Mechanical and neural oscillators with time delays

这个项目的动机是一些特定的生物和机械系统：瞳孔光反射，大脑神经网络和金属切割。虽然物理上无关，这些系统的行为是由相同的因素影响：非线性和时间延迟的存在。 时间延迟表示信息从系统的一部分传播到另一部分所花费的时间。非线性表示这样的事实，即在这些系统中，状态的微小变化往往会导致行为的巨大变化。 振荡器是任何执行重复运动的物体。一个熟悉的例子是钟摆。 这个项目考虑了几个振荡器，它们分为两个主题，基于它们是如何数学建模的。[I]具有时间延迟连接的神经振荡器。大脑是由称为神经元的细胞组成的，它们通过电信号相互交流。众所周知，这些信号是振荡的，并且振荡的性质取决于单个神经元的特性、神经元如何彼此连接以及连接中存在的时间延迟。 我们的目标是回答以下问题：为什么不同大脑结构中的神经细胞具有不同的特征？ 为什么神经细胞以某些方式连接，而不是其他方式？ 时间延迟如何影响这些问题的答案？[II]具有延时反馈的开关振荡器。这里考虑的振荡包括金属切削中使用的工具的振动和眼睛瞳孔的扩张和收缩。 这些系统的一个值得注意的方面是，它们的定义属性根据系统的状态快速切换。 我们将研究时滞反馈在这些系统中产生、消除或减小振荡的作用，这项工作的目标是双重的：获得对系统模型更深层次的数学理解，并根据应用解释这些知识。

This project is motivated by some specific biological and mechanical systems: the pupil light reflex, brain neural networks and metal cutting. Although physically unrelated, the behaviour of these systems is influenced by the same factors: the presence of nonlinearities and time delays.  The time delays represent the time it takes information to propagate from one part of the system to another. The nonlinearity represents the fact that in these systems a small change in the state can often result in a large change in behaviour.  An oscillator is any object that performs a repetitive motion. A familiar example is the pendulum of a clock.  This project considers several oscillators which fall into two themes, based on how they are modelled mathematically.[I] Neural Oscillators with Time Delayed Connections. The brain is made up of cells called neurons which communicate with each other through electrical signals. It is well known that these signals are oscillatory and that the properties of the oscillations depend on the characteristics of the individual neurons, how the neurons are connected to each other and the presence of time delays in the connections.  We will aim to answer the questions: Why do neural cells in different brain structures have different characteristics?  Why are neural cells connected in some ways and not others?  How do time delays affect the answers to these questions?[II] Switching Oscillators with Time Delayed Feedback.The oscillations considered here include the vibrations of a tool used in metal cutting and the dilation and constriction of the pupil of the eye.  A notable aspect of these systems is that their defining properties rapidly switch depending on the state of the system.  We will examine the role of time delayed feedback in creating, eliminating or reducing the size of oscillations in these systems.The goal of this work is twofold: to obtain a deeper mathematical understanding of the models for the systems, and to interpret this knowledge in light of the applications.