Shearing-induced asymmetry in entorhinal grid cells

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Abstract

Grid cells are neurons with periodic spatial receptive fields (grids) that tile two-dimensional space in a hexagonal pattern. To provide useful information about location, grids must be stably anchored to an external reference frame. The mechanisms underlying this anchoring process have remained elusive. Here we show in differently sized familiar square enclosures that the axes of the grids are offset from the walls by an angle that minimizes symmetry with the borders of the environment. This rotational offset is invariably accompanied by an elliptic distortion of the grid pattern. Reversing the ellipticity analytically by a shearing transformation removes the angular offset. This, together with the near-absence of rotation in novel environments, suggests that the rotation emerges through non-coaxial strain as a function of experience. The systematic relationship between rotation and distortion of the grid pattern points to shear forces arising from anchoring to specific geometric reference points as key elements of the mechanism for alignment of grid patterns to the external world.

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Figure 1: Grid alignment in a 1.5 m square environment.
Figure 2: Relationship between orientation and deformation of grid patterns.
Figure 3: Grid alignment in a 2.2 m square environment.
Figure 4: Grids can have multiple alignment solutions.

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Acknowledgements

We thank M. Mørreaunet, T. Solstad, B. Dunn, Y. Roudi, A. Treves and A. Witoelar for discussions. The work was supported by two Advanced Investigator Grants from the European Research Council (‘CIRCUIT’, grant agreement number 232608; ‘GRIDCODE’, grant agreement number 338865), the European Commission’s FP7 FET Proactive programme on Neuro-Bio-Inspired Systems (grant agreement 600725), the Kavli Foundation, the Louis-Jeantet Prize for Medicine, and the Centre of Excellence funding scheme of the Research Council of Norway (Centre for the Biology of Memory, grant number 145993; Centre for Neural Computation, grant number 223262).

Author information

T.S., H.S., M.-B.M. and E.I.M. planned experiments and analyses, H.S. collected data, T.S. performed analyses, and T.S. and E.I.M. wrote the paper, with inputs from the other authors.

Correspondence to Tor Stensola or Edvard I. Moser.

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Competing interests

The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Alignment of individual grid axes to box geometry.

a, Interpretation of alignment configurations. Bimodal distributions (red curves) around the two cardinal axes (bottom) can be mapped as unique alignment solutions to the cardinal axes x and y of a square box (top row). Orange triangles show how the basic unit of the grid pattern can be uniquely aligned to either cardinal axis (thick solid black lines). b, Distribution of angular offset for all 3 grid axes of all cells recorded in the 1.5 m box. Amount of data are shown for the two symmetric alignment configurations 0° and 15° (symmetry between box geometry and grid) and for the offset where overlap between box and grid axes is minimized (7.5°). For each cell, grid axes were sorted according to angular distance from the nearest wall (labelled axis 1–3 with increasing wall-distance (top inset)). The absolute offset from the nearest multiple of 60° was calculated for each axis as a measure of the deviation from a perfectly hexagonal grid parallel to the east–west axis (bottom inset). Frequency plots show these distributions with 3° wide bins (x axes). The offset for grid axis 1 (Amin) was consistently larger than for axes 2 and 3. For axis 1, 33.6% of the absolute values for Amin fell within 7.5 ± 1.5°, 5.8% within 0–1.5°, and only 2.9% within 15 ± 1.5°. c, Schematic illustrating that the farthest away any axis of a perfect grid can be from any wall in the environment is 15° because of the symmetries inherent in the grid pattern and the square box. Because the offset distribution is constrained to [0, 15]°, tests against uniformity were performed by multiplying the offsets by 24 to achieve a 360° range. d, Frequency plot of grid orientation for axes 1–3 for all cells in the 1.5 m environment after correcting for alignment direction (all cells with Amin < 0 were multiplied by −1 across the alignment axis (reflected)). Distance to WN is indicated by red lines for each axis. e, Frequency plot showing distribution width, or variance, for grid axes as a function of distance from WN in the 1.5 m environment. Distributions of the 3 grid axes were sorted by distance to WN (as in b) and are shown in distinct colours (as in Fig. 1g). f, Polar scatterplot (same data as in Fig. 1b) showing axis-specific offset from WN or 60° multiple in the 1.5 m box. All grids with Amin < 0 have been reflected around the Amin wall axis in order to re-align axes to one absolute offset solution. Colours as in Fig. 1g. Orange dashed lines show peak angular offset from 60° multiples (black lines) of parallel wall alignment (0°) for each axis. Orange wedges highlight the angle between each axis and its nearest 60° multiple of 0°. g, Relationship between expected angular offset from WN for a perfectly hexagonal grid with 7.5° angular offset on axis 1, and observed offset from 60° multiple for each grid axis in the 1.5 m box (mean of all cells). Least square regression line is indicated. Note that the observed offset decreases as the offset from WN approaches its maximum at the diagonal (45°). h, Frequency plots showing only slightly larger angular offset along axis 1 (the axis nearest WN) than axes 2 and 3 in the 2.2 m environment. For axis 1, a total of 7.7% of the cells from the 2.2 m box had minimal angular offsets within 0–1.5° of the nearest wall, 26.8% had offsets of 7.5 ± 1.5°, and only 6.8% were rotated 15 ± 1.5°. In 203/220 cells, grid orientation could be referred back to either of 3 distinct anchoring solutions. The mean offsets from the walls were similar, all centering around 7.5° (mean ± s.e.m. Amin: north–south wall negative offset: (n = 108), 7.54 ± 0.03°; positive offset (n = 58), 6.35 ± 0.07°; east–west wall negative offset (n = 37), 8.16 ± 0.12°). Offsets were largely unimodal in the 2.2 m box (Fig. 3c), in contrast to the 1.5 m environment (Fig. 1c).

