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). Red dots denote exemplar intermediate locations along the navigational trajectory (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 illustrating the 6-fold modulation for localizing grid cell activity. Participants’ movement directions (Top) of each trial were extracted from the original trajectories (Left) by connecting the starting and the ending locations. These trajectories were further categorized into either “aligned” or “misaligned” conditions relative to grid cell axes (Right & Bottom). The original trajectories of each participant were referred to Fig. S3. b-c) Voxel- and ROI-based sinusoidal analysis revealed significant 6-fold periodicity within the right EC (Voxel- based analysis: initial threshold: p = 0.05, two-tailed; cluster-based SVC correction for multiple comparisons: p < 0.05; Cohen’s d = 0.63; Peak MNI coordinate: 32, −6, −30; ROI-based analysis: t(32) = 3.56, p = 0.006, two tailed, corrected for multiple comparisons; Cohen’s d = 0.62). The black line indicates the outline of the EC. d) Schema illustrating the 6-fold periodicity reconstructed by the GLM in the 1D directional space (Left) and 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 within the bilateral MTL 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 outlines indicate the HPC and EC. d-e) 3-fold periodicity in the bilateral HPC revealed by sinusoidal analysis. (initial threshold: p = 0.05, two-tailed; Cluster-based SVC correction within the bilateral MTL for multiple comparisons: p < 0.05; Cohen’s d = 0.68; Peak MNI coordinate: −24, −18, −12; ROI analysis: t(32) = 3.94, p= 0.002, corrected for multiple comparisons; Cohen’s d = 0.70). f) Schematic illustration of the 3-fold periodicity reconstructed by the GLM in the 1D directional space (Left) and 2D Greeble space (Right).

Phase synchronization of BOLD signals between the EC and HPC in the directional domain.

a) Cross-participant phase coupling analysis revealed a significant correlation between the phases of the EC and HPC estimated from odd-numbered sessions using sinusoidal analysis (r = 0.41, p = 0.019). b) Amplitude-phase modulation analysis showed significantly stronger coupling strength between BOLD signals of the EC and HPC compared to surrogate data (t(32) = 7.24, p < 0.001; Cohen’s d = 1.14). c) Schematic illustration of a hypothetical peak-overlapping pattern (blue ellipse) between the HPC and EC activity in corresponding to spatial phase (one-cycle). d) The phases of HPC activity were classified into four bins. Significantly stronger coupling strength was found in the bins near phase 0 of the HPC compared to bins farther from phase 0 (F(1, 130) = 218.99, p < 0.001). Error bars indicate SEM. ***: p < 0.001.

The 3-fold periodicity in behavioral performance.

a) Visuomotor brain circuit potentially inheriting HPC’s 3-fold periodicity. b) Participants’ behavioral performance, measured by a composite index of trajectory length and deviation from the goal to ending locations, fluctuated as a function of movement 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 were activated by visiting discrete locations (black circles in 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 trajectory code V represented in the HPC. A trajectory code V is symmetrical for movement direction φ and φ +180°, representing a unique spatial orientation Ψ ranged from 0° to 180°. The δ value (y-axis) indicates the degree to which the spatial orientation Ψ of a trajectory code V aligns with the grid axes, with larger δ values (e.g., the spatial orientation ) 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 (blue dots) (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 movement directions across trajectory (i.e., trial).

The original movement 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 extracted from three consecutive movement steps within trajectories. On average, human participants moved in 3.8 directions, suggesting a “Radial Adjustment” strategy and excluding the possibility of the “horizontal-vertical movement” strategy, which results in 2 movement directions (black dashed line) for each trajectory (t(32) = 15.76, p < 0.001; two-tailed; Cohen’s d = 2.78). Error bars indicate SEM.

Learning effect of the object matching task.

The two-day design effectively eliminated the learning effect in 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). Dots indicate individual participants. Error bars indicate SEM.

Movement trajectories of human participants during morphing Greebles in MRI experiment.

The orange and red dots represent the starting and ending locations, respectively. The black lines indicate the movement trajectories. Participants’ behavioral performance, reflected by the summation of trajectory length and error size, are shown in the title, with superior performance indicated by smaller score.

Distribution of grid orientations in the EC.

Grid orientations ranging from −30° to 30° were calculated for each voxel. The uniformity of grid orientations was assessed using the Rayleigh test. Thirty out of thirty-three participants exhibited significantly clustered grid orientations, indicated by * (* p < 0.05, ** p < 0.01, *** p < 0.001, Bonferroni corrected). The length of the bars represents the number of voxels. The black line denotes the mean of orientations.

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 within the MTL 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.

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 within the MTL 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.

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

The significant clusters (Red) within the Default Mode Network were identified as representing 3-fold periodicity, while 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). Grey lines represent the cortical parcellation based on the DKT Atlas.