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Grid cell symmetry is shaped by environmental geometry

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Abstract

Grid cells represent an animal’s location by firing in multiple fields arranged in a striking hexagonal array1. Such an impressive and constant regularity prompted suggestions that grid cells represent a universal and environmental-invariant metric for navigation1,2. Originally the properties of grid patterns were believed to be independent of the shape of the environment and this notion has dominated almost all theoretical grid cell models3,4,5,6. However, several studies indicate that environmental boundaries influence grid firing7,8,9,10, though the strength, nature and longevity of this effect is unclear. Here we show that grid orientation, scale, symmetry and homogeneity are strongly and permanently affected by environmental geometry. We found that grid patterns orient to the walls of polarized enclosures such as squares, but not circles. Furthermore, the hexagonal grid symmetry is permanently broken in highly polarized environments such as trapezoids, the pattern being more elliptical and less homogeneous. Our results provide compelling evidence for the idea that environmental boundaries compete with the internal organization of the grid cell system to drive grid firing. Notably, grid cell activity is more local than previously thought and as a consequence cannot provide a universal spatial metric in all environments.

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Figure 1: Grid orientation aligns to walls in squares.
Figure 2: Relationship between different grid modules across geometrically different enclosures.
Figure 3: Grid pattern is distorted in trapezoids.
Figure 4: Grid pattern is inhomogeneous in trapezoids.

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Acknowledgements

We thank J. Poort for discussions. The research was supported by grants from the Wellcome Trust and the Gatsby Charitable Foundation. J.K. is a Wellcome Trust Sir Henry Welcome Fellow. C.B. is a Royal Society and Wellcome Trust Sir Henry Dale Fellow. This work was conducted in accordance with the UK Animals (Scientific Procedures) Act (1986).

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Authors

Contributions

J.K. and M.B. planned experiments and analyses. J.K., S.B. and C.B performed the experiments. J.K. and M.B. analysed the data. J.K., M.B., C.B. and J.O’K. wrote the manuscript. All authors discussed the results and contributed to the manuscript.

Corresponding author

Correspondence to Julija Krupic.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Sagittal Nissl-stained brain sections showing the recording locations in superficial layers II-III of mEC and PaS.

as,Yellow dots indicate the dorsal-ventral region where grid cells were recorded in the trapezoids. PAS marked with asterisk symbols. Scale bar, 500 µm.

Extended Data Figure 2 Sagittal Nissl-stained brain sections showing the recording locations in superficial II–III and deep V–VI layers of mEC.

an,Yellow dots indicate the dorsal-ventral region where grid cells were recorded in the trapezoids. Scale bar, 500 µm.

Extended Data Figure 3 Orientation clustering.

a, The autocorrelogram of the distribution of grid orientations in squares (shown in Fig. 1b). b, The Fourier spectrogram of autocorrelogram in a, left, and a typical example of the Fourier spectrogram of the autocorrelogram of shuffled orientations, right. Note the absence of a low-frequency peak in the latter. c, The distribution of maximum normalized Fourier power of 10,000 data surrogates (as shown in b, right). Red line indicates 95 percentile of the shuffled data. Blue line indicates the maximum normalized Fourier power of our data.

Extended Data Figure 4 Directional and velocity sampling in square and circular enclosures.

a, b, Mean directional (left) and velocity (right) sampling in square (a) and circular (b) enclosures. c, d, Typical examples of directional (left) and velocity (right) sampling profiles in square (c) and circle (d). e, Directional sampling in the left (left, black) and right (right, blue) parts of the trapezoid. The directional sampling in the left part of the trapezoid was significantly more biased than in the right (P = 0.01; two-sample t-test; means ± s.e.m. = 0.025 ± 0.004 (left) and 0.012 ± 0.002 (right). f, The velocity sampling bias was also significantly larger in the left part of the trapezoid (left, black) than the right (right, blue) (P = 0.006; two-sample t-test; mean velocity sampling ± s.e.m. = 9,803 ± 791 (cm/s)2 (left) and 5,413 ± 1,176 (cm/s)2 (right). g, h, Typical examples of the directional sampling (g) and velocity sampling (h) on the 2 sides of the trapezoid; rat 2104. The left side trapezoid direction circular score is 0.035 (black), the right, 0.009 (blue). The left side trapezoid velocity circular score is 8,268 (cm/s)2 (black) and the right, 3,904 (cm/s)2 (blue). i, j, Absence of a significant correlation ρ between the similarities of the left and right sides of the trapezoid and the difference in directional (i), P = 0.22 or velocity (j) scores, P = 0.82.

