Translational differentiation of vertically displaced surfaces by grid cells

Summary

Navigation is commonly associated with two-dimensional (2D) representations of space. Recordings from place and grid cells in the rodent and bat brain have largely upheld this association. Recent studies have investigated how these 2D representations might extend into the three-dimensional (3D) world. One unexplored question is whether grid cells represent vertically separated horizontal surfaces as a single 3D space or distinct planar environments. To address this issue, we recorded grid cells as rats foraged in both an open-field environment and one with a transparent floor suspended directly above the open-field environment. Rats either actively locomoted up a ramp to the elevated environment, or they were passively moved between the two environments, to test how differences in path integration may affect grid cell firing. We found that grid cell firing patterns in the elevated environment were translated (but not rotated) relative to those in the floor environment, and were consistent across active and passive sessions. The translation of the grid pattern on the elevated surface was consistent among co-recorded grid cells, but differed between animals and between different groups of grid cells recorded from the same animal. Non-grid spatially modulated cells also rearranged their location preferences between the two surfaces. Overall, we did not observe any evidence that the two surfaces were represented with a single 3D representation, but instead were treated as two distinct surfaces connected by a common orientation signal. These findings suggest that grid cell representations on visually distinct, vertically displaced horizontal surfaces are planar rather than volumetric.

eTOC Blurb

LaChance et al. record from grid cells as rats explore two vertically separated horizontal surfaces, one above the other, connected by a ramp. Grid cells differentiate the two surfaces by translating, but not rotating, their 2D firing patterns, suggesting that grid cells encode the surfaces as two separate contexts linked by a common orientation.

Introduction

Animals occupy and navigate a three-dimensional (3D) world. Despite this occurrence, spatially modulated neurons in the entorhinal-hippocampal circuit have largely been characterized in terms of their two-dimensional (2D) firing properties. For example, place cells in the hippocampus tend to fire preferentially when an animal occupies a specific location in a 2D arena¹, while grid cells in the medial entorhinal cortex (MEC) and adjacent parasubiculum (PaS) show multiple hexagonally arranged firing fields that tile 2D space^2-4. The periodic firing properties of grid cells are thought to reflect an entorhinal path integration mechanism, through which the allocentric distance and direction of travel along 2D paths can be calculated⁵.

Recent studies have attempted to extend our understanding of neural place representations into 3D space. For example, place cells possess coherent 3D place fields in volumetric environments, in both bats^6,7 and rats⁸. However, both place and grid cells display anisotropy in their representations of horizontal and vertical surfaces, such that firing fields are vertically elongated (and therefore contain less information) during vertical climbing^9,10. Other studies also suggest that grid cells may not be equipped to extend their regular 2D representations into the third dimension. One study of rat grid cells during locomotion in a 3D volumetric lattice attempted to determine if grid fields would tile volumetric space with a regular 3D lattice structure, but instead found that the fields were irregularly distributed¹¹. A complementary study in flying bats found that 3D grid fields showed some degree of local order but lacked a global lattice structure¹². Grid cells recorded from rats on a sloped surface largely fired as if the surface was horizontal¹³. Finally, grid cells recorded as rats ran up a helical structure showed repeating firing fields with each turn of the helix, suggesting that, again, a repeating planar representation was simply generalized onto each floor of the helix⁹, although this could potentially be interpreted as a ‘columnar’ arrangement of grid fields.

One facet of grid cell firing that has not been investigated is their response to pure vertical displacement between horizontal planar environments. That is, if an animal were to forage in a 2D open field environment, and was then allowed to move to a second planar environment that occupied the same horizontal location but was vertically translated, would grid cells create a single 3D representation spanning the two surfaces (columnar or otherwise), or would they treat them as two disparate environments by translating and/or rotating their grid patterns between the surfaces? Further, given the apparent path integration properties of grid cells^5,14-17, would grid cells treat the elevated surface differently if the animal were allowed to actively locomote to it or if the animal were passively transported to the elevated surface by an experimenter? This situation is similar to the condition human subjects encounter when orienting in a multi-story building – how do the spatial representations between different floors relate to one another (or between an inside floor and a rooftop)? To answer these questions, we recorded grid cells from the MEC or PaS as animals foraged in an open field on the floor, as well as an elevated wall-less environment with a transparent floor that was suspended above the original open field. With a transparent floor separating the two planar open fields, the rats should be able to perceive the spatial relationship between the two environments with one planar surface immediately above the second one. We found that grid patterns maintained their orientation and spacing but exhibited random translation of their firing fields between the two surfaces instead of following a columnar or otherwise 3D pattern, with no difference between active and passive movement onto the elevated surface. We also analyzed MEC/PaS cells with non-grid spatial firing, which similarly distinguished between the two surfaces with uncorrelated rate maps. Overall, these results suggest a planar rather than volumetric encoding of visually distinct, vertically displaced surfaces, and provide a crucial bridge between traditional investigations of grid coding in 2D arenas and more recent experiments in fully 3D environments.

Results

Grid cells differentiate between vertically separated surfaces

We recorded single neurons from the MEC or PaS of 7 rats as they foraged for sugar pellets in a 1.2 x 1.2 m square enclosure. Based on the locational firing properties of cells during the baseline session, we classified 60 cells (out of 177 total cells) as grid cells. Of these, 53 cells were recorded during the subsequent three sessions of the vertical displacement experiment (Figure 1A, C; Data S1A; Table S1, S2). The first session (Floor 1 session) involved foraging in a 1 x 1 m enclosure that had a ramp (~10 cm in width) placed along the east wall. The ramp was blocked so that animals could not travel up it during the floor session. After this session, the ramp block was removed, and the animals were allowed to actively locomote up the ramp and onto a transparent Plexiglas sheet that was slid into place atop the walls of the 1 x 1 m enclosure (Raised session). The floor of the elevated surface was transparent to maximize perceived visual continuity between the Floor and Raised environments¹⁸. The different sensory properties of the Raised environment (no walls, Plesiglas floor) were designed to help the animals distinguish between the two surfaces. Distal cues were limited by a floor-to-ceiling circular black curtain that surrounded the environment, which could be seen more readily on the Raised surface than the Floor because of the enclosure’s surrounding walls (like being on a rooftop). The presence of locomotor cues when the rat traveled between the two surfaces should also have aided the perception that the two environments were distinct with one directly above the other. After foraging on the elevated surface, the transparent floor was slid horizontally to allow access to the ramp, and the animal was allowed to actively locomote down the ramp for a final recording session in the 1 x 1 m enclosure (Floor 2 session). Thus, the rat had available both visual and locomotor cues to perceive the differences between the Floor and Raised Platform environments. Data for the Raised session was filtered to only include locations that had a direct analog on the Floor surface (Figure S1). Animals were highly familiar with the baseline environment, but were only exposed to the Floor and Raised environments once grid cells were isolated.

Figure 1. Grid cells on vertically separated surfaces.

