Transition and pattern formation in physical and physiological systems

将数学思想应用于物理和生物中的开放问题是一个不断发展的研究领域。这种增长的催化剂是科学计算，它使我们能够解决直到最近还遥不可及的复杂问题。在目前的提案中，动力系统的数学理论被用于研究人脑动力学和流体湍流。尽管表面上没有关联，但这些应用程序有许多共同的特点。例如，它们都是已知的强非线性，它们都能产生在一系列空间尺度和时间尺度上相干的信号。脑电图仪是一种常用的脑动力学测量方法，它通常会显示出首选频率的振荡，比如8-13赫兹的阿尔法频率，它还会显示“脑波”，即分布在大脑皮层上的同步活动。在流体湍流中，我们可以看到涡旋结构的形成、相互作用和分解。这项工作旨在解释这种连贯行为的动力学和起源。在过去的几十年里，计算动力系统理论的应用揭示了部分解释。在流体力学中，一些关于湍流如何从光滑流动中产生的问题已经用平衡流和时间周期流等构建块来回答。在大脑模型中，分叉分析揭示了从正常状态到癫痫等病理状态的转变。然而，要使这种类型的分析更接近现实世界的问题，还有很多工作要做。我提出了几个试图弥合这一差距的项目。其中一个项目涉及对人类大脑皮层电信号的生理上可信的模型的分析。我们将尝试辨别模型的稳健行为，因为我们改变了与生理过程相关的许多参数，并使用分歧理论对其进行分类。在流体湍流中，我们将尝试计算在空间和时间上局部化的解，就像在实验中经常观察到的流动一样，并计算具有Kolmogorov相似谱的新的时间周期解，这是湍流的标志。

The application of mathematical ideas to open problems in physics and biology is an ever growing area of research. The catalyst of this growth is scientific computing, which allows us to attack problems of a complexity that was, until recently, out of reach. In the current proposal, the mathematical theory of dynamical systems is used to study human brain dynamics and fluid turbulence. Although apparently unrelated, these applications have many traits in common. For instance, they are both known to be strongly nonlinear, and they both give rise to signals that can be coherent over a range of spatial scales and time scales. One common measurement of brain dynamics, the electroencephalograph, often shows oscillations at preferred frequencies, like the 8-13Hz alpha frequency, and also shows "brain waves", synchronized activity that spreads out over the cortex. In fluid turbulence we see vortical structures form, interact, and break down. The work proposed here aims to explain the dynamics and origin of such coherent behaviour.Over the last decades, the application of computational dynamical systems theory has unveiled part of the explanation. In fluid dynamics, some questions on how turbulent motion emerges from smooth flow have been answered in terms of buidling blocks like equilibrium and time-periodic flows. In brain modelling, bifurcation analysis has shed some light on transitions from normal to pathological states like seizures. However, there is still a lot of work to be done to bring this type of analysis closer to real-world problems. I propose several projects that attempt to close the gap.One project concerns the analysis of a physiologically plausible model of electrical signals in the human cortex. We will try to discern robust behaviour of the model as we vary its many parameters, linked to physiological processes, and classify it using bifurcation theory. In fluid turbulence, we will attempt to compute solutions that are localized in space and time, as are flows often observed in experiments, and to compute novel time-periodic solutions that have a Kolmogorov similarity spectrum, the hallmark of turbulence.