Mathematical Sciences: Dynamic Bifurcation Phenomena in Excitable Systems

动态分岔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，直到不稳定导致过渡到新的观察到的状态或行为。 该项目将使用数值和计算方法研究空间不受限制的轴突和有髓神经节段的调节现象。 主要目标是建立神经细胞中的被动负载（传导性和电容性）和调节之间的相互关系以及它如何影响信号传导。 该项目需要对边界和界面条件缓慢变化的反应扩散方程进行计算和分析分岔研究。 另一个目标是分类和探索不同类型的随机波动如何影响神经模型的调节。 将采用随机扰动方法进行退出时间估计和数值模拟。 在神经科学中，细胞水平的阈值和调节实验是通过逐渐增加或增加施加的电流来缓慢去极化可兴奋膜来进行的。 这种研究空间不受限制的轴突和有髓神经节段的调节现象的实验对于理解部分脱髓鞘神经传导失败的原因和方式非常重要；多发性硬化症等使人衰弱的疾病可引起髓鞘部分脱髓鞘的病变。 此外，该项目中开发的许多工具可用于研究科学和工程许多其他领域中发生的动态分岔现象。

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.