A) Schematic view of the recording arena across the four sessions of the active experiment. Grid cells are first recorded and identified during a baseline recording session in a 1.2 x 1.2 m square enclosure (Baseline session). Following this session, the animals forage in a visually similar 1 x 1 m square enclosure that has a ramp (10 cm width) placed along the east wall (Floor 1 session). Animals are then allowed to actively locomote up the ramp onto a transparent surface elevated 50 cm above the floor surface for another recording session (Raised session). Finally, the animals are allowed to locomote back down the ramp for a final floor recording session (Floor 2 session). B) Predicted grid patterns on the Floor and Raised surfaces for different volumetric and planar hypotheses of grid cell firing. Dark blue filled circles represent the locations of grid fields for a given grid cell in the Floor environment, while open circles show the predicted locations of grid fields in the Raised environment. C) Path and spike plots and firing rate heat maps for three example grid cells recorded across the sessions of the active experiment. Note that the grid cell firing patterns were similar between the Floor 1 and Floor 2 sessions, but differed in the Raised session. Numbers above each rate map indicate the peak firing rate for the session. D) Comparison of rate map correlations between the Floor 1 and Raised sessions and between the Floor 1 and Floor 2 sessions. Note that while the Floor 1 vs. Floor 2 correlations were significantly positive, the Floor 1 vs. Raised correlations did not differ from a 95^th percentile shuffle mean. E) Comparison of grid field spacings across the Floor and Raised sessions. No significant differences were found. F) Comparison of grid field radii across the Floor and Raised sessions. Radii were significantly increased in the Raised session. G) Comparison of rate map spatial coherence across the Floor and Raised sessions. Spatial coherence was significantly decreased on the raised surface. H) Comparison of grid scores across the Floor and Raised sessions. Grid scores were significantly decreased on the raised surface. I) Comparison of rate map peak firing rates across the Floor and Raised sessions. Peak firing rates were significantly decreased on the raised surface. * indicates significance (P < 0.05). See also Figures S1, S2, S3, Tables S1, S2, and Data S1.

We considered several possible frameworks by which grid cells could represent the vertically displaced surfaces (Figure 1B). One possibility was that they would construct a single volumetric representation of the environment, such that the firing pattern on the top surface would be continuous with the firing pattern on the bottom surface. For example, the 2D lattice pattern of circular grid fields on the arena floor could actually represent a 2D slice through a 3D lattice of spherical firing fields^19-22, which would indicate that grid cells function to efficiently represent 3D volumetric space. The most efficient spherical packing schemes are known as hexagonal close packed (HCP) and face-centered cubic (FCC) arrangements. While both of these arrangements would predict consistent field spacings and orientations on the Raised surface compared to the Floor, the predicted placement of the firing fields on the Raised surface differ depending on the packing scheme and where the Raised 2D surface intersects the 3D lattice. The HCP scheme predicts that the second vertical layer of grid fields would occupy the empty spaces between the grid fields of the Floor environment (e.g., negative rate map correlation between Floor and Raised), while the third layer would be identical to the first layer (positive correlation between Floor and Raised). In contrast, FCC would predict that the first three layers are offset from each other (negative correlation between Floor and Raised). Firing fields could also be arranged in an irregular volumetric pattern (as observed in truly volumetric environments^11,12), such that the fields on the Raised surface would be expected to have a random, non-periodic arrangement (~0 rate map correlation between Floor and Raised), implying that grid cells encode 3D volumetric space but lack a global lattice structure (though they could still maintain local order¹²). The final volumetric scheme we considered was a columnar arrangement of firing fields, which would predict identical firing patterns on the Floor and Raised surfaces (positive rate map correlation) and would imply that grid cells encode two vertically displaced surfaces with identical representations (as observed in a helix⁹).

We also considered several possibilities for planar coding by grid cells (Figure 1B). One possibility, which would be indistinguishable from the volumetric columnar scheme noted above, is that the Floor and Raised surfaces would show identical planar grid patterns (positive rate map correlation). Alternatively, grid cells could differentiate between the two surfaces by translating and/or rotating the planar grid pattern on the Raised surface relative to the Floor pattern. If the Raised pattern were the result of pure translation and not rotation, the grid orientation and spacing would be consistent between the two surfaces, but the field locations would be offset (~0 rate map correlation). If the grid cells follow this ‘pure translation’ schema, two further possibilities are that 1) the translation is a result of a continuous planar representation of the navigable space involving the Floor surface, the ramp, and the Raised surface; or, 2) the translation is random. If the translation were the result of a continuous planar representation of the environment, the distance and direction of translation would be expected to be consistent across animals and across different groups of grid cells recorded from the same animal. In contrast, a random translation would not be expected to produce this consistent result. Finally, the pattern on the Raised surface could be both rotated and translated relative to the Floor surface (~0 rate map correlation), such that the spacing but not the orientation nor placement of grid fields would be consistent between the two surfaces. These planar frameworks imply that grid cell representations are more strongly tied to local behavioral affordances (i.e., the navigable surfaces) than the continuous 3D space that contains them.

We first computed correlations between firing rate maps for grid cells recorded in the Floor and Raised sessions (Figure 1C, D; Data S1D). While firing rate maps were positively correlated between the two Floor sessions (repeated measures ANOVA: F(1, 52) = 389.88, P = 8.06e-26; mean Pearson $r=0.71\pm 0.021$; $t$-test against 95^th percentile shuffle mean: $t(52)=29.14$, P = 6.96e-34), there was no overall correlation between the Floor 1 session and the Raised session (mean Pearson $r=0.036\pm 0.023$; $t$-test against 95^th percentile shuffle mean: $t(52)=1.31$, P = 0.19; Figure 1D, suggesting against regular volumetric lattice, volumetric columnar, or planar repetition accounts of grid cell firing on the two surfaces (Figure 1B). A small number of grid cells (n = 7) were recorded during this experiment on two separate days, and these cells showed the same spatial firing patterns on both days (Data S1E). We were additionally interested in potential differences between grid cells with different field spacings, as those with larger spacings have been shown to be more likely to display columnar firing fields in a complex 3D environment¹¹. We found that most of the grid cells fell into one of two “modules” that clustered around a particular field spacing: one with a mean spacing of 58.22 cm (N = 33 cells), and the other with a mean spacing of 82.53 cm (N = 12 cells; Figure S2A). However, neither group displayed significantly positive correlations between the Floor 1 and Raised sessions (Module 1: mean Pearson $r=\text{-}0.002\pm 0.021$; t(32) = −3.68, P = 8.58e-4; Module 2: mean Pearson $r=0.11\pm 0.066$; t(11) = −0.15, P = 0.88; Figure S2B), suggesting that differences in grid field spacing did not affect the rearrangement of field locations in response to movement between the two surfaces.

We also assessed several additional aspects of spatial coding by grid cells on the Floor and Raised surfaces. There was no observed change in field spacing across sessions (mean number of detectable fields per session: Floor 1 = 2.26, Raised = 1.91, Floor 2 = 2.62; n = 30 cells with at least two detectable fields in all sessions; repeated measures ANOVA: F(2, 58) = 1.35, P = 0.27; Figure 1E). However, there was a significant increase in field radius in the Raised session (n = 50 cells with at least one detectable field on in all sessions; ANOVA: F(2, 98) = 5.86, P = 0.0039; paired $t$-tests, Floor 1 vs. Raised: $t(49)=\text{-}2.88$, P = 0.018; Floor 1 vs. Floor 2: $t(49)=0.25$, P > 0.99; Figure 1F), along with decreases in spatial coherence (orderliness of firing rates across adjacent location bins; see Methods; repeated measures ANOVA: F(2, 104) = 27.15, P = 3.26e-10; paired $t$-tests, Floor 1 vs. Raised: $t(52)=7.29$, P = 5.16e-9; Floor 1 vs. Floor 2: $t(52)=\text{-}0.77$, P > 0.99; Figure 1G), grid score (repeated measures ANOVA: F(2, 104) = 14.35, P = 3.13e-6; paired $t$-tests, Floor 1 vs. Raised: $t(52)=4.71$, P = 5.74e-4; Floor 1 vs. Floor 2: $t(52)=-0.35$, P > 0.99; Figure 1H), and peak firing rate (repeated measures ANOVA: F(2, 104) = 6.82, P = 1.65e-3; paired $t$-tests, Floor 1 vs. Raised: $t(52)=2.54$, P = 0.042; Floor 1 vs. Floor 2: $t(52)=1.32$, P = 0.58; Figure 1I). None of these measures exhibited any correlation with the number of exposures the animal had to the two-level environment (up to 11 exposures; Figure S2C), implying that the changes in field size, spatial coherence, grid score, and peak firing rate were associated with differences between the Floor and Raised surfaces instead of being a response to environmental novelty (which can cause grid field expansion²³). These results suggest that, while grid spacing was overall preserved, spatial coding by grid cells may have been less precise on the transparent wall-less surface than in the standard condition with opaque walls and floor.

