Author response:
The following is the authors’ response to the original reviews
Public Reviews:
Reviewer #1 (Public Review):
Summary:
This work studies representations in a network with one recurrent layer and one output layer that needs to path-integrate so that its position can be accurately decoded from its output. To formalise this problem, the authors define a cost function consisting of the decoding error and a regularisation term. They specify a decoding procedure that at a given time averages the output unit center locations, weighted by the activity of the unit at that time. The network is initialised without position information, and only receives a velocity signal (and a context signal to index the environment) at each timestep, so to achieve low decoding error it needs to infer its position and keep it updated with respect to its velocity by path integration.
The authors take the trained network and let it explore a series of environments with different geometries while collecting unit activities to probe learned representations. They find localised responses in the output units (resembling place fields) and border responses in the recurrent units. Across environments, the output units show global remapping and the recurrent units show rate remapping. Stretching the environment generally produces stretched responses in output and recurrent units. Ratemaps remain stable within environments and stabilise after noise injection. Low-dimensional projections of the recurrent population activity forms environment-specific clusters that reflect the environment's geometry, which suggests independent rather than generalised representations. Finally, the authors discover that the centers of the output unit ratemaps cluster together on a triangular lattice (like the receptive fields of a single grid cell), and find significant clustering of place cell centers in empirical data as well.
The model setup and simulations are clearly described, and are an interesting exploration of the consequences of a particular set of training requirements - here: path integration and decodability. But it is not obvious to what extent the modelling choices are a realistic reflection of how the brain solves navigation. Therefore it is not clear whether the results generalize beyond the specifics of the setup here.
Strengths:
The authors introduce a very minimal set of model requirements, assumptions, and constraints. In that sense, the model can function as a useful 'baseline', that shows how spatial representations and remapping properties can emerge from the requirement of path integration and decodability alone. Moreover, the authors use the same formalism to relate their setup to existing spatial navigation models, which is informative.
The global remapping that the authors show is convincing and well-supported by their analyses. The geometric manipulations and the resulting stretching of place responses, without additional training, are interesting. They seem to suggest that the recurrent network may scale the velocity input by the environment dimensions so that the exact same path integrator-output mappings remain valid (but maybe there are other mechanisms too that achieve the same).
The clustering of place cell peaks on a triangular lattice is intriguing, given there is no grid cell input. It could have something to do with the fact that a triangular lattice provides optimal coverage of 2d space? The included comparison with empirical data is valuable, although the authors only show significant clustering - there is no analysis of its grid-like regularity.
First of all, we would like to thank the reviewer for their comprehensive feedback, and their insightful comments. Importantly, as you point out, our goal with this model was to build a minimal model of place cell representations, where representations were encouraged to be place-like, but free to vary in tuning and firing locations. By doing so, we could explore what upstream representations facilitate place-like representations, and even remapping (as it turned out) with minimal assumptions. However, we agree that our task does not capture some of the nuances of real-world navigation, such as sensory observations, which could be useful extensions in future work. Then again, the simplicity of our setup makes it easier to interpret the model, and makes it all the more surprising that it learns many behaviors exhibited by real world place cells.
As to the distribution of phases - we also agree that a hexagonal arrangement likely reflects some optimal configuration for decoding of location.
And we agree that the symmetry within the experimental data is important; we have revised analyses on experimental phase distributions, and included an analysis of ensemble grid score, to quantify any hexagonal symmetries within the data.
Weaknesses:
The navigation problem that needs to be solved by the model is a bit of an odd one. Without any initial position information, the network needs to figure out where it is, and then path-integrate with respect to a velocity signal. As the authors remark in Methods 4.2, without additional input, the only way to infer location is from border interactions. It is like navigating in absolute darkness. Therefore, it seems likely that the salient wall representations found in the recurrent units are just a consequence of the specific navigation task here; it is unclear if the same would apply in natural navigation. In natural navigation, there are many more sensory cues that help inferring location, most importantly vision, but also smell and whiskers/touch (which provides a more direct wall interaction; here, wall interactions are indirect by constraining velocity vectors). There is a similar but weaker concern about whether the (place cell like) localised firing fields of the output units are a direct consequence of the decoding procedure that only considers activity center locations.
