Experimental design.

a) Depiction of the Greeble prototype (Gauthier and Tarr, 1997) and its two defining features, namely “Loogit” and “Vacso.” b) Inside the MRI scanner, participants adjusted the length of Loogit and Vacso to match the prototype by stepwise button presses, within a 10-s time limit. c) Conceptual object space. Each orange dot within the ring-shaped area represents a Greeble variant, while the central blue dot indicates the Greeble prototype (i.e., the goal location). The red dots denote exemplar intermediate locations along the navigational path (i.e., the black line). d) Density distribution of participants’ ending locations indicated an overall superior behavioral performance for detecting the periodic activity of the HPC.

6-fold periodicity in the EC.

a) Schematic defining path directions in the 6-fold modulation. Participants’ path directions (Top) were extracted from the original paths (Left) by connecting the starting and ending locations, with 0° arbitrarily set to the East as the reference, and were then classified as “aligned” or “misaligned” (Right & Bottom). The original paths of each participant were referred to Fig. S4. b) Voxel-based sinusoidal modulation revealed significant 6-fold periodicity within the right EC (Voxel-based analysis: initial threshold: p = 0.05, two-tailed; cluster-based small volume correction (SVC) for multiple comparisons: p < 0.05; Cohen’s d = 0.63; Peak MNI coordinate: 32, -6, -30). Volumetric results are displayed in radiological orientation; numbers below the brain slices indicate MNI coordinates. c) ROI-based analysis, using a functional mask derived from the significant right EC cluster and constrained within the anatomical EC, confirmed a significant 6-fold periodicity (t(32) = 3.56, p = 0.006, two-tailed, corrected for multiple comparisons across rotational symmetries; Cohen’s d = 0.62). The black line indicates the boundary of the EC. d) Schematic illustrating the 6-fold directional tuning curve reconstructed from sinusoidal modulation (Left) and its representation in the 2D Greeble space (Right).

3-fold periodicity in the HPC.

a) Schematic illustration of the spectral analysis procedure. b-c) Voxel-based spectral analysis revealed significant 3-fold periodicity in the bilateral HPC and 6-fold periodicity in the bilateral EC (initial threshold: p = 0.05, two-tailed; Cluster-based SVC correction for multiple comparisons: p < 0.05; For the HPC: Cohen’s d = 1.06; Peak MNI coordinate: -24, -20, -18; For the EC: Cohen’s d = 1.27; Peak MNI coordinate: -22, -14, -30). The black lines indicate the boundaries of the HPC and EC. d-e) 3-fold periodicity in the bilateral HPC identified using sinusoidal modulation (initial threshold: p = 0.05, two-tailed; cluster-based SVC correction: p < 0.05; Cohen’s d = 0.68; peak MNI coordinate: −24, −18, −12). ROI-based analysis using a functional mask of this cluster within the anatomical HPC confirmed the effect (t(32) = 3.94, p = 0.002; Cohen’s d = 0.70). Volumetric results are displayed in radiological orientation; numbers below the brain slices indicate MNI coordinates. f) Schematic illustrating the 3-fold directional tuning curve reconstructed from sinusoidal modulation (Left) and its representation in the 2D Greeble space (Right).

Phase synchronization between the HPC and EC activity in the directional domain.

a) Cross-participant circular–circular correlation analysis revealed a significant coupling between the EC and HPC phases (r = 0.42, p < 0.001). The green (outer ring) and purple (inner ring) dots denote HPC and EC phases, respectively. Each line connecting the EC and HPC phases represents one participant, with its color indicating the HPC phase (0–120°). b) Schematic illustration of a hypothetical peak-overlapping pattern (blue ellipse) between the HPC and EC activity in corresponding to spatial phase (one-cycle). c) Amplitude-phase modulation analysis revealed significant coupling between the 3-fold HPC activity and 6-fold EC activity in the bin centered at phase 0 (t(32) = 2.57, p = 0.02, Bonferroni-corrected across tests; Cohen’s d = 0.45). The coupling strength was computed as the difference between the observed modulation index (M) and the mean surrogate modulation index (M′). No significant coupling was observed in the control analyses testing phase synchronization between the 3-fold HPC and the 9-fold or 12-fold EC periodicities (p > 0.05). The error bars indicate SEM. ***: p < 0.001.

3-fold periodicity in behavioral performance.

a) Schematic illustration of visuospatial task performance potentially inheriting HPC’s 3-fold periodicity. b) Participants’ behavioral performance, measured by a composite index of path length and deviation from the goal to ending locations, fluctuated as a function of path directions. The shaded area denotes SE. c) Spectral analysis revealed significant power at the 3-fold of participants’ behavioral performance (p < 0.05, corrected for multiple comparisons). The red dashed line represents the baseline derived from permutation. d) Significantly higher phase-locking values were observed between participants’ behavioral performance and HPC activity compared to surrogate dataset (t(32) = 8.10, p < 0.001; Cohen’s d = 1.14).

