Synchrony and canards in two coupled FitzHugh--Nagumo equations

We describe the fast-slow dynamics of two FitzHugh--Nagumo equations coupled symmetrically through the slow equations. We use symmetry arguments to find a non-empty open set of parameter values for which the two equations synchronise, and another set with antisynchrony -- where the solution of one equation is minus the solution of the other. By combining the dynamics within the synchrony and antisynchrony subspaces, we also obtain bistability -- where these two types of solution coexist as hyperbolic attractors. They persist under small perturbation of the parameters. Canards are shown to give rise to mixed-mode oscillations. They also initiate small amplitude transient oscillations before the onset of large amplitude relaxation oscillations. We also discuss briefly the effect of asymmetric coupling, with periodic forcing of one of the equations by the other. We illustrate our results with numerical simulations.