Thank you for raising this point; we absolutely agree that the navigation task is somewhat niche. However, this was a conscious decision, to minimize any possible confounding from alternate input sources, such as observations. In part, this experimental design was inspired by the suggestion that grid cells support navigation/path integration in open-field environments with minimal sensory input (as they could, conceivably do so with no external input). This also pertains to your other point, that boundary interactions are necessary for navigation. In our model, using boundaries is one solution, but there is another way around this problem, which is conceivably better: to path integrate in an egocentric frame, starting from your initial position. Since the locations of place fields are inferred only after a trajectory has been traversed, the network is free to create a new or shifted representation every time, independently of the arena. In this case, one might have expected generalized solutions, such as grid cells to emerge. That this is not the case, seems to suggest that grid cells may somehow not be optimal for pure path integration, or at the very least, hard to learn (but may still play a part, as alluded to by place field locations). We have tried to make these points more evident in the revised manuscript.
As for the point that the decoding may lead to place-like representations, this is a fair point. Indeed, we did choose this form of decoding, inspired by the localized firing of place cells, in the hope that it would encourage minimally constrained, place-like solutions. However, compared to other works (Sorscher and Xu) hand tuning the functional form of their place cells, our (although biased towards centralized tuning curves) allows for flexible functional forms such as the position of the place cell centers, their tuning width, whether or not it is center-surround activity, and how they should tune to different environments/rooms. This allows us to study several features of the place cell system, such as remapping and field formation. We have revised to make this more clear in the model description.
The conclusion that 'contexts are attractive' (heading of section 2) is not well-supported. The authors show 'attractor-like behaviour' within a single context, but there could be alternative explanations for the recovery of stable ratemaps after noise injection. For example, the noise injection could scramble the network's currently inferred position, so that it would need to re-infer its position from boundary interactions along the trajectory. In that case the stabilisation would be driven by the input, not just internal attractor dynamics. Moreover, the authors show that different contexts occupy different regions in the space of low-dimensional projections of recurrent activity, but not that these regions are attractive.
We agree that boundary interactions could facilitate the convergence of representations after noise injection. We did try to moderate this claim by the wording “attractor-like”, but we agree that boundaries could confound this result. We have therefore performed a modified noise injection experiment, where we let the network run for an extended period of time, before noise injection (and no velocity signal), see Appendix Velocity Ablation in the revised text. Notably, representations converge to their pre-scrambled state after noise injection, even without a velocity signal. However, place-like representations do not converge for all noise levels in this case, possibly indicating that boundary interactions do serve an error-correcting function, also. Thank you for pointing this out.
As for the attractiveness of contexts, we agree that more analyses were required to demonstrate this. We have therefore conducted a supplementary analysis where we run the trained network with a mismatch in context/geometry, and demonstrate that the context signal fixes the representation, up to geometric distortions.
The authors report empirical data that shows clustering of place cell centers like they find for their output units. They report that 'there appears to be a tendency for the clusters to arrange in hexagonal fashion, similar to our computational findings'. They only quantify the clustering, but not the arrangement. Moreover, in Figure 7e they only plot data from a single animal, then plot all other animals in the supplementary. Does the analysis of Fig 7f include all animals, or just the one for which the data is plotted in 7e? If so, why that animal? As Appendix C mentions that the ratemap for the plotted animal 'has a hexagonal resemblance' whereas other have 'no clear pattern in their center arrangements', it feels like cherrypicking to only analyse one animal without further justification.
Thank you for pointing this out; we agree that this is not sufficiently explained and explored in the current version. We have therefore conducted a grid score analysis of the experimental place center distributions, to uncover possible hexagonal symmetries. The reason for choosing this particular animal was in part because it featured the largest number of included cells, while also demonstrating the most striking phase distribution, while including all distributions in the supplementary. Originally, this was only intended as a preliminary analysis, suggesting non-uniformity in experimental place field distributions, but we realize that these may all provide interesting insight into the distributional properties of place cells.
