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.