Contribution
Rotating waves had been seen locally in cortex (spindles, turtle and marmoset LFP, monkey prefrontal cortex, rodent slices) and one global rotation in arousal-averaged fMRI. This paper finds many small, mobile spirals coexisting across the whole human cortex in moment-to-moment fMRI, characterises their motion and pairwise interactions as defect dynamics, and links their configuration to tasks: it decodes task condition from them, and reads task-specific routing of activity flow from their arrangement.
Key insight
Treat the phase of slow cortical fluctuations as a field on a sheet and look for its topological defects. A defect is a spiral, and its rotation sense is a charge of ±1. Charges move, collide and annihilate as defects do in excitable and turbulent media. Because a spiral sets the direction of flow around it, flipping the sense of a few spirals at region boundaries reverses the flow through a whole region without building a new pattern. On this picture the brain switches the direction of large-scale traffic, bottom-up or top-down, by re-signing its defects.
Assumptions
- Data: HCP minimally pre-processed fMRI, 2 mm CIFTI grayordinates (about 32,000 vertices per hemisphere) on the HCP flat map. Three random cohorts of 100 (rest, language, language replication) and 100 subjects doing the working-memory task.
- Filtering: 4th-order zero-phase Butterworth, 0.01–0.1 Hz; difference-of-Gaussian spatial band-pass at scales from geometric eigenmode wavelengths. The analysis scale, 70–138 mm, is the band where spiral-based task classification is highest (Methods).
- Phase field: Hilbert phase per voxel; phase vector field V = ∇φ by circular central differences; vorticity ω = ∇ × V; spiral cores where |curl| > 1 and clusters of at least 3 × 3 voxels; boundaries grown while phase vectors stay within 45° of the circle’s tangent; boundaries then manually inspected.
- Null model: 3-D (x, y, t) Fourier phase randomisation, preserving spatial and temporal autocorrelation. Spatial filtering makes spirals appear in the null too, so a spiral time step counts only if its filtered–unfiltered phase similarity (share of voxels within 30°) exceeds the null’s 95th percentile.
- Flattening: the authors concede that the flat map does not preserve distances and that a wave reverberating round a sulcus could appear as a spiral; they test spiral density against sulcal depth and find no bias.
- Model: a 30 × 30 lattice of Kuramoto-type phase oscillators with nearest-neighbour sine coupling and random initial phases; spirals are reversed by φ → π − φ inside their boundaries.
Key results
- Prevalence and size (Fig. 2). 16.33 ± 2.22 spirals per time step in the left hemisphere at rest (18.8 ± 2.6 during the task); radius 10.46 ± 4.77 mm; 62% span more than one of 22 regions; amplitude falls towards the singularity; density highest on region and network boundaries, not dependent on gyri, sulci or inflections (Extended Data Fig. 2).
- Motion (Fig. 3). Angular speed 0.36 ± 0.31 rad/s, heavy-tailed; propagation 2.19 ± 1.80 mm/s; MSD ∝ τ^β with β between 1 and 1.9 (mean 1.5) for 96% of trajectories; centres travel up to about 10 cm.
- Interactions (Fig. 4). Full annihilation 51.04%, partial annihilation 46.40%, repulsion 2.55%. Full annihilation leaves a plane wave travelling opposite to the pre-collision flow. Cross-scale intersection ratios exceed the null for all scale pairs (P < 0.001) and peak late in a spiral’s life.
- Task specificity (Figs. 5–6). Story versus math listening reverses dominant rotation at the PCC/dorsal-stream/superior-parietal border, with the opposite sense in the homologous right-hemisphere region. Contrast maps are hemispherically symmetric (r = 0.63 listening, 0.66 answering) and strongest in frontoparietal, dorsal-attention and default-mode networks (auditory too when answering).
- Decoding. Four conditions at 48.33 ± 0.31% (replication 50.35%), null 24.32%, raw-amplitude decoder with matched input 30.2%. Working memory: stimulus type 43.72% (chance 25%), 0-back versus 2-back 66.72%, correct versus incorrect 58.96% (chance 50%).
- Unfiltered signals (Fig. 7). Trial-averaged, unfiltered task-evoked fMRI shows spirals at the same places and senses as the filtered data; PCA of the signal round a spiral traces a rotation in the first two components, which explain over 80% of variance.
