Pattern Formation in Semi-Discrete Excitable Media

Hastings 9703630可激发介质的半离散模型由常微分方程组的耦合系统组成，每个单元描述单个“细胞”中的电动力学。它们介于连续(PDE)模型和纯离散模型或细胞自动机之间。研究人员研究了为元胞自动机模型开发的原理是否适用于更现实的半离散情况。他考虑了从实验数据开发的特定模型，以给出对神经行为的定量描述。一个特别有趣的模型是Morris和LeCar的模型，该模型描述了藤壶肌肉纤维的电压振荡。该模型具有与著名的Fitzhugh-Nagumo系统不同的特征，这些特征在许多实验环境中都可以找到，而且似乎可以进行数学分析。研究人员探索了系统中的各种参数在空间模式传播中的作用。要考虑的因素包括激发阈值、耦合强度、快过程和慢过程的相对速率、介质的空间几何以及在激发态和难解态花费的相对时间。这个项目的目标是发展对半离散可激发介质中图案形成的理解。它的主要用途是神经生物学，尽管在许多生物和化学系统中都有可兴奋的介质。如果孤立的细胞对短暂的外部刺激表现出阈值现象，但本身不能支持持续的振荡，则称为可兴奋。半离散模型描述了由线性或非线性扩散机制耦合的可兴奋神经元的集合。例子包括神经、心脏和肌肉组织，支持现象，如波沿着有髓轴突传播，电刺激波席卷心肌，以及传播性皮质抑制，这是一种脑波现象。

Hastings 9703630 Semi-discrete models of excitable media consist of coupled systems of ordinary differential equations, with each unit describing the electrical kinetics in a single "cell". They are intermediate between continuous (pde) models and purely discrete models, or cellular automata. The investigator studies whether principles developed for cellular automata models apply to the more realistic semi-discrete case. He considers specific models developed from experimental data to give quantitative descriptions of neural behavior. A model of special interest is that of Morris and Lecar, describing voltage oscillations in barnacle muscle fiber. This model has different features from the well-known FitzHugh-Nagumo system, features that are found in a number of experimental settings and that, moreover, appear amenable to mathematical analysis. The investigator explores the role of various parameters in the system in the propagation of spatial patterns. Among the factors to be considered are excitation threshold, coupling strength, relative rates of fast and slow processes, spatial geometry of the medium, and the relative time spent in the excited and refractory states. The goal of this project is to develop an understanding of pattern formation in semi-discrete excitable media. The main intended application is to neurobiology, though excitable media are found in many biological and chemical systems. Isolated cells are called ``excitable'' if they exhibit a threshold phenomenon in response to a brief external stimulus, but cannot support continued oscillations on their own. A semi-discrete model describes collections of excitable neurons coupled by a linear or nonlinear diffusion mechanism. Examples include nerve, cardiac, and muscle tissue, supporting phenomena such as wave propagation down a myelinated axon, waves of electrical stimulation that sweep through heart muscles, and spreading cortical depression, a brain wave phenomenon.