To determine the extent to which variability of grid properties in individual modules exists, and to what extent this variability is a robust property of the grid code, we analyzed previously published MEC recordings (Gardner et al., 2022), which include tens to hundreds of grid cells simultaneously recorded. This allows us to characterize the distribution of grid properties within a single grid module, to an extent not possible with other data sets.

For each grid cell, we compute the grid spacing (λ) and orientation (θ) by measuring properties associated with the six hexagonally distributed peaks of the spatial autocorrelogram (SAC), as traditionally performed (Sargolini et al., 2006; Figure 2A; see Materials and methods). For clarity, we begin by focusing on a single module, recorded from an open field environment (recording identifier: Rat R, Day 1, Module 2 – R12). This module was picked for its long recording time (approximately 130 min of recorded activity, as compared to the other open field recordings that have 30 min or less of recorded activity) and for its large number of simultaneously recorded putative grid cells (N=168). In this module, we find that grid cells with high grid scores (>0.85;N=74) have a range of grid orientation and spacing (example cells shown in Figure 2B; distributions across module shown in Figure 2E and F), with λ ranging from approximately 65 cm to 90 cm and θ ranging from approximately 2∘ to 9∘. Overlaying the SACs of pairs of grid cells with similar grid spacing and different grid orientation (Figure 2C) or vice versa (Figure 2D) enables visualization of the extent of this variability.

Because individual grid fields can be cut-off by the boundaries of the environment, it is possible that computing the grid spacing from the SAC (which considers all grid fields) could lead to an under-estimate of λ for some grid cells. To verify that the broad distribution of grid spacing that we see within the same module is not due to the specifics of the SAC, we recomputed the grid spacing for all grid cells directly from their rate maps, considering only the three grid fields closest to the center of the environment (Figure 2—figure supplement 1A; see Materials and methods). We find that while the grid spacing estimated from the SAC tends to be larger than the grid spacing estimated from the rate maps, a similarly broad range of λ is again present (Figure 2—figure supplement 1B, C).

To assess whether the heterogeneity of grid properties present in a single grid module is a robust feature or attributable to noise (either in the recording or the grid property fitting procedure), we measure the variability in grid orientation and spacing within a single grid cell and between pairs of grid cells. If the heterogeneity is explainable by noise then we expect that the within-cell variability will be of the same magnitude as the between-cell variability. In contrast, if the heterogeneity is a robust feature of the grid code, then we expect the within-cell variability will be significantly smaller than the between-cell variability.

To measure the within- and between-cell variability, we split the recording into evenly spaced 30 s bins, randomly assigning each temporal bin to one of two equal length halves and computing the grid properties for each half of the recording (Figure 3A; see Materials and methods). We set inclusion criteria to filter out cells that do not have consistent hexagonally structured SACs across splits of the recording (see Materials and methods). Although these requirements are strict (see Figure 3—figure supplement 1 for percent of cells rejected), they set a conservative estimate on the amount of grid property variability, ensuring that we do not artificially inflate the variability due to inclusion of unreliable grid cells. We found that the length of the temporal bin used to split the data does not have a large impact on the percentage of cells accepted by this criteria (Figure 3—figure supplement 2; see Materials and methods).

