Bard Ermentrout

• 
  Wandering bumps in stochastic neural fields
  Bard Ermentrout
  We study the effects of noise on stationary pulse solutions (bumps) in spatially extended neural fields. The dynamics of a neural field is described by an integrodifferential equation whose integral term characterizes synaptic interactions between neurons in different spatial locations of the network. Translationally symmetric neural fields support a continuum of stationary bump solutions, which may be centered at any spatial location. Random fluctuations are introduced by modeling the system as a spatially extended Langevin equation whose noise term we take to be additive. For nonzero noise, bumps are shown to wander about the domain in a purely diffusive way. We can approximate the associated diffusion coefficient using a small noise expansion. Upon breaking the (continuous) translation symmetry of the system using spatially heterogeneous inputs or synapses, bumps in the stochastic neural field can become temporarily pinned to a finite number of locations in the network. As a result, the effective diffusion of the bump is reduced, in comparison to the homogeneous case. As the modulation frequency of this heterogeneity increases, the effective diffusion of bumps in the network approaches that of the network with spatially homogeneous weights. We end with some simulations of spiking models which show the same dynamics (This is joint work with Zachary Kilpatrick, UH)
• 
  Data mining the Kuramoto Equations on regular graphs
  Bard Ermentrout
  In this talk, I will describe the dynamics of a system of sinusoidally coupled phase oscillators on cubic graphs. The synchronous solution is always an attractor. However, as the graphs get larger (more nodes), it is possible to get other stable attractors. We study the basins, energy, and degree of stability of these non-synchronous attractors for all cubic graphs up to a certain number of nodes. We also use some techniques from computational algebraic geometry to show that for some graphs, the only attractor is synchrony.