Temporal and Spatio-Temporal Forcing of Oscillatory and Excitable Systems

提案：DMS-0309667PI: David Mary Silber [m-silber@northwestern.edu]机构：Northwestern university标题：振荡和可激系统的时空强迫摘要研究者与学生和同事研究了振荡或可激系统的时空强迫的三个重要问题：(1)参数激发表面波模式；(2)模式形成系统中的时空局部反馈；(3)基于Hopf分岔的内耳毛细胞声放大机制。在流体的自由表面上激发的法拉第波，根据流体的性质和周期强迫函数的形式，会形成各种各样的图案。研究者的研究计划侧重于三波和四波相互作用的分岔分析，当一个周期序列的δ函数脉冲应用于流体容器。这种理想化的强迫函数允许在应用于或接近不稳定开始的线性和弱非线性制度中取得前所未有的分析进展。这个项目探讨了如何设计周期强迫函数来支持特定的模式。在第二个研究项目中，利用时空反馈来探测非线性模式的形成过程，并对其进行主动控制。本文将通过线性稳定性分析、等变分岔理论和数值模拟研究局部时滞和空间变换反馈对时空模式的控制。在第三个项目中，对内耳毛细胞模型进行了分析，内耳毛细胞负责将声音引起的运动转化为电信号。最初的研究重点是两栖动物的毛细胞，已经确定了两种不同的机制来促进细胞的频率选择性——一种是由于毛束的主动机械运动，另一种是由毛细胞体内离子通道的电化学模型捕获的。在每个模型中，接近Hopf分岔有助于毛细胞的扩增特性。研究者的研究项目使用动力系统方法，从现有的两种Hopf分岔机制的详细生理模型中得出一个可靠的简化模型，并注意这两级放大对增益和频率选择性的影响。本项目为进一步研究毛细胞束耦合效应奠定了基础。许多空间扩展的非线性系统，包括水动力系统和激光系统，在受到外力作用时表现出时空混沌行为。研究者的研究计划将导致对如何消除不规则行为以支持时空规则模式的更深层次的理解。在水动力波的情况下，这是通过适当设计时间强迫函数来实现的，或者在非线性光学和化学系统的情况下，通过时空反馈来实现。将对理论结果和实验研究结果进行仔细的比较，为本研究工作提供有价值的反馈。研究者对内耳毛细胞生物物理模型的分析有助于更好地理解两种提出的频率选择性和放大机制的非线性如何共同作用以获得更大的增益。对应用数学研究生和博士后进行跨学科研究活动的培训是研究工作的一个组成部分。

Proposal: DMS-0309667PI: David Mary Silber [m-silber@northwestern.edu]Institution: Northwestern UniversityTitle: Temporal and Spatio-Temporal Forcing of Oscillatory and Excitable SystemsABSTRACTThe investigator, together with students and colleagues, studies three problems in which temporal or spatio-temporal forcing of oscillatory or excitable systems is important: (1) parametrically excited surface wave patterns, (2) spatio-temporal local feedback in pattern forming systems, and (3) Hopf bifurcation based mechanisms for amplification of sound by inner ear hair cells. Faraday waves, excited on the free surface of a fluid, form in a wide variety of patterns depending on the fluid properties and the form of the periodic forcing function. The investigator's research program focuses on a bifurcation analysis of three- and four-wave interactions when a periodic sequence of delta-function impulses is applied to the fluid container. This idealized forcing function admits unprecedented analytic progress to be made in the linear and weakly nonlinear regimes that apply at or near onset of instability. This project probes how the periodic forcing function may be designed to favor particular patterns. In the second research project spatio-temporal feedback is used to probe the nonlinear pattern formation process, as well as to actively control it. The control of spatio-temporal patterns by local time-delayed and spatially-transformed feedback will be investigated through linear stability analysis, equivariant bifurcation theory, and numerical simulation. In the third project, models of inner ear hair cells, responsible for translating sound-induced motion into electrical signals, are analysed. The initial focus is on amphibian hair cells, for which two separate mechanisms that contribute to the cells' frequency selectivity have been identified - one due to active mechanical motions of the hair bundle and the other captured by an electrochemical model of ion channels in the hair cell body. In each model proximity to a Hopf bifurcation contributes to the amplification properties of the hair cells. The investigator's research project uses dynamical systems methods to derive a reliable reduced model, from existing detailed physiological models of the two Hopf bifurcation mechanisms, with attention to the effects of this two-stage amplification on gain and frequency selectivity. This project lays the foundation for further investigation of the effects of coupling the hair cell bundles.Many spatially extended nonlinear systems, including hydrodynamic and laser systems, exhibit spatio-temporal chaotic behavior when subjected to external forcing. The investigator's research program will lead to a deeper understanding of how to eliminate irregular behavior in favor of spatio-temporally regular patterns. This is done through appropriate design of the temporal forcing function in the case of hydrodynamic waves, or through spatio-temporal feedback in the case of nonlinear optical and chemical systems. Careful comparison between theoretical results and results of experimental investigations will be made, providing valuable feedback to this research effort. The investigator's analysis of biophysical models of inner ear hair cells contributes to a greater understanding of how the nonlinearities in two proposed mechanisms of frequency selectivity and amplification might work together to achieve greater gain. The training of applied mathematics graduate students and postdoctoral fellows in interdisciplinary research activities is an integral part of the research effort.