We find that the distribution, across 100 random shuffles of the data into two halves, of within-cell variability of grid orientation and spacing is more concentrated around 0 than the between-cell variability (Figure 3B and C). Comparing the average within- and between-cell variability of grid spacing and orientation reveals that nearly all of the grid cells that passed the criteria for inclusion (N=82) exhibit more between-cell than within-cell variability (Figure 3D and E): 95.1% grid cells for orientation (⟨Δθ¯within⟩=1.2∘, ⟨Δθ¯between⟩=2.0∘;⟨⋅⟩ denotes mean across cells) and 100% of grid cells for spacing (⟨Δλ¯within⟩=1.9cm, ⟨Δλ¯between⟩=6.9cm). A Wilcoxon-Signed-Rank Test indicates that between-cell variability is significantly greater than within-cell variability, for both orientation and spacing (Δθ:p<0.001, Δλ:p<0.001). We perform a number of control analyses to confirm that our results are robust, finding that: (1) the average grid field width does not significantly affect the between-cell variability (Figure 3—figure supplement 3), (2) the amount of between-cell variability does not change significantly across the recording duration (Figure 3—figure supplement 4), (3) the between-cell variability is not significantly impacted by the boundaries of the arena (Figure 3—figure supplement 5), and (4) the amount of between-cell variability is not driven by the presence of cells that have both head direction and grid cell like tuning ‘conjunctive’ cells (Sargolini et al., 2006; Figure 3—figure supplement 6).

In keeping with convention, the reported grid cell properties are the average of those computed for each of the three independent axes in the SAC (Axis 1: aligned to ≈ 0°; Axis 2: aligned to ≈ 60°; Axis 3: aligned to ≈ 60° Stensola et al., 2015). To ensure that this averaging is not contributing to greater between-cell variability, we repeated the analysis above, restricting ourselves to each axis separately. The results again demonstrate that grid properties are significantly more robust within-cell than between-cell (Figure 4A and B). For each axis, the average between-cell variability for every cell was significantly higher than the average within-cell variability (Figure 4C and D), as reported by the Wilcoxon-Signed-Rank Test for orientation (p < 0.001, for all three axes) and for spacing (p < 0.001, for all three axes).

Having demonstrated that grid cell properties are robustly heterogeneous in a single module, we proceed to analyze the remaining recordings in the data set (N=420 cells across 8 modules; one module in the data set had no cells that passed our criteria; Gardner et al., 2022). Although there are differences across recordings, we find larger between- than within-cell variability for grid orientation and spacing is present across all recordings (Figure 5A and B; 80.0% of grid cells have greater between- than within-cell variability for orientation, ⟨Δθ¯within⟩=1.4∘ and ⟨Δθ¯between⟩=1.9∘;87.9% of grid cells have greater between- than within-variability for spacing, ⟨Δλ¯within⟩=1.7cm and ⟨Δλ¯between⟩=3.6cm). A Wilcoxon-Signed-Rank Test finds that these differences are significant (Δθ:p<0.001;Δλ:p<0.001). We do not find evidence suggesting that the within-cell variability is influenced by grid score (Figure 5C and D; linear regression for θ:R2=0.03,p=0.55 Wald Test; linear regression for λ:R2=0.07,p=0.14 Wald Test).

To understand whether the observed grid cell variability may be a heretofore unknown property of MEC computational models, we trained recurrent neural networks (RNN) to perform path integration. These RNN models have previously been shown to develop grid responses (Banino et al., 2018; Cueva and Wei, 2018; Sorscher et al., 2019) and have been argued to develop continuous attractor network structure (Sorscher et al., 2023), making them normative models for MEC function. We find that the RNNs develop grid responses with a distribution of grid spacing and orientation (Figure 3—figure supplement 7A-D), much like the neural data we analyzed (Figure 2). Changing the sparsity of the recurrent layer by increasing the strength of weight decay regularization reduces the variability (Figure 3—figure supplement 7E, F).

Taken together, our analysis of large-scale MEC recordings demonstrates that grid cells in the same grid module do not have a single grid spacing and orientation, but instead have a restricted range of values around the module mean. This property is a robust feature that cannot be explained by noise from the recording or fitting procedures and emerges in existing normative models of MEC.

The variability of grid cell properties within individual grid modules, while statistically significant, is small in magnitude. Can a computational benefit in the encoding of local space be gained from this level of inhomogeneity? How sensitive might such a computational benefit be to the exact amount of variability present?