Extended Data Figure 5 Schematic representation of grid re-alignment in different shape enclosures.

a, The relative orientation of two representative grid modules in a square. The grid cell with the smaller scale (green) is aligned 9° from the horizontal wall and the grid cell with the larger scale (blue) is aligned 9° from the vertical wall with the relative orientation between them equal to 30°. b, If grid cells respond independently and if the geometry of the environment determines their orientation in non-polarized environments such as a circular enclosure, grid cells in a should randomly realign (in the current example by 15°). c, Similarly, in the hexagonal enclosure both grid cells should start to align (left) or become 18° offset from each other (right). This was not observed in our experimental data. d, Simultaneously recorded grid cells from two different modules (rat r2183) in a square (left), a circle (middle) and a hexagon (right) maintained their relative orientation. Rate maps are shown on the top row and spatial autocorrelograms on the bottom.

Extended Data Figure 6 Distribution of grid components on the left and right sides of trapezoidal and square enclosures.

ad, The distribution of the individual grid cell components on the left (a) and right (b) sides of the trapezoid and square (left (c) and right (d)). Components 1, 2 and 3 are shown in blue, green and orange, respectively. Data from 8 rats, 10 grid modules, 26 grid cells. Note the similarity of all 4 blue components and of the green and orange components on both sides of the square and on the right side of the trapezoid.

Extended Data Figure 7 Simulated grid cells.

a, Grid cells were simulated using the orientations and wavelengths of the three main grid components (1, 2, 3) taken from the spatial autocorrelogram (a, right) of a real grid cell (a, left) as shown in this typical example. b, Simulated grid pattern was generated by summing three grid components to retrieve the rate map and spatial autocorrelogram (c) which well approximated the real data shown in a. f, g, examples of the procedure applied to the two cells in d and e, respectively. Grid cells were generated from the autocorrelogram in the square. h, Correlation coefficient between right and left side of the simulated grid data in trapezoid (left) and square (right). i, Grid scores of simulated grids on left and right side of trapezoids and squares. The lower grid score in the left side of the trapezoid is due to under-sampling. j, k, Change in orientation (j) and wavelength (k) of 3 components of simulated grids in trapezoid (black bars) and square (blue bars). All means ± s.e.m.

Extended Data Figure 8 Temporary rescaling versus permanent change in grid symmetry.

a, Expected rescaling of a grid cell (left) in a rectangle (middle). The number of grid fields stay the same (n = 10 fields) but the fields become more elliptical and farther apart along the x direction. After a few days of experience the fields converge to their initial size and shape, and the number of grid fields increases (right). The grid symmetry on the left and right belong to the same sixfold symmetry group (that is, hexagonal symmetry). b, Place cell response to rescaling of the environment. The place field initially becomes more elliptical (middle) and finally remaps to a random location (or stops firing altogether) as the animal becomes more experienced. c, Expected change in grid symmetry as the animal goes from square (a, left) to trapezoid if the change in pattern occurred due to the rescaling of the environment. Note that the number of fields remains unchanged. d, As the environment becomes more familiar the grid cell pattern should converge to hexagonal symmetry and the number of fields would increase. In general we did not observe this in our data. e, Instead, grid symmetry became more elliptical (not grid fields) and the scale increased towards the narrow side of the trapezoid.

Extended Data Figure 9 Grid cell pattern in trapezoid cannot be predicted by a simple transformation of a square to a trapezoid.

a, Panel shows 4 grid cells from 4 rats recorded in a square (left) and trapezoid (middle). Predicted grid pattern (right) obtained by transforming a grid recorded in a square (left) into a trapezoid. b, Correlation coefficients between the predicted morphed grid pattern and the one actually recorded in the trapezoid. The distribution is not significantly different from the normal distribution with a mean equal to zero (P = 0.74, t = −0.33, df = 36, one-sample t-test).

Extended Data Table 1 The main grid cell properties in hexagonal, square and circular enclosures

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Krupic, J., Bauza, M., Burton, S. et al. Grid cell symmetry is shaped by environmental geometry. Nature 518, 232–235 (2015). https://doi.org/10.1038/nature14153

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