Extended Data Figure 2 Rate map and spatial autocorrelation diagram for representative cells from all animals.

The 587 cells from the 1.5 m environment were ranked according to time of recording for each animal and every tenth cell on the list was then selected. Cells are sorted by animals. For each cell, the rate map is shown at the top and the autocorrelogram at the bottom. Colour scale, 0 Hz to peak rate for rate maps; [−1,1] for autocorrelation maps. Peak rate is indicated above the rate map and gridness below the autocorrelogram. Correlation is indicated by scale bar. Peak rate is indicated at the bottom of each rate map; angular offset at the top right of each autocorrelogram. Grid orientation and grid ellipticity were determined from the innermost hexagon of vertices in the autocorrelogram, as in Fig. 1a.

Extended Data Figure 3 Distribution of grid orientation (Amin) across individual animals in the 1.5 m environment.

a, Frequency distributions showing clustering of grid orientation around each of the 3 axes defined by 60° multiples of the east–west box axis (number of cells as a function of orientation). Each row shows one animal; animal number is indicated at the top left. Rat 13960 had two distinct data sets (1 and 2, respectively). b, Polar scatterplots showing distribution of grid orientation and grid spacing for the entire sample of grid cells in each animal. Grid spacing is indicated on the radial axis. For each cell, the location of the 6 innermost fields in the spatial autocorrelogram is shown, as in Fig. 1a, b. The cardinal axes of the recording environment are shown as black lines. Note similar offset from the east–west axis in at least 6 out of 7 data sets (all animals except 14146). The deviation from the parallel configuration (0°) was consistent across animals. Excluding cells with fewer than 6 grid fields did not cause any major change of the angular offset (mean ± s.e.m. of remaining cells: 7.5 ± 3.3°).

