Irregular distribution of grid cell firing fields in rats exploring a 3D volumetric space

Abstract

We investigated how entorhinal grid cells encode volumetric space. On a horizontal surface, grid cells usually produce multiple, spatially focal, approximately circular firing fields that are evenly sized and spaced to form a regular, close-packed, hexagonal array. This spatial regularity has been suggested to underlie navigational computations. In three dimensions, theoretically the equivalent firing pattern would be a regular, hexagonal close packing of evenly sized spherical fields. In the present study, we report that, in rats foraging in a cubic lattice, grid cells maintained normal temporal firing characteristics and produced spatially stable firing fields. However, although most grid fields were ellipsoid, they were sparser, larger, more variably sized and irregularly arranged, even when only fields abutting the lower surface (equivalent to the floor) were considered. Thus, grid self-organization is shaped by the environment's structure and/or movement affordances, and grids may not need to be regular to support spatial computations.

Fig. 1. Grid cells produced firing fields in a 3D climbing lattice.

a, Hypothetical grid field packings: standard horizontal hexagonal field configuration (i); exploded close-packed lattice, in this case HCP (layers color coded for clarity) (ii); units of the two optimal packings: HCP (left) alternates two layer-arrangements whereas FCC (right) has three (iii); columnar field configuration (iv); and random field configuration (v). b, Lattice maze schematic. c, Lattice maze photographs. See Extended Data Fig. 1a for arena photographs. d, Example coverage in a lattice session. Color denotes normalized (Norm.) dwell time in each region. e, Recording protocol. f, Example histology. Data for all animals can be seen in Supplementary Fig. 2. g, Three representative grid cells in the arenas (left) and lattice (right). Left–right: arena spike plots (gray shows coverage; red dots show spikes), arena rate maps, arena autocorrelations, volumetric spike plots, volumetric firing rate maps, rate maps as projected on to each of the three coordinate planes and projected autocorrelations. Color bars from top to bottom correspond to volumetric rate maps, autocorrelations and planar rate maps. All grid cells can be seen in Supplementary Fig. 1. Source data

Fig. 2. Grid cells mapped the lattice with large and widely spaced but stable fields.

a, Grid cell firing was spatially correlated between session halves (red lines denote medians; black lines denote 1st and 3rd quantiles; see also Extended Data Fig. 4). Colored data points represent grid cells, gray represents shuffled values (n = 5,000). b, Z-scored spatial information was higher than chance in all environments but reduced in the lattice. Text gives the proportion of cells exceeding the shuffle 95th percentile (z = 1.96; gray line). c, Arena and lattice spatial information was significantly positively correlated (n = 46 cells). LLS, linear least squares line fit. d, Position and size of every grid field (left) and proportion of fields in every lattice layer (right; see also Supplementary Fig. 7b) (n = 133 fields). e, Grid field radius was significantly larger in the lattice than in the first arena. f, The size of grid fields in the arena and lattice was significantly positively correlated (n = 46 cells). g, The number of fields per grid cell in the arena and lattice was positively correlated. h, Grid cells exhibited significantly fewer fields per m³ in the lattice maze. i, Grid spacing was significantly larger in the lattice (n = 47, 43 and 47 cells for Arena 1, Lattice and Arena 2, respectively). j, Grid spacing (maximum 120 cm) in the arena and lattice was uncorrelated, and arena grid modules (bottom histogram) were disrupted in the lattice (n = 43 cells). Cells for which no lattice spacing could be estimated are not shown in i or j. a,b,f,g, n = 47, 46 and 47 cells. b,f,g,i, Markers represent cells, black open circles denote mean and error bars denote s.e.m. For multiple comparisons: ***P < 0.001, **P < 0.01, **P < 0.05, all two-sided tests with Dunn–Sidak correction. See Supplementary Fig. 4 for schematic and validation of procedures in g–j. Source data

Fig. 3. Grid fields were randomly distributed in the lattice.

