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Golden ratio organization in human EEG is associated with theta-alpha frequency convergence: a multi-dataset validation study.

Code ↔ Paper

11 matches between paragraphs of the paper and lines of its authors' code, computed by the harvester (lexical-v1). Click a colored paragraph or line to see its counterpart.

The 11 matches
  1. [1] § Methods › Spectral analysis ↔ analysis/compute_pci.py, lines 164–216 · score 0.93 · Power spectral density, frontal channels, posterior channels, frontal theta validation, posterior alpha, Spectral centroids
  2. [2] § Methods › Aperiodic sensitivity analysis ↔ analysis/config.py, lines 40–47 · score 0.78 · peak width limits, aperiodic mode, peak height, FOOOF
  3. [3] § Methods › Datasets ↔ analysis/analyze_combined.py, lines 1–16 · score 0.74 · LEMON Mind Brain, PhysioNet, Body, EEGBCI, Cross
  4. [4] § Methods › Spectral analysis ↔ analysis/spectral_centroids.py, lines 27–67 · score 0.67 · Power spectral density, Spectral centroids, Hann, windows, overlap, Welch
  5. [5] § Methods › Datasets ↔ analysis/config.py, lines 60–66 · score 0.66 · closed baseline, PhysioNet, 160 Hz, eyes
  6. [6] § Methods › Phi coupling index ↔ analysis/compute_pci.py, lines 105–137 · score 0.64 · f_alpha, f_theta, Phi Coupling, harmonic, centroids, closer
  7. [7] § Methods › Validation procedures ↔ analysis/compute_pci.py, lines 164–216 · score 0.63 · Frontal theta validation, posterior channels, posterior alpha, PCI
  8. [8] § Methods › Theta-alpha convergence ↔ analysis/compute_pci.py, lines 140–161 · score 0.59 · Theta alpha convergence, f_alpha, f_theta, metric
  9. [9] § Methods › Pre processing ↔ analysis/preprocessing.py, lines 90–131 · score 0.59 · MNE, segmented, rejected, thresholds, exceeding, epochs
  10. [10] § Methods › Aperiodic sensitivity analysis ↔ analysis/spectral_centroids.py, lines 114–166 · score 0.58 · 1–40 Hz, spectral centroids, FOOOF, peak, power
  11. [11] § Methods › Pre processing ↔ analysis/config.py, lines 25–31 · score 0.58 · 1–45 Hz, bandpass, rejected, thresholds, epochs, preprocessed

