arxiv: 2605.03606 · v1 · submitted 2026-05-05 · 🧮 math.DS · q-bio.NC· q-bio.QM

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

Morten Gram Pedersen

Pith reviewed 2026-05-07 12:47 UTC · model grok-4.3

classification 🧮 math.DS q-bio.NCq-bio.QM

keywords cusped singularitiesmixed-mode oscillationsslow-fast systemsmutual inhibitionsingular Hopf bifurcationgeometric singular perturbation theoryneuronal modelsblow-up methods

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The pith

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

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that cusped singularities serve as organizing centers for mixed-mode oscillations in systems of mutually inhibiting slow and fast components. These singularities are folded singularities at cusp points on the critical manifold and generically lead to small-amplitude oscillations when analyzed with singular perturbation and blow-up methods. The small oscillations then interact with a global return mechanism to produce the mixed patterns of large and small cycles. This mechanism connects directly to the appearance of a singular Hopf bifurcation in the full system and is demonstrated in both population rate models and conductance-based neuron models with synaptic inhibition.

Core claim

Cusped singularities, defined as folded singularities located at cusp points of critical manifolds, provide a universal organizing mechanism for mixed-mode oscillations in coupled slow-fast systems with mutual inhibition. The geometric setup of these systems generically satisfies the conditions required by geometric singular perturbation theory and blow-up methods, guaranteeing that such cusped singularities yield small-amplitude oscillations. Mixed-mode oscillations appear from these small-amplitude oscillations combined with an appropriate return mechanism. The geometric presence of a cusped singularity is strictly related to the occurrence of a nearby singular Hopf bifurcation. In the two

What carries the argument

The cusped singularity, a folded singularity located at a cusp point of the critical manifold, which generates small-amplitude oscillations through local canard dynamics in the singular limit before the system returns to large excursions.

Load-bearing premise

The mutual inhibition and slow-fast separation generically produce a critical manifold with cusp points where folded singularities form and satisfy the conditions for geometric singular perturbation theory.

What would settle it

In the Curtu rate model or coupled Morris-Lecar neurons, moving the full-system equilibrium near the predicted location of the cusped singularity produces no small-amplitude oscillations or fails to yield mixed-mode patterns linked to the singular Hopf bifurcation.

Figures

Figures reproduced from arXiv: 2605.03606 by Morten Gram Pedersen.

**Figure 1.** Figure 1: Mixed-mode oscillations and the cusped singularity in the Curtu model. (A) Time () view at source ↗

**Figure 2.** Figure 2: Mixed-mode oscillations and the cusped singularity in the Morris-Lecar model. (A) Figure 2: Mixed-mode oscillations and the cusped singularity in the coupled Morris-Lecar model. view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Cusped singularities give a clean geometric account of MMOs under mutual inhibition, with the link to singular Hopf and the two model checks as the main concrete advances.

read the letter

The paper's central point is that in mutually inhibitory slow-fast systems the critical manifold often has a cusp, and placing a folded singularity there produces small-amplitude oscillations that, combined with a return mechanism, generate mixed-mode oscillations. This is framed as generic under the usual GSPT and blow-up assumptions, and the authors tie the cusp directly to a nearby singular Hopf bifurcation. They then show the pattern in the Curtu rate model and in coupled Morris-Lecar neurons with synaptic inhibition, where pushing the equilibrium near the cusp triggers the expected SAOs followed by spiraling away from the saddle-focus and the large excursions that close the MMO cycle. That is the new piece: a focused application of existing geometric tools to the mutual-inhibition case, with an explicit Hopf connection and two working examples rather than a purely abstract construction. The demonstrations are straightforward and match the predicted passage near the cusp, which is useful for readers who want to see the geometry in concrete neuronal models. The argument avoids circularity and rests on standard nondegeneracy conditions that the abstract claims hold generically. The main soft spot is that the genericity statement and the blow-up details are asserted rather than derived at length in the material I could check; a referee will want to see the precise verification that the cusp-folded singularity setup satisfies the required transversality and nondegeneracy conditions without hidden degeneracies. The return mechanism for the large-amplitude part is sketched but not analyzed as fully as the local passage near the cusp. Overall this is a paper for people already working in geometric singular perturbation theory applied to neural or coupled oscillator models. Readers who care about MMO mechanisms in inhibitory networks will get a usable organizing picture and two ready-to-check examples. It is coherent on its own terms and the evidence presented is sufficient to merit referee time rather than a desk reject.

