanical activity that it directs. In addition, work is proposed on issues arising from small invertebrate network C.P.G.'s. The focus of that work is to relate properties of the cells to emergent properties of the network. All work is to be done in coordination with experimentalists. Finally, some purely mathematical topics are proposed.