Rulkov Neural Maps Cross Coupling

---
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
1. **Coupling Strength**: Controls transition between uncoupled and strongly coupled regimes
2. **Perturbation Magnitude**: Determines slow variable dynamics and biological interpretation
3. **Network Size**: Scalable from 2-neuron pairs to arbitrary network sizes
4. **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