Supplementary Figure 4: Signatures of hexagonal geometry in grid-cell coupling.

a: A simulated triangular grid pattern. Each grey dot represents a vertex (receptive field) of the grid. The hexagonal tiles surrounding the vertices are Voronoi cells which form the phase space of the grid pattern. The black arrow marked “s” denotes the distance between adjacent vertices (the grid spacing). Within the central tile are drawn a series of concentric rings, indicating zones corresponding to different values of ϕ_G, the magnitude of a grid phase offset vector. Note that the phase tile border encroaches on the outermost ϕ_G zone, meaning that the largest values of ϕ_G have a nonuniform radial distribution. b: Illustration of the four spatial kernel types used to simulate grid firing fields. c: Correlation of simulated pairs of grid patterns as a function of the phase offset between the two patterns. Each position on the hexagonal phase tile represents a particular phase offset of one grid from the other. The colour at each point indicates the Pearson correlation r-value for two grid patterns with that phase offset. Each plot shows the result of simulations using a different spatial kernel type. In each case, the normalized field width parameter w was 0.19, equal to the empirically estimated value. Contour lines follow equal correlation values. Note the circle formed by the phases where correlation is zero (dashed black line), and the non-concentric deviations in the correlation pattern near the phase tile’s edge. Inset: examples of simulated rate maps. d: Relationship of grid pattern Pearson correlation r-values with ϕ_G and the grid field width. For all kernel types, the field width is defined as the radius enclosing 50% of the field’s mass, as a proportion of the grid spacing. The dashed line traces the path of the value of ϕ_G at which RMS crosses zero (ϕ_G0) Note the convergence of ϕ_G0 on a value of approximately 0.32 as field width increases. The black circle on the first plot indicates ϕ_G0 at the empirically estimated average field width for the sample of grid cells (see i). e: Negative correlations predominate on the grid phase tile. Plotted is the proportion of the area of the grid phase tile (as shown in (c)) containing positive Pearson r-values, for grid patterns of different field widths and spatial kernels. Note that for all widths and kernel types, the fraction of positive r-values is below 50%. f: Empirical relationship between ϕ_G and spatial rate-map Pearson correlation r-values among grid–grid pairs (black dots). The grey line indicates the values obtained from simulated rate maps for grid–grid pairs with uniform-randomly distributed phases. g: The empirical transition between positive and negative couplings occurs at a similar phase offset during all states. Plotted are Matthews correlation coefficient (MCC) values which indicate how accurately the distribution of grid–grid pair GLM β₀-values is separated into positive and negative values by splitting at different values of ϕ_G (see Methods). The vertical dotted grey line indicates the ϕ_G0-value of 0.32 determined from simulations, as shown in (d). The shaded regions show 95% bootstrap confidence intervals. h: ϕ_G -values that maximise the MCC shown in (g), with 95% bootstrap confidence intervals. The ϕ_G0 values for each state (RUN 0.32, SWS 0.33, REM 0.33) are similar to the value of 0.32 obtained from simulations (see d) (P > 0.05 in all cases, bootstrap, n = 135 grid–grid pairs). i: Estimated spacing and field width (w) for all grid cells. Field width was estimated with an algorithm which fitted a mixture of 2D Gaussian functions to a grid cell’s 2D firing rate map. The field width was estimated as the value of the Gaussian sigma parameter which achieved the best fit to the cell’s rate map, as a proportion of the cell’s grid spacing.