The EC-HPC PhaseSync Model.

a) Schematic illustrating the population activity of grid cells during mental planning. The simulated grid cell population was activated by visiting discrete locations (the black circles in the right panel) in the Greeble space along directions either aligned (purple) or misaligned (grey) relative to the grid axes. b) The 3-fold periodicity of the path code V represented in the HPC. A path code V is symmetrical for path direction ϕ and +180°, representing a unique spatial orientation ψ ranging from 0° to 180°. The δ value (y-axis) indicates the degree to which the spatial orientation ψ of a path code V aligns with the grid axes, with larger δ values (e.g., the spatial orientation (1, 3, and 5)) indicating a perfect match between ϕ and grid axes. c) Simulated vectorial representation of the HPC for centered-goal-based navigation. The 3-fold periodicity is driven by vectorial gradients inherited from δ. d) Random-goal-based navigational simulation. Significant spectral power of model performance was observed at 3-fold across 100 randomly selected goal locations (the blue dots and gray lines) (P<0.05, corrected for multiple comparisons). The red dashed line indicates the significance threshold derived from permutation. The blue shaded areas denote the standard error. Grey lines represent the spectral powers of goal-dependent simulations.

The averaged number of path directions across all paths (i.e., trial).

The original path directions towards East (e.g., key 1 on the response box), West (e.g., key 2), North (e.g., key 3), and South (e.g., key 4) were identified based on three consecutive movement steps within each path. On average, participants moved in 3.8 directions, suggesting a “Radial Adjustment” strategy and excluding the possibility of the “horizontal-vertical movement” strategy, which would result in 2 path directions (black dashed line) per path (t(32) = 15.76, p < 0.001; two-tailed; Cohen’s d = 2.78). The error bars indicate SEM.

Histograms of path directions for each participant.

Path directions were binned into 36 bins of 10° each. The uniformity of the distributions was assessed using the Rayleigh test. For all participants, no significant deviations from uniformity were observed (all p > 0.05, Bonferroni-corrected across participants). Bar height indicates the number of samples per bin.

Learning effect of the object matching task.

The two-day design effectively eliminated the learning effect from the MRI experiment. On day 1, participants’ behavioral performance increased as a function of experimental sessions, resulting a significantly negative slope (grey bar, t(32) = -2.46, p = 0.019, two-tailed; Cohen’s d = 0.44). In contrast, no significant learning effect was found on day 2 (red bar, t(32) = -0.74, p = 0.462, two-tailed; Cohen’s d = 0.13). The dots indicate individual participants. The error bars indicate SEM.

Movement paths of human participants during Greeble morphing in the MRI experiment.

Orange and red dots represent the starting and ending locations, respectively. The black lines indicate the movement paths. Participants’ behavioral performance, calculated as the sum of path length and error size, is shown in the title, with smaller score indicating better performance.

Distribution of grid orientations in the EC.

Grid orientations ranging from -30° to 30° were estimated for each voxel. The uniformity of grid orientations was assessed using both the Rayleigh test and PPC. Significantly clustering was observed in 30 out of 33 participants, indicated by * (* p < 0.05, ** p < 0.01, *** p < 0.001, Bonferroni-corrected across participants). Bar length indicates the number of voxels. Black line denotes the mean orientation.

3-fold periodicity of the HPC examined using 20° bin.

Significant clusters were revealed in the right HPC (initial threshold: p = 0.05, two-tailed; clustered based SVC correction for multiple comparisons: p < 0.05; Cohen’s d= 1.18; Peak MNI coordinate: 22, -24, -14). The black lines indicate the outline of the HPC and EC. Volumetric results are displayed in radiological orientation; numbers above the brain slices indicate MNI coordinates.

3- and 6-fold periodicity of participant groups revealed by spectral analysis.

All individual groups showed significant clusters of 3- and 6-fold periodicity in the HPC and EC, respectively (initial threshold: p = 0.05, two-tailed. Cluster-based SVC correction for multiple comparisons: p < 0.05). The peak MNI coordinates are shown on the bottom (3- fold: Group 1-3, t(9) = 8.0, t(9) = 7.6, t(12) = 10.8, Cohen’s d= 0.67, 0.87, 1.21; 6-fold: Group 1-3, t(9) = 13.8, t(9) = 8.7, t(12) = 6.7; Cohen’s d= 0.83, 0.63, 0.96). The black lines indicate the outline of the HPC and EC. Volumetric results are displayed in radiological orientation; numbers below the brain slices indicate MNI coordinates.

Whole brain representations of 6- and 3-fold periodicity revealed by spectral analysis.

Significant clusters (Red) within the DMN were identified as representing 3-fold periodicity, whereas 6-fold periodicity revealed clusters within the Salience Network, including the ACC and INS (initial threshold: p = 0.05, two-tailed. Whole brain cluster-based correction for multiple comparisons: p < 0.05). The grey lines represent the cortical parcellation based on the DKT Atlas.

Spectral analysis of human behavior.

a & b) Behavioral periodicity of path length and error size (deviation from the goal to ending locations). The shaded areas denote SE. c & d) Spectral analysis revealed significant 3-fold power for path length (p < 0.05, corrected for multiple comparisons), whereas no significant periodicity was observed for error size (p > 0.05). The red dashed line indicates the permutation-based baseline.