We next asked if grid cells maintained the orientation of their firing patterns between the two surfaces. We created spatial autocorrelograms for the Floor 1 and Raised rate maps, and computed a spatial correlation between the autocorrelograms following systematic rotations (from −90° to 90°) of the Floor 1 autocorrelogram. If a cell’s orientation was consistent between the two surfaces, the correlations should be highest at rotations of −60°, 0°, and 60° (Figure 2A). This result is what we found (Figure 2B), suggesting that the grid pattern on the Raised surface generally had the same orientation as the Floor surface. We next assessed the translation of the grid pattern between the two surfaces by computing a cross-correlogram between the Floor 1 and Raised rate maps and computing the distance and direction from the origin to the closest peak ²⁴; Figure 2C). Translations of the grid pattern were highly consistent among simultaneously recorded grid cells, although they differed strongly between animals (Figure 2D, E) and between different groups of grid cells recorded on different days from the same animal (ANOVA: F(2, 1375) = 47.70, P = 9.35e-21; $t$-tests, same session vs. different session: $t(103.24)=\text{-}4.10$, P = 2.50e-4; same session vs. different animal: $t(87.40)=\text{-}7.30$, P = 3.81e-10; different session vs. different animal: $t(761.36)=\text{-}6.55$, P = 3.08e-10; Figure 2C, E). Overall, the fact that grid orientation and spacing were consistent between the Floor and Raised surfaces suggests that neither irregular volumetric packing nor planar rotation and translation can explain grid cell firing on the two surfaces (Figure 1B), leaving pure planar translation as the most likely explanation. Further, as the specific translation differed between animals and between different groups of grid cells recorded from the same animal, it is unlikely that the grid cells encoded a single continuous planar representation of the global environment (which would cause a systematic and consistent shift in grid field locations following locomotion between the two surfaces), but rather that they represented the Floor and Raised environments as distinct contexts by exhibiting a random translation between the two surfaces. However, the consistent grid orientation suggests that the two surfaces were encoded as separate contexts within a single global environment, which is consistent with the firing of head direction cells during locomotion on multiple horizontal surfaces within the same room^25,26.

Figure 2. Grid cells translate but do not rotate their firing patterns between vertically displaced surfaces.

A) Analysis of grid orientation between the Floor and Raised surfaces for an example grid cell. Left, the autocorrelogram for the Floor session was systematically rotated and correlated with the unrotated Raised autocorrelogram. Right, correlations peaked at −60°, 0°, and 60°, suggesting that the orientation of the grid pattern was consistent between the Floor and Raised surfaces. B) Mean result of correlation analysis for all grid cells (± standard deviation (light gray), standard error (dark gray)). Note that peaks occur at −60°, 0°, and 60°, suggesting consistent grid orientation between the two surfaces. C) Floor 1 and Raised firing rate maps, along with their cross-correlations, for two separate sets of simultaneously recorded grid cells recorded from the same animal one month apart. Note that the translation of the grid pattern (red line on cross-correlogram) was consistent among co-recorded cells, but differed between the two distinct sets of cells recorded one month apart. D) Floor 1 and Raised rate maps, along with cross-correlations, for three grid cells recorded from three different animals (different from the animal in C). Note that the grid pattern translations differed between those three animals, as well as from the cells shown in (C). E) Histogram of Euclidean distance between grid pattern translations for pairs of cells recorded in the same recording session (top), cells recorded from the same animal but in different recording sessions (middle), and cells recorded from different animals (bottom). Note that translations were highly consistent between cells recorded in the same session. See also Data S1.

Grid cells do not differentiate between active and passive movement to the elevated surface

Of the 53 grid cells recorded in the vertical displacement experiment, we recorded 39 of them (n = 4 rats) in two subsequent recording sessions (Figure 3A). Following the Floor 2 session, we passively moved the animals from the floor to the elevated surface (Passive Raised session), skipping active locomotion up the ramp. Following this session, the animals were then passively placed back into the Floor environment for a final recording session (Floor 3 session). As grid cells have been implicated in path integration mechanisms, we expected grid patterns to differ on the elevated surface following passive movement compared to actively locomoting, as the animals’ movement patterns differed between the two conditions. In agreement with the active-only experiment, pairs of Floor sessions showed significantly positive rate map correlations (repeated measures ANOVA: F(4, 152) = 232.15, P = 2.68e-45; Floor 1 vs. Floor 2, mean $r=0.71\pm 0.025$, $t(38)=23.69$, P = 2.29e-24; Floor 2 vs. Floor 3, mean $r=0.69\pm 0.027$, $t(38)=20.32$, P = 5.22e-22), while pairs of Floor and Raised sessions did not (Floor 1 vs. Active Raised, mean $r=\text{-}0.0076\pm 0.021$, $t(38)=\text{-}3.84$, P = 4.53e-4 (significantly lower than 95^th percentile shuffle mean); Floor 2 vs. Passive Raised, mean $r=\text{-}0.014\pm 0.024$, $t(38)=\text{-}3.64$, P = 7.86e-4 (significantly lower than 95^th percentile shuffle mean); Figure 3B, C; Data S1D). Critically, rate maps showed significant positive correlations between the Active Raised and Passive Raised sessions (mean $r=0.58\pm 0.028$, $t(38)=16.30$, P = 9.69e-19). Thus, despite the expected differences in path integration between active and passive movement to the elevated surface, grid patterns were largely consistent across the two raised sessions, suggesting that allocentric elements of the environment were sufficient to maintain a consistent grid pattern across the two conditions. We did not detect any significant differences in field spacing (21 cells with at least two detectable fields across all sessions; repeated measures ANOVA, F(4, 80) = 1.95, P = 0.15; Figure 3D) or field radius (35 cells with at least one detectable field across all sessions; repeated measures ANOVA, F(4, 136) = 3.194, P = 0.015; all pairwise comparisons P > 0.05; Figure 3E) across the five sessions. However, rate map spatial coherence and peak firing rates were both significantly reduced in the Raised sessions compared to the Floor sessions (coherence: repeated measures ANOVA, F(4, 152) = 23.85, P = 2.52e-15; paired $t$-tests Floor vs. Raised, all P < 0.001; Floor vs. Floor or Raised vs. Raised all P > 0.05; peak firing rate: repeated measures ANOVA, F(4, 152) = 8.95, P = 2.19e-5; paired $t$-tests, Floor vs. Raised, all P < 0.05; Floor vs. Floor or Raised vs. Raised, all P > 0.05; Figure 3F, G). Grid scores were also decreased in the Active Raised session compared to the Floor (repeated measures ANOVA, F(4, 152) = 8.51, P = 3.20e-6; paired $t$-test, Floor 1 vs. Active Raised, $t(38)=3.83$, P = 4.68e-3) although the decrease in the Passive Raised condition did not reach significance (paired $t$-tests, Floor 2 vs. Passive Raised, $t(38)=2.75$, P = 0.091; all Floor vs. Floor or Raised vs. Raised P > 0.05; Figure 3H). Overall, grid cells appeared to represent the Raised surface as a single environment, distinct from the Floor environment, regardless of active or passive movement to the raised environment. However, their firing patterns were somewhat less coherent and grid-like on the wall-less transparent surface than in the Floor environment with opaque walls and floor.

