The Primary Function of MTL is Memory, not Navigation: Grid Cells are Non-spatial (what) and Place Cells are Memories (what and where) that Cause Grid Fields through Retrieval

Reviewer #1 (Public Review):

Huber proposes a theory where the role of the medial temporal lobe (MTL) is memory, where properties of spatial cells in the MTL can be explained through memory function rather than spatial processing or navigation. Instantiating the theory through a computational model, the author shows that many empirical phenomena of spatial cells can be captured, and may be better accounted through a memory theory. It is an impressive computational account of MTL cells with a lot of theoretical reasoning and aims to tightly relate to various spatial cell data.

In general, the paper is well written, but likely due to the complexity, there are various aspects of the paper that are difficult to understand. One point is that it is not entirely clear to me that it is a convincing demonstration of purely memory rather than navigation, but rather an account of the findings through the lens of memory. Below, I raise several big-picture theoretical questions. I also have some clarification questions about the model (where I also have some theoretical question marks - due to not achieving a full understanding).

(1) Although the theory is based on memory, it also is based on spatially-selective cells. Not all cells in the hippocampus fulfill the criteria of place/HD/border/grid cells, and place a role in memory. E.g., Tonegawa, Buszaki labs' work does not focus on only those cells, and there are certainly a lot of non-pure spatial cells in monkeys (Martinez-Trujillo) and humans (iEEG). Does the author mainly focus on saying that "spatial cells" are memory, but do not account for non-spatial memory cells? This seems to be an incomplete account of memory - which is fine, but the way the model is set up suggests that *all* memory is, place (what/where), and non-spatial attributes ("grid") - but cells that don't fulfil these criteria in MTL (Diehl et al., 2017, Neuron; non-grid cells; Schaeffer et al., 2022, ICML; Luo et al., 2024, bioRxiv) certainly contribute to memory, and even navigation. This is also related to the question of whether these cell definitions matter at all (Luo et al., 2024).

The authors note "However, this memory conjunction view of the MTL must be reconciled with the rodent electrophysiology finding that most cells in MTL appear to have receptive fields related to some aspect of spatial navigation (Boccara et al., 2010; Grieves & Jeffery, 2017). The paucity of non-spatial cells in MTL could be explained if grid cells have been mischaracterized as spatial." Is the author mainly talking about rodent work?

(2) Related to the last point, how about non-grid multi-field mEC cells? In theory, these also should be the same; but the author only presents perfect-look grid cells. In empirical work, clearly, this is not the case, and many mEC cells are multi-field non-grid cells (Diehl et al., 2017). Does the model find these cells? Do they play a different role?

As noted by the author "Because the non-spatial attributes are constant throughout the two-dimensional surface, this results in an array of discrete memory locations that are approximately hexagonal (as explained in the Model Methods, an "online" memory consolidation process employing pattern separation rapidly turns an approximately hexagonal array into one that is precisely hexagonal). "

If they are indeed all precisely hexagonal, does that mean the model doesn't have non-grid spatial cells?

(3) Theoretical reasons for why the model is put together this way, and why grid cells must be coding a non-spatial attribute: Is this account more data-driven (fits the data so formulated this way), or is it theoretical - there is a reason why place, border, grid cells are formulated to be like this. For example, is it an efficient way to code these variables? It can be both, like how the BVC model makes theoretical sense that you can use boundaries to determine a specific location (and so place cell), but also works (creates realistic place cells).

But in this case, the purpose of grid cell coding a non-spatial attribute, and having some kind of system where it doesn't fire at all locations seems a little arbitrary. If it's not encoding a spatial attribute, it doesn't have to have a spatial field. For example, it could fire in the whole arena - which some cells do (and don't pass the criteria of spatial cells as they are not spatially "selective" to another location, related to above).

(4) Why are grid cells given such a large role for encoding non-spatial attributes? If anything, shouldn't it be lateral EC or perirhinal cortex? Of course, they both could, but there is less reason to think this, at least for rodent mEC.

(5) Clarification: why do place cells and grid cells differ in terms of stability in the model? Place cells are not stable initially but grid cells come out immediately. They seem directly connected so a bit unclear why; especially if place cell feedback leads to grid cell fields. There is an explanation in the text - based on grid cells coding the on-average memories, but these should be based on place cell inputs as well. So how is it that place fields are unstable then grid fields do not move at all? I wonder if a set of images or videos (gifs) showing the differences in spatial learning would be nice and clarify this point.

