| name | cusped-singularity-mmo-analysis |
| description | Geometric singular perturbation analysis of mixed-mode oscillations (MMOs) in inhibitory neural networks using cusped singularities. Activation triggers: mixed-mode oscillations, MMO, cusped singularity, slow-fast neural system, mutual inhibition oscillation, singular perturbation neural, blow-up method neural, neural oscillation mechanism, slow-fast system analysis. |
Cusped Singularity Mixed-Mode Oscillation Analysis
Geometric singular perturbation theory establishing cusped singularities as universal organizing mechanism for mixed-mode oscillations in mutually inhibitory neural networks.
Metadata
- Source: arXiv:2605.03606
- Authors: Morten Gram Pedersen
- Published: 2026-05-05
- Subjects: Dynamical Systems (math.DS), Neurons and Cognition (q-bio.NC)
Core Methodology
Key Innovation
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. This is distinct from standard saddle-node induced MMOs and yields distinctive alternating oscillation patterns.
Technical Framework
- Geometric Setup: Mutually inhibitory slow-fast systems generically satisfy conditions for geometric singular perturbation theory (GSPT) and blow-up methods
- Cusped Singularity → SAOs: Geometric presence guarantees small-amplitude oscillations (SAOs) as system passes near the cusp
- Singular Hopf Connection: Cusped singularity is strictly related to nearby singular Hopf bifurcation — full-system saddle-focus
- SAO + Return → MMO: Small-amplitude oscillations combine with return mechanism to produce full MMOs
- Alternating Pattern: Spiraling away from saddle-focus creates distinctive alternating patterns, contrasting with standard saddle-node MMOs
Analysis Steps
- Identify the critical manifold and its fold/cusp structure
- Locate cusped singularities on the critical manifold
- Apply blow-up method to resolve dynamics near the cusp
- Verify GSPT conditions hold for the specific system
- Analyze singular Hopf bifurcation near the cusp
- Construct return mechanism connecting SAOs to large-amplitude oscillations
- Validate with specific neuronal models (Curtu rate model, Morris-Lecar with synaptic inhibition)
Applications
- Analyzing complex oscillatory patterns in mutually inhibitory neural circuits
- Understanding neural rhythm generation mechanisms
- Predicting oscillation transitions in inhibitory network models
- Distinguishing cusped-singularity MMOs from saddle-node induced MMOs
- General slow-fast inhibitory system analysis
Pitfalls
- Requires expertise in geometric singular perturbation theory and blow-up methods
- Applicable only to systems with mutual inhibition motif
- Mathematical analysis is non-trivial — requires careful manifold analysis
- Numerical validation may be needed for specific parameter regimes
Related Skills
- attractor-metadynamics-neural
- kuramoto-brain-network
- spiking-oscillation-mapping
- neural-dynamics-universal-translator
