Temporal and Spatio-Temporal Forcing of Oscillatory and Excitable Systems

提议：DMS-0309667PI：David Mary Silber[m-Silber@northwestern.edu]机构：西北大学标题：振荡和可兴奋系统的时空强迫和时空强迫研究人员与学生和同事一起，研究了振荡或可兴奋系统的时间或时空强迫是重要的三个问题：(1)参数激发的表面波模式，(2)模式形成系统中的时空局部反馈，以及(3)基于Hopf分叉的内耳毛细胞声音放大机制。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性强迫函数的形式，形成了各种各样的模式。研究人员的研究计划集中于对流体容器施加周期性的增量函数脉冲序列时，三波和四波相互作用的分叉分析。这种理想化的强迫函数允许在不稳定开始或接近开始时应用的线性和弱非线性区域中取得前所未有的分析进展。这个项目探索周期性强迫函数如何被设计成有利于特定的模式。在第二个研究项目中，利用时空反馈来探索非线性模式的形成过程，并对其进行主动控制。通过线性稳定性分析、等变分叉理论和数值模拟，研究了局部时滞和空间变换反馈对时空斑图的控制。在第三个项目中，分析了负责将声音诱导的运动转换为电信号的内耳毛细胞的模型。最初的焦点是两栖毛细胞，已经确定了两种有助于细胞频率选择性的独立机制-一种是由于毛束的主动机械运动，另一种是由毛细胞体中离子通道的电化学模型捕获的。在每个模型中，靠近Hopf分叉的位置有助于毛细胞的放大特性。研究人员的研究项目使用动态系统方法，从现有的两种Hopf分叉机制的详细生理模型中推导出可靠的简化模型，并注意这种两级放大对增益和频率选择性的影响。该项目为进一步研究毛细胞束的耦合效应奠定了基础。许多空间扩展的非线性系统，包括流体力学和激光系统，在外力作用下表现出时空混沌行为。研究人员的研究计划将使人们更深入地理解如何消除不规则行为，转而支持时空规则模式。这是通过在水动力波的情况下适当地设计时间强迫函数，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的。将对理论结果和实验调查结果进行仔细的比较，为本研究工作提供有价值的反馈。研究人员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拟议的频率选择性和放大机制中的非线性如何协同工作，以实现更大的增益。对应用数学研究生和博士后研究员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.