Construction of the cortical connectivity.

(a) Visualization of all neurons in the hemispherical dataset we use to build the cortical connectivity. The hemisphere contains about 1 million neurons in total, including about 300,000 cortical neurons. (b) Algorithm of building the connectome: i) First, we identify neuronal excitatory/inhibitory information based on the spatial transcriptomic data. ii) We then identify the voxelized position for all the neurons. iii) Based on the voxelized projection data, we calculate the number of connections between every pair of voxels (see Use of the voxelized connectivity data). iv) Connections are created randomly between the neurons in each pair of voxels. v) Repeat for all voxels in regions of interest. vi) Complete the connectivity. (c) The connectivity matrix shows the number of connections between different cortical regions(in log 10 scale). A full version of the connectivity matrix with annotated regions is shown in figure Supplement 1. (d) Visualization of outgoing connections from sampled neurons in the primary visual cortex (VISp, left), primary auditory cortex (AUDp, middle), and the ventral anterior cingulate areas (ACAv, right).

Construction of cortical connectivity

Macroscopic traveling waves emerge in response to random layer-4 stimulation through Allen connectivity.

(a) Stimulation protocol: a 10 Hz random Poisson spike train delivers a fixed-amplitude voltage bump Vstim to all layer-4 excitatory neurons; no other external input is applied. (b) Two rows of five spatial snapshots of the local-mean (KNN = 300) intracellular voltage, sampled every 5ms across the window highlighted in (c). A coherent wavefront propagates along the anterior-to-posterior axis. (c) Global mean intracellular voltage of all neurons (dark blue) overlaid on per-region mean voltages (light traces). The red dashed lines mark the snapshot window visualized in (b). (d) Region-sorted raster plot of ∼40,000 neurons sampled evenly from major cortical regions, anterior-to-posterior. See Videos for the movie of the simulation.

Top: schematics of Allen (a), local (b), and uniform connectivity (c); bottom: Visualization of outgoing connections from the same 50 randomly selected neurons in the primary visual cortex (VISp) for the three connectivity.

Allen connectivity produces a higher level of macroscopic wave activity than local and uniform connectivity.

(a) Theta-band (4-8 Hz) phase snapshots from three representative simulations using Allen (top), local (middle) and uniform (bottom) connectivity under matched Poisson stimulation, sampled at −10, −5, 0, +5 and +10ms relative to the peak of phase gradient directionality (PGD). Colors represent the theta-band generalized phase of each neuron, normalized between 0 and 1. (b) Theta-band PGD of the three simulations versus time, aligned to the PGD peak. (c) Per-band comparison of the maximum PGD of the three representative simulations shown in (a), one bar per band per connectivity. Please see Videos for animations and Quantitative measurement of neuronal activity (illustrated in figure Supplement 1) for the calculation of PGD.

Coupling strength, connectivity, and network synchrony jointly shape macroscopic-wave activity.

(a) Schematic of the three dynamic regimes that emerge across the (gAMPA, Vstim) parameter plane for Allen connectivity. At weak-to-intermediate coupling, the network supports coherent macroscopic traveling waves; at medium coupling, increased synaptic drive pushes the network into a disordered (asynchronous irregular) state where wave structure collapses; at strong coupling, the network reorganizes into a globally synchronous regime that again supports wave propagation. Insets show representative phase snapshots from each regime. (b) Per-band max PGD (mean ± SEM across the Poisson gAMPA × Vstim grid), plus the across-bands aggregate (‘All’), for Allen, local and uniform connectivity, with significance stars; full per-band (gAMPA, Vstim) heatmaps in figure Supplement 6. (c) Max PGD (alpha band) versus stimulus magnitude, error bars across gAMPA. (d) Max PGD (alpha band) versus excitatory conductance gAMPA, error bars across Vstim.(e) Kuramoto order parameter versus gAMPA in the alpha band. (f) Synchrony-PGD scatter, one point per (gAMPA, Vstim) combination, with Pearson r per connectivity (Allen r= 0.88, local r= 0.82, uniform r= 0.71 in alpha; all bands positive and significant at p < 10−3). Per-band versions of (c)-(e) are shown in figure Supplement 1, and per-band synchrony-PGD scatters in figure Supplement 2.

Macroscopic waves are maintained by the local field potential estimate.

(a) Paired snapshots of the local-average voltage (top, clipped to [−65, −50] mV) and the synaptic-LFP estimate (bottom, z-scored) across five consecutive 5-ms frames in a representative Allen Poisson simulation. The same propagating wavefront sweeps across the cortex in both modalities. (b) Voltage versus LFP max PGD scatter for Allen connectivity, one point per (gAMPA, Vstim, band); Pearson r = 0.78-0.89 per band, all p < 10−25. (c) Mean max PGD per band, voltage versus LFP, for Allen connectivity (paired p < 10−13 for every band; LFP carries ≈ 2-3× higher PGD than the intracellular voltage). (d) LFP-based mean max PGD per band, three connectivities; Allen wins in every band against both local and uniform. (e) LFP max PGD versus gAMPA for Allen connectivity, one line per band; the same three-regime profile (peak, dip, recovery) seen in voltage is preserved in LFP. See LFP estimation from intracellular voltage for the LFP estimation pipeline.