Referee Report

0 major / 4 minor

Summary. The paper claims that cusped singularities—folded singularities located at cusp points of critical manifolds—provide a universal organizing mechanism for mixed-mode oscillations (MMOs) in mutually inhibitory slow-fast systems. It uses geometric singular perturbation theory (GSPT) and blow-up methods to argue that the geometric setup of these systems generically satisfies the required conditions, producing small-amplitude oscillations (SAOs) that combine with a return mechanism to yield MMOs. The presence of a cusped singularity is linked to a nearby singular Hopf bifurcation. The framework is demonstrated in the Curtu rate model of mutually inhibitory populations and in coupled Morris-Lecar neurons with synaptic inhibition, where equilibria near the cusped singularity trigger SAOs followed by large-amplitude oscillations and distinctive MMO patterns.

Significance. If the central claims hold, the work is significant because it identifies a new, geometrically grounded mechanism for MMOs that is distinct from saddle-node-induced MMOs and directly tied to mutual inhibition, a common motif in neural systems. The reliance on established GSPT and blow-up techniques, combined with explicit demonstrations in two different neuronal models showing passage near the cusp triggering SAOs and spiraling away from the saddle-focus, provides concrete support. This could explain alternating MMO patterns observed in inhibitory networks and offers a biologically relevant organizing center with potential for broader application to other slow-fast inhibitory systems.

minor comments (4)

The introduction would benefit from an early, self-contained definition or diagram distinguishing cusped singularities from standard folded singularities and from cusp points of the critical manifold alone.
In the model demonstrations, explicit parameter values or bifurcation diagrams showing how the full-system equilibrium is pushed near the cusped singularity would improve reproducibility and clarity of the numerical evidence.
The precise geometric relation between the cusped singularity and the singular Hopf bifurcation (e.g., via the slow flow or blow-up coordinates) is stated but could be expanded with a short schematic or reference to the relevant coordinate chart.
Figure captions should specify the time scales, initial conditions, and which variables are plotted to make the MMO patterns and SAO passages easier to interpret without consulting the main text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work on cusped singularities as an organizing mechanism for MMOs in mutually inhibitory slow-fast systems. The referee's description accurately reflects the manuscript's use of GSPT and blow-up methods, the link to singular Hopf bifurcations, and the demonstrations in the Curtu and Morris-Lecar models. We appreciate the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external GSPT and blow-up methods

full rationale

The paper's derivation chain begins from the geometric setup of mutually inhibitory slow-fast systems and invokes established geometric singular perturbation theory plus blow-up methods to show that cusped singularities (folded singularities at cusp points) generically produce SAOs, which combine with a return mechanism to yield MMOs. It further derives the relation to a nearby singular Hopf bifurcation directly from the same geometry and nondegeneracy conditions. Concrete demonstrations in the Curtu rate model and coupled Morris-Lecar neurons confirm passage near the cusp triggers the predicted SAOs followed by spiraling away from the saddle-focus. No step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the cited theory is external, the nondegeneracy conditions are stated as generically satisfied rather than assumed from the target result, and the models serve as independent verification rather than circular input. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard tools from geometric singular perturbation theory and blow-up methods without introducing new free parameters or postulated entities; the cusped singularity is a geometric feature defined within the existing framework.

axioms (1)

domain assumption The geometric setup of these systems generically satisfies the conditions required by established geometric singular perturbation theory and blow-up methods
Invoked in the abstract to guarantee that cusped singularities yield small-amplitude oscillations.

pith-pipeline@v0.9.0 · 5546 in / 1323 out tokens · 77687 ms · 2026-05-07T12:47:44.765825+00:00 · methodology

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Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

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