To address these questions, we generate populations of synthetic grid cells (see Materials and methods), where we have complete control over the number of grid cells and their firing properties. For simplicity, we assume that all grid cells in our population have grid orientation and spacing sampled from Gaussian distributions, with means λ^ and θ^ and standard deviations σλ and σθ, respectively. Assigning each grid cell in our population a grid orientation and spacing, we are able to generate ‘ideal’ rate maps (Solstad et al., 2006). Sampling from a Poisson process on these ideal rate maps, we generate noisy grid cell rate maps (Figure 6A). Using a simple linear decoder (see Materials and methods), we can examine how decoding error of local space is affected by the number of grid cells in the population and the amount of variability (σλ and σθ).

We begin by investigating the decoding capabilities of a synthetic module with properties similar to that of the experimentally recorded module that was analyzed in detail (recording identifier R12; Figures 2 and 3). We therefore set λ^=85cm and θ^=6∘. To determine an appropriate value for σλ and σθ, we subtract the mean between-cell variability by the mean within-cell variability (⟨Δθ¯between⟩−⟨Δθ¯within⟩=2.0∘−1.2∘ and ⟨Δλ¯between⟩−⟨Δλ¯within⟩=6.9cm−1.9cm=5.0cm). We interpret these values as the amount of variability in grid spacing and orientation that is not due to noise. Therefore, we set σθ=1∘ and σλ=5cm. This amount of variability leads to similar sampled distributions of λ and θ as was found in the real data (compare Figure 6B and C with Figure 2E and F).

Applying the linear decoder to populations of synthetic grid cells, we find that decoding error decreases as the number of grid cells is increased (Figure 6D, blue line). When N=1024, the decoding error is ≈ 20 cm, substantially better than random decoding (Figure 6D, dashed black line). As expected, this result is not seen in a synthetic population of grid cells with zero variability in grid properties (σθ=0∘ and σλ=0cm; Figure 6D, black line). In particular, while the restricted number of grid firing fields enables a decoding error smaller than chance, the population with fixed grid properties exhibits little change in decoding error with increasing numbers of grid cells. This demonstrates that the improved encoding is specific to populations with inhomogeneity in their grid spacing and orientation.

To validate that the synthetic population is a reasonable surrogate to the experimentally recorded data, we perform the same decoding analysis on grid cells from recording ID R12 (see Materials and methods). As the real data is limited in the number of cells, we are only able to compare up to populations of N=64 (Figure 6E). However, in this restricted range, we find agreement between the synthetic population with grid property variability and the recorded data (Figure 6E, compare red and blue lines). This supports our hypothesis that the observed amounts of inhomogeneity in grid spacing and orientation can lead to improved decoding of local space.

To determine the extent to which the decrease in decoding error depends the exact variability of grid spacing and orientation, we perform a grid search over 36 pairs of (σλ,σθ) values in [0°, 1°, ... ,5°]×[0 cm, 1 cm, ... , 5 cm]. As expected, when the variability of both grid properties is large, we find nearly 0 cm decoding error of local space (Figure 6E, bottom right). However, there is additionally a range of σλ and σθ values that lead to improved decoding, including values smaller than the values matched to the experimental data (Figure 6E, above and to the left of the white star). We perform the same grid search on synthetic populations of grid cells with different spacings, λ^=50cm and λ^=100cm (Figure 6—figure supplement 1A, B), finding again that the decoding error drops below that of the fixed population with sufficient grid property variability. For the synthetic module with smaller spacing (λ^=50cm), the existence of more grid fields in the 1.5 m × 1.5 m arena leads to greater complexity of interference patterns (Figure 1), enabling the same amount of variability in grid spacing and orientation to lead to a sharper decrease in decoding error. Additionally, for the synthetic module with smaller spacing, a given value of σλ is a larger percentage of λ^, as compared to the synthetic module with larger spacing.

