Fig 1.
Unique representation and toroidal encoding in a grid-cell module with varying phases.
a) Illustration of grid cells modelled as a superposition of three plane waves. b) Visualization of three distinct grid cells from the same module, each with their respective unit cells superimposed, demonstrating the periodic patterns. The right-hand image shows the phases of the three grid cells within a shared unit cell. c) Population vector correlation for 1, 2, and 3 cells’ activity (left to right) relative to the activity at the red cross. White lines and dots highlight ambiguous points, defined as locations where the Euclidean distance between the activity vector at each location and that at the red cross is less than . For populations of 1 and 2 cells, ambiguities are present within the unit cell, while with 3 cells these ambiguities are resolved. The unit cell border colours (yellow, green, and pink) correspond to the phases from panel b) included in the population. d) Grid modules with varying numbers of cells (n = 1 , 2 , 3 , and 15) and randomly distributed phases, shown within a unit cell. e) Persistence barcodes of the grid module from d), indicating the lifetime of zero-, one-, and two-dimensional holes in the population representation. With a larger cell number (here n = 15), we observe one persistent 0D, two 1D, and one 2D bar, suggesting a toroidal topology. f) Population representation of the grid cells from d). The first three plots display the population activity of the grid cells along their firing rate axes. The final plot presents the UMAP projection of the activity of 15 grid cells in three dimensions, resembling a torus. Colours represent the population vector correlation relative to the centre of the unit cell.
Fig 2.
Distortions in spatial representation are encoded in the metric tensor.
a) An illustration of a standard football field (top) and a distorted version (bottom), where vertical distances compress as one moves to the right. b) The metric tensor components show how distances in a) are locally deformed in each direction. For example, the component shows that as one goes to the right, distances in the vertical direction will shrink, whereas the
component shows that as one goes up or down, distances to the right will expand. c) Phase arrangements of 100 grid cells inside a unit cell (upper left in each set of four figures), along with a three-dimensional UMAP projection of the generated activity coloured by the metric tensor components (indicated by labels). Unit cell plots are coloured by the determinant of the metric tensor. The example on the first row shows a low-dimensional projection of a conformally isometric torus, and trajectories are undistorted. The example on the second row shows a projection of the torus, when the phases are sampled densely around a small region. In this case, trajectories crossing phase-dense regions appear more dense on the manifold.
Fig 3.
Optimal phases for seven cells (a-h) and 100 cells (i-m).
a) Learned phase positions for seven grid cells within the unit cell (blue) and their inferred hexagonal arrangement (orange). b) Three-dimensional UMAP projection of the seven-cell population activity, colour-coded by Voronoi diagram (shown in d). c) A persistence diagram illustrating the activity topology from a). d) Voronoi diagram showing spatial partitioning of phases across an extended grid, with the primitive (rhombus) cell superimposed. e) L2 norm of population activity across the unit cell. Colour range spans from 0 to , the maximum norm for seven cells. f) Conformal isometry grid search when varying the angle and magnitude of the hexagonal phase solution in a). Colours are log C + 1 where C is the conformal isometry metric defined in Eq 5. g) Common phase translation grid search, showing invariance to phase shifts (colour scale from f). h) Grid search focusing on varying just one (marked as a red dot) of the phases, using the same colour scale as f). i) Ripley’s H-function analysis comparing the dispersion of optimised (blue) and random (orange) phases. The analysis was conducted on 20 evenly spaced radial distances up to but not including the unit cell radius (r = 2 ∕ 3). Stars indicate significant differences (p < 0 . 01), as determined by a permutation test (details in methods). j) Grid score analysis of learned versus random phase distributions, inferred through Gaussian kernel density estimation, plotted against varying kernel bandwidths in the range of (not including) zero to half the radius length of the unit cell r ∕ 2 = 1 ∕ 3. j) Grid score comparisons of optimised versus random phase distributions across different kernel bandwidths for Gaussian kernel density estimation (KDE). Bandwidths range up to half the unit cell radius (r ∕ 2 = 1 ∕ 3), highlighting the hexagonality of the optimised phase arrangement. k) Kernel density estimate (KDE) of the 15 replications of the phase solution from panel a), each with an additional random phase shift. Bandwidth for KDE is set at 0.1, with the resulting grid score shown in the title. The KDE is normalized within the unit cell to integrate to unity. l) (top) KDE visualisation of optimised phase distributions overlaid with phase locations, with titles indicating grid scores. Four bandwidths were selected: r ∕ 20, 5r ∕ 20, 10r ∕ 20, and 15r ∕ 20. Standard deviations for the KDEs are
, reflecting the KDE colour variability across bandwidths. All KDEs are normalized to have a unit integral. (bottom) Autocorrelograms of the KDEs to confirm hexagonal spatial periodicity. m) KDEs for random phase distributions at the same bandwidths as panel l), providing a baseline comparison. KDE standard deviations are
.