Figure 3. Grid cells do not differentiate between active and passive movement to the elevated surface.

A) Schematic view of the recording arena used in the active vs. passive experiment (baseline recording session not shown). After the initial Floor 1 session, animals actively locomoted up the ramp to the elevated surface for the Active Raised session, after which they actively locomoted down the ramp for the Floor 2 session. Following this session, the animals were passively moved by an experimenter from the floor to the elevated surface for a Passive Raised session, after which they were passively moved back to the floor for the Floor 3 session. B) Path and spike plots and firing rate heat maps for three example grid cells recorded across the five sessions of the active vs. passive experiment. Note that rate maps were similar across the floor sessions and across the raised sessions, but differed between the floor and raised sessions. Numbers above each rate map indicate the peak firing rate for the session. C) Comparison of correlations between subsequent pairs of Floor and Raised sessions. Note that the correlations between pairs of Floor sessions and between Active Raised and Passive Raised sessions were significantly positive, while Floor vs. Raised correlations did not exceed a 95^th percentile shuffle mean. D) Comparison of grid field radii across the five sessions of the experiment. No significant differences were observed. E) Same as (D) but for grid field spacing. F) Comparison of rate map spatial coherence across the five sessions of the experiment. Spatial coherence was significantly lower in the Raised sessions than in the Floor sessions. G) Comparison of rate map peak firing rates across the five sessions of the experiment. Peak firing rates were significantly lower in the Raised sessions than in the Floor sessions. H) Comparison of grid scores across the five sessions of the experiment. Grid scores were generally decreased in the Raised sessions than in the Floor sessions. * indicates significance (P < 0.05). See also Data S1.

Non-grid spatial cells on vertically separated surfaces

In addition to grid cells, we also recorded neurons across these manipulations that could be classified as border cells (active experiment, n = 5 cells from 2 rats; passive experiment, n = 1 cell from 1 rat) or non-grid spatial cells (active experiment, n = 37 cells from 6 rats; passive experiment, n = 25 cells from 4 rats; Data S1B, C). Non-grid spatial cells were defined as location-modulated neurons that did not pass criteria for grid or border tuning²⁷. These cells could resemble place cells, or could have more diffuse or multi-peaked firing patterns that did not meet grid cell criteria. Due to the small number of border cells, we focused on the firing properties of non-grid spatial cells, although rate maps for all border cells are provided in Data S1F for completeness. As with grid cells, non-grid spatial cells recorded in the active experiment showed higher rate map correlations between the two Floor sessions than predicted by a shuffle distribution (mean $r=0.55\pm 0.025$, $t(36)=17.81$, P = 1.97e-19), but did not exhibit a significant correlation between the Floor and Raised sessions (mean $r=0.13\pm 0.033$, $t(36)=1.26$, P = 0.22; Figure 4A, B; Data S1G). As with grid cells, rate map spatial coherence was decreased in the raised session compared to the floor sessions (F(2, 72) = 9.45, P = 4.98e-4; paired $t$-tests, Floor 1 vs. Raised: $t(36)=3.60$, P = 2.83e-3; Floor 1 vs. Floor 2: $t(36)=\text{-}0.12$, P > 0.99; Figure 4C), although peak firing rates did not differ across sessions (F(2, 72) = 0.27, P = 0.76; Figure 4D). To further investigate whether the transparent wall-less platform disrupted the firing of non-grid spatial cells, we also analyzed the spatial information content of the non-grid spatial cells, which did not differ across sessions (F(2, 72) = 0.088, P = 0.92; Figure 4E).

Figure 4. Non-grid spatial cells on vertically separated surfaces.

A) Path and spike plots and firing rate heat maps for an example non-grid spatial cell recorded across sessions of the active-only experiment. B) Comparison of rate map correlations between Floor 1 and Floor 2 sessions and between Floor 1 and Raised sessions for all non-grid spatial cells. Note that only the two Floor sessions were significantly correlated with each other. C) Comparison of rate map spatial coherence across the Floor and Raised sessions, showing decreased coherence in the Raised session. D) Comparison of rate map peak firing rate across the Floor and Raised sessions. No significant differences were found. E) Comparison of spatial information content across the Floor and Raised sessions. No significant differences were found. F) Path and spike plots and firing rate heat maps for two example non-grid spatial cells recorded in the active vs. passive experiment. G) Comparison of rate map correlations between subsequent pairs of Floor and Raised sessions. Note that the correlations between pairs of Floor sessions and between Active Raised and Passive Raised sessions were significantly positive, while Floor vs. Raised correlations did not differ from a 95^th percentile shuffle mean. H) Comparison of rate map spatial coherence across the Floor and Raised sessions. Spatial coherence was generally decreased in the Raised sessions. I) Comparison of rate map peak firing rates across the Floor and Raised sessions. No significant differences were found. J) Comparison of spatial information content across the Floor and Raised sessions. No significant differences were found. For (A) and (F), numbers above each rate map indicate the peak firing rate for that session. * indicates significance (P < 0.05). See also Data S1.

For the 25 non-grid spatial cells recorded in the passive experiment, significant positive correlations were observed between pairs of Floor sessions and between the Active and Passive Raised sessions, but not between Floor and Raised sessions (Floor 1 vs. Floor 2, mean $r=0.54\pm 0.026$, $t(24)=15.99$, P = 2.68e-14; Floor 2 vs. Floor 3, mean $r=0.52\pm 0.029$, $t(24)=13.51$, P = 1.03e-12; Floor 1 vs. Active Raised, mean $r=0.14\pm 0.034$, $t(24)=1.29$, P = 0.21; Floor 2 vs. Passive Raised, mean $r=0.092\pm 0.031$, $t(24)=\text{-}0.12$, P = 0.90; Active Raised vs. Passive Raised, mean $r=0.27\pm 0.042$, $t(24)=4.20$, P = 3.16e-4; Figure 4F, G; Data S1G). As with the active-only cells, spatial coherence was generally decreased in both Raised sessions compared to the Floor sessions (repeated measures ANOVA, F(4, 96) = 11.61, P = 9.96e-8; all Raised vs. Floor pairwise comparisons P < 0.05, except for Active Raised vs. Floor 3: P = 0.12; no other significant comparisons; Figure 4H). There were no pairwise differences between sessions in terms of peak firing rate (repeated measures ANOVA, F(4, 96) = 0.74, P = 0.57; Figure 4I) or spatial information content (repeated measures ANOVA, F(4, 96) = 3.23, P = 0.030; no significant pairwise comparisons; Figure 4J) for the non-grid spatial cells. Overall, non-grid spatial cells appeared to represent the two vertically displaced surfaces with distinct location preferences, similar to grid cells.