We have explained these choices in the revised text, and expanded analyses on all animals to showcase these results more clearly.
Reviewer #2 (Public Review):
Summary:
The authors proposed a neural network model to explore the spatial representations of the hippocampal CA1 and entorhinal cortex (EC) and the remapping of these representations when multiple environments are learned. The model consists of a recurrent network and output units (a decoder) mimicking the EC and CA1, respectively. The major results of this study are: the EC network generates cells with their receptive fields tuned to a border of the arena; decoder develops neuron clusters arranged in a hexagonal lattice. Thus, the model accounts for entorhinal border cells and CA1 place cells. The authors also suggested the remapping of place cells occurs between different environments through state transitions corresponding to unstable dynamical modes in the recurrent network.
Strengths:
The authors found a spatial arrangement of receptive fields similar to their model's prediction in experimental data recorded from CA1. Thus, the model proposes a plausible mechanisms to generate hippocampal spatial representations without relying on grid cells. This result is consistent with the observation that grid cells are unnecessary to generate CA1 place cells.
The suggestion about the remapping mechanism shows an interesting theoretical possibility.
We thank the reviewer for their kind feedback.
Weaknesses:
The explicit mechanisms of generating border cells and place cells and those underlying remapping were not clarified at a satisfactory level.
The model cannot generate entorhinal grid cells. Therefore, how the proposed model is integrated into the entire picture of the hippocampal mechanism of meｍory processing remains elusive.
We appreciate this point, and hope to clarify: From a purely architectural perspective, place-like representations are generated by linear combinations of recurrent unit representations, which, after training, appear border-like. During remapping, the network is simply evaluated/run in different geometries/contexts, which, it turns out, causes the network to exhibit different representations, likely as solutions to optimally encoding position in the different environments. We have attempted to revise the text to make some of these interpretations more clear. We have also conducted a supplementary analysis to demonstrate how representations are determined by the context signal directly, which helps to explain how recurrent and output units form their representations.
We also agree that our model does not capture the full complexity of the Hippocampal formation. However, we would argue that its simplicity (focusing on a single cell type and a pure path integration task), acts as a useful baseline for studying the role of place cells during spatial navigation. The fact that our model captures a range of place cell behaviors (field formation, remapping and geometric deformation) without grid cells also point to several interesting possibilities, such that grid cells may not be strictly necessary for place cell formation and remapping, or that border cells may account for many of the peculiar behaviors of place cells. However, we wholeheartedly agree that including e.g. sensory information and memory storage/retrieval tasks would prove a very interesting extension of our model to more naturalistic tasks and settings. In fact, our framework could easily accommodate this, e.g. by decoding contexts/observations/memories from the network state, alongside location.
Reviewer #3 (Public Review):
Summary:
The authors used recurrent neural network modelling of spatial navigation tasks to investigate border and place cell behaviour during remapping phenomena.
Strengths:
The neural network training seemed for the most part (see comments later) well-performed, and the analyses used to make the points were thorough.
The paper and ideas were well explained.
Figure 4 contained some interesting and strong evidence for map-like generalisation as environmental geometry was warped.
Figure 7 was striking, and potentially very interesting.
It was impressive that the RNN path-integration error stayed low for so long (Fig A1), given that normally networks that only work with dead-reckoning have errors that compound. I would have loved to know how the network was doing this, given that borders did not provide sensory input to the network. I could not think of many other plausible explanations... It would be even more impressive if it was preserved when the network was slightly noisy.
Thank you for your insightful comments! Regarding the low path integration error, there is a slight statistical signal from the boundaries, as trajectories tend to turn away from arena boundaries. However, we agree, that studying path integration performance in the face of noise would make for a very interesting future development.
Weaknesses:
I felt that the stated neuroscience interpretations were not well supported by the presented evidence, for a few reasons I'll now detail.
