Neural spiking and bursting rhythms in space-clamped (i.e., ODE) models are typically driven by either canard dynamics or slow passage through Hopf bifurcations. In both cases, solutions which are attracted to quasi-stationary states (QSS) sufficiently before a fold or Hopf bifurcation remain near the QSS for long times after the states have become repelling, resulting in a significant delay in the loss of stability and hence in the onset of oscillations. In this work, we present the spatio-temporal analogues of these delayed bifurcation phenomena in multi-time-scale reaction-diffusion equations. We show the existence of canard-induced bursting rhythms in a spatially extended model of the electrical activity in pituitary cells. We then derive asymptotic formulas for the space-time boundaries that act as buffers beyond which solutions cannot remain near the repelling QSS (and hence at which the delayed onset of oscillations must occur) for slow passage through Hopf bifurcations in reaction-diffusion equations.
This is joint work with Tasso J. Kaper (Boston University).