Connectivity matrix with detail region legend and degree distribution.

(a) The same connectivity matrix shown in Fig. 1c, but with detailed region legend. (b) The out and in-degree distribution of all neurons in log 2 scale.

Single-neuron-resolution view of the simulation in Fig. 2.

(a) Region-mean voltage traces (offset for clarity), one trace per major cortical region in anterior-to-posterior order. (b) Two rows of five voltage snapshots taken every 5 ms over the window high-lighted in (a), showing the wavefront propagation at single-neuron rather than KNN-averaged resolution.

Comparison of the three connectivity matrices: a Allen, b local and c uniform connectivity.

For local connectivity, the diagonal terms represent the connections within the same region. And the off-diagonal terms exist because these regions are physically adjacent to each other.

Illustration of the three-dimensional phase-gradient pipeline used to quantify macroscopic wave activity (Quantitative measurement of neuronal activity), shown for a single representative snapshot.

(a) The per-neuron intracellular voltage is linearly interpolated onto the regular CCFv3 Cartesian grid (color: membrane voltage in mV). (b) After band-pass filtering and the Hilbert transform, every grid point carries a generalized phase Φ(x, t) (color: phase from − π to π). (c) The spatial gradient ∇Φ of the phase field is computed at each grid point (white arrows); the phase gradient directionality (PGD) summarizes how well aligned these gradient vectors are and is our scalar measure of macroscopic traveling-wave activity.

delta-band (0.5-4 Hz) version of Figure 4 (which shows the theta band).

(a) 3 × 5 spatial snapshots of the delta-band generalised phase (rows: Allen / Local / Uniform; columns: five frames centered on the Allen-PGD peak); phase is computed on the voltage signal with 300-NN spatial smoothing in the complex (Hilbert) domain, with no Cartesian-grid interpolation. (b) delta-band PGD(t) over a 100 ms window centered on the Allen peak, all three connectivities. (c) Per-band max PGD bars for the same simulation (identical across the four band figures, included to match the main-text Figure 4 layout).

alpha-band (8-12 Hz) version of Figure 4; layout and conventions as in figure Supplement 2.

beta-band (12-30 Hz) version of Figure 4; layout and conventions as in figure Supplement 2.

gamma-band (30-100 Hz) version of Figure 4; layout and conventions as in figure Supplement 2.

Per-band PGD heatmaps and Allen advantage across the Poisson (gAMPA, Vstim) sweep.

(a) 3 × 5 grid of max-PGD heatmaps over the Poisson (gAMPA, Vstim) sweep; rows are the three connectivities (Allen / Local / Uniform), columns the five frequency bands (delta, theta, alpha, beta, gamma). Every heatmap shares the same axes (gAMPA on x, Vstim on y) and the turbo colour scale on the right (Max PGD). (b) Per-band Allen-minus-opponent mean PGD gap (mean ± SEM across the 10 × 12 grid), with grouped bars for Allen − Local (orange) and Allen − Uniform (blue) and two-sided significance stars (*: p < 0.05; **: p < 0.01; ***: p < 10−4); panel (b) occupies the bottom-right slot of the 3 × 5 grid.

Per-band versions of Figure 5c-e.

A 3 × 5 grid combining (a) max PGD versus Vstim (averaged over the gAMPA sweep; supplement to Figure 5c); (b) max PGD ver-sus gAMPA (averaged over the Vstim sweep; supplement to Figure 5d); and (c) maximum Kuramoto synchrony R(t) = eiΦ(x,t) x versus gAMPA (averaged over the Vstim sweep; supplement to Figure 5e). Columns are the five frequency bands (delta, theta, alpha, beta, gamma); error bars are SEM across the orthogonal sweep parameter. The peak-dip-recovery coupling profile of Figure 5d and the matching synchrony dip are visible in every band for Allen and uniform connectivity; local connectivity decays monotonically across all bands.

Synchrony-PGD scatter across all bands.

One point per (gAMPA, Vstim) combination on the 10 × 12 sweep grid, with x = max Kuramoto synchrony and y = max PGD, coloured by connectivity (Allen / Local / Uniform). Per-connectivity Pearson r values are shown in each panel’s legend. Synchrony and PGD are positively correlated in every band and every connectivity (all p < 10−3). This extends Figure 5f from the alpha band to all five canonical bands.