Temporal and Spatio-Temporal Forcing of Oscillatory and Excitable Systems

提案：DMS-0309667 PI：大卫玛丽西尔伯[m-silber@northwestern.edu]机构：西北大学名称：振荡和可激发系统的时间和时空强迫摘要研究者与学生和同事一起研究了三个问题，其中振荡或可激发系统的时间或时空强迫是重要的：（1）参数激发表面波图案，（2）图案形成系统中的时空局部反馈，（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性质和周期性强迫函数的形式以各种各样的图案形成。研究人员的研究计划侧重于三波和四波相互作用的分叉分析时，三角函数脉冲的周期序列施加到流体容器。这个理想化的强迫函数承认前所未有的分析进展，在线性和弱非线性制度，适用于或接近发病的不稳定。 本计画探讨如何设计周期性的强迫函数以利于特定的模式。第二个研究计画是利用时空回馈来探讨非线性模式的形成过程，并主动地控制它，透过线性稳定性分析、等变分歧理论、以及数值模拟，来探讨局部时间延迟与空间转换回馈对时空模式的控制。在第三个项目中，分析了负责将声音引起的运动转化为电信号的内耳毛细胞模型。 最初的重点是两栖动物的毛细胞，其中两个独立的机制，有助于细胞的频率选择性已被确定-一个是由于积极的机械运动的毛束和其他捕获的电化学模型的离子通道在毛细胞体。 在每个模型中接近一个霍普夫分叉有助于毛细胞的放大性能。 研究人员的研究项目使用动力系统的方法来推导出一个可靠的简化模型，从现有的详细的生理模型的两个霍普夫分岔机制，注意这两个阶段的放大增益和频率选择性的影响。 许多空间扩展的非线性系统，包括流体动力学系统和激光系统，在受到外力作用时都表现出时空混沌行为。 研究人员的研究计划将导致更深入地了解如何消除不规则的行为，有利于时空规律的模式。 这是通过适当的设计的时间强制功能的情况下，流体动力学波，或通过时空反馈的情况下，非线性光学和化学系统。理论结果和实验研究结果之间的仔细比较，将提供有价值的反馈，这项研究工作。 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共同作用以实现更大的增益。 应用数学研究生和博士后研究员在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.