Dear Mr. Baumann,

Thank you very much for submitting your manuscript "Metric information in cognitive maps: Euclidean embedding of non-Euclidean environments" for consideration at PLOS Computational Biology. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. The reviewers appreciated the attention to an important topic. Based on the reviews, we are likely to accept this manuscript for publication, providing that you modify the manuscript according to the review recommendations.

In particular, the authors should make use of Bayesian statistics or information criteria for model selection (e.g. AIC or BIC) to make a more formal comparison between their embedding model and a topological framework; and - if possible - attempt to relate their embedding model to a wider range of published data sets, to ensure that their conclusions are robust. 

Please prepare and submit your revised manuscript within 30 days. If you anticipate any delay, please let us know the expected resubmission date by replying to this email.

When you are ready to resubmit, please upload the following:

[1] A letter containing a detailed list of your responses to all review comments, and a description of the changes you have made in the manuscript. Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out

[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).

Important additional instructions are given below your reviewer comments.

Thank you again for your submission to our journal. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.

Sincerely,

Daniel Bush

Academic Editor

PLOS Computational Biology

Thomas Serre

Section Editor

PLOS Computational Biology

***********************

A link appears below if there are any accompanying review attachments. If you believe any reviews to be missing, please contact ploscompbiol@plos.org immediately:

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: In this paper the authors propose a model that can account for deviation from Euclidean space in cognitive maps holding on the hypothesis that cognitive maps are fundamentally Euclidean. Basically, they suggest that, instead of learning and representing the position of objects/events in space using a non-euclidean graphical structure, what people usually do is to embed a non-metric model in 2D Euclidean coordinates. By re-analyzing the data from a previous behavioral experiment, they show that the embedded model and the graph model (non Euclidean) explain the data equally well. Although the embedded model, thus, is not necessarily better than a graph model, the former is preferable because it allows to integrate local repeated measures into a global structure (whereas in a non-metric graph-like cognitive map, each edge is independent from the others).

I liked the paper, it investigates a very important as well as difficult question: Whether humans navigate the environment constructing an Euclidean map. The author suggest that this is the case, proposing a hybrid model in which a non-Euclidean graph is embedded in a 2d Euclidean coordinates and re-analyzing an important dataset that, putatively, provide evidence for non-Euclidean navigation. I find the paper well written and, to the level of understanding that I can provide, which does not cover all the mathematical details as well as all the relevant literature on this specific argument (cognitive maps vs cognitive graphs), it seems solid: I could not find any specific problem in the analysis or the construction of the model.

Although, at the end, the paper cannot adjudicate between the labeled graph and the embedded graph model, I think it will be a good addition to the literature and the current debate. But I have some suggestions:

(1) The author can report Bayesian statistics to support the lack of difference between models

(2) Is there another dataset on which the two models can be compared? It would be more convincing to see a replication of the results using different data (with different non-Euclidean environments)

(3) P. 16 “The embedding may possibly be somewhat better at predicting the within-subject angular deviation in the rips and folds dataset, but the results did not pass the selected significance threshold” – I suggest to delete this sentence from the manuscript. The difference is not significant and it should not be commented as a hint of a possible difference.

Reviewer #2: The study utilized data from Warren et al. (2017) to investigate whether embedding a graph into Euclidean coordinates can explain data from a wormhole experiment better than a labeled graph. They used a numerical optimization method to derive embeddings, which are representations of the cognitive map. For their primary dataset, they derived shortcut predictions from both the non-metric labeled and embedded graph models. These predictions were then compared to human shortcut estimates from the Warren et al. (2017) study. The authors found that both the Euclidean embedding and the non-metric model predicted the data with similar accuracy. They conclude that since the embedding graph is simpler, it is a better model for the cognitive map than the non-metric models.

The authors employed a unique approach by using a numerical optimization method to derive embeddings, and the use of both non-metric labeled and embedded graph models to predict human behavior in navigating non-Euclidean environments is a solid approach. This approach takes the question about non-metric graphs into a new phase of direct comparisons between models. Given that most of the data were derived from another paper, design questions are minimal. I do have a few concerns that are important to address.

Major concerns:

The biggest concern is about the model comparison. The two models predicted the data with similar accuracy, which could lead to ambiguity in comparing the two models. The authors assume that the Euclidean embedded model is simpler, but there is not really much to support that claim. Conducing AIC or BIC to more directly compare the models would help, or at least providing more justification as to why metric embedding is simpler. A topological graph seems like a fairly simple idea to me. The metric embedding yields multi-level paths (e.g., Figures 4 and 5), which seems like it could be far more complex than the topological graph, and is also non-Euclidean.

Is the stress function simpler than a set of heuristics, such as regularizing to 90 degrees?

How does the embedding explain inconsistencies in regular Euclidean judgments, e.g., Diwadkar & McNamara, Tversky 1992, etc.? If the metric embedding simplifies to Euclidean structure in normal environments (see comment below), then how can it explain other violations of Euclidean geometry observed in the literature?

Minor concerns:

Metric embedding is central to this paper, but I didn’t see a really clear definition that could be operationalized (or tested/falsified in the future). It starts to come out on line 202, but an earlier definition would allow the reader to follow the arguments in the introduction. Similarly, details about the non-metric labeled graph model only come out on page 12. Much of those details are in the other paper, but a brief description (e.g., averaging of paths found by vector addition) would be useful.

When the authors say the embedding is not a simple averaging and so it won’t end up in the middle, it is not clear how that calculation is made to get the new vector.

How accurate are the measurements of the distances and angles for each triplet (e.g. lines 265)? If they are all very accurate, wouldn’t that lead to a Euclidean structure? Or is that the case if you are dealing with a metric map, but not with a wormhole? I think this might have been brought up later on but was not completely clear to me.

Abstract line 20 “so” should probably be “so-called”?

**********

Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data and code should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data or code —e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

Figure Files:

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org.

Data Requirements:

Please note that, as a condition of publication, PLOS' data policy requires that you make available all data used to draw the conclusions outlined in your manuscript. Data must be deposited in an appropriate repository, included within the body of the manuscript, or uploaded as supporting information. This includes all numerical values that were used to generate graphs, histograms etc.. For an example in PLOS Biology see here: http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.1001908#s5.

Reproducibility:

To enhance the reproducibility of your results, we recommend that you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. Additionally, PLOS ONE offers an option to publish peer-reviewed clinical study protocols. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols

References:

Review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript.

If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

Dear Mr. Baumann,

We are pleased to inform you that your manuscript 'Metric information in cognitive maps: Euclidean embedding of non-Euclidean environments' has been provisionally accepted for publication in PLOS Computational Biology.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow up email. A member of our team will be in touch with a set of requests.

Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.

IMPORTANT: The editorial review process is now complete. PLOS will only permit corrections to spelling, formatting or significant scientific errors from this point onwards. Requests for major changes, or any which affect the scientific understanding of your work, will cause delays to the publication date of your manuscript.

Should you, your institution's press office or the journal office choose to press release your paper, you will automatically be opted out of early publication. We ask that you notify us now if you or your institution is planning to press release the article. All press must be co-ordinated with PLOS.

Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Computational Biology. 

Best regards,

Daniel Bush

Academic Editor

PLOS Computational Biology

Thomas Serre

Section Editor

PLOS Computational Biology

***********************************************************

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: The authors replies are convincing and I think this paper will be a valuable contribution to the debate.

Reviewer #2: The authors have responded well to all of my comments.

**********

Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data and code should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data or code —e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: None

**********

PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No