Paper

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The authors' code

Python · 263 lines · 7 KB · MIT · 4 matches

  1. """
  2. compute_pci.py - Core functions for Phi Coupling Index analysis
  4. Golden Ratio Organization in Human EEG
  5. Author: Andrei Condrea
  6. ORCID: 0009-0002-6114-5011
  7. """
  9. import numpy as np
  10. from scipy import signal
  12. # Constants
  13. PHI = 1.618034 # Golden ratio
  14. EPSILON = 0.1 # Regularization parameter
  15. THETA_BAND = (4, 8)
  16. ALPHA_BAND = (8, 13)
  19. def compute_psd_welch(data, sfreq, fmin=1, fmax=45):
  20. """
  21. Compute Power Spectral Density using Welch's method.
  23. Parameters
  24. ----------
  25. data : array
  26. EEG time series (samples,)
  27. sfreq : float
  28. Sampling frequency in Hz
  29. fmin, fmax : float
  30. Frequency range of interest
  32. Returns
  33. -------
  34. freqs : array
  35. Frequency values
  36. psd : array
  37. Power spectral density values
  38. """
  39. nperseg = min(int(4 * sfreq), len(data)) # 4-second windows
  40. freqs, psd = signal.welch(data, sfreq, nperseg=nperseg, noverlap=nperseg//2)
  41. mask = (freqs >= fmin) & (freqs <= fmax)
  42. return freqs[mask], psd[mask]
  45. def compute_spectral_centroid(psd, freqs, fmin, fmax):
  46. """
  47. Compute spectral centroid (center of mass) within frequency band.
  49. Parameters
  50. ----------
  51. psd : array
  52. Power spectral density
  53. freqs : array
  54. Corresponding frequencies
  55. fmin, fmax : float
  56. Band limits in Hz
  58. Returns
  59. -------
  60. float
  61. Centroid frequency in Hz
  62. """
  63. mask = (freqs >= fmin) & (freqs <= fmax)
  64. freqs_band = freqs[mask]
  65. psd_band = psd[mask]
  67. if psd_band.sum() == 0:
  68. return np.nan
  70. return np.sum(freqs_band * psd_band) / np.sum(psd_band)
  73. def compute_peak_frequency(psd, freqs, fmin, fmax):
  74. """
  75. Find peak frequency within frequency band.
  77. Parameters
  78. ----------
  79. psd : array
  80. Power spectral density
  81. freqs : array
  82. Corresponding frequencies
  83. fmin, fmax : float
  84. Band limits in Hz
  86. Returns
  87. -------
  88. float
  89. Peak frequency in Hz
  90. """
  91. from scipy.ndimage import uniform_filter1d
  93. mask = (freqs >= fmin) & (freqs <= fmax)
  94. freqs_band = freqs[mask]
  95. psd_band = psd[mask]
  97. if len(psd_band) == 0:
  98. return np.nan
  100. # Smooth to avoid noise peaks
  101. psd_smooth = uniform_filter1d(psd_band, size=3)
  102. return freqs_band[np.argmax(psd_smooth)]
  105. def compute_pci(f_alpha, f_theta, epsilon=EPSILON):
  106. """
  107. Compute Phi Coupling Index.
  109. PCI = log((|R - 2| + ε) / (|R - φ| + ε))
  111. Where R = f_alpha / f_theta
  113. Parameters
  114. ----------
  115. f_alpha : float
  116. Alpha frequency (centroid or peak) in Hz
  117. f_theta : float
  118. Theta frequency (centroid or peak) in Hz
  119. epsilon : float
  120. Regularization parameter (default: 0.1)
  122. Returns
  123. -------
  124. float
  125. PCI value
  126. - PCI > 0: closer to φ (1.618) than to 2:1
  127. - PCI < 0: closer to 2:1 than to φ
  128. """
  129. if np.isnan(f_alpha) or np.isnan(f_theta) or f_theta == 0:
  130. return np.nan
  132. ratio = f_alpha / f_theta
  134. distance_to_harmonic = np.abs(ratio - 2.0)
  135. distance_to_phi = np.abs(ratio - PHI)
  137. return np.log((distance_to_harmonic + epsilon) / (distance_to_phi + epsilon))
  140. def compute_convergence(f_alpha, f_theta):
  141. """
  142. Compute theta-alpha convergence metric.
  144. Convergence = 1 / |f_alpha - f_theta|
  146. Parameters
  147. ----------
  148. f_alpha : float
  149. Alpha frequency in Hz
  150. f_theta : float
  151. Theta frequency in Hz
  153. Returns
  154. -------
  155. float
  156. Convergence value (higher = frequencies closer together)
  157. """
  158. diff = np.abs(f_alpha - f_theta)
  159. if diff == 0:
  160. return np.nan
  161. return 1 / diff
  164. def analyze_subject(data, sfreq, posterior_channels=None, frontal_channels=None):
  165. """
  166. Complete analysis pipeline for a single subject.
  168. Parameters
  169. ----------
  170. data : array
  171. EEG data (channels x samples)
  172. sfreq : float
  173. Sampling frequency
  174. posterior_channels : list
  175. Indices of posterior channels for averaging
  176. frontal_channels : list, optional
  177. Indices of frontal channels for theta validation
  179. Returns
  180. -------
  181. dict
  182. Dictionary with all computed metrics
  183. """
  184. results = {}
  186. # Average posterior channels
  187. if posterior_channels is not None:
  188. posterior_data = data[posterior_channels].mean(axis=0)
  189. else:
  190. posterior_data = data.mean(axis=0)
  192. # Compute PSD
  193. freqs, psd = compute_psd_welch(posterior_data, sfreq)
  195. # Compute spectral metrics
  196. results['theta_centroid'] = compute_spectral_centroid(psd, freqs, *THETA_BAND)
  197. results['alpha_centroid'] = compute_spectral_centroid(psd, freqs, *ALPHA_BAND)
  198. results['theta_peak'] = compute_peak_frequency(psd, freqs, *THETA_BAND)
  199. results['alpha_peak'] = compute_peak_frequency(psd, freqs, *ALPHA_BAND)
  201. # Compute derived metrics
  202. results['ratio'] = results['alpha_centroid'] / results['theta_centroid']
  203. results['PCI'] = compute_pci(results['alpha_centroid'], results['theta_centroid'])
  204. results['convergence'] = compute_convergence(results['alpha_centroid'], results['theta_centroid'])
  206. # Frontal theta analysis (if channels provided)
  207. if frontal_channels is not None and len(frontal_channels) >= 2:
  208. frontal_data = data[frontal_channels].mean(axis=0)
  209. freqs_f, psd_f = compute_psd_welch(frontal_data, sfreq)
  210. results['theta_frontal'] = compute_spectral_centroid(psd_f, freqs_f, *THETA_BAND)
  212. # Frontal-based PCI (frontal theta, posterior alpha)
  213. results['PCI_frontal'] = compute_pci(results['alpha_centroid'], results['theta_frontal'])
  214. results['convergence_frontal'] = compute_convergence(results['alpha_centroid'], results['theta_frontal'])
  216. return results
  219. def epsilon_sensitivity(f_alpha, f_theta, epsilons=None):
  220. """
  221. Test PCI stability across epsilon values.
  223. Parameters
  224. ----------
  225. f_alpha : float
  226. Alpha frequency
  227. f_theta : float
  228. Theta frequency
  229. epsilons : list, optional
  230. Epsilon values to test (default: [0.001, 0.01, 0.1, 0.5, 1.0])
  232. Returns
  233. -------
  234. dict
  235. Epsilon -> PCI mapping
  236. """
  237. if epsilons is None:
  238. epsilons = [0.001, 0.01, 0.1, 0.5, 1.0]
  240. return {eps: compute_pci(f_alpha, f_theta, epsilon=eps) for eps in epsilons}
  243. if __name__ == "__main__":
  244. # Example usage
  245. print("Phi Coupling Index Analysis")
  246. print("="*50)
  247. print(f"Golden ratio (φ): {PHI}")
  248. print(f"Theta band: {THETA_BAND} Hz")
  249. print(f"Alpha band: {ALPHA_BAND} Hz")
  250. print()
  252. # Example calculation
  253. f_theta = 6.0 # Hz
  254. f_alpha = 10.0 # Hz
  256. ratio = f_alpha / f_theta
  257. pci = compute_pci(f_alpha, f_theta)
  258. conv = compute_convergence(f_alpha, f_theta)
  260. print(f"Example: θ={f_theta} Hz, α={f_alpha} Hz")
  261. print(f" Ratio: {ratio:.3f}")
  262. print(f" PCI: {pci:.3f} ({'φ-organized' if pci > 0 else '2:1 organized'})")
  263. print(f" Convergence: {conv:.3f}")