(6) Other predictions. Clearly, the model makes many interesting (and quite specific!) predictions. But does it make some known simple predictions?
• More place cells at rewarded (or more visited) locations. Some empirical researchers seem to think this is not as obvious as it seems (e.g., Duvellle et al., 2019; JoN; Nyberg et al., 2021, Neuron Review).
• Grid cell field moves toward reward (Butler et al., 2019; Boccera et al., 2019).
• Grid cells deform in trapezoid (Krupic et al., 2015) and change in environments like mazes (Derikman et al., 2014).

Reviewer #2 (Public Review):

The manuscript describes a new framework for thinking about the place and grid cell system in the hippocampus and entorhinal cortex in which these cells are fundamentally involved in supporting non-spatial information coding. If this framework were shown to be correct, it could have high impact because it would suggest a completely new way of thinking about the mammalian memory system in which this system is non-spatial. Although this idea is intriguing and thought-provoking, a very significant caveat is that the paper does not provide evidence that specifically supports its framework and rules out the alternate interpretations. Thus, although the work provides interesting new ideas, it leaves the reader with more questions than answers because it does not rule out any earlier ideas.

Basically, the strongest claim in the paper, that grid cells are inherently non-spatial, cannot be specifically evaluated versus existing frameworks on the basis of the evidence that is shown here. If, for example, the author had provided behavioral experiments showing that human memory encoding/retrieval performance shifts in relation to the predictions of the model following changes in the environment, it would have been potentially exciting because it could potentially support the author's reconceptualization of this system. But in its current form, the paper merely shows that a new type of model is capable of explaining the existing findings. There is not adequate data or results to show that the new model is a significantly better fit to the data compared to earlier models, which limits the impact of the work. In fact, there are some key data points in which the earlier models seem to better fit the data.

Overall, I would be more convinced that the findings from the paper are impactful if the author showed specific animal memory behavioral results that were only supported by their memory model but not by a purely spatial model. Perhaps the author could run new experiments to show that there are specific patterns of human or animal behavior that are only explained by their memory model and not by earlier models. But in its current form, I cannot rule out the existing frameworks and I believe some of the claims in this regard are overstated.

In addition to the broader concerns noted above regarding the absence of any specific behavioral data that are explained by their model and not by existing spatial models, I am additionally concerned that this manuscript does not explain a number of important key empirical results in the rodent grid cell literature.

* The paper does not fully take into account all the findings regarding grid cells, some of which very clearly show spatial processing in this system. For example, findings on grid-by-direction cells (e.g., Sargolini et al. 2006) would seem to suggest that the entorhinal grid system is very specifically spatial and related to path integration. Why would grid-by-direction cells be present and intertwined with grid cells in the author's memory-related reconceptualization? It seems to me that the existence of grid-by-direction cells is strong evidence that at least part of this network is specifically spatial.

* I am also concerned that the paper does not do enough to address findings regarding how the elliptical shape of grid fields shifts when boundaries of an environment compress in one direction or change shape/angles (Lever et al., & Krupic et al). Those studies show compression in grid fields based on boundary position, and I don't see how the authors' model would explain these findings.

* Are findings regarding speed modulation of grid cells problematic for the paper's memory results?

* A further issue is that the paper does not seem to adequately address developmental findings related to the timecourses of the emergence of different cell types. In their simulation, researchers demonstrate the immediate emergence of grid fields in a novel environment, while noting that the stabilization of place cell positions takes time. However, these simulation findings contradict previous empirical developmental studies (Langston et al., 2010). Those studies showed that head direction cells show the earliest development of spatial response, followed by the appearance of place cells at a similar developmental stage. In contrast, grid cells emerge later in this developmental sequence. The gradual improvement in spatial stability in firing patterns likely plays a crucial role in the developmental trajectory of grid cells. Contrary to the model simulation, grid cells emerge later than place cells and head direction cells, yet they also hold significance in spatial mapping.

* The model simulations suggest that certain grid patterns are acquired more gradually than others. For instance, egocentric grid cells require the stabilization of place cell memories amidst ongoing consolidation, while allocentric grid cells tend to reflect average place field positions. However, these findings seemingly conflict with empirical studies, particularly those on the conjunctive representation of distance and direction in the earliest grid cells. Previous studies show no significant differences were found in grid cells and grid cells with directional correlates across these age groups, relative to adults (Wills et al., 2012). This indicates that the combined representation of distance and direction in single mEC cells is present from the earliest ages at which grid cells emerge.