a, Structure scores (χFCC, χHCP and χCOL) for grid cells (markers) and simulations (simulated fields shown as shaded polygons; HCP, FCC, COL and RND are HPC, FCC, columnar and uniformly random field simulations respectively). b, The 2D side projections of a showing that grid cells overlapped the most with random configuration scores. c, Grid cells categorized based on which field region they fell into (top; 30% of cells are uncategorized) or which configuration score was maximal (bottom). d, Interfield distance CV for grid field distances in the arena and lattice compared with their respective shuffled data. Lattice data did not differ from spike-train shuffles (n = 45, 45 and 46 cells). Markers represent cells, open circles denote mean, and error bars denote s.e.m. e, Schematic of simulated HCP (top) and FCC (bottom) arrangements. Distance between repetitions of the same layer differ between layer repetitions in HCP (top) and FCC (bottom). f, The analysis shown in e finds the expected peak correlation patterns in simulated configurations (left; colors same as a) but not real grid cells (right; mean and s.e.m.). g, Grid cells exhibited low grid scores in all Cartesian coordinate planes of the lattice. Open circles denote average across cells and error bars denote s.e.m. a,b,f,g, n = 46 cells. For multiple comparisons: ***P = < 0.001, **P < 0.01, **P < 0.05; all two-sided tests with Dunn–Sidak correction. Source data

Fig. 4. Grid fields in the lattice were vertically elongated and some formed hexagonal columns.

a, Filled markers represent fields. Top: field elongation in the arena was calculated as the ratio of the largest (‘a’) and second largest (‘b’) axes. In the lattice, ‘b’ was instead calculated as the average of the two smallest axes. Bottom left: fields were significantly more elongated in the lattice. Right: z-scored elongation relative to 100 shuffles; text gives the proportion of fields exceeding the shuffle 95th percentile (that is, nonspherical; z = 1.96, gray line; n = 124, 133 and 120 fields for Arena 1, Lattice and Arena 2 trials respectively). b, Top: spherical heatmaps of the direction of all grid field principal axes (see Supplementary Fig. 9 for a schematic). Bottom: mean and 95% confidence interval of fields with a principal axis parallel to the X, Y or Z axis relative to chance (red area denotes 99th percentiles). Asterisks denote significant deviation; the numbers give effect size (Cohen’s d). c, Markers represent cells. Left: grid scores for all grid cells in the arena XY plane (n = 47) and each projected plane of the lattice maze (n = 46). Right: proportion of grid cells with a grid score exceeding the 95th percentile of a chance distribution in each lattice plane. d, Examples of significant XY grid cells recorded from two rats. Top: volumetric firing rate map. Bottom: autocorrelation of the XY projected firing rate map. See Extended Data Fig. 6 for square grid scores and Extended Data Fig. 7 for further analyses on significant XY grid cells. a,c, open circles denote mean and error bars denote s.e.m. For multiple comparisons: ***P < 0.001, **P < 0.01, **P < 0.05; all two-sided tests with Dunn–Sidak correction. Source data

Extended Data Fig. 1. Animals explored the whole lattice maze with a bias for horizontal movements.

(a) Side and top-down views of the arena. (b) Different views of the 3D positions for a representative arena session. (c) Different views of the 3D positions for a representative lattice maze session. (d) Top) averaged dwell time histogram for all arena sessions (n=42); Bottom) mean ± SEM proportion of time spent at different positions within the arena averaged across sessions. Rats explored the entire arena homogenously. (e) Top) averaged dwell time histogram for all lattice maze sessions (n=42); bottom) Mean ± SEM proportion of time spent in each layer of the maze, averaged across sessions. Rats explored the entire lattice homogenously in X and Y but with a strong vertical bias for the bottom layer. (f) Left) spherical heatmaps showing the time spent moving at every possible yaw × pitch angle in the arena and lattice (n=42 sessions). Inset schematics give the maze shape and the corresponding axes shown extending through the spheres. Right) Filled markers represent sessions (n=42), open circles denote mean, error bars denote SEM. As reported previously, rats were biased towards horizontal movements in both mazes and were far more likely to move parallel to the maze axes: the walls of the arena or the sides and bars of the lattice (F(2,123) = 417.2, p = 1.56 × 10⁻⁵⁵, η² = 0.872; one-way ANOVA).

