Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh Nagumo system

We investigate the organisation of mixed-mode oscillations in the self-coupled FitzHugh Nagumo system. This type of oscillations can be explained as a combination of relaxation oscillations and small-amplitude oscillations controlled by canard solutions that are associated with a folded singularity on a critical manifold. The self-coupled FitzHugh Nagumo system has a cubic critical manifold for a range of parameters, and an associated folded singularity of node type. Hence, there exist corresponding attracting and repelling slow manifolds that intersect in canard solutions. We present a general technique for the computation of two-dimensional slow manifolds (smooth surfaces). It is based on a boundary value problem approach where the manifolds are computed as one-parameter families of orbit segments. Visualisation of the computed surfaces gives unprecedented insight into the geometry of the system. In particular, our techniques allow us to identify canard solutions as the intersection curves of the attracting and repelling slow manifolds.