Spectral Differentiation: A Log-Frequency Coordinate System for Human EEG Bands Reveals a Within-Band Developmental Axis

EEG frequency bands have been defined by clinical convention rather than principled theory. We present the largest formal model comparison of EEG frequency scaling to date — 8.5 million oscillatory peaks from 6 datasets (N = 5,322, ages 5–88), with HBN's 11 data-collection releases serving as internal replications.

The study makes two contributions, and they are asymmetric: the coordinate system is the primary result, and the within-band developmental axis it reveals is the empirical one.

1. A coordinate system (primary). Log-frequency organization is decisively superior to linear (ΔBIC > 48). Five genuine density troughs — real spectral voids, not 1/f artifacts — are recovered in 100% of subject-level bootstrap iterations (each significant at p < 0.0001 against a 1/f-matched Poisson null), with a global geometric-mean inter-trough ratio of 1.629 (95% CI [1.616, 1.636]). Among named constants proposed in the frequency-scaling literature, only φ = 1.618 falls inside the interval; √2, e−1, √3, the octave, and e are excluded (p < 0.0001). This identifies φ as the best-supported named coordinate constant, not an exact or ontological claim — the point estimate (1.629) sits just above φ (p = 0.062), and the data are equally compatible with nearby unnamed values in 1.62–1.64. A φ-lattice coordinate system (boundaries at f₀ × φⁿ, f₀ = 7.60 Hz) produces the simplest within-band peak-density profiles (R² = 0.980), with each interior boundary within 4% of its locally simplicity-maximizing position. The grid's value is its boundary placement, not any privileged status of its interior coordinates: no fine structure exists at φ-specific positions within bands. We reframe φ from a statement about neural dynamics to a statement about measurement.

2. A within-band measure (empirical). Using this coordinate system, we define spectral differentiation — the within-band concentration of a subject's oscillatory peaks toward the interior versus the edges — as an individual-differences measure that the fixed clinical bands cannot express. It is orthogonal to band power, IAF, aperiodic slope, and connectivity.

• Reliability: Stable over five years (cross-band median test–retest ICC = 0.42, rising to 0.746 for the most reliable cognitively-relevant feature, beta-low ushape; overall maximum 0.77, alpha asymmetry; Dortmund longitudinal, N = 208).
• Lifespan trajectory and two-component decomposition: A developmental rise, a plateau (~15–40 years), a mid-life decline, and late-life degradation, substantially coupled to alpha peak-frequency maturation (ρ = +0.76 to +0.78). A per-subject IAF-anchored reanalysis resolves the signal into two separable layers — an IAF-coupled alpha-mountain component and a smaller IAF-independent within-band-reorganization component — with an asymmetry across life stages: pediatric development retains most FDR survivors under IAF-anchoring (77/90 population, 73/90 IAF in HBN, N = 2,856), whereas adult aging is predominantly IAF-slowing (Dortmund 36/90 → 13/90, N = 516). A power-matched HBN subsample (drawn to N = 516, 100 resamples) confirms the asymmetry is biological, not a sample-size artifact (IAF-FDR median 42; never at or below Dortmund's 13).
• Internal replication: The developmental effect vector replicates across all eleven independent HBN releases (mean inter-release effect-vector r = 0.82).
• Selectivity — psychopathology dissociation: Externalizing and internalizing (bifactor-specific dimension scores, residualized of a general p-factor) dissociate in a method-independent manner. The robust form is an alpha-band Steiger dissociation — the four strongest features (asymmetry, inv_noble3, ramp_depth, noble6) differ at p < 10⁻⁶ under both FOOOF and IRASA. The raw FDR survivor counts (30 externalizing and 43 internalizing under FOOOF; 26 and 18 under IRASA) are method-sensitive and not load-bearing.
• Selectivity — cognition (exploratory): The same features track cognition (36 of 720 feature-by-test correlations survive FDR in LEMON, N = 196), anchored by a beta-low reasoning association (center-depletion × LPS, ρ = −0.248). We treat this as an exploratory, FOOOF-specific, age-coupled discovery layer rather than a validated biomarker: the anchor is not method-independent (IRASA ρ ≈ −0.23, p_FDR = 0.11) and does not survive age control.
• Informative nulls: Silent for personality (0/11,970 FDR in LEMON, replicated at 0/450 in a clinical sample), for medical and metabolic markers (0/4,950), for handedness, and for psychiatric diagnosis after age control (ADHD vs. MDD, 0/72 in TDBRAIN). The contrast — 36 cognitive survivors versus zero for personality, on identical features and subjects — is the structural payload.
• External replication / lifespan extension: TDBRAIN (N = 1,272, clinical psychiatric referrals) replicates the population-level findings and extends the trajectory to age 88, including late-life α/β boundary shallowing.

The principal structural and developmental findings replicate under IRASA, a non-parametric alternative to FOOOF that makes no 1/f shape assumption; the beta-low cognitive association is the notable FOOOF-specific exception. We report which findings survive these method-independence and decomposition checks and which do not.

Keywords: EEG, frequency bands, logarithmic scaling, log-frequency coordinate system, golden ratio, spectral parameterization, oscillatory architecture, spectral differentiation, cognitive development, FOOOF, IRASA, IAF-anchoring

Related identifiers:

• Supersedes: https://doi.org/10.5281/zenodo.18765773 (EEGMMIDB preprint)
• Supersedes: https://doi.org/10.5281/zenodo.18856274 (LEMON preprint)
• Data: https://doi.org/10.5061/dryad.1vhhmgr8t
• Code: https://github.com/neurokinetikz/research

License: CC BY 4.0