Phi Coupling Index: Golden Ratio Organization in Human EEG

Overview

This repository contains analysis code and processed data for investigating golden ratio (φ ≈ 1.618) organization in human EEG theta-alpha frequency architecture.

Key Finding: The Phi Coupling Index (PCI) strongly predicts theta-alpha convergence across 314 subjects (r = 0.638, p < 10⁻³⁷), suggesting that proximity to φ in cross-frequency ratios has functional relevance for neural dynamics.

Bonus Discovery: Population mean theta-alpha ratio (1.7221) approximates e-1 ≈ 1.7183 (difference: 0.0038), suggesting transcendental mathematical constants may characterize default brain states.

Mathematical Framework

Constant	Value	Role in Brain Dynamics
φ (Golden Ratio)	1.618	Optimal flexibility zone - maximal desynchronization
e-1	1.718	Population default attractor - "resting" state
2:1 (Harmonic)	2.000	Rigid phase-locking - processing mode

The Phi Coupling Index (PCI) quantifies where an individual falls on this spectrum:

PCI = log(|ratio - 2.0| / |ratio - φ|)

• PCI > 0 → φ-organized (closer to golden ratio)
• PCI < 0 → Harmonic-organized (closer to 2:1)

Repository Contents

Analysis Scripts

Script	Description
phi_specificity_analysis.py	Core PCI computation and φ-specificity testing
phi_prediction_test.py	Correlation analysis: PCI vs theta-alpha convergence
cross_frequency_analysis.py	Cross-frequency coupling metrics
comprehensive_analysis.py	Full pipeline with all metrics
meditation_baseline_analysis.py	Meditation vs baseline comparisons
coherence_analysis.py	Inter-electrode coherence
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split_half_validation.py	Split-half cross-validation (rules out circularity)

1cecdf1 (Update project)

Data Files

File Pattern	Description	N
alpha_s*.mat	Processed spectral data (Dataset 1)	20
alpha_subj_*.mat	Processed spectral data (Dataset 2)	18
subject_*.mat	Raw spectral matrices	5
meditation_spec.mat	Meditation condition spectra	-
boxplot_*.csv	Summary statistics for visualization	-

Generated Figures

• phi_specificity_analysis.png - φ vs 2:1 distance comparison
• phi_prediction_test.png - PCI correlation with convergence
• cross_frequency_coupling.png - CFC visualization
• comprehensive_phi_analysis.png - Multi-panel summary figure <<<<<<< HEAD =======
• figure5_split_half_validation.png - Split-half cross-validation proof

1cecdf1 (Update project)

Key Results

1. Population Distribution

• 67.2% of subjects show φ-organization (PCI > 0)
• Mean ratio: 1.7221 (SE: 0.012)
• Approximates e-1 = 1.7183 (Δ = 0.0038)

2. Predictive Power

Predictor	r	p-value	95% CI
PCI (φ-index)	0.638	2.6×10⁻³⁷	[0.580, 0.690]
Distance from φ	-0.745	<10⁻²⁴	[-0.79, -0.69]
Distance from 2:1	+0.687	<10⁻²⁰	[0.62, 0.74]

3. Aperiodic Robustness

• 99.6% of subjects retained φ-organization after 1/f slope correction
• Effect is property of true oscillations, not spectral artifact

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4. Split-Half Validation (Rules Out Circularity)

To address concerns about mathematical circularity between PCI and convergence, we performed split-half cross-validation on subjects with raw EEG epochs:

• N = 37 subjects with raw epochs
• PCI computed from ODD epochs → Convergence computed from EVEN epochs
• r = 0.843, p < 10⁻¹⁰, 95% CI [0.72, 0.92]

This demonstrates that the PCI-convergence relationship reflects stable individual differences, not mathematical coupling.

1cecdf1 (Update project)

Installation

git clone https://github.com/ExeqTer91/open-neuro-data.git
cd open-neuro-data
pip install numpy scipy matplotlib pandas

Usage

# Basic PCI computation
from phi_specificity_analysis import compute_pci

theta_centroid = 5.2  # Hz
alpha_centroid = 9.8  # Hz
pci = compute_pci(theta_centroid, alpha_centroid)
print(f"PCI: {pci:.3f}")  # Positive = φ-organized

Citation

If you use this code or data, please cite:

@article{ursachi2026phi,
  title={Phi Coupling Index Predicts Theta-Alpha Convergence in Human EEG},
  author={Ursachi, Andrei},
  journal={NeuroImage},
  year={2026},
  note={Under review}
}

Related Work

• Klimesch, W. (2013). An algorithm for the EEG frequency architecture of consciousness. Frontiers in Human Neuroscience, 7, 766.
• Pletzer, B., Kerschbaum, H., & Klimesch, W. (2010). When frequencies never synchronize: the golden mean and the resting EEG. Brain Research, 1335, 91-102.

License

MIT License - see LICENSE for details.

Contact

• Author: Andrei Ursachi
• Email: contact@andreiursachi.eu
• Preprint: [bioRxiv - pending]

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"The golden ratio is not merely a description of default EEG architecture but reflects a functional mechanism for neural flexibility."