Finally, we consider how decoding within a single module compares to decoding across two modules. When the two modules are consecutive (e.g. modules 1 and 2), their mean grid spacing, λ^1 and λ^2, has been experimentally found to be related via λ^2≈2λ^1 (Stensola et al., 2012). In such a case, decoding activity from populations with fixed grid properties leads to nearly 0 cm error (Figure 6—figure supplement 2A, black line). The addition of variability in grid properties does not significantly change the behavior of the decoding (Figure 6—figure supplement 2A, blue line). This suggests that small amounts of inhomogeneity may not disrupt the previously achieved theoretical bounds on decoding from multiple grid modules (Mathis et al., 2012a; Mathis et al., 2012b; Stemmler et al., 2015; Wei et al., 2015). However, if the two modules being decoded from are non-consecutive (e.g. modules 1 and 3) then the mean grid spacing can be related by an integer multiple, λ^3≈2λ^1. In such a setting, the grid fields of the larger module are a subset of the grid fields of the smaller module, up to a rotation (due to the difference in orientation between the two modules), and we again find that variability in grid properties can improve decoding accuracy (Figure 6—figure supplement 2B, compare black and blue lines). Similar results are found when using experimentally found values of λ^ (Stensola et al., 2012) and not assuming an exact λ^n+1=2λ^n relationship (Figure 6—figure supplement 2C, D).

Taken together, these results suggest that individual grid modules can exhibit significant encoding of local space via heterogeneity in their grid properties, even when the extent of the variability in θ and λ is similar to that found in the analysis of the experimental recordings. This benefit can also improve encoding in cases when multiple, non-consecutive modules are considered.

While the data set we analyzed (Gardner et al., 2022) represents an advance in the ability to simultaneously record from tens to hundreds of putative grid cells, across grid modules, the MEC remains a challenging brain region to access for large-scale neurophysiological experiments. Indeed, with our conservative inclusion criteria, we were ultimately limited by having only 420 grid cells included in our analysis. Future work can perform more detailed and complete characterizations of grid property heterogeneity, as new neurotechnologies that enable larger yield of grid cells are developed (Low et al., 2014; Zong et al., 2022).

Our decoding analysis, while demonstrating the possibility that variability in grid properties can be used by individual grid modules to enhance the encoding of local spatial information, made several simplifying assumptions, including: (1) independent Poisson noise for neural activity, (2) linear decoding, and (3) normal distribution of grid properties. Although comparison to real data showed that these assumptions are reasonable (Figure 6E), future work can assess the extent to which these restrictions can be lifted (e.g. to incorporate correlated neural noise), while still enabling individual grid modules to have low decoding error.

The heterogeneity in grid properties we characterize motivates the investigation of several new lines of research. Because these are directions that we believe to be fruitful for the field as a whole, we outline them below, with our hypotheses for possible answers.

The neural activity analyzed in this paper comes from a publicly available data set, which has previously been described in detail (Gardner et al., 2022). We provide brief summary of the methodology and the experimental paradigms used during the recordings.

Three male rats (Long Evans – Rats Q, R, and S) were implanted with Neuropixels silicon probes (Jun et al., 2017; Steinmetz et al., 2021). These probes were targeted at the MEC-parasubiculum region and the surgery was performed as described previously (Gardner et al., 2019; Steinmetz et al., 2021). After three hours of recovery, recordings were performed.

In the recordings analyzed, the rats foraged for randomly dispersed corn puffs in a 1.5 × 1.5m² square open field arena, with walls of height 50 cm. The rats were familiar with the environment and task, having trained 10–20 times prior to the implantation. The rats were food restricted to motivate their foraging, being kept at a minimum of 90% of their original body weight (300–500 g).

All procedures in the original study were approved by the Norwegian Food and Safety Authority and done in accordance with the Norwegian Animal Welfare Act and the European Convention for the Protection of Vertebrate Animals used for Experimental and Other Scientific Purposes. Protocols were approved by the Norwegian Food Safety Authority (FOTS ID 18011 and 18013).

