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

TL;DR

This paper demonstrates that cusped singularities serve as a universal organizing principle for mixed-mode oscillations in inhibitory neural systems, linking geometric singularities to complex oscillatory patterns.

Contribution

It introduces cusped singularities as a new, generic mechanism for organizing MMOs in coupled slow-fast systems with mutual inhibition, supported by theoretical and neuronal model analysis.

Findings

Cusped singularities guarantee small-amplitude oscillations in inhibitory systems.

MMOs arise from SAOs combined with a return mechanism near cusped singularities.

Presence of cusped singularities is linked to nearby singular Hopf bifurcations.

Abstract

Mutual inhibition is a common motif in neural systems. Here, we establish that cusped singularities - folded singularities located at cusp points of critical manifolds - provide a universal organizing mechanism for mixed-mode oscillations (MMOs) in coupled slow-fast systems with mutual inhibition. We show that the geometric setup of these systems generically satisfies the conditions required by established geometric singular perturbation theory and blow-up methods, guaranteeing that such cusped singularities yield small-amplitude oscillations (SAOs). MMOs appear from the SAOs combined with an appropriate return mechanism. Further, we show that the geometric presence of a cusped singularity is strictly related to occurrence of a nearby singular Hopf bifurcation. We demonstrate the efficacy of this framework 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 full system equilibrium near the cusped singularity triggers SAOs as the system passes near the cusp and approaches a full-system saddle-focus related to the singular Hopf bifurcation. Large-amplitude oscillations appear as the system spirals away from the saddle-focus, leading to MMOs, which may exhibit distinctive alternating patterns, in contrast to standard saddle-node induced MMOs. Our results establish cusped singularities as a generic, biologically relevant mechanism for complex oscillatory dynamics in inhibitory neural networks as well as for other inhibitory slow-fast systems.