Extended Data Figure 4 Modular organization of grid alignment.

ac, Data from 1.5 m environment. a, Polar scatterplot of all cells recorded in the 1.5 m box (grey), as in Fig. 1b. Superimposed are all points from one animal (orange points with multiplicative colour). Green ellipses indicate 3 different modules. Different modules assumed either of the two alignment configurations around the east–west box axis, that is, distributions for individual modules were always unimodal. b, Kernel smoothed density curve showing frequency of observations across values of Amin (all cells; kernel width: 2.5°). Coloured lines show means of individual grid modules. Colours correspond to different animals. Module means showed the same distribution trend as the pooled data. The average of the smallest angular offset (Amin) with grid modules as the unit of analysis was similar to the average of the pooled data (mean ± s.e.m.: 6.3 ± 0.2°). Both within and across animals, the distribution was bimodal. c, To visualize cross-animal clustering of grid orientation, for each grid module we counted the number of other modules with mean grid orientations that distributed within a narrow range of the mean orientation of the reference module (± 2.5°). Left plot shows for each module (in individual rows) which other module means fell within ± 2.5°. Modules were sorted according to mean grid orientation and assigned a rank value (module ID). Each module has a unique colour based on module ID (colour-bar). The grid orientation of a module was typically shared by several other modules (4.2 ± 1.9 module means within ± 2.5°), as indicated by many long rows as well as consistent clustering of colour across rows). Bottom plot shows cumulative distribution of the same data. Taken together, the data suggest that the number of anchoring solutions is limited and that solutions are reused across modules. df, Data from 2.2 m environment. d, Polar scatterplot with an individual animal highlighted in purple. e, Distribution of mean values for grid orientation across modules, as in b. The average minimal orientation offset across module means in the 2.2 m box was 7.2 ± 0.3° (mean ± s.e.m.). f, Clustering of mean grid orientation across modules. Notation as in c. g, h, Distortion and rotation of grid pattern are independent of grid spacing. g, Scatterplots showing relationship between module identity and grid offset (left), optimal shearing parameter (middle), and grid ellipticity (right). Each circle corresponds to one cell. Dark red lines show means ± s.d. for successive modules based on pooled data of individual cells. Blue lines show means ± s.d. for mean values of each module (slightly offset in x direction for clarity). Data from 1.5 m and 2.2 m environments are combined. Note lack of effect of module identity on angular offset, shearing parameter or ellipticity. h, Same plots as in g, but binned by grid spacing. Spacing range was partitioned into 4 equal parts. The values underneath each panel show the centre value of each partition. The absolute offset decreased only minimally with grid spacing (repeated-measures Kruskal–Wallis test with 4 bins of grid spacing as the within-subjects factor: H = 0.8, d.f. = 3, P = 0.85; mean offset for modules M1–M2: 7.3°; mean offset for M3–M4: 6.9°).

Extended Data Figure 5 Grid deformation is constrained by environmental geometry.

