Mathematical Biology Seminar

Carson Chow
Wednesday, April 22, 2009
3:05pm in LCB 215
Kinetic Theory of Coupled Oscillators

Abstract: Networks of coupled oscillators have been used to model a wide range of phenomena such as interacting neurons, flashing fireflies, chirping crickets and coupled Josephson junctions. Typically, these networks have been studied analytically in regimes where the number of oscillators are small or in the "mean field" infinite size limit. The dynamics of networks that are large but not infinite is not well understood, although this is where many of the interesting applications lie. I will present a formalism to analyze large but finite-sized networks using an approach borrowed from the kinetic theory of gases and plasmas. The result is an infinite number of coupled equations for the moments of the probability density function for the dynamics (i.e. moment hierarchy) that can be truncated to estimate finite-sized fluctuation and correlation effects. In addition, it can be shown that the moment hierarchy is equivalent to a path integral formulation where diagrammatic methods can be employed to assist in the analytical calculations.