Connectome harmonics · computational neuroscience

Brain activity rides the connectome's harmonics — and LSD bends which ones

Three reproducible experiments on real human neuroimaging (plus one exploratory aside). We turn the brain's structural wiring into a graph, take the eigenmodes of its graph Laplacian (the connectome's harmonics), project resting-state fMRI onto them, and ask what changes under a psychedelic. Every result below is interactive.

Structural connectomeHCP-YA group average · Schaefer-400 parcellationHCP-YA·Schaefer atlas

Functional dataOpenNeuro ds003059 · 12 subjects · LSD vs placebo restOpenNeuro ds003059

Replication cohortOpenNeuro ds006072 · 7 subjects · psilocybin vs active placeboOpenNeuro ds006072

Cortical geometryfsLR-32k surface eigenmodes (LaPy) · NSBLabNSBLab BrainEigenmodes·LaPy

Experiment 1

Does LSD reorganize the harmonic spectrum?

Method: project ds003059 BOLD onto the HCP connectome harmonics (parcellated, CHAP-style) and compare power per mode, LSD vs placebo.

Yes. Low-frequency harmonics lose power under LSD — the direction reported by Atasoy 2017. low-band LSD/placebo ≈ 0.77high-band ≈ 0.85

Experiment 2

Is there a geometric “latent space” in the brain?

Method: use the lowest harmonics as coordinate axes (a Laplacian eigenmap) and embed all 400 regions — no functional labels given to the algorithm.

Yes. The seven Yeo networks fall into a continuous layout that emerges from connectivity alone. Default-mode ↔ Visual along mode 1unsupervised

Experiment 3

Shape vs wiring — and does LSD move the winner?

Method: reconstruct BOLD from the first N vectors of three bases (connectome, cortical geometry, distance rule); paired test of the geometric−connectome advantage, LSD vs placebo.

A tie at rest — but LSD shifts it toward geometry. The geometric basis reliably gains on the connectome under LSD. ΔAUC +0.006paired t(11)=4.60p=0.0008d_z=1.3312/12 subjectsholds after GSR + permutation

Replicates in psilocybin. An independent cohort (ds006072, n=7) shows the same geometry-gains direction. ΔAUC +0.010perm p=0.031Wilcoxon p=0.031d_z=0.87t p=0.061 · marginal after GSR

Exploratory aside

Does the psychedelic brain become more “harmonious”? (STV)

A niche, speculative theory of consciousness — included as a falsification probe, not a core claim.

Method: score the consonance / spectral symmetry of the connectome-harmonic spectrum (Sethares dissonance, entropy) — paired test, drug vs placebo.

No — it gets less consonant. Against the naive STV prediction, the harmonic spectrum becomes more dissonant and broader under LSD (the “entropic brain” direction). dissonance ↑ p=0.001d_z=1.24entropy ↑ p=0.007holds after GSR + permpsilocybin: same direction, n.s.

Bottom line: under LSD the harmonic repertoire reorganizes — less low-frequency, wiring-bound structure and a measurable drift toward smooth cortical-shape patterns. The psychedelic shift is a within-subject effect significant at p<0.001 across all 12 subjects, and it survives nuisance regression (global-signal + detrend + motion-proxy censoring) and an exact permutation null — so it is not merely a motion or global-signal artifact. It then replicates in direction in an independent psilocybin cohort (ds006072, n=7: same geometry-gains shift, d_z=0.87, exact-permutation p=0.031) — a different drug, scanner, and pipeline. As a speculative aside, a harmonic-consonance probe (a falsification test of the niche Symmetry Theory of Valence) finds the spectrum becomes less consonant and broader under LSD (p=0.001, survives GSR + permutation) — the opposite of a naive “psychedelic = more symmetry” reading, and a reminder that “drifting toward the smooth geometric basis” is not the same as “becoming more consonant.” It remains a coarse, parcellated result, and the psilocybin replication is underpowered (n=7, active placebo).

