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arXiv · 2605.03606

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

Morten Gram Pedersen
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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.

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Morten Gram Pedersen. 2026-05-05. Cusped singularities organize mixed-mode oscillations in mutually inhibitory slow-fast systems. https://arxiv.org/abs/2605.03606

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