Extended Data Fig. 2. Grid cells were more stable than chance throughout recordings.

For panels A, D, E & F: n=47 cells. For panels a & d: filled markers represent cells, open circles denote mean, error bars denote SEM. (a) Grid cell firing rates did not differ between the mazes (F(2,138) = 2.0, p = .135, η² = 0.029, one-way ANOVA). (b) Example grid cell ratemaps and autocorrelations for the two arena sessions. In addition to ratemap autocorrelations we also calculated the cross-correlation between arenas 1 and 2. Only cells with a grid score exceeding the 95^th percentile of a shuffle in both arena sessions were analysed. (c) Total number of cells with a grid score exceeding the 95^th percentile of a shuffle in each arena session. The purple bar sections represent the cells categorised as grid cells and included in the main analyses, grey bars represent unstable cells that met our grid cell criteria in only one arena session. Main results were also replicated using only these unstable grid cells (Extended Data Fig. 5). (d) Grid scores of all grid cells calculated for both arenas and the cross-correlation between arena maps. These scores did not differ (F(2,138) = 1.1, p = .337, η² = 0.016, one-way ANOVA). (e) Left) example firing of a grid cell in both arenas. Middle) to determine if grid cells were stable between arenas we correlated their arena firing rate maps (blue area; blue triangle denotes median) and compared this distribution to correlations between 5000 shuffled ratemaps (grey area; grey triangle denotes 95^th percentile). Right) cumulative density curves of the same distributions. Shaded areas denote 95% confidence intervals. The blue vertical line marks the 95^th percentile of the shuffle on the x-axis and the blue horizontal line (very close to the x-axis) marks the y-intercept of the observed correlation distribution with this. Grid cells were more stable than chance (shuffled arena correlations 95^th percentile: 0.23, grid cell median correlation: 0.59, D = 0.96, p < .001, two-sample Kolmogorov-Smirnov test). (f) The Euclidean distance between waveforms in different session pairs for all grid cells (Methods: Waveform stability). Red horizontal lines denote medians; black horizontal lines denote 1^st and 3^rd quantiles, error bars denote data range. Distances did not change across recordings (F(2,124) = 2.6, p = .0787; one-way ANOVA) suggesting grid cells were stably recorded throughout the experiment. Although distances were smallest when comparing the lattice to each arena (Group average Lattice vs Arena 1: 25.4, Lattice vs Arena 2: 25.5, Arena 1 vs Arena 2: 38.1) which is consistent with a gradual decrease in stability over time. In each case the distances between recording pairs were also significantly lower than chance which was estimated using pyramidal cell pairs co-recorded on the same tetrodes (p < .0001 in all cases, two-sample t-tests; black distributions).

Extended Data Fig. 3. Grid cells were more stable than chance within sessions, especially in the XY plane.

See Methods: Spatial stability within sessions. For panels a & b: n=47 cells. (a) Left) representative grid cell volumetric firing rate maps used for the volumetric correlation analysis; each represents one half of the same lattice session. Note that these maps have much larger voxels than ones used or shown elsewhere. Middle) raincloud plots showing the distribution of correlation values found for the arena and lattice sessions. Grey distributions represent correlations between random grid cells recorded in each maze. Red lines denote medians, black lines denote 1^st and 3^rd quantiles. Right) cumulative distribution functions of the same distributions. Shaded areas denote 95% confidence intervals. In all sessions grid cells were more stable than chance (p < .0001 in all cases; two-sample t-tests) and the mazes did not differ (F(2,137) = 1.6, p = .203, η² = 0.023; one-way ANOVA). (b) Left) same cell as a but shown using projected maps used for planar correlations. Middle) raincloud plots as in a, showing the correlations found for each projected plane of the lattice for all grid cells. Right) cumulative distribution functions of these data. Shaded areas denote 95% confidence intervals. All projections were more stable than chance (p < .001 in all cases; two-sample t-tests) but horizontal (XY) projections yielded significantly higher correlations than vertical ones (F(2,135) = 5.8, p = .00377, η² = 0.079; one-way ANOVA) which is consistent with previous findings in place cells.