- Activity flow (Fig. 8). Opposite pairs make S-shaped “gates”, like pairs saddle-shaped “walls”. During math listening four spiral clusters route flow from primary auditory to lateral temporal cortex; during answering seven route it the other way, through DMN, FPN and DAN back to primary auditory cortex. A local phase-vector classifier separates the two sessions best inside this “region of coordination”. In the lattice model, annihilation is again the commonest interaction, and reversing selected spirals reverses the flow between them (Extended Data Fig. 9).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Slow cortical fMRI phase fields contain many coexisting spirals around phase singularities, beyond what filtering of autocorrelated noise produces | moderate to strong: a phase-randomised null with a filtered–unfiltered similarity test, replicated in unfiltered task averages; flat-map distortion argued against, not excluded | Figs. 2, 7; Extended Data Figs. 1–2 |
| C2 | Spiral centres move superdiffusively and interact as topological defects (annihilation of opposite charges, repulsion of like ones) | moderate: descriptive statistics over 100 subjects; interaction categories defined by the authors | Figs. 3–4 |
| C3 | Spiral locations and rotation senses carry task information beyond raw fMRI amplitude | moderate: above chance and above a matched amplitude decoder, replicated in a second cohort, but the analysis scale was chosen to maximise this decoding | Fig. 6; Methods |
| C4 | Interacting spirals coordinate correlated activation and deactivation and route task activity bottom-up or top-down | weak to moderate: read from trial-averaged streamlines and a phenomenological model; no perturbation of spirals in the brain | Fig. 8; Extended Data Figs. 7–9 |
| C5 | Transitions between functional network states are continuous, not discrete switches between metastable states | weak: an interpretation of C4, not tested against a state-switching model | Discussion |
| C6 | Brain spirals are “essential” to brain function and computation | speculative: stated as “might be” | Discussion |
Concepts
- brain spiral: a localised rotational wave in the phase field, with high vorticity at its centre and a phase singularity there.
- phase singularity / topological charge: a point round which the phase winds by ±2π; charge +1 or −1 by rotation sense (the paper calls anticlockwise positive vorticity).
- phase vector field: the spatial gradient of the phase map, read as the local direction of activity flow.
- full annihilation / partial annihilation / repulsion: the three pairwise interaction types (Methods).
- region of coordination (ROC): a region whose task-specific flow is set by the spirals that encircle it.
- superdiffusion: mean-squared displacement growing as τ^β with β > 1.
Connections
- Belousov–Zhabotinsky (LIT-531, NOTE-414) and Cross & Hohenberg (LIT-527, NOTE-418). Annihilating waves and spirals at broken fronts in BZ, and defects of broken phase symmetry in non-equilibrium patterns, are the physics these objects belong to. The paper cites Kuramoto and Coullet et al., not these.
- Turing 1952 (LIT-535, NOTE-427). Travelling waves are Turing’s case (e) and need at least three morphogens; the analogy to cortex is only that propagating patterns come from a reaction–diffusion-type medium, which this paper does not model.
- Vohryzek et al. 2022 (LIT-080, NOTE-054). Rival description of non-stationarity (C5): discrete substates with dwell times versus continuous spiral-organised flow.
- Kelso 2021 (LIT-601). Phase-oscillator coordination; its metastability is relative coordination, compatible with C5.
- Verzhbinsky et al. 2026 (LIT-633). Coordination by synchronous bursts rather than propagation, at a timescale three orders faster.
- Muller et al. 2026 (LIT-647). The companion review on travelling waves, unread.
Bearing on the record
- First reading in the record to claim that defect dynamics of the kind documented in BZ and in pattern-formation theory organise brain activity. A THEORY on large-scale cortical dynamics as an excitable medium could cite it, with a second source from electrophysiology.
- Bears on LIT-080’s landscape picture as a rival (C5); no THEORY in the record yet states either side.
- No ML instruction. Nothing here belongs in the anthology.
Note of 2026-10-16: THEORY-088 now cites LIT-638 as a source.
Limitations
- Scale chosen on the outcome. The spatial band was fixed where spiral decoding is maximal; decoding accuracy is therefore an optimistic estimate, though the replication cohort used the same band.
- Flat-map and filtering artefacts. The null controls filtering; the flat map’s distortion is argued against with a sulcal-depth test only.
- Haemodynamic signal. Spirals are in BOLD at 0.01–0.1 Hz; the paper cites, but does not show, that infra-slow fMRI reflects infra-slow LFP.
- Flow is phase gradient. “Activity flow” is the phase-vector field, not measured transfer of activity or information.
- No perturbation. Reversing spirals is done only in a 30 × 30 lattice of phase oscillators.
- Uncorrected contrast maps. Task-contrast maps threshold P < 0.05 “no adjustment is made for multiple comparisons” (Fig. 5 caption).
- Manual steps. Spiral boundaries are inspected by hand; network boundaries were annotated manually from a published figure.
Open questions
- Do spirals of the same kind appear in electrophysiology at the scale of the whole cortex, and are they the same events?
- Does a discrete-state model of the same scans (for example, phase-locking states) lose its explanatory power once spiral positions and charges are regressed out, or the reverse?
- What mechanism makes cortical defects superdiffusive where fluid defects diffuse normally?
Corrections
- none to a seeded skim (there was no seed)
- Sign convention. Results (p. 2) assign topological charge “+1 and –1” to clockwise and anticlockwise spirals respectively; Methods (spiral density map) assign −1 to clockwise and +1 to anticlockwise, matching positive vorticity for anticlockwise. The Methods convention is the one the maps use; nothing downstream depends on which.