Dimension reduction: From Soliton Dynamics to Neural Networks

Eli Shlizerman, University of Washington
April 20th, 2011 at 4PM–5PM in 939 Evans Hall [Map]

The interplay between coherence and decoherence through subtle dynamics is a widespread phenomenon across applications and is often modeled by nonlinear PDEs or as large systems of ODEs. Derivation of low dimensional models that retain these dynamics can give rise to the identification of the responsible mechanisms for these phenomena. In this talk I will present the dimension reduction methods that we have developed for near-integrable, dissipative nonlinear PDEs and neural network models. The analysis of the outcome models allows us to characterize synchrony, localized structures and instabilities in these systems. In particular, I will describe the route to spatio-temporal chaos in the forced nonlinear Schrodinger equation, the onset of multi-pulsing in mode-locked lasers, and the mean-voltage for coupled conductance-based neurons. The common ground for these problems is that the reduced models faithfully capture the bifurcation structure. A detailed study of these models leads to an explanation of the observed behavior and reveals novel regimes.