Mathematical Sciences: Dynamic Bifurcation Phenomena in Excitable Systems

动态分叉现象的建模和分析在科学和工程中非常重要，因为实验通常是通过动态调整一个或多个参数来执行的，直到失稳导致向新的观察状态或行为的转变。利用数值和计算方法，这个项目将研究空间不受限制的轴突和有髓神经节的调节现象。一个主要的目标是建立被动负荷(传导和电容)和神经细胞调节之间的相互关系，以及它是如何影响信号的。该项目需要对具有缓慢变化的边界和界面条件的反应扩散方程的计算和解析分支进行研究。另一个目标是对不同种类的随机波动如何影响神经模型中的适应进行分类和探索。退出时间估计和数值模拟将采用随机摄动法。在神经科学中，细胞水平的阈值和调节实验是通过增加或增加施加的电流来缓慢地去极化可兴奋的膜来进行的。这些实验研究空间不受限制的轴突和有髓神经节中的调节现象，对于了解部分脱髓鞘神经为什么以及如何发生传导故障非常重要；多发性硬化症等衰弱疾病可以诱导部分脱髓鞘的病变。此外，这个项目中开发的许多工具对于研究动态分叉现象很有用，这些现象发生在科学和工程的许多其他领域。

The modeling and analysis of dynamic bifurcation phenomena is important in science and engineering because experiments are often performed by dynamically tuning one or more parameters until destabilization causes a transition to a new observed state or behavior. Using numerical and computational methods, this project will investigate accommodation phenomena in spatially-unrestricted axons and myelinated nerve segments. A primary goal is to establish the interrelationship between passive loading (both conductive and capacitive) and accommodation in nerve cells and how it affects signaling. The project requires computational and analytic bifurcation studies of reaction-diffusion equations with slowly varying boundary and interface conditions. Another objective is to classify and explore how different kinds of random fluctuations influence accommodation in nerve models. Stochastic perturbation methods for exit time estimates and numerical simulations will be employed. In neuroscience, threshold and accommodation experiments at the cellular level are performed by slowly depolarizing an excitable membrane by ramping or stepping up an applied current. Such experiments to investigate accommodation phenomena in spatially-unrestricted axons and myelinated nerve segments are important for understanding why and how conduction failures occur in partially demyelinated nerve; debilitating diseases such as Multiple Sclerosis can induce lesions that partially demyelinate the myelin sheath. In addition, many of the tools developed in this project are useful for studying dynamic bifurcation phenomena, which occur in many other areas of science and engineering.