Mathematical Sciences: Networks of Coupled Oscillators

研究人员研究了模拟化学突触的耦合松弛型振荡器的大型网络。这些导致了包含许多物理参数和多个时间尺度的大型微分方程系统。这些模型通常表现出令人难以置信的丰富的动态行为结构；这在很大程度上取决于各种参数。研究者系统地研究在给定网络中可能出现的集体行为类型，确定导致这种丰富振荡行为的数学机制，将模型中参数变化时可能发生的分岔类型进行分类，并确定潜在数学结构的物理后果。本项目中考虑的网络类型出现在许多应用中。这些包括神经系统，呼吸，运动，分泌和场景分割模型的几个领域。该项目检查胰腺细胞爆发振荡的模型。这些细胞负责胰岛素的分泌。它还分析了神经系统各个部分的振荡模型，包括海马体和丘脑。研究者还继续研究场景分割和视觉处理的其他方面。这项工作有助于对胰腺功能和各种神经生理问题的基本理解。

Terman The investigator studies large networks of relaxation-type oscillators with coupling that mimics chemical synapses. These lead to large systems of differential equations that include numerous physical parameters as well as multiple time scales. The models typically display an incredibly rich structure of dynamic behavior; this depends in a highly nontrivial way on the various parameters. The investigator systematically studies what sorts of collective behavior are possible in a given network, determines the mathematical mechanisms responsible for this rich oscillatory behavior, classifies the sorts of bifurcations that can take place as parameters in the models are varied, and determines the physical consequences of the underlying mathematical structures. The types of networks considered in this project arise in numerous applications. These include several areas of the nervous system, respiration, movement, secretion, and models for scene segmentation. The project examines models for bursting oscillations in pancreatic beta cells. These cells are responsible for the secretion of insulin. It also analyzes models for oscillations in various parts of the nervous system, including the hippocampus and the thalamus. The investigator also continues a study of scene segmentation and other aspects of visual processing. This work contributes to fundamental understanding of pancreatic functions and a variety of neurophysiological questions.