The neural activity analyzed in this paper was post-processed, before made publicly available. We describe, in brief, the post-processing performed (Gardner et al., 2022), as well as the post-processing we performed on the downloaded data.

Spike sorting, via KiloSort 2.5 (Steinmetz et al., 2021), was applied to the data recorded from the Neuropixel probes. Individual units were deemed putative cells if their average spike rate was in the range of 0.5–10 Hz, and 99% of their interspike intervals were greater than 2ms.

For each putative cell, rate maps were constructed by averaging the activity at binned spatial positions in the open field arena. This raw rate map was smoothed, using a Gaussian kernel. The autocorrelation of these rate maps were computed, and a grid score calculated, as described previously (Sargolini et al., 2006).

From the downloaded spike trains, we constructed rate maps and autocorrelograms in a similar manner, using code made publicly available (Banino et al., 2018; Figure 2).

The public data set we analyzed (Gardner et al., 2022) contained the module identity of all putative grid cells. In brief, these module identities were assigned by first projecting the 2D autocorrelogram of every recorded unit onto a 2D space using the non-linear dimensionality reduction algorithm UMAP (McInnes et al., 2018). Then, the DBSCAN algorithm (Ester et al., 1996) was used to cluster the units, based on their position in the dimensionally reduced space. Cluster membership served as the basis for grid module classification, with the largest cluster being removed as it was found to not contain spatially selective autocorrelograms. This non-supervised module assignment yielded clusters of high grid scores and similar grid spacing and orientation within each cluster-module. More details can be found in Gardner et al., 2022. We performed no further analysis regarding module identity.

Grid spacing and grid orientation were computed according to standard methods described in detail previously (Stensola et al., 2012; Butler et al., 2019). Briefly, the goal of the procedure is to identify the location of the six nearest fields in the spatial autocorrelogram (SAC; Figure 2A). This was achieved by performing the following steps. First, the SAC was smoothed using a 2D Gaussian filter with σ=1. Second, the center of the SAC was excluded by applying a mask. Because the size of the SAC’s central peak changes for every module, the radius of such mask was re-computed for each module. Third, we thresholded the SAC by applying an extended-maxima transform ndimage.maximum_filter. Fourth, we identified the center of every field by using the function scipy.stats.find_peaks.

Once the peaks of every field had been found, we computed the location of every peak in polar coordinates. We then selected the 6 peaks that were closest to the center of the SAC, based on the computed radial components. Because every SAC is symmetric, we considered for further analysis the 3 peaks closest to the X axis in angular distance (Axis 1, 2, and 3 Stensola et al., 2015). Grid spacing was computed as the arithmetic mean of the radial component of the 3 peaks (except when each peak was analyzed separately – Figure 4). Given that the SAC dimensions are twice of that of the real arena, we multiplied the SAC radial mean by a factor of 2. Grid orientation was computed as the angular mean of orientations (relative to the x-axis) of the three peaks (except when each peak was analyzed separately – Figure 4).

In order to ensure that subsequent analysis was performed only on cells whose SAC’s could be well described by hexagonal structure, we imposed the following constraints: (1) the relative angle between two peaks could not be < 30°; (2) the relative angle between two peaks could not be > 90°; (3) the values for grid spacing between the peaks could not be substantially different (the ratio of spacings between any two peaks must be > 0.5 and < 2). Cells that did not meet these criteria were determined to have “Poor grid fit” and were rejected from all subsequent analysis. The percentage of all cells that were removed by this inclusion criteria is shown in Figure 3—figure supplement 1. We note that cells from one of the nine modules in the publicly available data set that we analyzed had 98.4% of cells rejected by this criteria (recording identifier – R23). We therefore did not include it in any subsequent analysis.

To compute the grid score of recorded MEC cells, we made use of previously published code (Banino et al., 2018), that is based on metrics that have become standards in quantifying grid cell properties (Sargolini et al., 2006).

