Title: Spatio-temporal canards in neural field equations,
Abstract: Canards are special solutions to ordinary differential equations that follow
invariant repelling slow manifolds for long time intervals. In realistic
biophysical single cell models, canards are responsible for several complex
neural rhythms observed experimentally, but their existence and role in
spatially-extended systems is largely unexplored. We describe a novel type of
coherent structure in which a spatial pattern displays temporal canard
behaviour. Using interfacial dynamics and geometric singular perturbation
theory, we classify spatio-temporal canards and give conditions for the
existence of folded-saddle and folded-node canards. We find that
spatio-temporal canards are robust to changes in the synaptic connectivity and
firing rate. The theory correctly predicts the existence of spatio-temporal
canards with octahedral symmetries in a neural field model posed on the unit
sphere.