LSD vs placebo connectome-harmonic power spectra, 12 subjects — **Experiment 1.** Harmonic power vs spatial frequency, LSD vs placebo (n=12). Low-frequency modes are suppressed under LSD.

Reconstruction accuracy of connectome, geometric and EDR bases, with per-subject paired shift — **Experiment 3.** Reconstruction accuracy for each basis (left/middle) and the per-subject paired shift of the geometric−connectome advantage (right): nearly every subject rises toward geometry under LSD.

Psilocybin vs methylphenidate reconstruction accuracy and per-subject paired shift — **Replication (psilocybin, ds006072, n=7).** Same per-subject paired test on an independent drug/scanner/pipeline. Activity again shifts toward geometry under psilocybin vs active placebo (Δ=+0.010, d_z=0.87, exact-permutation p=0.031; marginal after GSR).

Harmonic consonance, spectral entropy and centroid under LSD and psilocybin — **Exploratory aside (STV probe).** Harmonic consonance/symmetry, drug vs placebo. Under LSD the spectrum becomes more dissonant (p=0.001, d_z=1.24), broader (entropy p=0.007) and higher-frequency — the opposite of a naive “psychedelic = more symmetry” reading; psilocybin trends the same way (n=7, n.s.). *STV is a speculative, niche theory; this is a falsification probe, not a core result.*

↑ Play with the live simulator above — explore the latent space and inspect every result · 📄 Read the preprint (PDF) Group-average, parcellated (400 regions) research/education artifacts — not individual or clinical data, and not medical advice. The 400-region resolution is exactly where the geometry-vs-wiring gap is weakest, so Exp. 3 tests the LSD shift, not the broader debate (cf. Mansour 2024).

Connectome Harmonics Simulator

Graph Laplacian eigenmodes as standing waves on a brain-like network

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presented atEdge City

↓ findings, figures & the paper

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What this demonstrates

An interactive study of connectome harmonics on real human data — the HCP group-average structural connectome, a 12-subject LSD fMRI cohort (OpenNeuro ds003059), and an independent 7-subject psilocybin cohort for replication (OpenNeuro ds006072). Three questions are tested (plus one exploratory aside), each with a result you can inspect below:

1. Does LSD reorganize the brain's harmonic spectrum? Low-frequency harmonics lose power under LSD (LSD/placebo ≈ 0.77) — the direction reported by Atasoy 2017. see the spectra ↓
2. Is there a geometric “latent space” in the brain? Yes — the seven functional networks separate into a continuous layout in harmonic coordinates, with no labels given to the algorithm. explore it ↓
3. Shape vs wiring — which explains activity, and does LSD change it? All bases are close; cortical shape (and a distance surrogate) do at least as well as wiring. Under LSD the geometric basis gains on the connectome. see the curves ↓
Exploratory aside. Does the psychedelic brain get more “harmonious” (the niche, speculative symmetry theory of valence)? No — the harmonic spectrum becomes more dissonant and broader under LSD (p=0.001, survives GSR), the opposite of the naive prediction; psilocybin trends the same way. Included as a falsification probe, not a core claim. see the consonance probe ↓

Connectome data

Switch graphs to compare a teaching toy model with a real group-average human structural connectome.

Mode tour

Auto-cycle through harmonic modes with short explanations. Low modes are broad global waves; high modes are finer patterns.

Press start tour to walk through modes 1–40. Each mode is an eigenvector of the graph Laplacian — a “standing wave” allowed by the connectome geometry.

Simulation

field model harmonic mode: 1

speed: 0.25× amplitude: 1.00 coupling K (oscillator view): 1.2

2π

What this simulates

Nodes are brain regions; edges are structural connections. The normalized graph Laplacian turns connectivity into a geometry. Its eigenvectors v_k are spatial harmonics — the connectome’s allowed standing-wave patterns.

x_i(t) = a_k v_k,i cos(√λ_k · t)

This demo animates those basis patterns. Real fMRI activity can be decomposed into the same harmonics (see literature below).