ac, Data from 1.5 m environment. a, Frequency plot showing orientation (tilt) of the semi-major axis of an ellipse fit to the inner hexagon of firing vertices in each cell’s spatial autocorrelogram (all 587 cells that were recorded in the 1.5 m box). For clarity, ellipse values are shown over the entire 360° range (each cell thus contributes 2 data points as unique ellipse orientation values distribute within a 180° range). The superimposed curve shows a kernel smoothed density estimate of the same data (Gaussian kernel width: 7.5°). The distribution of ellipse orientation was highly non-uniform both within and between animals (Rayleigh test in the entire cell sample: Z = 12.8, P = 2.5 × 10−6 for (ellipse orientation distribution) × 2). Two unique modes within [−90, 90]° were identifiable. Modes were defined as all the values that fell between the two neighbouring troughs (red lines that mark the boundary of each mode-distribution, labelled 1 and 2, respectively). Cells within modules shared ellipse orientation. On average, for one mode, the long axis of the ellipse was offset by 27.7 ± 1.1° from the east–west axis of the recording box; for the other mode, the offset was 17.9 ± 1.4° from the north–south axis (means ± s.e.m.). To determine if the spread of ellipse orientation was smaller than expected by chance, we drew, for each module, m random ellipse orientation values from the pool of all cells (n = 587), and determined the circular spread (standard deviation) of that distribution. m was the number of cells in the module. The procedure was repeated 300 times per module to allow calculation of z scores. The circular spread of ellipse orientation within modules was significantly lower than expected by chance (mean ± s.e.m. z = 4.53 ± 1.33; t(17) = 3.37 P = 0.006, one sample t-test). b, Frequency of the same data as in a, but modulo 90° to control for differential ellipse alignment (modulo 90° brings out trends common to each of the 4 walls in the box). A single peak was detectable at 65.4° (red line). c, Minimal wall offset (Amin) as a function of ellipse orientation for each cell. Values are shown for an ellipse range of 360°, such that each cell contributes 2 points. Each mode of the distribution of ellipse orientations consisted primarily of cells from one of the two configurations for grid orientation (positive or negative offset). dg, Comparison of deformation in the 1.5 and 2.2 m boxes. d, Circular histograms showing ellipse orientation for all cells recorded in the 1.5 m box (left) and the 2.2 m box (right). Ellipse orientations were multiplied by 2 to achieve a 360° range. Although there was clear clustering around two modes in the smaller box, no clustering was apparent in the data from the large box. e, Same as in d, but for ellipse orientation modulo 90° and multiplied by 4 to achieve a 360° range (left, 1.5 m box; right, 2.2 m box). The modulo operation brings out distribution trends that are similar for the 4 walls in the box. No such trends were apparent for the 2.2 m environment. Mean vector length for the distributions is shown in the centre of each plot (red solid line and axis). f, Kernel smoothed density curves from ellipse orientation in all cells recorded in the 1.5 m box (left) and the 2.2 m box (right). We identified all peaks (orange lines) within [−90, 90]° (light areas), and then for each peak calculated the log ratio between the peak and the smaller of its two neighbouring troughs (green lines). While only 2 peaks were identified for ellipse orientation in the 1.5 m box (a), 5 peaks were present in the 2.2 m box. g, Log peak-trough ratios from the values calculated in f (means ± s.e.m.). Values were significantly lower in the 2.2 m environment than the 1.5 m environment (two sample t-test, t(5) = 3.51, P = 0.02). h, Scatterplot showing, for all cells in the 2.2 m environment, grid orientation from the grid axis second furthest away from the nearest wall (WN) versus absolute ellipse orientation. There was a strong linear correlation between these parameters (r = 0.41, P = 3.82 × 10−10), as expected if interactions from a second shear axis had occurred (linear regression is shown as red line). i, j, Distribution of angular offset after shearing along each of the cardinal axes of the 1.5 m box. i, Frequency plots showing minimal absolute offset from multiples of 60° of the parallel (0°) configuration for all cells before shearing (left) and after shearing in the north–south (y) and east–west (x) directions (middle and right, respectively; insets). For each box axis, we used the shear transform that minimized ellipticity. Before shearing, the distribution showed a clear offset (left). After shearing in the y direction (north–south; middle panel), the angular offset was eliminated. Shearing along the x axis (east–west) did not reduce the offset (right), despite similar reduction of the ellipticity. Note that, in simulated grids, the shear parameter γ needed to rotate a perfectly hexagonal grid from 0° to 7.5° (γ = tan(π/24) induced ellipticity of 1.16, a value almost identical to the mean ellipticity observed in the data (1.17). j, Kernel smoothed density estimates of Amin (minimal offset from any wall) for all cells (Gaussian kernel width: 2.5°) after shearing along both the x and y axis (left and right panel, respectively). There were two distinct modes after shearing in the x direction. The distribution for the y direction, in contrast, became unimodal and centred on the east–west box axis (0°), suggesting that the bimodal distribution of angular offset was caused by shearing in this direction.

Extended Data Figure 6 Observed grid deformation and rotational offset cannot be reproduced by mere diagonal compression, where forces from the corners move fields perpendicularly towards the diagonal.