compute_pci.py at commit c64a0da, under MIT · at the source

Overview

Authors: Andrei Ursachi1
ORCID iDs: Andrei Ursachi
  1. Independent Researcher, Bucharest, Romania
Journal: Frontiers in human neuroscience, volume 20, article 1781338
Dates: received 5 January 2026; accepted 11 February 2026; published online 4 March 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.3389/fnhum.2026.1781338 · PMID 41859481 · PMCID PMC12996120 · OpenAlex W7133531784
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: EEG (modality), human (organism), methods / tools (subfield)
Methods: Spectral & time-frequency, Connectivity, Preprocessing, Statistics
Keywords: cross-frequency coupling, EEG, golden ratio, individual differences, oscillations, spectral analysis, theta-alpha
Topic: Functional Brain Connectivity Studies (Cognitive Neuroscience, Neuroscience), according to OpenAlex
Citations: not cited yet (Europe PMC); 28 references in the paper

Abstract

Background: The golden ratio (ϕ ≈ 1. 618) has been proposed as an organizing principle for EEG frequency bands, potentially minimizing spurious cross-frequency synchronization. However, whether individual differences in ϕ-organization have functional correlates remains unexplored.

Objective: We investigated whether proximity of theta-alpha frequency ratios to ϕ is associated with theta-alpha frequency convergence near the 8 Hz boundary.

Methods: We developed the Phi Coupling Index (PCI), which quantifies spectral frequency ratio proximity (note: “coupling” here refers to frequency ratio relationships, not phase-amplitude coupling), quantifying proximity to ϕ vs. harmonic 2:1 organization. Spectral centroids were computed from eyes-closed resting-state EEG across two independent datasets (N = 320): PhysioNet EEGBCI (N = 109) and LEMON Mind-Brain-Body (N = 211). We performed comprehensive validation including: (1) null model simulation, (2) per-dataset replication, (3) robust statistics, (4) ϕ-specificity parameter sweep, (5) epsilon sensitivity analysis, and (6) frontal theta validation.

