**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.