Title: Signal transmission properties of unidirectional chains of phase oscillators.
Speaker: Stan Mintchev
Affiliation: Cooper Union
Date and Time: Friday, April 6, 11:00am-12:00pm
Place: Phillips 640
Abstract. Phase oscillator ensembles exhibiting a preferred direction of coupling can mimic a spatially distributed communication line with effective mesoscale properties. In mathematical neuroscience, such networks provide a framework for understanding the generation and propagation of electrical impulses across brain tissues. Motivated by such applications, we consider signal transmission problems in this idealized setting. We focus on generation and stability of traveling wave solutions (TW) in feedforward chains of idealized neural oscillators featuring a pulse emission / Type-I phase response (PRC) interaction. Our prior investigations examined a smooth version of this model by way of a combination of numerical and analytical techniques: an iterative fixed point scheme was used to verify existence of TW, and these findings motivated an abstract hypothesis that turns out sufficient to establish global stability of the solution. We have since completed a full mathematically-rigorous study of a piecewise affine version of this model, establishing all of the observed phenomenology (both existence and stability of TW) analytically. In addition, we now have an understanding of how a robust TW solution in this setting may be generated with a variety of external forcing stimuli, including such that are structurally distinct from the consequent wave.