For analysis of the distribution of grid properties (Figure 2E and F), we included only grid cells with grid scores greater than 0.85. This was done to demonstrate that variability was present even in cells that exhibit robust grid cell properties. In the subsequent analyses, an alternative criteria is used, which considers the reliability of the grid responses (see below).

In order to characterize the spacing with a method that is less susceptible to the effects of having grid fields at the boundaries of the environment (as the SAC method could be), we compute the spacing using the rate maps of individual grid cells (Figure 2—figure supplement 1A). Once the rate map was computed, we smoothed it with a Gaussian filter, setting σ=1.5. Then, using the function scikit-image.feature.peak_local_max we extract the position of the center of each firing field. Because our goal with this analysis is to show that the fields in the border do not impact our analysis with the SAC, we restrict ourselves to the 3 peaks that are closest to the center of the arena (Figure 2—figure supplement 1A). We compute the distances between those three peaks to each other and report the spacing as the average of the distances measured.

To characterize the within- and between-cell variability of grid spacing and orientation, we employed the following approach. First, we split the data into bins of fixed length (30 s). From this, we randomly assigned each interval to one of two blocks (denoted as blocks A and B), with exactly half the total number of intervals in each block. For each grid cell, we computed the grid spacing [λA(i) and λB(i)] and orientation [θA(i) and θB(i)], from the data in each block. The within-cell differences of grid spacing and orientation was determined as Δθwithin(i)=θA(i)−θB(i) and Δλwithin(i)=λA(i)−λB(i). Each grid cell’s properties were also compared those of another grid cell, with the match being made using random sampling without replacement. These comparisons were determined as Δθbetween(i,j)=θA(i)−θB(j) and Δλbetween(i,j)=λA(i)−λB(j). This process was repeated 100 times (referring to each iteration as a ‘shuffle’), per recording. Examples of splits of the data, for different shuffles, is schematically illustrated in Figure 3A. The resulting distributions of Δθwithin, Δλwithin, Δλwithin, and Δλbetween, across grid cells and splits of the data were then compared. If the within-cell variability in grid spacing and orientation was smaller than between-cell variability, we concluded that the variability of grid cell properties was a robust feature of the data and not due to noise.

When performing this shuffle analysis, we found that some cells, despite having a good grid fit when all the data was considered, did not have SACs that were well described by hexagonal structure when the data was split in half. We viewed this a manifestation of unreliable grid coding and a possible confound in our quantification of variability. As such, we introduced a new inclusion criteria (replacing that of requiring a grid score of > 0.85), only considering cells that had poor grid fits on < 5% of all shuffles. The percentage of all cells that were removed by this inclusion criteria is shown in Figure 3—figure supplement 1. In general, we found that cells with high grid score were reliable, although there were exceptions. Additionally, we found that size of the bin used for splitting the data did not significantly affect the percent of cells with good grid fits (passed the prior inclusion criteria) that were considered reliable (Figure 3—figure supplement 2).

To study how variability in grid cell properties might endow the grid code with computational advantages, we generated synthetic grid cell rate maps, so that we could have complete control over the distribution of their properties. These synthetic grid cell rate maps were constructed as follows.

First, the lengths of each dimension the ‘arena’ within which the simulated grid cells exist (Lx and Ly) were set. Then, for N∈N grid cells, the grid spacing and orientation were sampled via λ(i)∼N(μλ,σλ) and θ(i)∼N(μθ,σθ) where N(μ,σ) is a normal distribution with mean μ and variance σ2. Grid phase was sampled as ϕ(i)∼U([0,Lx]×[0,Ly]), where ϕ is a two dimensional vector, with first component uniformly sampled from [0,Lx] and second component uniformly sampled from [0,Ly]. To construct a population with no variability in grid properties (to use as a control), we set σλ=0m. and σθ=0∘.