Reviewer #3 (Public Review):

The author presents a novel theory and computational model suggesting that grid cells do not encode space, but rather encode non-spatial attributes. Place cells in turn encode memories of where those specific attributes occurred. The theory accounts for many experimental results and generates useful predictions for future studies. The model's simplicity and potential explanatory power will interest others in the field, though there are a number of concerns that should first be addressed.

A crucial assumption of the model is that the content of experience must be constant in space. It's difficult to imagine a real-world example that satisfies this assumption. Odors and sounds are used as examples. While they are often more spatially diffuse than an objects on the ground, odors and sounds have sources that are readily detectable. Animals can easily navigate to a food source or to a vocalizing conspecific. This assumption is especially problematic because it predicts that all grid cells should become silent when their preferred non-spatial attribute (e.g. a specific odor) is missing. I'm not aware of any experimental data showing that grid cells become silent. On the contrary, grid cells are known to remain active across all contexts that have been tested, including across sleep/wake states. Unlike place cells, grid cells do not seem to turn off. Since grid cells are active in all contexts, their preferred attribute must also be present in all contexts, and therefore they would not convey any information about the specific content of an experience.

The proposed novelty of this theory is that other models all assume that grid cells encode space. This isn't quite true of models based on continuous attractor networks, the discussion of which is notably absent. More specifically, these models focus on the importance of intrinsic dynamics within the entorhinal cortex in generating the grid pattern. While this firing pattern is aligned to space during navigation and therefore can be used as a representation of that space, the neural dynamics are preserved even during sleep. Similarly, it is because the grid pattern does not strictly encode physical space that grid-like signals are also observed in relation to other two-dimensional continuous variables.

The use of border cells or boundary vector cells as the main (or only) source of spatial information in the hippocampus is not well supported by experimental data. Border cells in the entorhinal cortex are not active in the center of an environment. Boundary-vector cells can fire farther away from the walls but are not found in the entorhinal cortex. They are located in the subiculum, a major output of the hippocampus. While the entorhinal-hippocampal circuit is a loop, the route from boundary-vector cells to place cells is much less clear than from grid cells. Moreover, both border cells and boundary-vector cells (which are conflated in this paper) comprise a small population of neurons compared to grid cells.

Author Response

OVERVIEW OF RESPONSE TO REVIEWS

I thank the three anonymous reviewers for providing well-informed, constructive feedback on the initial version of this manuscript. Based on their comments I will revise the manuscript and hopefully improve it in several ways. I expected a great deal of resistance to the ideas proposed in this model because they break from traditional approaches. One of my goals in developing this model was to argue for a paradigm shift regarding the concept of a “receptive field”. Experimentally, the receptive field is defined as the set of preferred environmental sensory circumstances that cause a neuron to become highly active. Traditional interpretation of receptive fields implicitly assumes that the environmental circumstances that give rise to the receptive field do so in a purely bottom-up fashion (the cell is “receiving” its field), in which case the receptive field specifies the function of the cell. In other words, the receptive field is what the cell does. However, some brain regions (e.g., entorhinal cortex) receive substantial feedback from downstream regions (e.g., hippocampus), and feedback can play an important role in determining the receptive field. As applied to a memory account of MTL, this feedback is memory retrieval and reactivation. Thus, the multifield spatial response of grid cells doesn’t necessarily mean that their function is spatial. Consideration of bottom-up versus top-down signals gives rise to the proposal that the bottom-up preference of many grid cells is some non-spatial attribute even though they exhibit a spatial receptive field owing to retrieval in specific locations.

One thing I will emphasize in a revision is that this model can address findings in the vast literature on learning, memory, and consolidation. The question asked in this study is whether a memory model can also explain the rodent navigation literature. This is not an attempt to provide definitive evidence that this is a better account of the rodent navigation literature. Instead, the goal is to model the rodent navigation literature even though this is a memory model rather than a spatial/navigation model. Nevertheless, within the domain of rodent spatial/navigation, this model makes different predictions/explanations than spatial/navigation models. For instance, this is the only model predicting that many grid cells with spatial receptive fields are non-spatial (see predictions in Box 1). As reviewed in Box 1, this is the only model that can explain why head direction conjunctive grid cells become head direction cells in the absence of hippocampal feedback and it is the only model that can explain why some grid cells are also sensitive to sound frequency (see several other unique explanations in Box 1).