First, I was unconvinced by the interpretation of the reported recurrent cells as border cells. An equally likely hypothesis seemed to be that they were positions cells that are linearly encoding the x and y position, which when your environment only contains external linear boundaries, look the same. As in figure 4, in environments with internal boundaries the cells do not encode them, they encode (x,y) position. Further, if I'm not misunderstanding, there is, throughout, a confusing case of broken symmetry. The cells appear to code not for any random linear direction, but for either the x or y axis (i.e. there are x cells and y cells). These look like border cells in environments in which the boundaries are external only, and align with the axes (like square and rectangular ones), but the same also appears to be true in the rotationally symmetric circular environment, which strikes me as very odd. I can't think of a good reason why the cells in circular environments should care about the particular choice of (x,y) axes... unless the choice of position encoding scheme is leaking influence throughout. A good test of these would be differently oriented (45 degree rotated square) or more geometrically complicated (two diamonds connected) environments in which the difference between a pure (x,y) code and a border code are more obvious.
Thank you for pointing this out. This is an excellent point, that we agree could be addressed more rigorously. Note that there is no position encoding in our model; the initial state of the network is a vector of zeros, and the network must infer its location from boundary interactions and context information alone. So there is no way for positional information to leak through to the recurrent layer directly. However, one possible reason for the observed symmetry breaking, is the fact that the velocity input signal is aligned with the cardinal directions. To investigate this, we trained a new model, wherein input velocities are rotated 45 degrees relative to the horizontal, as you suggest. The results, shown and discussed in appendix E (Learned recurrent representations align with environment boundaries), do indicate that representations are tuned to environment boundaries, and not the cardinal directions, which hopefully improves upon this point.
Next, the decoding mechanism used seems to have forced the representation to learn place cells (no other cell type is going to be usefully decodable?). That is, in itself, not a problem. It just changes the interpretation of the results. To be a normative interpretation for place cells you need to show some evidence that this decoding mechanism is relevant for the brain, since this seems to be where they are coming from in this model. Instead, this is a model with place cells built into it, which can then be used for studying things like remapping, which is a reasonable stance.
This is a great point, and we agree. We do write that we perform this encoding to encourage minimally constrained place-like representations (to study their properties), but we have revised to make this more evident.
However, the remapping results were also puzzling. The authors present convincing evidence that the recurrent units effectively form 6 different maps of the 6 different environments (e.g. the sparsity of the code, or fig 6a), with the place cells remapping between environments. Yet, as the authors point out, in neural data the finding is that some cells generalise their co-firing patterns across environments (e.g. grid cells, border cells), while place cells remap, making it unclear what correspondence to make between the authors network and the brain. There are existing normative models that capture both entorhinal's consistent and hippocampus' less consistent neural remapping behaviour (Whittington et al. and probably others), what have we then learnt from this exercise?
Thanks for raising this point! We agree that this finding is surprising, but we hold that it actually shows something quite important: that border-type units are sufficient to create place-like representations, and learns several of the behaviors associated with place cells and remapping (including global remapping and field stretching). In other words, a single cell type known to exist upstream of place cells is sufficient to explain a surprising range of phenomena, demonstrating that other cell types are not strictly necessary. However, we agree that understanding why the boundary type units sometimes rate remap, and whether that can be true for some border type cells in the brain (either directly, or through gating mechanisms) would be important future developments. Related to this point, we also expanded upon the influence of the context signal for representation selection (appendix F)
Concerning the relationship to other models, we would argue that the simplicity of our model is one of its core strengths, making it possible to disentangle what different cell types are doing. While other models, including TEM, are highly important for understanding how different cell types and brain regions interact to solve complex problems, we believe there is a need for minimal, understandable models that allows us to investigate what each cell type is doing, and this is where we believe our work is important. As an example, our model not only highlights the sufficiency of boundary-type cells as generators of place cells, its lack of e.g. grid cells also suggest that grid cells may not be strictly necessary for e.g. open-field/sensory-deprived navigation, as is often claimed.