Extended Data Fig. 4. Grid cells did not remap according to movement direction.

See Methods: Stability between movement epochs. For panels c & d: n=47 cells. (a) An example lattice trajectory filtered to show only vertical movements. (b) Same as a but for horizontal movements. (c) Left) example grid cell volumetric firing rate maps used for the movement direction analysis; each map represents the same grid cell activity filtered for vertical (left) or horizontal (right) movement epochs. Note that these maps have much larger voxels than ones used or shown elsewhere. Middle) raincloud plots showing the distribution of correlation values found for grid cells in the lattice (blue) and between random grid cells (grey). Red lines denote medians, black lines denote 1^st and 3^rd quantiles. Right) cumulative distribution functions of the same distributions. Shaded areas denote 95% confidence intervals. Grid cells were more stable than chance (t(4834) = 12.8, p = 6.49 × 10⁻³⁷; two-sample t-test). (d) Left) example grid cell projected maps used for planar correlations. Middle) raincloud plots as in a, showing the correlations found for each projected plane of the lattice for all grid cells. Right) cumulative distribution functions of these data. Shaded areas denote 95% confidence intervals. All projections were more stable than chance (p < .0001 in all cases; two-sample t-tests) and they did not differ (F(2,135) = 2.1, p = .1249, η² = 0.0303; one-way ANOVA).

Extended Data Fig. 5. Main results were similar in unstable grid cells.

These plots supplement main figures but focus on unstable grid cells (that is cells that met our grid cell criteria in only one arena session; grey areas in Extended Data Fig. 2c). For panels a-f, h (right) & i (right): open circles denote mean, error bars denote SEM and text gives the results of one-way ANOVAs. For multiple comparisons: *** = p < .001, ** = p < .01, ** = p < .05, all two-sided tests with Dunn-Sidak correction. (a) Supplement to Fig. 2g; Grid field radius was similar in the lattice and arena sessions. n=40, 35, 28 & 27 cells. (b) Supplement to Fig. 2i; grid spacing was significantly larger in the lattice. n=40, 25, 28 & 20 cells. (c) Supplement to Fig. 2b; Z-scored spatial information was higher than chance in all environments but reduced in the lattice. n=40, 36, 28 & 28 cells. (d) Z-scored sparsity was also lower than chance in all environments but was higher in the lattice. n=40, 36, 28 & 28 cells. (e) Supplement to Fig. 2f; grid cells exhibited significantly fewer fields per m³ in the lattice maze. n=76, 82, 68 & 74 cells. (f) Supplement to Fig. 4a; fields were significantly more elongated in the lattice. n=157, 233, 166 & 188 cells. (g) Supplement to Fig. 3a; structure scores (χFCC, χHCP and χCOL) for grid cells (n=47, black markers), unstable grid cells (n=68, red markers) and simulations (convex hulls shown as shaded polygons). (h) Left) Supplement to Fig. 3c; All grid cells (stable & unstable) categorized based on which convex hull they fell into. Right) configuration specific scores for stable (n=47, black markers) and unstable (n=68, red markers) grid cells. (i) Left) Supplement to Fig. 3c; unstable grid cells categorized based on which convex hull they fell into. Right) configuration specific scores for unstable grid cells (n=68) only.

Extended Data Fig. 6. Weak evidence of square pattern activity in grid cells and non-grid cells.

