Mathematical Biology: Nonlinear Dynamics of Oscillator Networks

许多种群的生物振荡器可以表现出显著的收集行为。尽管它们各自的自然频率不同，但振荡器会自发地同步到一个共同的频率。例如，蟋蟀齐声鸣叫，萤火虫齐声闪烁，心脏起搏器细胞同步放电。在科学和技术的其他几个分支中，人们想要模仿大自然在设计能够自我同步的网络方面的成功。例如，半导体激光器阵列在同步时会产生更大的集体功率，但众所周知，这种锁相操作在实践中很难实现。Pi和他的学生使用动力系统理论、分叉理论和统计力学的数学方法以及数值模拟来研究振荡器网络的动力学。其中一个项目研究了最近在细胞振荡器通过信号分子交换相互通信的模型中发现的一种特殊的数学现象。另外三个项目冒险进入振子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。每一种情况的目标都是回答一个在现实世界中很重要的数学上有趣的问题。具体来说：(1)如何使用合成基因电路来构建多细胞生物钟的模拟？(2)通过同步许多不精确的部件可以制造出精确的生物钟，这是发生在我们自己的生物钟中的吗？(3)是什么导致伦敦千禧大桥在开幕当天摇晃？这些项目中的每一个都得到了相关领域顶尖实验的合作。振荡器网络出现在整个科学和技术领域，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会受到同步振荡的威胁，心脏起搏器细胞的同步发射保持心脏跳动。研究人员从数学上探索振荡器网络的行为，研究网络如何自我同步。预计将在以下方面受益：了解有节奏的活动细胞如何在组织和器官中协同工作；描述行人人群可能无意中导致人行桥摇晃的潜在危险方式；以及衍生出涉及振荡器阵列的技术应用，如激光、微波振荡器和超导约瑟夫森结。通过在这里提供的研究机会培训三名毕业生，该项目还有助于开发对我们国家在科学和工程方面的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.