Complex Oscillations and Invariant Manifolds

这个项目研究具有多个时间尺度的动力系统的数学理论，并开发新的计算方法来将该理论应用于生物现象的模型。本研究采用几何方法对这些问题进行研究。特别是，它研究了在组织复杂振荡中起关键作用的不变流形。计算这些流形的新的计算方法是研究的重点之一。事实上，目前普遍缺乏计算机研究流形的方法。创造一个全面的“光滑计算几何”是这项研究的长期目标。该项目还寻求开发将模型与数据进行拟合的方法。一个复杂的动力学模型的所有参数都可以测量，或者从经验时间序列数据中使用系统的方法来估计这些参数，这是很少见的。对于多时间尺度系统，这是一个特别困难的优化问题，因为动态的突变不容易被平滑优化算法所基于的二次模型所拟合。这个项目试图确定这些突然变化发生在哪里。该方法还实现了准确的灵敏度分析，该分析描述了模型轨迹随参数变化的变化率。它们的设计目的是为设计周期性运行状态的工程系统而不是稳定运行状态的工程系统提供可用的方法工具包。动力系统理论在关联在种群动力学、化学反应、激光等领域观察到的广泛不同的现象方面取得了惊人的成功。这个项目沿袭了这一传统，试图解释在节奏过程中观察到的普遍动力学行为，其中许多过程表现出复杂的振荡。呼吸、心跳、昼夜节律、月经周期和动物运动是这些方法应用于生物节律的几个例子。高等动物的所有主要运动方式：行走、奔跑、滑行、游泳和飞行都是身体周期性运动的结果。神经系统中普遍存在的爆发性振荡体现了时间的复杂性：神经元活跃放电的时期与静止期交替。在非平衡化学反应器的混合模式振荡中，大振幅振荡和小振幅振荡交替出现。在这些复杂的振荡中，多个时间尺度是固有的。因此，这个项目开发了新的方法来分析具有多个时间尺度的动力系统，其结果产生了更深层次的数学理解，即系统中的快速变化是如何由慢速分量的变化引起的。几何模型将这种变化的机制简化为最简单的形式，并为从具有多个时间尺度的系统的数值模拟获得的神秘结果提供数学解释。

This project studies the mathematical theory of dynamical systems with multiple time scales and develops new computational methods for bringing this theory to bear upon models of biological phenomena. The research employs geometric approaches to study these problems. In particular, it investigates invariant manifolds that play a key role in organizing complex oscillations. New computational methods for computing these manifolds are one focus of the research. Indeed, there is a general lack of methods for computer investigation of manifolds. Creation of a comprehensive ``smooth computational geometry'' is a long term goal of the research. The project also seeks to develop methods for fitting models to data. It is rare that all of the parameters of a complex dynamical model can be measured or that systematic methods are used to estimate these parameters from empirical time series data. With multiple time scale systems, this is a particularly difficult optimization problem because abrupt changes in the dynamics are not readily fit by the quadratic models upon which smooth optimization algorithms are based. This project seeks to identify where these abrupt changes occur. The methods also enable accurate sensitivity analysis that describes the rates of change of model trajectories as parameters are varied. They are designed to contribute to the toolkit of methods available for designing engineered systems with periodic operating states rather than ones which are steady.Dynamical systems theory is astonishingly successful in relating widely disparate phenomena observed in population dynamics, chemical reactions, lasers and much more. This project follows this tradition, seeking to explain universal dynamical behaviors observed in rhythmic processes, many of which display complex oscillations. Respiration, the heartbeat, circadian rhythms, menstrual cycles and animal locomotion are a few examples of biological rhythms to which the methods apply. All the primary modes of locomotion of higher animals: walking, running, slithering, swimming and flying result from cyclic motions of the body. Bursting oscillations that are ubiquitous in the nervous system exemplify temporal complexity: epochs of active firing of neurons alternate with quiescent periods. In mixed mode oscillations of non-equilibrium chemical reactors, epochs of large and small amplitude oscillations alternate. Multiple time scales are inherent in these complex oscillations. Thus, this project develops new methods for the analysis of dynamical systems with multiple time scales and the results yield a deeper mathematical understanding of how rapid changes in a system can result from variations of slow components. Geometric models reduce the mechanisms for such changes to their simplest forms and provide mathematical explanations for enigmatic results obtained from numerical simulations of systems with multiple time scales.