PCSY-D-24-00137

Neuronal networks quantified as vector fields

PLOS Complex Systems

Dear Dr. Jörntell,

Thank you for submitting your manuscript to PLOS Complex Systems. After careful consideration, we feel that it has merit but does not fully meet PLOS Complex Systems's publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript within 60 days Feb 18 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at complexsystems@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pcsy/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

* A rebuttal letter that responds to each point raised by the editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. This file does not need to include responses to any formatting updates and technical items listed in the 'Journal Requirements' section below.

* A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

* An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, competing interests statement, or data availability statement, please make these updates within the submission form at the time of resubmission. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

We look forward to receiving your revised manuscript.

Kind regards,

Juan Gonzalo Barajas-Ramirez

Academic Editor

PLOS Complex Systems

Journal Requirements:

1. We ask that a manuscript source file is provided at Revision. Please upload your manuscript file as a .doc, .docx, .rtf or .tex.

2. Please provide a complete Data Availability Statement in the submission form, ensuring you include all necessary access information or a reason for why you are unable to make your data freely accessible. If your research concerns only data provided within your submission, please write "All data are in the manuscript and/or supporting information files" as your Data Availability Statement.

Additional Editor Comments (if provided):

The reviews consider the work rudimentary yet with potential and give a considerably large set of recommendations and arguments that the authors need to addressed point by point with improvements that satisfies their requirements. Please prepare a revised version of the submitted manuscript and a point-by-point replay to the reviewers.

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Reviewers' Comments:

Reviewer's Responses to Questions

Comments to the Author

1. Does this manuscript meet PLOS Complex Systems’s publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Partly

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: No

Reviewer #2: Yes

Reviewer #3: N/A

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3. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data 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—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: No

Reviewer #2: Yes

Reviewer #3: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS Complex Systems does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The authors show that the brain's function is determined by interactions among neurons, which form a highly complex dynamical system. They propose a new approach to describe the functional properties of neuronal networks using the weights of synaptic connections and the current activity of neurons, without relying on dimensionality reduction.

Could you elaborate further on how dimensionality reduction specifically impacts the understanding of neuronal interactions? It would be helpful to include a brief example to clarify its limitations and why this new approach is more effective.

The concept of ‘state space’ is introduced, but it might be beneficial to provide a concrete example or visualization to make this abstract idea more accessible to readers, especially those less familiar with the term.

The introduction mentions the goal of creating a "ground truth" framework for multi-neuronal interactions. Can you provide a clearer statement on how this approach differs from existing methods and its potential impact on neuroscience research?

Consider reorganizing the introduction to start with the core problem in simpler terms before diving into technical details. This could help readers grasp the motivation behind the study more quickly.

Could you provide more context or a brief explanation about the non-spiking neuron model used?

In Equation (1), could you explain more clearly the role of the static leak constant k and how it influences the overall activity of the postsynaptic neuron?

The term "critical point" is mentioned frequently. Could you provide a brief explanation of its significance in the context of neuronal dynamics, perhaps linking it to a broader concept in dynamical systems?

Could you provide a clearer explanation or an initial overview of the vector field concept as applied to neural interactions?

When extracting planes from higher-dimensional networks, what criteria are used to select the planes?

The discussion describes how the critical point can shift due to changes in neuron activity. Could you provide more detail on how these shifts might relate to specific neural processes or behaviors?

While the discussion touches on potential complexities like non-linear activation functions and varying synaptic types, it would be beneficial to explicitly outline limitations of the current model and possible directions for future research.

Reviewer #2: The paper explores a novel conceptual framework for analyzing multi-neuronal interactions in biological networks using high-dimensional vector fields. By leveraging activity-defining constituents such as synaptic weights and neuronal activity, the authors demonstrate how network dynamics can be represented as a vector field, elucidating critical properties of neuronal activity states. This method addresses limitations in traditional dimensionality-reduction techniques, providing insights into network structure and behavioral adaptability. The study introduces planar subspace extraction to visualize complex interactions and discusses critical point dynamics influenced by excitatory and inhibitory synaptic weight distributions. Overall, the work offers a promising tool to better understand the dynamical behaviors of neuronal networks. The paper is clear and well-written. However, the following issues may need to be resolved before the paper can be published.

(1) The proposed framework is solely based on computational models and lacks validation against experimental data, such as neuronal activity recordings or network-level interactions observed in biological systems. Incorporating or referencing experimental findings could significantly strengthen the claim that the proposed vector field approach accurately reflects biological neuronal networks.

(2) The exploration is limited to fixed or skewed synaptic weight distributions without considering the dynamic nature of synaptic plasticity (e.g., long-term potentiation or depression). This neglects the impact of temporal changes in connectivity on network behavior.

