Bard Ermentrout - Nonlocal Phase Waves

There is increasing evidence that much of the rhythmic activity seen in the brain is not synchronous, but rather organized into various types of waves such as plane waves and rotating waves. In many of these experiments, the quantities measured are the spatial phase of the oscillation. Thus, a natural approach is to study the dynamics of coupled phase oscillators. Since interactions in the brain are nonlocal, in this talk, I will describe some recent results on equations of the form:

d u(x,t)/dt = omega(x) + int_D K(x-y)H(u(y,t)-u(x,t)) dy

where u(x,t) is the local spatial phase, omega(x) is the intrinsic frequency at x, K(x) is a convolution kernel, and H(u) is a periodic interation function. The domain, D will be either a ring, a line interval, or an annulus. We first review the behavior on a ring when the frequency is constant. Next, we study the existence and stability of rotating waves on an annulus as the size of the hole changes. Finally, we consider the line interval when there is a frequency gradient. We apply several techniques including conversion to boundary value problems and singular perturbation. This work is joint with graduate student Yujie Ding.