a, Diagonal compression model showing forces acting on a shrinking pattern that is anchored to opposite corners. Top left, square lattice (no elongation) representing starting point in novel environment. Top right, vector field (red arrows) of proposed forces resulting from pattern shrinkage given anchoring to opposite corners (red dots). Without elongation in novelty, pattern shrinkage during familiarization to the environment is hypothesized to be radially uniform. However, corner anchoring will stop shrinkage along the diagonal of the anchoring corners. This results in a compression of the pattern along the opposite diagonal. Bottom left, a simulated grid pattern that was uniformly enlarged (as indicated by the perfect circle fit) to mimic a novelty response. Upon familiarization to the environment (shrinkage, bottom right), and in the presence of anchoring to opposite corners, the pattern undergoes diagonal compression. The resulting pattern has an angular offset (here 7.5°) and is accompanied by pattern deformation. Force vectors are shown as red arrows inside the box, and component vectors are shown as red arrows outside the box. The diagonal compression transform is described by:where, in order to offset an aligned grid by 7.5°,b, Comparison of grid pattern characteristics after shearing and after diagonal compression, both aimed at achieving a 7.5° angular offset. Black lines and dots show the original symmetric and aligned grid pattern. Red dots and ellipse denote pattern deformation imparted by shearing, while blue dots and ellipse show deformation caused by diagonal compression. 7.5° offset is indicated by a dashed black line. Note the pronounced elliptical deformation of the diagonally compressed pattern compared to the sheared pattern. c, Histogram of ellipticity values observed in 1.5 m and 2.2 m recording environments. The median is indicated by a black line. Red and blue lines show ellipticity after 7.5° offset induced by shearing and diagonal compression, respectively. The ellipticity induced by shearing matches the data almost perfectly, while the value for diagonal compression is much higher than observed. Together, these analyses suggest that although diagonal compression can induce both angular offset and elliptic deformation, the relationship between these variables does not render this transformation a likely candidate for explaining the observed grid alignment. In contrast, the relationship between ellipticity and offset following shearing closely matches observation (Extended Data Fig. 7), suggesting that this is indeed the process that underlies the alignment process.

Extended Data Figure 7 Experience-dependent changes in grid patterns can be explained by shearing of grids that are anchored to diagonally opposite corners.

