Cusped singularities organize mixed-mode oscillations in mutually inhibitory slow-fast systems

A new mathematical framework published on arXiv (2605.03606) by Morten Gram Pedersen establishes that cusped singularities—special folded points on critical manifolds—serve as universal organizers for mixed-mode oscillations (MMOs) in coupled slow-fast systems with mutual inhibition. The paper shows that these geometric structures produce small-amplitude oscillations (SAOs) via geometric singular perturbation theory and blow-up methods, which then combine with a return mechanism to create the alternating large-and-small amplitude patterns characteristic of MMOs.

Pedersen validates the theory in two distinct neuronal models: the Curtu rate model of mutually inhibitory neural populations and coupled Morris-Lecar neurons with synaptic inhibition. In both cases, pushing the system equilibrium near the cusped singularity triggers SAOs as the system passes the cusp and approaches a full-system saddle-focus linked to a singular Hopf bifurcation. Large-amplitude oscillations emerge as the system spirals away, producing alternating patterns distinct from standard saddle-node-induced MMOs. This work provides a generic, biologically relevant mechanism for complex oscillatory dynamics in inhibitory neural networks, with implications for understanding brain rhythms, seizure activity, and neural computation.