Transition and pattern formation in physical and physiological systems

数学思想在物理学和生物学中的应用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无法解决的复杂问题。在目前的提议中，动力系统的数学理论被用来研究人脑动力学和流体湍流。虽然表面上看起来不相关，但这些应用程序有许多共同的特征。例如，已知它们都是强非线性的，并且它们都产生可以在空间尺度和时间尺度范围内相干的信号。一种常见的大脑动力学测量方法，脑电图仪，经常显示在首选频率的振荡，如8- 13赫兹的阿尔法频率，也显示“脑波”，同步活动，分布在皮层。在流体湍流中，我们看到旋涡结构的形成、相互作用和分解。这里提出的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已经揭示了部分解释。在流体动力学中，关于湍流运动如何从光滑流中产生的一些问题已经在平衡流和时间周期流等方面得到了回答。在大脑建模中，分叉分析揭示了从正常状态到病理状态（如癫痫发作）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。我提出了几个项目，试图缩小差距。 其中一个项目涉及分析人类大脑皮层电信号的生理学合理模型。我们将尝试辨别模型的鲁棒行为，因为我们改变了与生理过程相关的许多参数，并使用分叉理论对其进行分类。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，并计算具有柯尔莫哥洛夫相似性谱（湍流的标志）的新颖的时间周期解。

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.