To address effects of experience, grid orientation was compared at two stages of acquaintance with the environment, during the first trial of exploration in a novel square box in a novel test room and during exploration of a similar but highly familiar box in a familiar room. We used a previously published set of 20 grid cells from 5 rats that ran in 1 m wide square boxes with shape and colour similar to the 1.5 and 2.2 m boxes in the main study2. 85% of the 20 grid cells had 6 fields or more. a, Rate maps for a representative grid cell recorded in a 1 m wide square box in a familiar room (left), a novel room (middle), and a second time in the familiar room (right). b, Cumulative distribution frequency plot showing distribution of orientation offsets in novel and familiar rooms for cells that were recorded in both environments. c, d, Ratio of grid spacing (c) and grid ellipticity (d) in the novel and the familiar environment (novel/familiar). Red line shows mean ( ± s.e.m.). P < 0.05. Grid spacing and grid ellipticity increased from familiar to novel by factors of 1.06 ± 0.03 and 1.08 ± 0.03, respectively (one-sample t-test of log ratios: t(12) = 2.25, P < 0.05 and t(12) = 2.30, P < 0.05; all cells recorded in both environments), consistent with previous observations of grid cells in novel environments18. e, Distribution of ellipse orientation in familiar and novel environments (left and right, respectively). Orientation is expressed in relation to the east–west wall (inset). Ellipse orientation was more sharply distributed in the novel environment, with a circular mean of 81.1° and s.d. of 16.7°, near orthogonal to the anchoring wall. In the familiar environment, the distribution was broader (circular s.d. of 37.4°), suggesting that the impact of the walls is relaxed with repeated experience. f, Analyses of simulated grid patterns showing that both de-elliptification and non-coaxial rotation can be induced also with elongated grid patterns as the starting point, if the grid is anchored to opposite corners of the recording box. Top left, square lattice parallel to the east–west wall axis, elongated along the north–south wall axis. An ellipse is fitted to indicate degree of elongation (red). Top right, same lattice but allowed to relax towards a less elongated state. If the grid is anchored to opposing corners (red dots), a compression shear (corner shear) vector displacement field is generated (red arrows). Bottom left, simulated grid pattern, aligned and elongated in the same way as the lattice above. The elongation produces an ellipse (red) oriented along the north–south axis. No rotation is present at this initial stage. Bottom right, relaxation towards a less deformed state, with the grid anchored to opposing corners, produces a sheared pattern with an angular offset. Shearing force vectors are shown as red arrows in each of the non-anchoring corners. The transform is defined by:ordepending on which corner pairs operate as anchors. So long as only one γ is non-zero, the transform produces shear-like displacement along one of the cardinal axes, but the shear axis is rotated 45° so as to leave the two anchor corners in place. Corner shearing minimized ellipticity and removed the angular offset of the pattern (see i). g, h, Analysis of relationship between ellipticity, angular offset and elongation level in novel environments. g, Optimal corner-shearing for minimizing ellipticity in an elongated grid. Ellipticity values are brightness-coded. Changes in grid orientation may take place after minimization of the elongation of the grid when the grid is anchored to opposite corners. For such changes to occur, shearing should minimize ellipticity given the initial elongated state. To test this idea, we systematically elongated simulated grid patterns along the wall axis perpendicular to the grid alignment axis (y axis in plot). For each elongation level, we next corner-sheared this pattern (with corners as anchoring points), through a range of shear parameters (x axis in plot, values correspond to the offsets associated with each shear parameter [offset = tan−1 (shear parameter)]. For all elongation factors >1, shearing to a level >0 produced less ellipticity than in the original pattern. This was not the case for the standard shear transform in which no corner anchoring occurs (not shown). Across elongation levels, we determined which shearing parameter minimized ellipticity (blue line). We next found the intersection between these optima and the shear parameter that produces a 7.5° offset (red line). The elongation factor at this intersection was 1.28. Thus, in order for a 7.5° offset to represent the optimal amount of shearing in simulated data with anchoring to diagonally opposite corners, the initial elongation factor of the grid pattern is 1.28, very close to the observed ellipticity in the novel environment (panel h). h, Cumulative distribution function (black line) of ellipticity in grid cells recorded in a novel room. The optimal elongation factor is indicated in blue. Note the proximity of this factor to the median ellipticity level. i, Frequency histograms showing actual distribution of orientation for each grid axis (top), and distribution after corner-shearing using the transform in f (middle and bottom). Middle panel, shearing to one pair of anchoring corners; bottom, shearing to the other pair. After corner-shearing to the first pair, the offset was minimized to the same extent as in the simple shearing paradigm (middle, red asterisk). The offset could be abolished almost completely by corner shearing (peak symmetry offset after shearing 0°, kernel smoothed density curve with Gaussian kernel width 1.35°, 60% of the data distributed within 0–5°). After corner shearing, however, the distribution of ellipticity displayed less variation than after simple shearing (s.d. of 0.0025 versus 0.009). j, Proposed model of deformation and rotation of grid patterns as a function of experience. From left to right, default minimum-energy state of grid, elongated grid in novel environment, sheared grid with rotation after experience, and reversal of sheared grid to default state by reverse corner-shearing analysis. We suggest that, in novel environments, grid cells may start out with an orientation that aligns one grid axis with a band along a wall defined by the activity of border cells. This initial alignment may disrupt the symmetry of the grid pattern. Through shearing, the grid may then be relaxed towards a lower-energy-state solution that is less dependent on the initial anchoring segments.

Extended Data Figure 8 Distribution of offset and ellipticity in a circular environment.

We have shown that, in square environments, the grid pattern is deformed and rotated by shear forces parallel to the walls of the environment. Here we show how grid orientation and grid ellipticity are distributed in a circular environment. We used a sample of 23 grid cells from 6 rats that had been recorded in a 2 m wide circular environment in a previous study2. 100% of the cells had 6 grid fields or more. a, Example grid rate maps (top) and associated spatial autocorrelograms (bottom). b, Histogram of grid orientation for all 3 grid axes and across all cells recorded in the circle. Grid orientation was more variable than in the square environments. The distribution of mean grid orientation was not significantly different from uniform (Rayleigh test (mean grid orientation was multiplied by 6 to achieve 360° range); Z = 1.23, P = 0.30). The ellipticity of the grid pattern was significantly increased compared to the ellipticity in the square boxes (1.24 ± 0.025 versus 1.17 ± 0.004; Z = −2.98 P = 0.003). This increase in ellipticity likely reflects grid pattern variability due to the absence of local geometric landmarks such as corners.

