## Abstract

The mammalian hippocampal formation provides neuronal representations of environmental location, but the underlying mechanisms are poorly understood. Here, we report a class of cells whose spatially periodic firing patterns are composed of plane waves (or bands) drawn from a discrete set of orientations and wavelengths. The majority of cells recorded in parasubicular and medial entorhinal cortices of freely moving rats belonged to this class and included grid cells, an important subset that corresponds to three bands at 60° orientations and has the most stable firing pattern. Occasional changes between hexagonal and nonhexagonal patterns imply a common underlying mechanism. Our results indicate a Fourier-like spatial analysis underlying neuronal representations of location, and suggest that path integration is performed by integrating displacement along a restricted set of directions.

Grid cells represent the animal’s location by firing in a hexagonally symmetric array of locations covering the entire environment (*1*). These cells are found in the medial entorhinal cortex (mEC) (*1*, *2*) and in pre- and parasubiculum (PaS) (*2*). This spatially periodic firing may provide the spatial metric for the hippocampal cognitive map (*3*–*6*). The hexagonal symmetry raises important questions: Is this an entirely unique pattern or one end of a continuum? Is this pattern required for spatial representation, or does it reflect properties such as stability (*7*) or coding efficiency (*8*)? We recorded 351 cells from superficial layers (II and III) of the medial part of dorsocaudal mEC (five implants) and adjacent PaS (two implants) in seven adult male rats (fig. S1) while they foraged for food in a square enclosure (1.69 m^{2}). Many of these showed the regular hexagonal pattern characteristic of grid cells [26% passed the standard “gridness” measure (*1*)] (Fig. 1A), but a surprisingly large portion (44%) had stable multipeaked patterns lacking the signature hexagonal symmetry (Fig. 1, B and C).

We used two-dimensional (2D) Fourier spectral analysis to identify cells whose spatial firing patterns showed significant spatial periodicity (*9*), being predominantly composed of a small number of Fourier components (that is, periodic spatial bands with different wavelengths and orientations that sum to produce the firing-rate map) (Fig. 1F). A cell was categorized as spatially periodic if its strongest Fourier component exceeded 95% of those in spatially shuffled data (fig. S2). Overall, 70% of cells were spatially periodic (Fig. 1E and fig. S3, all cells). Of these, 37% were grid cells, which usually have three main Fourier components with similar wavelengths, oriented at multiples of 60° from each other (Fig. 1, F and G, and figs. S4 to S8). The remaining spatially periodic cells had one to four main Fourier components with a greater range of relative orientations and wavelengths (Fig. 1G, figs. S6 and S7, and see fig. S8 for comparability of physiological properties). Spatial periodicity might reflect local rather than global spatial structure in the multipeaked firing fields. Accordingly, we used a second shuffling technique based on shuffling local peak-centered segments of the data. The results show that 144 of 154 (94%) spatially periodic nongrid cells and 89 of 91 (98%) grid cells had significantly more spatially periodic firing patterns than the *P* = 0.05 level in these shuffled data [that is, significantly more global periodicity than predicted by their local spatial structure (*P* < 0.01, binomial probability distribution) (fig. S9) (*9*)]. Several cells with one or two main Fourier components (for example, see Fig. 1C and fig. S10) were reminiscent of the band cells postulated as inputs to grid cells in some computational models [(*7*, *10*), see also (*11*–*13*)].

Could the spatially periodic nongrid cells provide a consistent metric for an environment comparable to the grid cells? First, we looked at the stability of their firing patterns in familiar environments, between successive trials on the same day and between trials on different days. Spatially periodic nongrid cell firing patterns were significantly more stable than chance, both within and between days. However, grid-cell firing patterns were even more stable than nongrid spatially periodic cells on both comparisons (Fig. 2A). The greater stability of grid-cell firing patterns corresponded to greater stability in the orientations of their Fourier components (Fig. 2B), suggesting a causal relationship between the two.

Some spatially periodic cells (11%) changed their firing patterns from grids to nongrids or vice versa between trials in the same environment (figs. S11 and S12 and table S1). Figure 2C shows an example of a cell whose firing pattern changed from gridlike to bandlike (a simultaneously recorded grid cell remained unchanged, see fig. S12 for details). Such transitions did not reflect unidirectional drift of the grid pattern (sliding time-window spatial autocorrelation analysis) (fig. S13). These transitions suggest a continuous population of spatially periodic cells, with grids and nongrids reflecting different combinations of a small set of elemental periodic bands.

A larger proportion of spatially periodic cells (32%) changed between grids and nongrids across different environments (Fig. 2D and table S1), although the majority did not change category (Fig. 2, E and F). These transitions corresponded to changing configurations of underlying periodic bands. They did not reflect simpler transformations previously reported for grid cells after environmental manipulations [rotations and translations (*14*), rescaling (*15*), or expansions (*16*)].