One striking result was figure 7, the hexagonal arrangement of place cell centres. I had one question that I couldn't find the answer to in the paper, which would change my interpretation. Are place cell centres within a single clusters of points in figure 7a, for example, from one cell across the 100 trajectories, or from many? If each cluster belongs to a different place cell then the interpretation seems like some kind of optimal packing/coding of 2D space by a set of place cells, an interesting prediction. If multiple place cells fall within a single cluster then that's a very puzzling suggestion about the grouping of place cells into these discrete clusters. From figure 7c I guess that the former is the likely interpretation, from the fact that clusters appear to maintain the same colour, and are unlikely to be co-remapping place cells, but I would like to know for sure!
This is a good point, and you are correct: one cluster tends to correspond to one unit. To make this more clear, we have revised Fig. 7, so that each decoded center is shaded by unit identity, which makes this more evident. And yes, this is, seemingly in line with some form of optimal packing/encoding of space, yes!
I felt that the neural data analysis was unconvincing. Most notably, the statistical effect was found in only one of seven animals. Random noise is likely to pass statistical tests 1 in 20 times (at 0.05 p value), this seems like it could have been something similar? Further, the data was compared to a null model in which place cell fields were randomly distributed. The authors claim place cell fields have two properties that the random model doesn't (1) clustering to edges (as experimentally reported) and (2) much more provocatively, a hexagonal lattice arrangement. The test seems to collude the two; I think that nearby ball radii could be overrepresented, as in figure 7f, due to either effect. I would have liked to see a computation of the statistic for a null model in which place cells were random but with a bias towards to boundaries of the environment that matches the observed changing density, to distinguish these two hypotheses.
Thanks for raising this point. We agree that we were not clear enough in our original manuscript. We included additional analyses in one animal, to showcase one preliminary case of non-uniform phases. To mitigate this, we have performed the same analyses for all animals, and included a longer discussion of these results (included in the supplementary material). We have also moderated the discussion on Ripley’s H to encompass only non-uniformity, and added a grid score analysis to showcase possible rotational symmetries in the data. We hope this gets our findings across more clearly
Some smaller weaknesses:
- Had the models trained to convergence? From the loss plot it seemed like not, and when including regularisors recent work (grokking phenomena, e.g. Nanda et al. 2023) has shown the importance of letting the regularisor minimise completely to see the resulting effect. Else you are interpreting representations that are likely still being learnt, a dangerous business.
Longer training time did not seem to affect representations. However, due to the long trajectories and statefulness involved, training was time-intensive and could become unstable for very long training. We therefore stopped training at the indicated time.
- Since RNNs are nonlinear it seems that eigenvalues larger than 1 doesn't necessarily mean unstable?
This is a good point; stability is not guaranteed. We have updated the text to reflect this.
- Why do you not include a bias in the networks? ReLU networks without bias are not universal function approximators, so it is a real change in architecture that doesn't seem to have any positives?
We found that bias tended to have a detrimental effect on training, possibly related to the identity initialization used (see e.g. Le et al. 2015), and found that training improved when biases were fixed to zero.
- The claim that this work provided a mathematical formalism of the intuitive idea of a cognitive map seems strange, given that upwards of 10 of the works this paper cite also mathematically formalise a cognitive map into a similar integration loss for a neural network.
We agree that other works also provide ways of formalizing this concepts. However, our goal by doing so was to elucidate common features across these seemingly disparate models. We also found that the concept of a learned and target map made it easier to come up with novel models, such as one wherein place cells are constructed to match a grid cell label.
