Mathematical Biology: Nonlinear Dynamics of Oscillator Networks

Cornell University, Ithaca NY

Investigators

Abstract

Strogatz Many populations of biological oscillators can exhibit remarkable collective behavior. Despite differences in their individual natural frequencies, the oscillators spontaneously synchronize to a common frequency. Examples include chorusing crickets, fireflies that flash in unison, and synchronous firing of cardiac pacemaker cells. In several other branches of science and technology, one would like to imitate nature's success at designing networks that can synchronize themselves. For instance, a semiconductor laser array generates greater collective power when it synchronizes, but such phase-locked operation is notoriously difficult to achieve in practice. The PI and his students study the dynamics of oscillator networks, using mathematical methods of dynamical systems theory, bifurcation theory, and statistical mechanics, along with numerical simulation. One project investigates a peculiar mathematical phenomenon that was recently discovered in a model of cellular oscillators communicating with one another through the exchange of a signaling molecule. Three additional projects venture into areas where oscillator theory has rarely been applied, namely molecular genetics, stochastic analysis, and the interplay of human behavior and civil engineering. The goal in each case is to answer a mathematically fascinating question that is important in the real world. Specifically: (1) How can one build the analog of a multicellular biological clock, using synthetic gene circuits? (2) Can a precise biological clock be made by synchronizing many imprecise components, and is this what takes place in our own circadian pacemakers? (3) What caused London's Millennium Bridge to wobble on opening day? Each of these projects enlists the collaboration of a top experimentalist in the relevant field. Networks of oscillators arise throughout science and technology and are ubiquitous in nature: lasers arrays work better when they oscillate synchronously, structures such as buildings and bridges can be endangered by synchronous oscillation, synchronous firing of cardiac pacemaker cells keeps hearts beating. The investigator mathematically explores the behavior of networks of oscillators, studying how networks synchronize themselves. Benefits are expected for the understanding of how rhythmically active cells work together in tissues and organs; for the characterization of the potentially dangerous ways that crowds of pedestrians can inadvertently cause footbridges to shake; and for spin-offs to technological applications involving arrays of oscillators, such as lasers, microwave oscillators, and superconducting Josephson junctions. By training three graduate students through the research opportunities offered here, this project also helps to develop human resources that are vital to our nation's success in science and engineering.