Modeling and Analyzing Pattern Formation Mechanisms Derived from Nonlinear Physiological and Neuronal Systems

这项研究旨在对细胞生理学和神经科学中的非线性系统产生的时间、空间和时空模式形成机制进行建模和分析。解析和数值方法被用来研究描述这些系统的非线性常微分方程组、偏微分方程组和积分微分方程组。与向无序增加的状态进化的封闭物理系统不同，生物系统通常是开放的，并进化成日益复杂的结构/模式。了解这些模式形成的机制是生物学研究的主要目标之一，对于揭示它们的功能和开发潜在的控制它们的方法至关重要。由于这些过程的非线性和动力学性质，数学方法在实现仅使用实验工具很难甚至不可能获得的理解方面往往是必不可少的。这项研究的主题是：(1)神经内分泌细胞的单细胞行为和激素的同步节律分泌；(2)耦合细胞中的一般同步机制；(3)可兴奋的单细胞和多细胞介质中的波传播；以及(4)噪声在生物模式形成中的作用。在开发模型时，我们使用最新实验的数据和结果。一旦开发出一个模型，我们就会深入了解每一种新奇行为背后的数学原理。一个描述弥漫分布的GnRH神经元中脉动性GnRH释放的同步和产生的模型就是一个很好的例子。这些项目与实验细胞生理学前沿发展之间的密切关系证明了拟议研究的新颖性和潜在影响。在对这些模型中出现的方程的数学理解方面的进展将加深我们对非线性微分方程的理解，这些方程以前没有得到很好的探索，并有可能被应用于解决相关领域中其他感兴趣的问题。

This research is aimed at modeling and analyzing temporal, spatial, and spatio-temporal pattern-formation mechanisms arising from nonlinear systems in cell physiology and neural sciences. Analytical and numerical methods are employed/developed in the study of the nonlinear ordinary, partial, and integral differential equations that describe these systems. Unlike closed physical systems that evolve toward a state of increased disorder, biological systems are typically open and evolving into structures/patterns of increasing complexity. Understanding the mechanisms driving the formation of these patterns is one of the major goals of biological research and is vital in revealing their functions and developing potential ways of controlling them. Due to the nonlinear and dynamical nature of these processes, mathematical methods are often essential in achieving an understanding that is otherwise difficult or even impossible to obtain using only experimental tools. The topics of this proposed research are: (1) single-cell behaviors and synchronized rhythmic secretion of hormones from neuroendocrine cells; (2) general synchronization mechanisms in coupled cells; (3) the wave propagation in excitable single- and multi-cellular media; and (4) effects of noise in the formation of biological patterns. When developing models, we use data and results from most recent experiments. Once a model is developed, we seek in-depth understanding of the mathematics behind each novel behavior. A model describing the synchronization and generation of pulsatile GnRH release in diffusely distributed GnRH neurons is a good example. The novelty and potential impact of the proposed research is evidenced by the close relationship between these projects and the frontier developments in experimental cell physiology. Progress in the mathematical understanding of the equations arising in these models will deepen our understanding of nonlinear differential equations that were poorly explored previously and has the potential of being applied to solving other problems of interest in related areas.