Extended Data Figure 9 Effect of shearing on deformation and reorientation of grid patterns in 1.5 m and 2.2 m environments.

a, b, Distribution of angular offset after shearing along each of the cardinal axes of the 2.2 m box. a, Kernel smoothed density estimates (Gaussian kernel width: 2.5°) of grid orientation for all 3 grid axes for all cells recorded in the 2.2 m box. The top panel shows orientation after shearing along the axis orthogonal to the alignment axis WN (solid black curve). Shearing was performed separately for cells aligned to the east–west and north–south axis of the box and then combined on the basis of which relationship the shear axis had to the alignment axis (orthogonal for top plot). Dashed grey lines show the original distribution before shearing. Red lines indicate the orthogonal box axes (0 ± 30°, 0° corresponds to the east–west box axis, 30° to the north–south axis). Peaks detected in the sheared distribution around the red lines are reported above each peak. Note reduction—but not complete elimination—of angular offsets after shearing orthogonal to the alignment direction. Bottom, same as above but for shear axes parallel to the alignment axes. Shearing along this axis did not result in reduction of the offset. b, Kernel smoothed density curves (Gaussian kernel width: 1.35°) for absolute offsets from nearest multiples of 60° (referred to the parallel configuration) after shearing along both axes: orthogonal to alignment axis (top) and parallel to the alignment axis (bottom). Methods for shearing were the same as in a. Peak for each distribution is indicated by red solid lines. Axes were sorted per cell according to their angular distance from the nearest wall (axis 1 = Amin; axis 3 had the largest angle from nearest wall). After shearing orthogonal to the alignment axis, the offset peak was reduced considerably. Shearing along the other axes did not reduce the offset. ce, The effect of adding a second shear interaction. c, Effects of anchoring to multiple walls in a simulated grid. A square lattice is used for illustrative purposes. Top row, square lattice aligned to the cardinal axes with no shear interactions from any wall (left). Simulated rate map of grid aligned to the east–west box axis (middle). Spatial autocorrelogram generated from rate map (right). Ellipse and inner fields are shown in black. Middle row, the square lattice after a shear transform along the north–south axis (left). Black line with arrow illustrates shearing interaction from wall. Shear origin (anchoring point) is shown in red. Simulated rate map after shearing is shown in the middle, and the resulting autocorrelogram is shown to the right. Bottom row: square lattice (left), simulated rate map (middle) and resulting autocorrelogram (right) after sequential shearing forces from two wall axes. Note that adding the second shear force has no impact on the smallest angular offset (Amin) but the orientation of axes 2 and 3 is changed and the ellipse orientation is altered accordingly. d, Grid axes and ellipses from autocorrelograms in c before (left) and after (right) the shearing that minimized both ellipticity and grid offset (illustrated with insets). The east–west axis and its 60° multiples are shown in grey. The axis nearest one of the walls is highlighted in red. Note the elimination of grid offset in the case with single shear interactions (middle), but inability of simple shearing to eliminate the offset (asterisk) when two shear interactions were operative (bottom). e, Effect of second shear interaction on grid orientation and deformation. We systematically applied shearing from a second shear axis to the simulated grid that was already sheared in one direction to induce a 7.5° offset. A range of shear factors was explored (−0.35 to 0.35). Left column, effect of second shear on grid orientation for individual grid axes. The middle panel shows the axis that was originally sheared to become offset by 7.5°. Note minimal change in offset in this axis, while systematic changes occurred in the two remaining axes (top and bottom respectively), resulting in further elliptic deformation of the grid pattern. Top right, grid orientation averaged across the 3 axes. Bottom right, ellipticity of the grid pattern as a function of the second shear parameter. f, Quadrant spatial autocorrelograms for grid cells with large spacing yielding fewer than 3 grid fields per quadrant. Left column, simulated rate maps with one, two or three grid fields. Middle column, spatial autocorrleograms from the rate maps in the left column. As expected, a single field was not informative in terms of grid features such as grid orientation or spacing, whereas two fields yielded information about one axis and three fields were informative about all axes. Taken together across all cells from a module, such contributions may average to overcome the effect of sparse field-sampling and recapitulate the original local grid geometry. Right column, simulation based on 20 grid cells with different grid phase (but similar spacing and orientation) showing that when spatial autocorrelograms from multiple maps (from one module) with varying number of fields from various axes are combined (averaged), grid features from all axes can be retrieved and used to determine grid geometry in subdivisions of the original environment. White asterisks show field peaks detected by the algorithm. Orange lines show the grid orientation used to generate the simulated grid cells within a quadrant. The average autocorrelogram faithfully captures the original grid geometry even if the average number of fields per cell is less than three.