This study is an attempt to unify the spatial/navigation and learning/memory literatures with a relatively simply model. Given the simplicity of the model, there are important findings that the model cannot address -- it is not that the model makes the wrong predictions but rather that it makes no predictions. The role of running speed is one such variable for which the model makes no predictions. Similarly, because the model is a rate-coded model rather than a model of oscillating spiking neurons, it makes no predictions regarding theta oscillations. The model is an account of learning and memory for an adult animal, and it makes no predictions regarding the developmental or evolutionary time course of different cell types. This model contains several purely spatial representations such as border cells, head direction cells, and head direction conjunctive grid cells. In evolution and/or in development, it may be that these purely spatial cell types emerged first, followed by the evolution and/or development of non-spatial cell types. However, this does not invalidate the model. Instead, this is a model for an adult animal that has both episodic memory capabilities and spatial navigation capabilities, irrespective of the order in which these capabilities emerged.

Grid cell models that are purely spatial are agnostic regarding the thousands of findings in the literature on memory, learning, and consolidation whereas this model can potentially unify the learning/memory and spatial/navigation literatures. The reason to prefer this model is parsimony. Rather than needing to develop a theory of memory that is separate from a theory of spatial navigation, it might be possible to address both literatures with a unified account. There are other grid cell models that can explain non-spatial grid-like responses (Mok & Love, 2019; Rodríguez‐Domínguez & Caplan, 2019; Stachenfeld et al., 2017; Wei et al., 2015) and these models may be similarly positioned to explain memory results. However, these models assume that grid cells exhibiting spatial receptive fields serve the function of identifying positions in the environment (i.e., their function is spatial). As such, these models do not explain why most of the input to rodent hippocampus appears to be spatial (these models would need to assume that rodent hippocampus is almost entirely concerned with spatial navigation). This account provides an answer to this conundrum by proposing that grid cells with spatial receptive fields have been misclassified as spatial. Below I give responses to some of the specific comments made by reviewers, grouping these comments by topic:

COMMENTS RELATED TO THE NEED/MOTIVATION FOR THIS MODEL

In a revision, I will clarify that the non-spatial MTL cell types that are routinely found in primate and human studies are fully compatible with this model. The reported simulations are focused on the specific question of how it can be that most mEC and hippocampal cell types in the rodent literature appear to be spatial. It is known that perirhinal cortex is not spatial. However, entorhinal cortex is the gateway to hippocampus. If the hippocampus has the capacity to represent non-spatial memories, it must receive non-spatial input from entorhinal cortex. These simulations suggest that characterization of the rodent mEC cortex as primarily spatial might be incorrect if most grid cells (except perhaps head direction conjunctive grid cells) have been mischaracterized as spatial.

Lateral entorhinal cortex also projects to hippocampus, and one reviewer asks about the distinction between lateral versus medial entorhinal cortex. From this memory perspective, the important question is which part of the entorhinal cortex represents the non-spatial attributes common to the entire recording session, under the assumption that the animal is creating and retrieving memories during recording. If these non-spatial attributes are represented in lateral EC, there would be grid cells in lateral EC (but these are not found). There is evidence that lateral EC cells respond selectively in relation to objects (Deshmukh & Knierim, 2011), but in a typical rodent navigation study there are no objects in the enclosure.

One reviewer asks whether this model is built to explain the existing data or whether the assumptions of this model are made for theoretical reasons. The BVC model (Barry et al., 2006), which is a precursor to this model, is a theoretically efficient representation of space that could support place coding. If the distances to different borders are known, it’s not clear why the MTL also needs the two-dimensional Fourier-like representation provided by grid cells. This gives rise to the proposal that grid cells with spatial receptive fields are serving some function other than representing space. In the proposed model, the precise hexagonal arrangement of grid cells indicates a property that is found everywhere in the enclosure (i.e., a “tiling” of knowledge for where the property can be found). This arrangement arises from the well-documented learning process termed “differentiation” in the memory literature (McClelland & Chappell, 1998; Norman & O’Reilly, 2003; Shiffrin & Steyvers, 1997), which highlights differences between memories to avoid interference and confusion.