Discussion

Our results demonstrate that grid cells represent vertical displacement between two visually distinct planar surfaces that are vertically separated from one another by translating (but not rotating) their planar grid patterns between the two surfaces to create two distinct maps. Further, grid patterns on the elevated surface were highly similar across exposures regardless of whether the animals actively locomoted up a ramp or were passively placed on the higher surface. Interestingly, grid cells appeared to encode space with less precision on the elevated surface with transparent floor and no walls than on the floor surface with opaque floor and walls. Non-grid spatial cells exhibited a similar pattern, changing their location preferences between the surfaces without strongly distinguishing between passive and active movement and displaying less coherent firing patterns on the Raised surface.

While the two surfaces were distinguished by translated grid patterns, grid cells maintained a consistent orientation between the two surfaces with respect to the global room reference frame. Global orientation cues have been shown to exert strong influence on grid orientation²⁸, which is consistent with the idea that head direction cells, which encode an animal’s orientation in a largely global reference frame^25,26,29, are responsible for setting the orientation of the grid pattern^30,31. Head direction cells have been shown to maintain consistent preferred directions with respect to a global reference frame on different horizontal surfaces within the same room^25,26. While we did not record head direction cells in the current study, it seems likely that they would maintain their global orientation preferences between the two surfaces.

Grid cells have been previously shown to either redistribute their field-specific firing rates^27,32 or translate and/or rotate their grid patterns in response to environmental changes^24,33. Generally, changes to the local environment in the same room tend to cause firing rate redistribution²⁷ or pure translation^24,33 of grid patterns, whereas recordings in different rooms lead to both rotation and translation^24,27. In one study, recordings in identical square arenas in two separate rooms caused rotation and translation of grid patterns between the environments, whereas changing from a square to a circular arena in the same room purely caused translation of the grid pattern²⁴. Similarly, nonmetric alterations to a single square arena (i.e., changes in color and/or scent but not shape or size) induced translation but not rotation of grid patterns³³. These results are similar to those of the current study, such that the orientation, but not the translational offset of the grid pattern, was preserved across different local contexts (i.e., different floors) within the same global environment. Thus, it is likely that grid cells represented the two surfaces as different contexts within the same overall room environment, providing insight into how the brain represents, for example, visually distinct floors within the same building (or, more accurate to our apparatus design, between a floor inside the building and the rooftop). Another study found that changes to arena shape or color in the same room induced new location preferences by nongrid spatial cells but only reorganization of field-specific firing rates by grid cells²⁷. In contrast, both grid and non-grid spatial cells distinguished between the two vertically separated surfaces by reorganizing their location preferences in the current study, suggesting that the Floor and Raised environments were perceived as sufficiently distinct surfaces to cause grid pattern translation, likely due to the vertical separation of the surfaces in addition to their floor and boundary differences.

How might the observed grid pattern translation between the two surfaces relate to the remapping of hippocampal place cells? While grid cells often shift the placement and orientation of their firing patterns in different environments or contexts, they do so coherently, such that colocalized grid cells with the same field spacing will always translate and rotate by the same amount^24,27,33-35. In contrast, ‘global’ or ‘complex’ remapping among hippocampal place cells refers to a phenomenon wherein exposure to two separate environments or contexts causes a random redistribution of place cell firing fields, such that the shift in one place cell’s firing field (or whether that place cell will be active at all) cannot be predicted from another place cell^24,36,37. Despite these distinct phenomena, translation and/or rotation of entorhinal grid patterns has been previously demonstrated to co-occur with global remapping of hippocampal place cells²⁴. It is therefore likely that hippocampal place cells would display global remapping between the two surfaces in the current experiment.

We considered the possibility that grid cells formed a continuous global representation of the Floor and Raised environments connected by the ramp, which should lead to a predictable and consistent translation of grid fields between the Floor and Raised environments. In contrast, while simultaneously recorded grid cells translated their firing fields by the same amount, comparisons between animals (or between different groups of grid cells recorded from the same animal) revealed an overall random assortment of translations, suggesting that the surfaces were encoded as distinct contexts rather than a single continuous planar environment. It is worth noting that grid cells are able to form a continuous global representation of connected environments when they lie in the same horizontal plane given prolonged experience (> 15 exposures³⁸), whereas the current study included a maximum of 11 exposures (Figure S2C). It is therefore possible that a more continuous representation could have emerged given additional experience with the experimental apparatus. It is also important to note that in some continuous environments (such as a hairpin maze) grid cells have been shown to form fragmented maps³⁹.

Our results imply that grid cell representations on visually distinct, vertically displaced horizontal surfaces are planar instead of volumetric. This result provides insight into how the entorhinal cortex represents vertically separated discrete horizontal surfaces, which has not been investigated previously, and indicates that rodent grid cells primarily encode the layout of navigable surfaces rather than the 3D space that contains them. This distinction may relate to the inherent navigational constraints of terrestrial animals such as rats, which are bound to travel along local surfaces^21,22, and is more consistent with grid cell models that emphasize planar coding of space^21,22 rather than truly volumetric 3D coding^19,20. This experiment also provides a necessary bridge between traditional grid cell recordings in 2D environments and more recent 3D experiments by demonstrating how grid cells represent and distinguish between 2D surfaces separated in 3D space. However, because the experimental surfaces were two discrete surfaces, we were unable to assess how grid cells may represent the space between those two surfaces. One interesting follow-up experiment would be to allow foraging on multiple intermediate levels between the floor and the most elevated surface. Grid patterns may 1) translate their patterns on each floor, 2) show an identical pattern on each elevated surface due to the same transparent floor (similar to findings in a helix⁹), or 3) because this experiment would approach more of a continuous change in elevation, they may show volumetric (though likely irregular¹¹) packing of their grid fields.

Given the proposed and experimentally observed influences of path integration on grid cell firing^5,14,17, we were surprised to find that grid cells exhibited the same firing pattern on the elevated surface regardless of whether the animals actively locomoted up a ramp or were passively moved by the experimenter. This result has implications for models of grid pattern formation that rely primarily on path integration processes^15,16, as the distinct patterns on the Floor and Raised surfaces could not be predicted from the animal’s path between the floors, and therefore other sensory or spatial properties of the environment appear to have dictated the placement of the grid fields, which other models of grid pattern formation have focused on^40,41. It is possible that the lack of opaque floor and walls on the top surface in the current study caused path integration to be less precise, and therefore grid cells became more reliant on cues that indicated the animal’s allocentric location on the surface. Alternatively, the sensory cues present on the two surfaces may have been sufficiently distinct to override path integration processes and induce a random translation of the grid pattern. It is worth noting that the lack of influence of path integration in the current study differs from a previous study of head direction cells during movement from a horizontal surface to a vertical surface, in which active movement up a ramp caused the cells to reference their directional firing to the global room reference frame, while passive movement to the vertical surface caused them to use the local surface reference frame⁴². Further research should investigate the accuracy of grid cell firing patterns on surfaces with different tactile and visual properties, and whether some surfaces support path integration mechanisms better than others, which might force the use of allocentric spatial coding.

