| name | rulkov-neural-maps-cross-coupling |
| title | Rulkov Neural Maps Cross Coupling |
| description | Novel coupling methodology for Rulkov neural maps preserving chaos and generating strange attractors |
| trigger_words | ["rulkov neural maps","cross coupling neural maps","neural map coupling","devaney chaos neural","strange attractor neural"] |
On a Cross Coupling of Rulkov Neural Maps
Overview
This skill implements the novel cross-coupling methodology for Rulkov neural maps introduced in arXiv:2607.22318. The approach provides a biologically-inspired coupling mechanism that preserves key dynamical properties while enabling complex emergent behaviors in coupled neural systems.
Core Contributions
Analytical Guarantees
- Boundedness Preservation: The coupling maintains bounded motion in the coupled system
- Chaos Preservation: Existence of snap-back repeller is preserved, ensuring Devaney chaos via Marotto theorem
- Biological Interpretation: Heuristic biological interpretation for transitions to non-small perturbations in slow variables
Dynamical Properties
- Global Strange Attractor: Coupled system exhibits a global strange attractor with fractal structure
- Non-integer Dimension: Kaplan-Yorke dimension computation confirms fractal nature
- Scalable Architecture: Generalization proposed for arbitrary numbers of coupled neurons
Numerical Validation
- Time Series Analysis: Comprehensive time series characterization of coupled dynamics
- Lyapunov Spectra: Full Lyapunov exponents spectra demonstrating chaotic behavior
- Bifurcation Diagrams: Bifurcation analysis across coupling parameters
- Basins of Attraction: Basin structure analysis for different initial conditions
Implementation Guidelines
When to Use
- Modeling coupled neural populations with preserved chaotic dynamics
- Studying emergence of complex attractors in neural networks
- Investigating biological plausibility of neural coupling mechanisms
- Exploring scalability of chaotic neural systems
Key Parameters
- Coupling Strength: Controls transition between uncoupled and strongly coupled regimes
- Perturbation Magnitude: Determines slow variable dynamics and biological interpretation
- Network Size: Scalable from 2-neuron pairs to arbitrary network sizes
- Initial Conditions: Affects basin of attraction and transient dynamics
Validation Protocol
- Compute Kaplan-Yorke dimension to confirm strange attractor structure
- Generate Lyapunov exponents spectra to verify chaotic behavior
- Perform bifurcation analysis across coupling parameters
- Analyze basins of attraction for robustness assessment
Mathematical Foundation
Rulkov Map Basics
The standard Rulkov map consists of fast variable x and slow variable y:
x_{n+1} = f(x_n, y_n)
y_{n+1} = y_n + μ(σ - x_n - y_n)
Cross-Coupling Mechanism
The novel coupling introduces interactions between slow variables of different neurons, with heuristic biological interpretation for perturbation transitions.
Chaos Preservation Proof
Using Marotto's theorem, the existence of snap-back repellers in the original system implies Devaney chaos preservation in the coupled system.
Applications
Computational Neuroscience
- Modeling neural population dynamics with realistic coupling
- Studying emergence of collective chaotic behavior
- Investigating information processing in chaotic neural systems
Dynamical Systems
- Strange attractor generation and characterization
- Fractal dimension analysis in coupled map systems
- Bifurcation theory applications to neural models
Machine Learning
- Chaotic reservoir computing with coupled Rulkov maps
- Dynamical system initialization for neural networks
- Emergent computation in chaotic neural architectures
References
- Primary Paper: Disca, S. (2026). On a cross coupling of Rulkov neural maps. arXiv:2607.22318
- Related Work: Marotto's theorem on snap-back repellers and Devaney chaos
- Applications: Kaplan-Yorke dimension for strange attractor characterization
Activation Keywords
Use this skill when working with:
- Coupled neural map systems
- Chaotic neural dynamics preservation
- Strange attractor generation in neural networks
- Rulkov map extensions and modifications
- Biological interpretation of neural coupling mechanisms
