Weakly coupled two-slow–two-fast systems, folded singularities and mixed mode oscillations

We study mixed mode oscillations (MMOs) in systems of two weakly coupled slow/fast oscillators. We focus on the existence and properties of a folded singularity, called folded saddle node of type II (FSN II), that allows the emergence of MMOs in the presence of a suitable global return mechanism. As FSN II corresponds to a transcritical bifurcation for a desingularized reduced system, we prove that, under certain non-degeneracy conditions, such a transcritical bifurcation exists. We then apply this result to the case of two coupled systems of FitzHugh–Nagumo type. This leads to a non-trivial condition on the coupling that enables the existence of MMOs.