STAR Methods

EXPERIMENTAL MODEL AND SUBJECT DETAILS

Subjects were 7 female Long-Evans rats aged 4-7 months and weighing 230-310 grams at the start of testing. Rats were individually housed in Plexiglas cages and maintained on a 12 hour light/dark cycle. Prior to surgery, food and water were provided ad libitum. All experimental procedures involving the rats were performed in compliance with institutional standards as set forth by the National Institutes of Health Guide for the Care and Use of Laboratory Animals and approved by the Dartmouth Institutional Animal Care and Use Committee.

METHOD DETAILS

Electrode construction

Animals were implanted with a movable microdrive consisting of a bundle of four tetrodes targeting the medial entorhinal cortex (MEC). The tetrodes were constructed by twisting together four strands of 17-μm nichrome wire. These twisted strands were subsequently threaded through a single 26-gauge stainless steel cannula, and the end of each wire was connected to a single pin of a Mill-Max connector. The two center pins of the connector were attached to the cannula, which acted as an animal ground. Three drive screws were secured around the connector using dental acrylic, making the electrode drivable in the dorsal-ventral plane.

Electrode implantation

Animals were anesthetized with isoflurane. They were subsequently placed in a stereotaxic frame, and an incision was made in the scalp to expose the skull. A single craniotomy was drilled above the target structure. Implant coordinates were 0.45 mm anterior to the transverse sinus, 4.6 mm lateral to lambda, and 0.5 - 1 mm ventral to the cortical surface. The electrodes were also angled 10° forward in the sagittal plane, such that the tetrodes were pointing anteriorly. All electrodes were secured to the skull using dental acrylic. One animal also had cannulae bilaterally implanted above the anterior thalamus as part of a separate experiment.

Recovery and behavioral training

Rats were allowed 7 days to recover from surgery, after which they were placed on food restriction such that their body weight reached 85-90% of its pre-surgical level. During this time, the rats were also trained to forage for randomly scattered sucrose pellets within a gray square box (1.2 × 1.2 m; 50 cm in height) surrounded by a uniform black curtain that formed a circle around the square box. The box itself was featureless except for a white cardboard sheet placed along the south inside wall. The floor was composed of gray photographic backdrop paper. Recording began when the animals’ walking paths showed uniform coverage (>80%) of the entire arena during 20-min foraging sessions.

Recording of neural data

Over the course of weeks to months, tetrodes were ‘screened’ for cells as the animals foraged for sucrose pellets in the open arena. Electrical signals were pre-amplified using unity-gain operational amplifiers on an HS-18-MM headstage. Signals from each tetrode wire were then differentially referenced against a quiet channel from a separate tetrode and bandpass filtered (600 Hz to 6 kHz) using a Cheetah 32 Data Acquisition System. If signals on a given tetrode crossed a predefined amplitude threshold (30 to 50 μV), they were time-stamped and digitized at 32 kHz for 1 ms. The headstage was also equipped with red and green light-emitting diodes (LEDs) spaced ~6 cm apart over the head and back of the animal, respectively. A color video camera positioned over the arena captured video frames with a sampling rate of 30 Hz, and an automated video tracker extracted the x- and y-positions of the LEDs as well as their angle in an allocentric frame. The tracking frames were timestamped so they could be matched up to the neural data. If clearly isolated waveforms were visually apparent, a 20-min baseline recording session in the 1.2 x 1.2 m square box took place. If no clear waveforms were detected, electrodes were advanced ~50 to 100 μm and screened again at least 2 hours later or the next day.

Spike sorting

Spike sorting was conducted offline. Spikes collected from a recording session were first automatically sorted into clusters using the automated clustering program Kilosort⁴³, after which manual cleanup was performed using the manual clustering program SpikeSort3D (Neuralynx). If cells were recorded across multiple sessions in a day (i.e., the vertical displacement experiment) automatic sorting was performed on a merged dataset to ensure cluster continuity, and then results were separated into individual sessions for manual corrections and analysis. For the manual step, waveform features including peak, valley, height, width, and principal components were used to visualize the characteristics of individual spikes across multiple wires of a tetrode simultaneously as a 3D scatter plot. Correction of automatically sorted clusters, which was not always required, was performed by drawing a polygon around the visually apparent boundaries of each cluster. Single-unit isolation was assessed using metrics such as L-ratio and isolation distance, as well as assessment of temporal autocorrelograms for the presence of a refractory period. Despite significant advancement of the tetrodes between recording sessions, we sometimes found that the same cells were recorded multiple times on the same tetrodes across recording sessions (based on analyzing waveform shape and location in cluster space); in these cases we only used the first recording session of the cell. For each well-isolated cluster, we saved the timestamps for each spike and then analyzed and matched them to the tracking data.

Elevated surface experiment

If a grid cell was isolated in the baseline session (classification criteria discussed below), the floor paper was changed, and the following series of recording sessions took place (Figure 1A):

Floor 1 session:

The enclosure for this session was visually similar to the 1.2 x 1.2 m baseline enclosure, but slightly smaller at 1 x 1 m. Walls were gray and 50 cm in height, with a large white cue card placed along the south wall. There was also a gray wooden ramp with width ~10 cm placed along the east wall, which extended from the floor to the top of the wall. A wooden block was placed on the ramp to keep the animals from accessing the ramp during the Floor session. Animals were allowed to forage for sugar pellets in this enclosure for a 10-min recording session.

Active Raised session:

The block was removed from the ramp, and the animal was allowed to actively locomote up the ramp to the top of the wall. At this point, a 1 x 1 m Plexiglas sheet was slid into place atop the walls of the Floor enclosure, and the animal was allowed to move from the ramp onto this surface before it was slid over the top of the ramp. The clear Plexiglas floor was used to maximize perceived visual continuity between the Floor and Raised environments. Both the Plexiglas floor and lack of physical boundaries on the Raised surface should have helped the animals to distinguish between the two environments, making them visually and tactilely distinct. Additionally, while distal cues were blocked by a black circular curtain during the entire experiment, this black curtain was more visually salient on the Raised surface due to the absence of surrounding walls compared to the walled enclosure when the rat was on the floor. These different visual properties, along with the rat’s proprioceptive cues when traveling between the two environments, should have enabled the rat to perceive that the two planar environments were distinct, but were located one above the other. Animals were allowed to forage for sugar pellets on this surface for a 10-min recording session.

Floor 2 session:

The Plexiglas sheet comprising the elevated surface was slid back slightly to reveal the ramp, at which point the animal was allowed to actively locomote down the ramp to the Floor surface as the Plexiglas sheet was completely removed. The block was placed back onto the ramp, and the animal was allowed to forage in the Floor enclosure for a 10-minute recording session.

In some cases, two subsequent recording sessions took place (Figure 2A):

Passive Raised session:

After the Floor 2 session, the animal was picked up by an experimenter, the Plexiglas sheet was slid into place on top of the walls, and the animal was placed onto the Plexiglas sheet. A 10-minute foraging session on this surface then took place.

Floor 3 session:

The animal was picked up by an experimenter, the Plexiglas sheet was removed, and the animal was placed onto the Floor surface for a final 10-min recording session.

Animals were only introduced to the two-level environment once a grid cell was isolated in the baseline recording session. Therefore, analyses regarding the number of exposures to the environment (Figure S2C) indicate the total number of times the animal had experienced the two-level environment when the recording took place.