Results: Across 320 subjects, 80.0% showed ϕ-organization (PCI > 0). PCI was strongly associated with theta-alpha convergence [r = 0.54, p < 10−25; Spearman ρ = 0.82 (higher rank correlation reflects monotonic association with bounded transform)]. This effect: (1) exceeded null model expectation by >5 SD; (2) replicated across both datasets (r = 0.50-0.63); (3) was robust to outliers; (4) showed ϕ-specificity in parameter sweep; (5) remained qualitatively consistent across epsilon values (r = 0.41-0.78); and (6) critically, frontal theta analysis yielded even stronger effects (r = 0.718, p ≈ 10−35), providing evidence against volume conduction artifacts. Demographic controls in LEMON (partial correlation controlling for age) yielded r = 0.490, virtually identical to uncorrected r = 0.497, indicating age does not substantially confound the effect. However, pre-planned exploratory subgroup analyses revealed meaningful individual differences: the association was stronger in younger adults (age < 40: r = 0.574, N = 142) than older adults (r = 0.344, N = 69; note: reduced statistical power in older subgroup), and notably stronger in females (r = 0.680, N = 77) than males (r = 0.429, N = 134), consistent with known sex differences in alpha oscillation characteristics. High-ϕ subjects (PCI > median) showed frequency profiles converging toward the 8 Hz boundary (θ = 6.24 Hz, α = 9.75 Hz) compared to low-ϕ subjects (θ = 5.85 Hz, α = 10.20 Hz), with no age confound (mean ages 37.7 vs. 35.7 years).

Conclusions: PCI demonstrates a robust association with theta-alpha convergence that exceeds structural expectations, replicates across datasets, and shows specificity to the golden ratio. The striking frontal theta validation (r = 0.718) provides converging evidence against a simple spatial-mixing explanation and is consistent with neurophysiological organization.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repository

Its files are read in the Code ↔ Paper reader above, with 11 matches between paragraphs and lines of code.

ExeqTer91/eeg-phi-coupling

License: MIT
State: the link answers, verified on 30 September 2026
Evidence: files inventoried
Commit: c64a0da59953b63403e57a74d6addc02f447ab12, 19 February 2026
Languages: Python (7)
Size: 16 files, 7 scripts
Software Heritage: not archived
Found in: “Data availability statement”
Holds: README, license file, CITATION.cff, environment (requirements.txt)
Not found: tests, continuous integration, documentation
Tools: NumPy (6 files), SciPy (4 files), pandas (3 files), MNE-Python (2 files), Matplotlib (1 file), seaborn (1 file), specparam (formerly FOOOF) (1 file)
Availability: 1 check, the latest on 30 September 2026: the link answers
  • 30 September 2026: the link answers
9 files
At the source: github.com/ExeqTer91/eeg-phi-coupling

The paper's code and data availability statement is in the Data section.

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Data

Datasets cited

Data availability statement

Publicly available datasets were analyzed in this study. PhysioNet EEGBCI: https://physionet.org/content/eegmmidb/. LEMON: https://fcon_1000.projects.nitrc.org/indi/retro/MPI_LEMON.html. Analysis code: https://github.com/ExeqTer91/eeg-phi-coupling.

Reproduced under the paper's license (CC BY), from the paper cited above.

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Version 1, 30 September 2026: the first record

Recorded: type, language, journal, volume, pages, dates, 1 author, 7 keywords, 28 references.

Cite

This paper

Ursachi, A. (2026). Golden ratio organization in human EEG is associated with theta-alpha frequency convergence: a multi-dataset validation study. Frontiers in human neuroscience, 20, 1781338. https://doi.org/10.3389/fnhum.2026.1781338

BibTeX

@article{ursachi2026golden,
author = {Ursachi, Andrei},
title = {{Golden ratio organization in human EEG is associated with theta-alpha frequency convergence: a multi-dataset validation study}},
journal = {Frontiers in human neuroscience},
year = {2026},
month = mar,
volume = {20},
pages = {1781338},
publisher = {Frontiers Media SA},
issn = {1662-5161},
doi = {10.3389/fnhum.2026.1781338},
url = {https://doi.org/10.3389/fnhum.2026.1781338},
pmid = {41859481},
pmcid = {PMC12996120}
}

RIS

TY - JOUR
AU - Ursachi, Andrei
TI - Golden ratio organization in human EEG is associated with theta-alpha frequency convergence: a multi-dataset validation study
T2 - Frontiers in human neuroscience
J2 - Front Hum Neurosci
PY - 2026
DA - 2026/03/04
VL - 20
SP - 1781338
SN - 1662-5161
PB - Frontiers Media SA
DO - 10.3389/fnhum.2026.1781338
UR - https://doi.org/10.3389/fnhum.2026.1781338
LA - en
ER -

CSL-JSON

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