Aim Achieved? Impact/Utility/Context of Work
Given the listed weaknesses, I think this was a thorough exploration of how this network with these losses is able to path-integrate its position and remap. This is useful, it is good to know how another neural network with slightly different constraints learns to perform these behaviours. That said, I do not think the link to neuroscience was convincing, and as such, it has not achieved its stated aim of explaining these phenomena in biology. The mechanism for remapping in the entorhinal module seemed fundamentally different to the brain's, instead using completely disjoint maps; the recurrent cell types described seemed to match no described cell type (no bad thing in itself, but it does limit the permissible neuroscience claims) either in tuning or remapping properties, with a potentially worrying link between an arbitrary encoding choice and the responses; and the striking place cell prediction was unconvincingly matched by neural data. Further, this is a busy field in which many remapping results have been shown before by similar models, limiting the impact of this work. For example, George et al. and Whittington et al. show remapping of place cells across environments; Whittington et al. study remapping of entorhinal codes; and Rajkumar Vasudeva et al. 2022 show similar place cell stretching results under environmental shifts. As such, this papers contribution is muddied significantly.
Thank you for this perspective; we agree that all of these are important works that arrive at complementary findings. We hold that the importance of our paper lies in its minimal nature, and its focus on place cells, via a purpose-built decoding that enables place-like representations. In doing so, we can point to possibly under explored relationships between cell types, in particular place cells and border cells, while challenging the necessity of other cell types for open-field navigation (i.e. grid cells). In addition, our work points to a novel connection between grid cells, place cells and even border cells, by way of the hexagonal arrangement of place unit centers. However, we agree that expanding our model to include more biologically plausible architectures and constraints would make for a very interesting extension in the future.
Thank you again for your time, as well as insightful comments.
Recommendations for the authors:
Reviewer #1 (Recommendations For The Authors):
Even after reading Methods 5.3, I found it hard to understand how the ratemap population vectors that produce Fig 3e and Fig 5 are calculated. It's unclear to me how there can be a ratemap at a single timestep, because calculating a ratemap involves averaging the activity in each location, which would take a whole trajectory and not a single timestep. But I think I've understood from Methods 5.1 that instead the ratemap is calculated by running multiple 'simultaneous' trajectories, so that there are many visited locations at each timestep. That's a bit confusing because as far as I know it's not a common way to calculate ratemaps in rodent experiments (probably because it would be hard to repeat the same task 500 times, while the representations remain the same), so it might be worth explaining more in Methods 5.3.
We understand the confusion, and have attempted to make this more clear in the revised manuscript. We did indeed create ratemaps over many trajectories for time-dependent plots, for the reasons you mentioned. We also agree that this would be difficult to do experimentally, but found it an interesting way to observe convergence of representations in our simulated scenario.
Fig 3b-d shows multiple analyses to support output unit global remapping, but no analysis to support the claim that recurrent units remap by rate changes. The examples in Fig 3ai look pretty convincing, but it would be useful to also have a more quantitative result.
We agree, and only showed that units turn off/become silent using ratemaps. We have therefore added an explicit analysis, showcasing rate remapping in recurrent units (see appendix G; Recurrent units rate remap)
Reviewer #2 (Recommendations For The Authors):
Some parts of the current manuscript are hard to follow. Particularly, the model description is not transparent enough. See below for the details.
Major comments:
(1) Mathematical models should be explained more explicitly and carefully. I had to guess or desperately search for the definitions of parameters. For instance, define the loss function L in eq.(1). Though I can assume L represents the least square error (in A.8), I could not find the definition in Model & Objective. N should also be defined explicitly in equation (3). Is this the number of output cells?
Thank you for pointing this out, we have revised to make it more clear.
(2) In Fig. 1d, how were the velocity and context inputs given to individual neurons in the network? The information may be described in the Methods, but I could not identify it.
This was described in the methods section (Neural Network Architecture and Training), but we realize that we used confusing notation, when comparing with Fig. 1d. We have therefore changed the notation, and it should hopefully be clearer now. Thanks for pointing out this discrepancy.
(3) I took a while to understand equations (3) and (4) (for instance, t is not defined here). The manuscript would be easier to read if equations (5) and (6) are explained in the main text but not on page 18 (indeed, these equations are just copies of equations 3 and 4). Otherwise, the authors may replace equations (3) and (4) with verbal explanations similar to figure legend for Fig. 1b.
(4) Is there any experimental evidence for uniformly strong EC-to-CA1 projections assumed in the non-trainable decoder? This point should be briefly mentioned.