Histology

Once recordings were complete, animals were deeply anesthetized with sodium pentobarbital and small marking lesions were made at the electrode tips by passing a small anodal current (15 μA, 15 to 20 s) through two active wires from separate tetrodes. Animals were then intracardially perfused with saline followed by 10% formalin solution, after which the brains were removed from the skull and postfixed in 10% formalin solution with 2% potassium ferrocyanide for at least 24 hr. The brains were then transferred to 20% sucrose solution for at least 24 hr, after which they were frozen and sliced in the sagittal plane (30 μm sections) using a cryostat. Sections were mounted on glass microscope slides and stained with thionin, after which electrode tracks were examined using a light microscope. Locations of recorded cells were determined by measuring backward from the most ventral location of the marking lesions or, if marking lesions were not visible, the electrode tracks (Figure S3). Delineations of parahippocampal regions were drawn mainly from ^4,44. We were not able to determine electrode placement for one rat; however, given the implant coordinates and the presence of grid cells, we are confident that cells recorded from this animal were in MEC or PaS.

Cell classifications with a generalized linear model

Cells were initially classified as encoding up to three behavioral variables using 10-fold cross-validation with a Poisson generalized linear model (GLM⁴⁵). The behavioral variables were: allocentric head direction, 2D location, and linear speed. Briefly, for a given model, the firing rate vector $r$ for a single cell over all time points was modeled as follows:

$$r=\text{exp}({\sum }_i{X}_i^T{\beta }_i)$$

where $X$ is a matrix containing animal state vectors for a single behavioral variable across time points $T$, $\beta$ represents the parameter vector for that behavioral variable (similar to a tuning curve), and $i$ indexes across behavioral variables included in the model. The parameter vectors for a given model are learned by maximizing the log-likelihood $l$ of the real spike train $n$ given the model’s estimated rate vector $r$:

$$l=\sum _t{n}_t\log (r_t)-r_t-\log (n_t!)$$

Where $t$ indexes over time points. In order to avoid overfitting for the cross-validation procedure, an additional smoothing penalty $P$ was added to the objective function which penalizes differences between adjacent bins of each parameter vector (similar to fused ridge regularization):

$$P=\sum _{i}S\sum _j\frac{1}{2}\ast {({\beta }_{i,j+1}-{\beta }_{i,j+1})}^2$$

Here, $S$ is a smoothing hyperparameter (20 for head direction and speed, 2 for 2D location), $i$ indexes over variables, and $j$ indexes over response parameters for a given variable. Response parameters were estimated by minimizing ($\mathrm{P}-l$) using SciPy’s optimize.minimize function. Thirty bins were used for allocentric head direction parameter vectors, ten bins were used for linear speed, and 400 bins (20 x 20) were used for 2D location.

For cross-validation, data for a session was split into training (9/10 of the session) and test (1/10 of the session) data ($k=10$ folds). Parameter vectors were estimated by minimizing the objective function on the training data using the full model with all four variables. Drawing parameter estimates from the full model helps to reduce correlation artifacts between variables⁴⁶ and makes models with different variable combinations more comparable. Log-likelihoods for models with all possible variable combinations were computed. This procedure was repeated until all portions of the data had been used as test data (10 folds).

To select the best model, the log-likelihood values from the best two-variable model were compared to those from the best one-variable model. If the two-variable model showed significant improvement from the one-variable model (using a one-sided Wilcoxon signed-rank test), then the best three-variable model was compared to the two-variable model, and so on. If the more complex model was not significantly better, the simpler model was chosen. If the chosen model performed significantly better than an intercept-only model, the chosen model was used as the cell’s classification. Otherwise, the cell was marked ‘unclassified’⁴⁵.

Linear speed calculation

For each time point in a session, instantaneous linear speed was calculated by first fitting a line across 5 adjacent x- or y-position values (x and y speeds computed separately) centered on that time point and then extracting the slope of the line as the x- or y-component of speed. The overall speed for each time point was then computed as the length of the vector sum of these components.

Allocentric location firing rate maps

The animal’s two-dimensional location throughout the recording session was divided into 2.5 cm × 2.5 cm bins. For each cell, the number of spikes fired when the animal occupied each bin was divided by the amount of time the animal spent in that bin to calculate a firing rate for each location in the environment. The resulting firing rate heat maps were smoothed with a Gaussian filter with a standard deviation of 3.75 cm. For the Floor and Raised sessions, we truncated the eastern 10 cm of the rate maps to exclude the placement of the ramp, which was not visited in the Floor sessions. Rate map spatial coherence, a measure of the orderliness in the firing rate map across adjacent location bins, was also calculated for each cell. This measure was computed by first estimating the firing rate for each location bin by taking the mean of its eight nearest neighbors, then computing the Pearson correlation between the true and estimated rate maps⁴⁷.

Grid cell classification

To classify grid cells, we computed a grid score^3,30 based on each cell’s firing rate map. Briefly, for each cell, a spatial autocorrelogram was computed for the smoothed allocentric location firing rate map which correlated the map with itself (Pearson $r$) at all possible spatial offsets. If a cell had firing fields that were hexagonally periodic (like a grid cell) this procedure should result in a ring around the center of the autocorrelogram with six evenly spaced peaks. The radius of this ring was used to estimate grid field spacing in the baseline recording session. For each cell, we identified the most probable inner and outer radii of this ring, and then correlated the ring with itself (Pearson $r$) at 3º rotational offsets from 0º to 180º. A hexagonally periodic signal would cause peaks at offsets of 60º and 120º and troughs at offsets of 30º, 90º, and 150º. The grid score was therefore calculated as the difference between the smallest correlation value at 60º and 120º and the largest correlation value at 30º, 90º, and 150º. A neuron was considered a grid cell based on the initial baseline recording session if it (i) passed the GLM classification procedure for location modulation, and (ii) had a grid score > 95^th percentile of a within-cell shuffle distribution.

Grid cell firing field analyses

To detect grid cell firing fields, each cell’s allocentric location firing rate map was thresholded to remove bins with firing rates lower than 30% of the rate map’s peak firing rate. Individual grid fields were then detected as contiguous groups of bins with an area of at least 150 cm². Grid field spacings on the two-level apparatus were then computed as the mean pairwise distance between the centers of mass of each grid field, and grid field sizes were computed as the mean radius of the detected fields. Based on the baseline recording session, grid cells were sorted into two apparent spacing “modules” by fitting a bimodal Gaussian mixture to a histogram of the field spacings, and then assigning grid cells to each “module” if they fell within three standard deviations of the mean of that module’s associated Gaussian fit (Figure S2A).

Grid cell rotation and translation analyses

To determine if grid cells rotated their firing patterns between the Floor and Raised environments, we first created spatial autocorrelograms for each cell’s rate map in each environment. Then, in steps of 1° from −90° to 90°, the Floor 1 autocorrelogram was rotated relative to the Raised autocorrelogram, and the correlation between the two (Pearson $r$) was computed. If grid patterns were consistent between the two surfaces, this correlation function would be expected to have peaks at −60°, 0°, and 60°.

To determine the extent to which grid cells translated their firing patterns across space between the Floor and Raised environments, we computed a spatial cross-correlation between the Floor 1 and Raised rate maps. We then identified individual fields of the cross-correlogram by thresholding the cross-correlogram to remove bins with correlation values lower than 20% of the peak correlation value and finding contiguous groups of bins with area of at least 150 cm². The translation of the cell’s firing pattern was then defined as the vector from the origin to the center-of-mass of the closest field²⁴.