CONCERNS RELATED TO LIMITATIONS AND CONFLICTING RESULTS

One reviewer points out that individual grid cells will typically reveal a grid pattern regardless of the environmental circumstances, which, according to this model, indicates that all such circumstances have the same non-spatial attribute. This might seem strange at first, but I suggest that there is a great deal of “sameness” to the environments used in the published rodent navigation experiments. For instance, as far as I’m aware, the animal is never allowed to interact with other animals during spatial navigation recording. Furthermore, the animal is always attached to wires during recording. The internal state of the animal (fear, aloneness, the noise of electronics, etc.) is likely similar across all recording situations and attributes of this internal state are likely represented in the hippocampus as well as in the regions that provide excitatory drive to hippocampus. The claim of this model is that the grid cells are “tagging” different navigation enclosures as places where these things happen (fear, aloneness, electronics, metal floor, no objects, etc.). The interesting question is what happens when the animal is allowed to navigate in a more naturalistic setting that includes varied objects, varied food sources, varied surfaces, other animals, etc. Do grid cells persist in such a naturalistic environment? Or do they lose their regularity, or even become silent, considering that there is no longer a uniformity to the non-spatial attributes? The results of Caswell Barry et al. (2012), demonstrate that the grid pattern expands and becomes less regular in a novel environment. Nevertheless, the novel environment in that study was uncluttered rather than naturalistic. It remains to be seen what will happen with a truly naturalistic environment.

One reviewer asks how this model relates to non-grid multifield cells found in mEC (Diehl et al., 2017; see also the irregularly arranged 3D multifield cells reported by Ginosar et al., 2021). A full explanation of these cells would require a new simulation study. In a revision, I will discuss these cells, which reveal a consistent multifield spatial receptive field and yet the multiple fields are irregular in their arrangement rather than a precise hexagonal lattice. On this memory account, precise hexagonal arrangement of memories is something that occurs when there is a non-spatial attribute found throughout the enclosure. However, in a typical rodent navigation study, there may be some non-spatial attributes that are not found everywhere in the enclosure. For instance, consider the set of locations within the enclosure that afford a particular view of something outside of the enclosure or the set of locations corresponding to remembered episodic events (e.g., memory for the location where the animal first entered the enclosure). For non-spatial characteristics that are found in some locations but not others within in the enclosure, the cells representing those non-spatial attributes should reveal multifield firing at irregular locations, reflecting the subset of locations associated with the non-spatial attribute.

One reviewer suggests that this model cannot explain the finding that grid fields become warped (e.g., grid fields arranged in an ellipse rather than a circle) in the same manner that the enclosure is warped when a wall is moved (Barry et al., 2007). The way in which I would simulate this result would be to assume that the change in the boundary location was too modest to be noticed by the animal. Because the distances are calculated relative to the borders, an unnoticed change in the border would not change the model in terms of the grid field as measured by proportional distances between borders. However, because the real-world Euclidean positions of the border are changed, the grid fields would be changed in terms of real-world coordinates. This is what I was referring to in the paper when I wrote “For instance, perhaps one egocentric/allocentric pair of mEC grid modules is based on head direction (viewpoint) in remembered positions relative to the enclosure borders whereas a different egocentric/allocentric pair is based on head direction in remembered positions relative to landmarks exterior to the enclosure. This might explain why a deformation of the enclosure (moving in one of the walls to form a rectangle rather than a square) caused some of the grid modules but not others to undergo a deformation of the grid pattern in response to the deformation of the enclosure wall (see also Barry et al., 2007). More specifically, if there is one set of non-orthogonal dimensions for enclosure borders and the movement of one wall is too modest as to cause avoid global remapping, this would deform the grid modules based the enclosure border cells. At the same time, if other grid modules are based on exterior properties (e.g., perhaps border cells in relation to the experimental room rather than the enclosure), then those grid modules would be unperturbed by moving the enclosure wall.” Related to the question of enclosure geometry, the irregularity that can emerge in trapezoid shaped enclosures was discussed in the section of the paper that reads “As seen in Figure 12, because all but one of the place cells was exterior when the simulated animal was constrained to a narrow passage, the hippocampal place cell memories were no longer arranged in a hexagonal grid. This disruption of the grid array for narrow passages might explain the finding that the grid pattern (of grid cells) is disrupted in the thin corner of a trapezoid (Krupic et al., 2015) and disrupted when a previously open enclosure is converted to a hairpin maze by insertion of additional walls within the enclosure (Derdikman et al., 2009).”