Extended Data Figure 10 Two-axis corner-shearing removed the offset in the 2.2 m environment.

Several observations point to shear-like forces from two perpendicular walls as more common in the 2.2 m environment than the 1.5 m environment (Fig. 3). Minimizing ellipticity by simple shearing completely removed the angular offset in the 1.5 m box. In the 2.2 m box, however, the effect was only moderate, reflecting the fact that the grid was anchored to two walls, not one. Here we sought to determine if the two optimal shear parameters associated with two-axis corner-shearing were recoverable with the same approach as for simple shearing (minimization of ellipticity). Specifically, we tried to recover the original configuration of a simulated grid pattern on which we had applied two-axis corner-shearing in advance. The starting (default) grid pattern was made perfectly symmetrical and aligned to the east–west axis. We then applied 2-axis corner-shearing using the reverse transform proposed to occur as the result of grid shrinkage with corner anchoring during familiarization (corner-shearing):We set , and . a, Simulated grid pattern before (black) and after (red) two-axis shearing. The offset from the first shear axis is 7.5° but the second shear axis introduces further rotation, combined with deformation of the two remaining axes. b, Ellipticity surface resulting from systematic exploration of the two-shear parameter dimensions. Height and colour indicate ellipticity. x and y axes denote the angular offset that the shearing parameters would cause in simple shearing (from −40 to 40°). Note the prominent minimum corresponding to a unique solution to the problem of recovering the original pair of shear parameters (arrow). c, Same as in b, but as pure colour map. The parameter set that was used to shear the grid initially is shown with green lines. The point of minimum ellipticity is shown as a white circle. In this case, the retrieved parameters were exactly those that were used in the original transform of the pattern. The example illustrates that the original shear parameter set could be recovered completely by two-axis reverse shearing. d, Colour map of angular offset resulting from two-axis shearing across the same range of shear parameter values as in a. Black corresponds to wall alignment (0° offset). The point of minimal ellipticity (white circle in c) is the same as for minimal offset (white circle in d). e, Cumulative distribution function showing angular offset of the data recorded in the 2.2 m box before (black) and after (red) two-axis corner shearing (Fig. 3h). Note consistent shift to the left after shearing. 2-axis corner shearing significantly reduced the original offset (Wilcoxon rank sum test: Z = 6.6, P = 4.4 × 10−11); the offset was abolished almost completely by corner shearing (peak symmetry offset after shearing 0°, kernel smoothed density curve with Gaussian kernel width 1.35°, 60% of the data distributed within 0–5°), while normal 2-axis shearing had little effect (Fig. 3h). f, Scatterplot of optimal corner-shear parameter and original grid offset for all data in the 2.2 m box. Shown is tan−1 of the shear parameter (in degrees) to illustrate offset that the parameter would yield in simple shearing). The two main scatter clusters correspond to the two distinct anchoring solutions in the 2.2 m environment (that is, the two wall-axes). Red dashed lines show the wall axes of the environment ([−30, 0, 30]°).

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Stensola, T., Stensola, H., Moser, M. et al. Shearing-induced asymmetry in entorhinal grid cells. Nature 518, 207–212 (2015). https://doi.org/10.1038/nature14151

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