Geometric latent space

The harmonics aren't just waves — they're coordinate axes. Each region gets a position (v_a,i, v_b,i) in a low-dimensional embedding (this is a Laplacian eigenmap / diffusion-map latent space). If a smooth geometry exists, the seven Yeo functional networks should separate into a continuous layout — without ever telling the algorithm what they are.

x-axis y-axis

LSD & connectome harmonics

The paper Atasoy et al. 2017 (Scientific Reports) — often discussed as “connectome-specific harmonic waves under LSD” — does not publish a downloadable harmonic-power dataset. It reports findings from decomposing LSD/placebo fMRI:

↓ Low-frequency harmonics (broad modes like those at the start of the tour) contribute less under LSD.

↑ High-frequency harmonics (fine-grained modes) contribute more.

Expanded repertoire — more distinct harmonic brain states become active, with correlated cross-frequency co-activation.

We ran a parcellated CHAP-style analysis on OpenNeuro ds003059. Spectra below are computed by projecting real resting-state fMRI onto the HCP connectome harmonics. Full vertex-level pipeline: CHAP.

Geometry vs connectivity

Is brain activity shaped more by the connectome's wiring or by the cortex's shape? Following Pang et al. 2023, we reconstruct the ds003059 fMRI from the first N vectors of three Schaefer-400 bases and measure accuracy vs N: connectome (real wiring, incl. long-range tracts), geometric (cortical shape, Laplace–Beltrami modes), and EDR (connectivity that decays with distance only — geometry-as-a-network, not real wiring). So the headline "toward geometry, away from wiring" is the geometry − connectome contrast; the geometry − EDR contrast compares two flavours of geometry and is a different question.

Vertex resolution (fsLR-32k, real wiring). Geometry vs the empirical high-resolution group-average connectome (Pang 2023), on the same fsLR-32k cortex as the psilocybin CIFTIs — no parcellation, no surrogate. The geometric (shape) basis reconstructs activity better than real wiring in both psilocybin and placebo; the drug shift toward geometry is directionally consistent but underpowered at vertex level (n=7, Δ=+0.004, d_z=0.47, perm p=0.20).

Harmonic consonance & the symmetry theory of valence Exploratory aside

A speculative side-probe, not a core result. The Symmetry Theory of Valence is a niche, non-mainstream theory of consciousness whose strong form is hard to falsify; we include it only because it yields one concrete, testable corollary — and that corollary fails in the predicted direction, which is itself the useful finding.

A natural follow-up: does the degree of geometric/harmonic order correlate with positive affect? The Symmetry Theory of Valence (Johnson 2023) proposes that how pleasant an experience feels tracks the symmetry of its neural representation; the Qualia Research Institute operationalizes this on connectome harmonics via consonance. We can't test the affect correlation directly — neither dataset ships per-subject mood ratings — but we can test the mechanism direction: does a psychedelic move the harmonic spectrum toward consonance/symmetry, or away?

For each subject and state we project BOLD onto the connectome harmonics, take the power per mode, and score the spectrum's Sethares dissonance (lower = more consonant), its spectral entropy (breadth), and its spectral centroid (high vs low spatial frequency), then run the same within-subject paired test as Experiment 3.

▸ How these numbers are computed (entropy, centroid, dissonance)

Everything starts from one object — the harmonic power spectrum of a scan — and each metric is just a different summary of it.

Step 0 — the spectrum. The fMRI is a matrix Y of 400 regions × time, z-scored per region. We project every frame onto the connectome harmonics (the orthonormal eigenvectors v_k of the graph Laplacian): C = Vᵀ·Y, so C[k,t] is "how much of mode k is active at time t." The power of mode k is its time-averaged squared coefficient, p_k = mean_t C[k,t]², normalized so Σ p_k = 1. So p is a distribution over the harmonics: what fraction of this brain's activity lives at each spatial frequency. Each mode also has a frequency ω_k = √λ_k (low mode = smooth/global, high mode = fine-grained; the √ is the drum-pitch analogy).