Fig 4.
Signs of conformal mapping and phase hexagonality in a module of 105 experimental grid cells.
a) Euclidean distance between the grid cell population vector at the red cross and population vectors at all other spatial locations. b) Metric tensor components for the experimental data, with values represented on a shared colour scale shown in the colour bar. c) Neural-physical distance plot comparing experimental data, phase-clustered data, and spatially shuffled neural distances, indicating how neural distances correspond to physical distances. d) 1D histogram of the metric tensor components from (b), with distributions from spatially shuffled ratemaps included as a comparative baseline. e) Same as (c), but restricted to short physical distances, defined as less than 25% of the maximum physical distance in (c). Linear regression analyses are conducted for subsets corresponding to 5%, 10%, 15%, 20%, and 25% of the total physical distance range. The legend includes the Pearson correlation coefficients (r-values) from the regressions, illustrating the strength of the linear relationship between neural and physical distances for shorter and longer distance ranges. f) Ripley’s H-function on the experimental phases, with random uniform sampling as a baseline. Shaded areas represent twice the standard deviation from 100 resampling trials, and significant deviations (two standard deviations from the baseline) are marked with red stars. A CI+noise module (105 phases total, derived from 15 noise-induced copies of a 7-cell CI solution) provides an additional comparison, with its radius set at 1.4 times that of the data radius. g) Unit cells and kernel density estimates (KDEs) from the grid modules in (e) with phases superimposed. The colour scale is shared across each KDE.
Fig 5.
Impact of the Number of Cells on Spatial Encoding.
a) One dimensional tuning curves for populations of 2, 3, and 10 neurons modelled by Eq 1. b) Population activity vectors corresponding to the waves from (a) form an isotropic ring with radius , embedded in a plane. Increasing the number of cells expands this ring, intersecting with more unique voxels (distinct population states) and enhancing spatial resolution by enabling finer distinctions between encoded positions. c) The conformal scale (
) from Eq 2 plotted against the firing rate (A) and number of cells (N), with light blue contour lines representing the square grid’s scale (
). d) Population vector norm, or energy, shown relative to the firing rate A and cell count N, illustrating how energy scales with module size. e) A schematic of the unit cell with overlaid meshes of different granularities to depict spatial resolution. The left side, with a coarser mesh, represents a module with fewer cells, while the right side, with a finer mesh, represents a module with more cells, highlighting the resolution enhancement with increased cell count. f) The average geodesic distance between initial and optimised phase positions, plotted against module size (N = 7 , 14 , 21 , … , 133). g) Illustration of how increasing cell count leads to larger toroidal structures and extended neural trajectories, highlighting spatial and topological expansion in larger modules.
Fig 6.
Model comparison: Three models, linear decoding (LIN, orange), conformal isometry (CI, green) and homology (HOM, dark orange), along with random phases (blue), evaluated on each other’s metrics. In addition, the purple inset shows the decoding error of a non-linear argmax decoder applied to populations with random phases.Zoomed in plots show the performance between 3 and 10 cells in a linear scale, with 6 (black) and 7 (red) cells highlighted by dashed lines.