Waves and persistent states in neural systems Due to
the massive recurrent interactions between neurons, networks are able
to sustain various types of persistent activity. By persistent, I mean
activity that remains after a stimulus is removed. Working memory is
believed to be such a state - a localized region of cortex remains
active until the memory task is completed. We have several types of
models involving combinations of multi-region interactions,
multi-layer interactions and facilitation. This work has been with Mark Bodner; we are currently investigating the relationship between gamma rhythms and working memory. With David van Mannen, we have begun to explore the transition to clustering in gamma oscillations by looking at a particular set of maps.
Another type of persistent activity comes in the form of propagating
waves. These are observed in brain slices, cell cultures, and in
vivo. We use spatial networks of neurons and singular perturbation
theory to study propagation of activity in one- and
two-dimensions. Jeremy Harris and I are looking at Discontinuous neural field models in order to study the transition from waves to fronts. The nature of the nonlinearities requires us to delicately handle the transistions across the discontinuities. We employ Filippov methods to resolve various bifurcations that can occur.