Thank you for raising this point. The decoding from EC (the RNN) to CA1 (the output layer) consists of a trainable weight matrix, and may thus be non-uniform in magnitude. The non-trainable decoding acts on the resulting “CA1” representation only. We hope that improvements to the model description also makes this more evident.
(5) The explanation of Fig. 3 in the main text is difficult to follow because subpanels are explained in separate paragraphs, some of which are very short, as short as just a few lines.
This presentation style makes it difficult to follow the logical relationships between the subpanels. This writing style is obeyed throughout the manuscript but is not popular in neuroscience.
Thanks for pointing this out, we have revised to accommodate this.
(6) Why do field centers cluster near boundaries? No underlying mechanisms are discussed in the manuscript.
This is a good point; we have added a note on this; it likely reflects the border tuning of upstream units.
(7) In Fig. 4, the authors presented how cognitive maps may vary when the shape and size of open arenas are modified. The results would be more interesting if the authors explained the remapping mechanism. For instance, on page 8, the authors mentioned that output units exhibit global remapping between contexts, whereas recurrent units mainly rate remapping.
Why do such representational differences emerge?
We agree! Thanks for raising this point. We have therefore expanded upon this discussion in section 2.4.
(8) In the first paragraph of page 10, the authors stated ".. some output units display distinct field doubling (see both Fig. 4c), bottom right, and Fig. 4d), middle row)". I could not understand how Fig. 4d, middle row supports the argument. Similarly, they stated "..some output units reflect their main boundary input (with greater activity near one boundary)." I can neither understand what the authors mean to say nor which figures support the statement. Please clarify.
This is a good point, there was an identifier missing; we have updated to refer to the correct “magnification”. Thanks!
(9) The underlying mechanism of generating the hexagonal representation of output cells remains unclear. The decoder network uses a non-trainable decoding scheme based on localized firing patterns of output units. To what extent does the hexagonal representation depend on the particular decoding scheme? Similarly, how does the emergence of the hexagonal representation rely on the border representation in the upstream recurrent network? Showing several snapshots of the two place representations during learning may answer these questions.
This is an interesting point, and we have added some discussion on this matter. In particular, we speculate whether it’s an optimal configuration for position reconstruction, which is demanded by the task and thus highly likely dependent on the decoding scheme. We have not reached a conclusive method to determine the explicit dependence of the hexagonal arrangement on the choice of decoding scheme. Still, it seems this would require comparison with other schemes. In our framework, this would require changing the fundamental operation of the model, which we leave as inspiration for future work. We have also added additional discussion concerning the relationship between place units, border units, and remapping in our model. As for exploring different training snapshots, the model is randomly initialized, which suggests that earlier training steps should tend to reveal unorganized/uninformative phase arrangements, as phases are learned as a way of optimizing position reconstruction. However, we do call for more analysis of experimental data to determine whether this is true in animals, which would strongly support this observation. We also hope that our work inspires other models studying the formation and remapping of place cells, which could serve as a starting point for answering this question in the future.
(10) Figure 7 requires a title including the word "hexagonal" to make it easier to find the results demonstrating the hexagonal representations. In addition, please clarify which networks, p or g, gave the results shown here.
We agree, and have added it!
Minor comments:
(11) In many paragraphs, conclusions appear near their ends. Stating the conclusion at the beginning of each paragraph whenever possible will improve the readability.
We have made several rewrites to the manuscript, and hope this improves readability.
(12) Figure A4 is important as it shows evidence of the CA1 spatial representation predicted by the model. However, I could not find where the figure is cited in the manuscript. The authors can consider showing this figure in the main text.
We agree, and we have added more references to the experimental data analyses in the main text, as well as expanded this analysis.
(13) The main text cites figures in the following format: "... rate mapping of Fig. 3a), i), boundary ...." The parentheses make reading difficult.
We have removed the overly stringent use of double parentheses, thanks for letting us know.
(14) It would be nice if the authors briefly explained the concept of Ripley's H function on page 14.
Yes, we have added a brief descriptor.
