Determining the Mathematical Principles of Daily Timekeeping

准确的昼夜节律对于几乎所有生物体的生存都是至关重要的。对于数以百万计出国旅行或工作时间不规律的美国人来说，更好地理解生物钟的计时尤为重要。最近对昼夜节律的研究发现了在视交叉上核(SCN)内数千个细胞中的每个细胞产生计时的复杂生物机制。视交叉上核是大脑中充当哺乳动物中心生物钟的区域。在这里，基于来自许多实验小组的数据，建立了SCN内神经元的详细数学模型。对这些模型进行了数学分析和简化，以确定管理计时的关键属性。新的数学和数值方法被开发出来，以允许研究复杂的神经元网络，并确定如何从耦合振荡器的集体行为中产生计时。这些结果应该适用于许多生物系统。生物钟被作为细胞生物学和电生理学的模型系统来研究，以确定可以应用于其他生理系统的设计原则，特别是那些由耦合振荡器组成的系统。电生理学的数学模型将遵循霍奇金-赫胥黎公式。细胞生物学的数学模型将遵循群体作用形式主义。最近对单个细胞的昼夜节律机制的实验研究将被纳入这些数学模型。了解控制昼夜节律的动力学将对昼夜节律领域非常有帮助，该领域试图确定大量蛋白质、离子通道和神经元如何共同形成人体的中央时钟。该方案的一般数学工作包括通过迭代映射研究生化反馈回路的吸引子，有效模拟大维神经元振荡器的种群密度方法，以及可将大量耦合振荡器的模型简化为二维模型的新的ANSATZ。这些数学方法将使用我们的昼夜计时模型进行测试。

Accurate circadian (daily) timekeeping is essential to the survival of almost all organisms. A better understanding of circadian timekeeping is particularly important for the millions of Americans who travel overseas or work on irregular schedules. Recent research on circadian rhythms has discovered the complex biological mechanisms that generate timekeeping in each of the thousands of cells within the suprachiasmatic nuclei (SCN), the region of the brain that acts as the central circadian clock in mammals. Here, detailed mathematical models of neurons within the SCN are developed based on data from many experimental groups. These models are mathematically analyzed and simplified to determine the key properties that govern timekeeping. New mathematical and numerical methods are developed to allow for the study of complex neuronal networks and determine how timekeeping emerges from the collective behavior of coupled oscillators. These results should be applicable to many biological systems. Circadian clocks are studied as a model system in cellular biology and electrophysiology to determine design principles that can be applied to other physiological systems, particularly those consisting of coupled oscillators. Mathematical models of electrophysiology will follow the Hodgkin-Huxley formalism. Mathematical models of cellular biology will follow the mass action formalism. Recent experimental research on the mechanisms of circadian timekeeping in individual cells will be incorporated into these mathematical models. Understanding the dynamics controlling circadian timekeeping will be extremely helpful for the field of circadian rhythms, which seeks to determine how a large number of proteins, ion channels and neurons work together to form the body's central clock. The general mathematical work in this proposal includes studying the attractors of biochemical feedback loops through iterative maps, a population density method for efficiently simulating large dimensional neuronal oscillators, and a new ansatz that can reduce a model of a large number of coupled oscillators to a two dimensional model. These mathematical approaches will be tested using our models of circadian timekeeping.