The three metrics (each one line of p and ω):

Spectral entropy = −Σ p_k ln p_k / ln K — Shannon entropy of p on a 0–1 scale. Near 0 = power packed into a few modes; near 1 = spread evenly across many (a broad repertoire). It only measures how spread out the power is.
Spectral centroid = Σ p_k ω_k — the power-weighted average frequency, the spectrum's "center of mass." Higher = energy has moved into finer, higher-frequency modes.
Consonance / dissonance — the only metric that looks at relationships between modes. We treat the spectrum as a musical chord: each mode is a tone with frequency f_k and loudness a_k = √p_k. For every pair of modes we add the Sethares (1993) "sensory dissonance," a curve that is 0 for identical tones, peaks when two tones are close but not equal (they "beat," like two out-of-tune notes), then fades when they're far apart. Total over all pairs = dissonance (lower = more consonant). Because that curve is calibrated in Hz, we rescale the ω's into the audible range (lowest mode → 220 Hz) — this keeps all frequency ratios and is identical across conditions, so it can't bias the LSD-vs-placebo comparison.

Entropy and dissonance both rise when power spreads out, but they differ: entropy ignores where the modes sit, while dissonance specifically penalizes power landing in neighbouring, "beating" modes. Each per-scan number then goes through the same paired drug-vs-placebo test as Experiment 3. Full code: harmonic_consonance.py.

Result — against the naive prediction. Under LSD the harmonic spectrum becomes more dissonant (Δ=+16.7, paired p=0.001, exact-permutation p=0.002, d_z=1.24), broader (entropy p=0.007) and shifted toward higher spatial frequencies (centroid p=0.015); all survive global-signal regression. Psilocybin (n=7) trends the same way but is not significant. This is the “entropic brain” direction — the opposite of a simple “psychedelic bliss = more symmetry/consonance” reading of STV.

Note this is not in tension with Experiment 3: activity can drift toward the smooth geometric basis relative to wiring while the overall spectrum still spreads to higher modes — “more geometric alignment” ≠ “more consonance.” The consonance metric is the sharper STV variable.

Caveats: this tests the mechanism direction, not the affect correlation (no per-subject valence labels exist in either release); it measures spatial-harmonic consonance, whereas QRI's “neural annealing” predicts a transient rise in energy/entropy before annealing, so an acute broadening is not a clean refutation of STV; n=12/7, parcellated, and the spatial-frequency→audio mapping is a modelling choice (the direction is robust across raw, GSR, and permutation, and for LSD across both the connectome and geometric harmonic substrates).

References

Atasoy et al. 2016 — Connectome-specific harmonic waves (foundational theory).
Atasoy et al. 2017 — Harmonic repertoire reorganization under LSD.
Sanchez et al. 2020 — Robustness of harmonics to connectivity changes.
Vohryzek et al. 2024 — Integrative, segregative, and degenerate harmonics.
Carhart-Harris et al. 2020 — LSD fMRI dataset (OpenNeuro ds003059).
Siegel et al. 2025 — psilocybin precision-imaging dataset (OpenNeuro ds006072), replication cohort.
Johnson 2023 — Qualia Formalism and a Symmetry Theory of Valence (updated from Principia Qualia, 2016; Experiment 4).
Sethares 1993 — Local consonance / sensory-dissonance model (used for the consonance metric).

Built on real data: the HCP-YA group-average Schaefer-400 structural connectome, the 12-subject LSD fMRI cohort from OpenNeuro ds003059, and the 7-subject psilocybin replication cohort from OpenNeuro ds006072. These are group-average / parcellated results for research and education — not individual or diagnostic measurements, and not medical advice. (The “toy synthetic graph” dataset is the only synthetic one.)