Mathematical Biology: Nonlinear Dynamics of Oscillator Networks

许多生物振子群体可以表现出显著的集体行为。尽管它们各自的固有频率不同，但振荡器自发地同步到一个共同的频率。例子包括合唱的蟋蟀，齐声闪光的萤火虫，以及心脏起搏器细胞的同步放电。在科学和技术的其他几个分支中，人们希望模仿大自然在设计可以自我同步的网络方面的成功。例如，半导体激光阵列在同步时产生更大的集体功率，但这种锁相操作在实践中是出了名的难以实现。该pi和他的学生研究振荡器网络的动力学，使用数学方法的动力系统理论，分岔理论，统计力学，以及数值模拟。其中一个项目研究了最近在一个细胞振荡器模型中发现的一种特殊的数学现象，该模型通过信号分子的交换相互通信。另外三个项目冒险进入振荡器理论很少应用的领域，即分子遗传学，随机分析，以及人类行为和土木工程的相互作用。每个案例的目标都是回答一个在现实世界中很重要的数学上引人入胜的问题。具体来说：(1)如何使用合成基因电路构建多细胞生物钟的模拟物？(2)精确的生物钟可以通过同步许多不精确的组成部分来制造吗？这是我们自己的昼夜节律起搏器所发生的吗？是什么原因导致伦敦千禧桥在开幕当天摇晃？每个项目都需要相关领域的顶尖实验家的合作。振荡器网络出现在整个科学和技术中，并且在自然界中无处不在：激光阵列在同步振荡时工作得更好，建筑物和桥梁等结构可能受到同步振荡的威胁，心脏起搏器细胞的同步放电使心脏保持跳动。研究者用数学方法探索振荡器网络的行为，研究网络如何自我同步。对节律活跃细胞如何在组织和器官中协同工作的理解有望带来益处；对于潜在的危险方式的特征，行人人群可能无意中导致人行天桥晃动；以及涉及振荡器阵列的技术应用的衍生产品，例如激光器，微波振荡器和超导约瑟夫森结。通过这里提供的研究机会培养三名研究生，该项目还有助于培养对我国在科学和工程方面取得成功至关重要的人力资源。

StrogatzMany populations of biological oscillators can exhibit remarkablecollective behavior. Despite differences in their individualnatural frequencies, the oscillators spontaneously synchronize toa common frequency. Examples include chorusing crickets,fireflies that flash in unison, and synchronous firing of cardiacpacemaker cells. In several other branches of science andtechnology, one would like to imitate nature's success atdesigning networks that can synchronize themselves. Forinstance, a semiconductor laser array generates greatercollective power when it synchronizes, but such phase-lockedoperation is notoriously difficult to achieve in practice. ThePI and his students study the dynamics of oscillator networks,using mathematical methods of dynamical systems theory,bifurcation theory, and statistical mechanics, along withnumerical simulation. One project investigates a peculiarmathematical phenomenon that was recently discovered in a modelof cellular oscillators communicating with one another throughthe exchange of a signaling molecule. Three additional projectsventure into areas where oscillator theory has rarely beenapplied, namely molecular genetics, stochastic analysis, and theinterplay of human behavior and civil engineering. The goal ineach case is to answer a mathematically fascinating question thatis important in the real world. Specifically: (1) How can onebuild the analog of a multicellular biological clock, usingsynthetic gene circuits? (2) Can a precise biological clock bemade by synchronizing many imprecise components, and is this whattakes place in our own circadian pacemakers? (3) What causedLondon's Millennium Bridge to wobble on opening day? Each ofthese projects enlists the collaboration of a top experimentalistin the relevant field.Networks of oscillators arise throughout science and technologyand are ubiquitous in nature: lasers arrays work better when theyoscillate synchronously, structures such as buildings and bridgescan be endangered by synchronous oscillation, synchronous firingof cardiac pacemaker cells keeps hearts beating. Theinvestigator mathematically explores the behavior of networks ofoscillators, studying how networks synchronize themselves. Benefits are expected for the understanding of how rhythmicallyactive cells work together in tissues and organs; for thecharacterization of the potentially dangerous ways that crowds ofpedestrians can inadvertently cause footbridges to shake; and forspin-offs to technological applications involving arrays ofoscillators, such as lasers, microwave oscillators, andsuperconducting Josephson junctions. By training three graduatestudents through the research opportunities offered here, thisproject also helps to develop human resources that are vital toour nation's success in science and engineering.