CONCERNS THAT WILL BE ADDRESSED WITH GREATER CLARIFICATION

One reviewer asks why a cell representing a non-spatial attribute found everywhere in the enclosure would not fire everywhere in the enclosure. In theory, cells could fire constantly. However, in practice, cells habituate and rapidly reduce their firing rate by an order of magnitude when their preferred stimulus is presented without cessation (Abbott et al., 1997; Tsodyks & Markram, 1997). After habituation, the firing rate of the cell fluctuates with minor variation in the strength of the excitatory drive. In other words, habituation allows the cell to become sensitive to changes in the excitatory drive (Huber & O’Reilly, 2003). Thus, if there is stronger top-down memory feedback in some locations as compared to others, the cell will fire at a higher rate in those remembered locations. In brief when faced with constant excitatory drive, the cell accommodates, and becomes sensitive to change in the magnitude of the excitatory drive.

One reviewer asks for greater clarification regarding the simulation result of immediate stability for grid cells but not place cells. In a revision, I will provide a video showing a sped-up birds-eye view of the place cell memories for the 3D simulations that include head direction, showing the manner in which memories tend to linger in some locations more than others as they consolidate. This behavior was explained in the text that reads “Because the non-spatial cell’s grid field reflects on-average memory positions during the recording session (i.e., the locations where the non-spatial attribute is more often remembered, even if the locations of the memories are shifting), the grid fields for the non-spatial are immediately apparent, reflecting the tendency of place cells to linger in some locations as compared to other locations during consolidation. More specifically, the place cells tend to linger at the peaks and troughs of the border cell tuning functions (see the explanation above regarding the tendency of the grid to align with border cell dimensions). By analogy, imagine a time-lapsed birds-eye view of cars traversing the city-block structure of a densely populated city; this on-average view would show a higher density of cars at the cross-street junctions owing to their tendency to become temporarily stuck at stoplights. However, with additional learning and consolidation, the place cells stabilize their positions (e.g., the cars stop traveling), producing a consistent grid field for the head direction conjunctive grid cells.” The text describing why some locations are more “sticky” than others reads “Additional analyses revealed that this tendency to align with border cell dimensions is caused by weight normalization (Step 6 in the pseudocode). Specifically, connection weights cannot be updated above their maximum nor below their minimum allowed values. This results in a slight tendency for consolidated place cell memories to settle at one of the three peak values or three trough values of the sine wave basis set. This “stickiness” at one of 6 peak or trough values for each basis set is very slight and only occurred after many consolidation steps. In terms of biological systems, there is an obvious lower-bound for excitatory connections (i.e., it is not possible to have an excitatory weight connection that is less than zero), but it is not clear if there is an upper-bound. Nevertheless, it is common practice with deep learning models include an upper-bound for connection weights because this reduces overfitting (Srivastava et al., 2014) and there may be similar pressures for biological systems to avoid excessively strong connections.”

One reviewer points out that Border cells are not typically active in the center of enclosure. However, the model can be built without assuming between-border cells (early simulations with the model did not make this assumption). Regarding this issue, the text reads “Unlike the BVC model, the boundary cell representation is sparsely populated using a basis set of three cells for each of the three dimensions (i.e., 9 cells in total), such that for each of the three non-orthogonal orientations, one cell captures one border, another the opposite border, and the third cell captures positions between the opposing borders (Solstad et al., 2008). However, this is not a core assumption, and it is possible to configure the model with border cell configurations that contain two opponent border cells per dimension, without needing to assume that any cells prefer positions between the borders (with the current parameters, the model predicts there will be two border cells for each between-border cell). Similarly, it is possible to configure the model with more than 3 cells for each dimension (i.e., multiple cells representing positions between the borders).” The Solstad paper found a few cells that responded in positions between borders, but perhaps not as many as 1 out of 3 cells, such as this particular model simulation predicts. If the paucity of between-border cells is a crucial data point, the model can be reconfigured with opponent-border cells without any between border cells. The reason that 3 border cells were used rather than 2 opponent border cells was for simplicity. Because 3 head direction cells were used to capture the face-centered cubic packing of memories, the simulation also used 3 border cells per dimensions to allow a common linear sum metric when conjoining dimensions to form memories. If the border dimensions used 2 cells while head direction used 3 cells, a dimensional weighting scheme would be needed to allow this mixing of “apples and oranges” in terms of distances in the 3D space that includes head direction.

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