Border cell classification

To classify border cells, we calculated a border score⁴⁸ based on each cell’s allocentric firing rate map. We first thresholded the rate maps to exclude all bins with firing rates < 20% of the maximum firing rate. We then identified firing fields as groups of contiguous bins with an area of at least 200 cm², after which we computed the maximum coverage of any single wall by any single firing field (expressed as a fraction $c$). We then computed the mean distance from the nearest wall of all bins belonging to any firing field (normalized by their firing rates), and normalized the resulting value by one half the longest length of any wall to calculate the value $d$. The border score was then computed as follows:

$$\text{Border score}=(c-d)∕(c+d)$$

A neuron was considered a border cell based on the initial baseline recording session if it (i) passed the GLM classification procedure for location modulation, (ii) could not be classified as a grid cell, and (iii) had a border score > 95^th percentile of a within-cell shuffle distribution.

Classification of non-grid spatial cells

To classify non-grid and non-border cells as spatially modulated, we calculated spatial information content for each cell. This was computed based on smoothed allocentric firing rate maps and smoothed spatial occupancy histograms using the following equation⁴⁹:

$$\text{Spatial information}={\sum }_i{P}_i\frac{R_i}{R}{\log }_2\frac{R_i}{R}$$

Where $i$ indexes across spatial bins, $P_i$ is the probability of the animal occupying bin $i$ (taken from the smoothed occupancy histogram), $R_i$ is the mean firing rate of the cell in bin $i$ (taken from the smoothed firing rate map), and $R$ is the mean firing rate across all spatial bins. A neuron was considered a non-grid spatial cell based on the initial baseline recording session if it (i) passed the GLM classification procedure for location modulation, (ii) could not be classified as a grid or border cell, and (iii) had spatial information content > 95^th percentile of a within-cell shuffle distribution.

Classification of head direction cells

To classify cells as being tuned to the animal’s allocentric head direction, we constructed head direction tuning curves using 6° bins. For each cell, the number of spikes associated with each head direction bin was divided by the total amount of time that bin was occupied. We then computed the mean vector length of the resulting tuning curve to indicate tuning strength. A cell was classified as an HD cell based on the initial baseline recording session if it: (i) passed the GLM classification procedure for head direction modulation, (ii) had a mean vector length > 95^th percentile of a within-cell shuffle distribution, and (iii) had a peak firing rate > 1 Hz in its head direction tuning curve.

Classification of speed cells

To classify cells as being tuned to linear speed, we constructed speed tuning curves using 4 cm bins from 0 cm/s up to a maximum of 40 cm/s. For each cell, the number of spikes associated with each bin was divided by the time that bin was occupied to obtain a tuning curve. The linear R² fit of this tuning curve was computed to indicate tuning strength. A cell was classified as a speed cell if it: (i) passed the GLM classification procedure for speed modulation, (ii) had a linear R² fit > 95^th percentile of a within-cell shuffle distribution, and (iii) had a peak firing rate > 1 Hz in its speed tuning curve.

Shuffling procedure for cell classifications

Each cell’s spike train was randomly shifted by at least 30 s, with entries beyond the end wrapped to the beginning, to offset the spike data from the behavioral data without interrupting its temporal structure. Relevant tuning scores were then computed based on the shifted spike train. This procedure was repeated 400 times for each cell, and a within-cell 95th percentile cutoff was used to determine tuning significance for individual cells.

Shuffling procedure for assessment of rate map correlations

As the ‘chance’ level for correlations between firing rate maps on the Floor and Raised surfaces may not be 0, given potential biases in animal behavior and firing preferences of grid and non-grid spatial cells, we assessed whether a given population of cells showed overall positive rate map correlations by comparing the correlations to a random shuffle distribution. This procedure involved randomly shuffling the rate map identities for each recording session and computing pairwise correlations between the shuffled rate maps. This procedure was repeated 1000 times to produce 1000 shuffled correlation distributions, after which the mean of each distribution was taken, and the 95^th percentile mean was used as a significance threshold. Correlations for a given group of cells were considered significantly positive if the mean of the distribution exceeded the 95^th percentile shuffle mean according to a one-sample $t$-test.

QUANTIFICATION AND STATISTICAL ANALYSIS

Statistical analyses were performed using Python code. All tests were two-sided (except for GLM classifier cross-validation comparisons^45,50) and used an α level of 0.05. Within-cell comparisons across multiple conditions were assessed using a one-way repeated-measures ANOVA. If samples violated sphericity (assessed using Mauchly's test), we applied a Greenhouse–Geisser correction. Unpaired comparisons were assessed using a one-way ANOVA. Post hoc pairwise comparisons were performed using Bonferroni-corrected $t$-tests (Python package Pingouin). Valueslisted inthe text indicate mean ± standard error unless otherwise specified.

Supplementary Material

1

Data S1. Additional plots for example cells, related to Figures 1-4.

A) 2D location firing rate maps (top row) and 2D spatial autocorrelograms (bottom row) from the baseline recording session in the 120 x 120 cm recording arena for all 53 grid cells recorded in the active experiment, ordered from highest to lowest grid score. Numbers above each rate map indicate the cell’s peak firing rate, while numbers above each autocorrelogram indicate the cell’s grid score.

B) 2D location firing rate maps from the baseline recording session in the 120 x 120 cm recording arena for all 37 irregular spatial cells recorded in the active experiment, ordered from highest to lowest spatial information content. Numbers above each rate map indicate the cell’s peak firing rate.

C) Same as (B) but for all five border cells recorded in the active experiment, ordered from highest to lowest border score. Numbers below each rate map indicate the cell’s border score.

D) Path and spike plots (top row) and 2D firing rate maps (bottom row) for ten additional example grid cells recorded across all sessions of the active and passive experiment.

E) Path and spike plots (top row) and 2D firing rate maps (bottom row) for seven grid cells recorded in the active experiment on two different days. Note the similarity of locational firing preferences across days.

F) Path and spike plots (top row) and 2D firing rate maps (bottom row) for all five border cells recorded in at least the active experiment. The top four cells were only recorded in the active experiment, while the last cell was recorded in both active and passive sessions.

G) Path and spike plots (top row) and 2D firing rate maps (bottom row) for eight example non-grid spatial cells recorded in at least the active experiment. The top two cells were recorded in only the active experiment, while the remaining six cells were recorded in both active and passive sessions.

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Experimental models: Organisms/strains
Long-Evans rats	Envigo
Software and algorithms
Python v3.9	Anaconda	https://www.anaconda.com/
Cheetah v6.3.2	Neuralynx	https://neuralynx.com/software/cheetah
SpikeSort3D v2.5.4	Neuralynx	https://neuralynx.fh-co.com/research-software/spikesort-3d/
KiloSort	Pachitariu et al., 2023 (ref. 43)	https://github.com/cortex-lab/KiloSort
Data and code from this paper	This paper	https://github.com/taube-lab/LaChance_Grid_Translation_2025
Other
Digital Lynx SX acquisition system	Neuralynx	https://neuralynx.fhco.com/research-hardware/data-acquisition/digital-lynx-sx/
S-18-MM headstage	Neuralynx	https://neuralynx.fhco.com/research-hardware/animal-interfaces/headstage-pre-amplifiers/hs-18-mm-led/
17-μm nichrome wire	California Fine Wire	https://calfinewire.com/

Highlights.

1. Rats explore two vertically separated horizontal surfaces connected by a ramp.

2. Grid cells translate, but do not rotate, their firing patterns between the surfaces.

3. Grid cells do not distinguish active vs. passive movement between the surfaces.

4. Non-grid spatial cells also remap their location preferences between the surfaces.