Are the spatially periodic cells in one animal composed from the same set of bands? All of these cells tended to have Fourier components clustered around a small number of orientations and wavelengths (figs. S5 to S7) (*15*). As expected, grid-cell components were aligned (*15*, *17*) and oriented at 60° to each other (Fig. 3, A to G; Rayleigh vector *R*_{0} = 0.58 ± 0.14 for clustering modulo 60°, *P* < 0.01), corroborated by a clustering of the angular separation of neighboring components around 60° (Fig. 3H; Rayleigh vector *R*_{0} = 1.1, *P* < 0.001). The orientations of components of spatially periodic nongrid cells tended to align with those of grid cells (Fig. 3, A to G, and fig. S5; mean correlation coefficient of 0.54 ± 0.08, *P* < 0.01 from shuffled data) but included a wider range of relative orientations (Fig. 3I).

If grid cells and spatially periodic nongrid cells are composed of subsets from a larger set of bandlike components, there should be examples of simultaneously recorded cells with different components. Figure 4, A and C, shows a grid and nongrid spatially periodic cell differing by ~30° in the orientation of one of their components (see fig. S14 for more examples). Figure 4, B and D, shows two simultaneously recorded grid cells whose three component orientations all differ by ~30° and also differ in wavelengths (see fig. S15 for details).

Finally, we asked whether the proportions of grid and spatially periodic nongrid cells differed between the two anatomical regions investigated. Although the overall percentage of spatially periodic cells was lower in mEC (156/239, 65% in mEC; 89/112, 79% in PaS), the proportion of grid cells in mEC (75/156 or 48%) was much higher than in PaS (16/89 or 18%), despite the close anatomical proximity of these regions (fig. S1). This observation suggests that both regions, with very distinct afferent and efferent connectivity (*2*, *18*, *19*), can generate spatially periodic cells, albeit with major differences in the prevalence of pure hexagonal symmetry. In both regions, cell firing shows strong theta modulation: 67% of all spatially periodic cells were theta-modulated in PaS (mean frequency 9.2 ± 0.1Hz; 65% of all cells were theta-modulated) and 56% in mEC (with similar mean frequency; 47% of all cells were theta-modulated), pointing to a common characteristic of spatially periodic cells regardless of their anatomical location (fig. S16).

The abrupt increase in the proportion of grid cells in mEC suggests that mEC has microcircuitry or anatomical inputs organized to prefer components at 60° angles, which, in turn, provides the most stable inputs to the hippocampus. In contrast, PaS, with a higher proportion of spatially periodic cells but more varied and less stable firing patterns, heavily projects to the superficial layers of mEC (*18*–*20*) and may represent an intermediate stage in constructing stable grids from periodic bands.

The heterogeneous spatially periodic firing patterns reported here could reflect a common process of self-organization acting on band cells upstream of the recorded cells (*7*, *10*). The most stable outcome of such a process would be hexagonal grids (*10*), though mixtures of orientations organized at multiples of around 30° and 60° can also be stable (*7*) and may support multiscale representation (*4*, *21*). What mechanism might underlie the bands? Theta-modulated firing reminds us that theta rhythmicity controls the scale of spatial representations (*22*–*24*) and that interference between theta oscillations could integrate self-motion to produce bandlike firing patterns that drive grid-cell firing (*10*–*12*). These band cells might correspond to phase-modulated firing from theta cells in the septo-hippocampal circuit (*13*) or may be generated by 1D ring attractors (*7*, *25*). Alternatively, in the attractor model (*4*, *6*, *26*), recurrent interactions between grid cells could produce a stable gridlike, as well as periodic bandlike Turing pattern (*6*, *26*, *27*) of firing across a sheet of cells, which translates into spatial firing as the animal moves.

In summary, the firing patterns of the majority of cells recorded in PaS and mEC showed a stable spatial periodicity described by superposition of a small number (up to four) of elemental periodic bandlike components drawn from a discrete set of orientations and wavelengths. Grid cells correspond to hexagonal configurations, which have the greatest spatial stability. Our results suggest that path integration is performed by integrating self-motion along a restricted set of directions, mediated by planar periodic representations (bands) of multiple scales along each direction, and that self-organization of these bandlike inputs may underlie the firing of all spatially periodic cells in the parahippocampal region.

## Supplementary Materials

www.sciencemag.org/cgi/content/full/337/6096/853/DC1

Materials and Methods

Figs. S1 to S16

Table S1

References

## References and Notes

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**Acknowledgments:**This work was supported by the European Union Framework 7 (SPACEBRAIN) grant, the UK Medical Research Council, the Gatsby Charitable Foundation, and the Wellcome Trust. J.K. received support from a CoMPLEX studentship. We thank M. Bauza, T. Wills, C. Barry, F. Cacucci, and M. Witter for useful discussions and S. Burton for help with histology. This work was conducted in accordance with the UK Animals (Scientific Procedures) Act (1986).