For each grid cell, we generated idealized grid responses by summing three two-dimensional sinusoids (Solstad et al., 2006), such that the activity at x=(x,y)∈[−Lx/2,Lx/2]×[−Ly/2,Ly/2] is given by(1)X(i)(x)=Ximax23(13∑j=13cos⁡[kj(θi)(x+ϕi)]+12),

where Ximax is the maximal firing rate and kj(θi) are the wave vectors with 0°, 60°, and 120° angular differences(2)k1(θi)=k2⋅[cos⁡(θi+π/12)+sin⁡(θi+π/12),cos⁡(θi+π/12)−sin⁡(θi+π/12)]k2(θi)=k2⋅[cos⁡(θi+5π/12)+sin⁡(θi+5π/12),cos⁡(θi+5π/12)−sin⁡(θi+5π/12)]k3(θi)=k2⋅[cos⁡(θi+3π/4)+sin⁡(θi+3π/4),cos⁡(θi+3π/4)−sin⁡(θi+3π/4)],

where k=4π/(3λi).

To match the recorded neural data, where individual grid cells have distinct maximal firing rates, Ximax∼N(13,8). We enforced Ximax to be within [2, 30], by setting any sampled values outside of this range to the boundary values (i.e. 2 or 30).

To determine the extent to which local spatial information can be decoded from the activity of populations of grid cells with different degrees of variability in their grid properties, we performed the following analysis.

We generated noisy synthetic spike rates of N grid cells by assuming a Poisson process and sampling using the idealized rate maps (Equation 1). More concretely, the activity of grid cell i at position (x,y) was assumed to be a random variable with a Poisson distribution, whose mean was X(i)(x,y). Thus, the probability of observing X~(i)(x,y) spikes, at position (x,y), is given by(3)X~(i)(x,y)∼P(X(i)(x,y)).

For a given resolution of the arena, we generated X~j(i)(x,y), where j=1,...,10. That is, we constructed 10 noisy rate maps. We performed cross-validated decoding by averaging across 9 of the 10 rate maps, to get an average rate map X^(i)(x,y). For sake of simplicity, consider X^(i)(x,y)=1/9∑j=19X~j(i)(x,y). To decode local position from the held out noisy rate map [e.g. X~10(i)(x,y)], we multiplied the activity at each position by all the positions in the average rate map, taking the sum and assigning the decoded position as that with the largest value. The Euclidean distance between the decoded position and the true position is considered the error. This was performed 10 times (holding out each rate map once) and the average error across all positions in the environment was then averaged across all 10 of the validation splits.

To decode the electrophysiological data (Gardner et al., 2022), we sampled subpopulations of grid cells from the N=82 units (recording ID R12) that passed the criteria described previously. For each subpopulation, we split the recorded activity into 10 evenly spaced temporal bins, and constructed average ratemaps for each bin. This is consistent with what was done in decoding the synthetic data. All ratemaps, for each cell, were normalized by the maximal activity. We then randomly chose 5 of the 10 time bins to serve as the ‘train’ data, the remaining 5 served as the ‘test’ data. The rate maps were averaged and then the linear decoder was applied. We sampled 10 different choices of splitting the data and 25 choices of subpopulation.

Code used for training and evaluating path integrating recurrent neural network (RNNs) (Sorscher et al., 2019; Sorscher et al., 2023) was pulled from https://github.com/ganguli-lab/grid-pattern-formation (Schaeffer et al., 2023). We used the same hyperparameters as are present on the repository (see Table 1), except when examining the effect of weight regularization on grid property variability, in which case we trained RNNs with weight decay magnitude from {10^-6, 10^-5, 10^-4, 2.5 . 10^-4}. The greater the value, the greater the enforced sparsity. These were chosen as they had previously been found to lead to good path integration performance and have high grid scores (Sorscher et al., 2023). We independently verified that this was true for the resulting RNNs. We computed the grid score, spacing, and orientation using similar code implementations as was used for the neural data. For each value of weight decay, we trained three independent RNNs, using different random seeds.