Dynamics of neuronal and genetic networks

数学在研究复杂的生理系统中越来越有价值，例如细胞功能的遗传控制和大脑和神经系统对运动的控制。这个研究项目涉及这种“网络”的数学模型，既帮助理解和解释它们的动态行为，并发现在任何相互作用的组件网络中起作用的潜在数学原理。研究计划的一个方面是使用大脑运动回路部分的神经元网络描述和非线性动力学方法来研究帕金森病导致的运动异常，震颤，以及植入特定大脑结构的高频电刺激器缓解这些症状的方法。（值得注意的是，尽管它们被广泛使用，但人们对它们的工作原理并不了解。）参与运动控制的大脑结构网络是复杂的，目前还不清楚这些刺激器是如何在物理上运作的，为什么它们可以有效地放置在几个不同的大脑结构中的一个，以及这些结构是如何交流的。基因控制蛋白质的产生，这些蛋白质执行细胞功能，但这些蛋白质调节其他基因的活性。因此，相互作用的基因网络是细胞发育和代谢的基础。一个动态有趣的基因调控过程是细胞中某些蛋白质的昼夜（每日）振荡。这种振荡在许多生物体中很常见，但涉及不同的生化成分和基因，因此，一个挑战是了解它们是如何产生和运作的，以及为什么它们如此强烈地持续存在（例如，时差只是我们自己的昼夜节律的暂时中断）。这是一个涉及相互作用的网络回路的问题，我的研究计划的另一个主要目标是从数学的角度来理解互连模式如何决定具有打开或关闭组件的一般网络的动态行为。这一框架允许分析方法相当深入地探索复杂动力学，从而有助于复杂系统的数学以及提供对生理学的洞察。

Mathematics is increasingly valuable in studying complex physiological systems, such as the genetic control of cell functions and control of movement by the brain and nervous system. This research project deals with mathematical models of such 'networks', both to help understand and explain their dynamical behaviour, and to discover underlying mathematical principles at work in any networks of interacting components.One aspect of the research program is to use neuronal network descriptions of parts of the brain's motor circuitry and methods of nonlinear dynamics to study movement abnormalities resulting from Parkinson's disease such as tremor, and the means by which high-frequency electrical stimulators implanted in particular brain structures alleviate these symptoms. (Remarkably, how they work is not well understood, despite their common use.) The network of brain structures involved in motor control is complex, and it is not yet clear how these stimulators operate physically, why they can be placed effectively in one of several different brain structures and how these structures communicate.Genes control the production of proteins that carry out cell functions, but these proteins regulate the activity of other genes. Thus, networks of interacting genes underly cell development and metabolism. One gene-regulated process that is dynamically interesting is the circadian (daily) oscillations of some proteins in cells. Such oscillations are common across many organisms but involve different biochemical components and genes, so a challenge is to understand how they arise and operate and why they are so strongly persistent (jet lag, for example, is only a temporary disruption in our own circadian rhythms). This is a question involving interacting network loops, and another main thrust of my research program is to understand from a mathematical point of view how the pattern of interconnection determines dynamical behaviour in general networks with components that switch on or off. This framework permits analytic methods to probe quite deeply into complex dynamics, and thus contributes to the mathematics of complex systems as well as providing insight into physiology.