Published: March 4, 2022

Bard Ermentrout, Department of Mathematics, University of Pittsburgh

Phase in Space: Spatiotemporal dynamics of nonlocally coupled oscillators

The ability of neuroscientists to record large regions of the brain at high temporal resolution has demonstrated that neuronal oscillations are not synchronized, but rather, organized into spatio-temporal patterns such as plane- and rotating waves.  In typical experiments phase gradients are computed from filtered local field potentials, thus, a natural mathematical framework is coupled phase equations. Indeed, when interactions in spatially distributed oscillatory media are "weak", it is possible to reduce the dynamics to a system of phase equations for the phase u(x,t):

u_t = w(x) + int_D K(x-y) H[u(y,t)-u(x,t)]

where w(x) represents heterogeneities, D, is some one- or two-dimensional domain, K(x) is a coupling kernel and H[u] is the phase-interaction function. In this talk, I will discuss the existence and stability of rotating waves when D is an annulus. I will show that as the inner radius shrinks, rigid rotating waves lose existence through a saddle-node and this results in the birth of co-called chimeras. I will also describe some recent work on boundary effects and how they are sufficient to lead to target-like patterns even when w(x)=0.   This work is joint with Yujie Ding and Andrea Welsh.