For panels a & c: Filled markers represent cells, open circles denote mean, error bars denote SEM. (a) Left) square grid scores for all grid cells (n=46) in each projected plane of the lattice maze. Square grid scores are low in each of the lattice planes and these did not differ (F(2,135) = 2.5, p = .0872, η² = 0.036, one-way ANOVA). Right) proportion of grid cells with a square grid score exceeding the 95th percentile of a chance distribution in each lattice plane. A small number of cells exhibited a significant square firing pattern when projected onto the XY and YZ planes. Grey line shows the value that would be expected by chance (5%). (b) Three examples of these significant XY square grid cells. Top row shows the volumetric firing rate map, text gives the rat number, date, tetrode and cluster. Bottom row shows the autocorrelation of the XY projected firing rate map, text gives the square grid score (SGS) of the autocorrelation. (c-d) same as a-b but for all non-grid cells (n=550, 551 & 550). Square grid scores were again low and did not differ between projected planes (F(2,1648) = 1.2, p = .2975, η² = 0.0015, one-way ANOVA). No plane exhibited more square grid cells than would be expected by chance (5%, grey line).

Extended Data Fig. 7. Cells with significant XY hexagonality in the lattice maze.

For panels c & d: Filled markers represent cells, open circles denote mean, error bars denote SEM. Six grid cells (1 from one animal, 5 from another across a total of 3 sessions) showed a significant hexagonal grid score in the XY plane of the lattice maze. (a) Both grid spacing (z = −3.3, p < .001) and field size (z = 4.5, p < .0001; permutation tests: Methods: Statistics and figures) in the arena were significantly higher in the XY grid cells (n=6) than the remaining grid cells (n=41) suggesting that the hexagonality of large scale grid cells may be better preserved in 3D space. (b) The layer of the mEC the cells were recorded in was not related to XY hexagonality. Text gives the result of a two-sided Chi-square test of expected proportions which indicates that the layers where significant XY grid cells (red area) were found could be drawn by chance from the underlying distribution of grid cells (black area). (c-d) Theta modulation (Methods: Spike phase and autocorrelation analyses) and the proportion of layer crossings that were vertical in the lattice maze (Methods: Behaviour and spherical heatmaps) were also not related to XY hexagonality. Text gives the result of permutation tests (Methods: Statistics and figures). n=41 & 6 cells.

Extended Data Fig. 8. Directional modulation was preserved in the lattice.

These analyses were conducted on head direction estimated from speed filtered displacement (Methods: Directional analyses and shuffles). (a) Markers represent cells, open circles denote mean, error bars denote SEM. Rayleigh vector lengths for all 40 directionally modulated cells (that is, any cells with a Rayleigh vector length exceeding the 95^th percentile of 100 spike train shuffles in both arena sessions, may include grid cells). Grey line denotes a commonly used arbitrary cut-off (0.3) and text gives the proportion of cells with a Rayleigh vector length greater than this. Vector lengths were lower in the lattice (F(2,117) = 7.1, p = .0013, η² = 0.108, one-way ANOVA). (b) Vector lengths were correlated between Arena 1 and the Lattice suggesting preserved directional modulation in this maze. LLS: linear least squares line fit. Text gives the result of a Pearson linear correlation and the slope of the LLS. (c) Left) normalized Arena 1 tuning curves for all directionally modulated cells sorted by their preferred firing direction (PFD; peak firing). Right) normalized lattice tuning curves sorted by their Arena 1 PFD. The preserved diagonal ordering suggests that cells maintained the same allocentric firing directions in the lattice; the population correlation between arena and lattice tuning curves (r = 0.85) exceeded the 95^th percentile of 1000 shuffles (r = 0.33; z = 8.3, p < .0001) confirming this. (d) Overlaid arena and lattice polar tuning curves for 8 example directional cells. Coloured text gives the Rayleigh vector length for the corresponding tuning curves. (e) Same as c but for all conjunctive grid × direction cells (n = 7). As before, the correlation between arena and lattice tuning curves (r = 0.91) exceeded the 95^th percentile of 1000 shuffles (r = 0.55; z = 3.7, p = .0001) confirming that conjunctive grid cells maintained the same allocentric firing directions in the lattice. (f) Example conjunctive grid cells, one per row, ratemap, spatial autocorrelation and overlaid tuning curves as in D.