(3) The study assumes identical, linear input-output functions for all neurons. This does not account for neuron-type-specific dynamics, such as non-linear threshold behavior in pyramidal neurons or sustained responses in metabotropic synapses, which may lead to inaccuracies when applying the framework to real neural circuits.

(4) The study predominantly examines critical points in vector fields but overlooks other significant features such as attractors, limit cycles, or transient trajectories that could provide a more complete picture of network dynamics.

(5) Despite claims about applicability to behavioral adaptability, the manuscript does not explicitly link its findings to specific functional phenomena such as decision-making or sensory integration.

Reviewer #3: Summary: Authors propose a rudimentary model of analysis of recurrently connected neural networks, with the ambition to capture the activity of biological brains. The model consists in constructing high-dimensional vectors field that describes the neural activity as a dynamical system. In the high-dimensional space, the activity of each neural unit defines a particular dimension, while synaptic weights of neural units define the vector field in this neural space. A neural unit can consist of single neurons or of a linear combination of activity of several neurons. Fixed points in the vector space occur when no neuron is active or when the excitation and inhibition match.

Major:

1) In biological brains, the activation of one neuron (an action potential) typically influences the subthreshold activity of another neuron, without necessarily causing the other neuron to fire an action potential and change its spiking output. In brain of mammals and beyond sensory periphery, typically, a certain number of presynaptic inputs is required to bring the postsynaptic neuron to its firing threshold. The activation of the postsynaptic neuron j will thus typically be directly impacted by the activation of the presynaptic neuron i (in the sense that it would lead to the threshold crossing of the postsynaptic neuron) only in case when the presynaptic neuron is excitatory and the membrane potential of the postsynaptic neuron is close to the threshold. This would in general be true for only a small proportion of presynaptic inputs. Another possibility is that the synaptic weight of the presynaptic neuron is very strong and by itself sufficient to trigger an action potential in the presynaptic neuron, however, this is not common in the cortex nor in the spinal cord. I believe that because of the lack of threshold in this model, the model cannot capture the activity of biological neurons over time, and also not in a single point in time.

However, I believe that the model can potentially capture the average effect of firing rate changes in a neural network in a steady state. This would imply that the variable a_i is interpreted as the time-averaged firing rate of the neuron i. The change in the average firing rate of presynaptic neurons can impact the average firing rate of postsynaptic neurons. Can Authors comment on this?

2) What is a “non-perpendicular” neuron? As far as I understand the model, all neurons (or their linear combinations) are by construction perpendicular to each other, but the observation plane can cut though the hypercube of neural activity space at an arbitrary angle.

3) The text of Results calls the Figure 2F-G, but I the Figure 2 does not seem to have these plots.

4) It is unclear why in the Eq. 1, the activity of the postsynaptic neuron is formulated as the ratio, in particular, it is not explained what is the meaning of the denominator. It seems like normalization, but it is not clear why normalization is necessary or how is it justified?

5) Authors use a non-spiking neuron model that emulates conductance-level neuron models and is reported to have a static leak component. It is unclear how conductance-based model can have a static leak, since a dynamic nature of conductance seems to be essential to conductance-based models.

6) Authors describe to some length the trivial solutions when no neuron is active, but more attention could be given to less trivial cases where fixed points occur due to the match of excitation and inhibition. I am not aware of any biological system that would consist of only excitatory or only inhibitory neurons.

7) In the last paragraph of discussion (starting line 240), authors expand on how their model has the potential to capture neural activity that underlies behavioral decisions. Since the model does not demonstrate convincingly how it would capture the temporal dynamics of neural networks, I am skeptical that it could capture transients of neural activity that are important for decision-making signals in biological brains. I suggest to rephrase this paragraph and tune down the applications of the model, because the results do not demonstrate the relevance of the model to describe decision-making.

8) Modeling neural networks as dynamical systems has been done before and constitutes a notable subfield of computational neuroscience (see for example Boerlin et al. PLOS CB 2013, Brendel et al. PLOS CB 2020, Koren et al. eLife 2024, Podlaski and Machens, Neural Computation 2024). The present manuscript might increase in relevance if authors would situate their work within the relevant literature. Authors seem to entirely ignore the literature where dynamical systems are used to model the activity of recurrent neural networks.

9) From my understanding, the model is linear, as the vector field in each dimension is defined as a weighted activity of the presynaptic neuron, or alternatively as a linear combination of neural activities. While such model might be a useful approximation of a physical system, it might not be able to capture non-linearities that might be, to some extent at least, important to describe biological neural systems (Thalmeier et al. PLOS CB 2016). Could Authors comment on that?

10) How does the frequency-current (f-I) curve of a neural model affect the vector field? I suggest that Authors investigate this dependency for some examples of the most relevant f-I curves.

Minor

-Eq. 4: it is not explained how this equation is derived. Such description could fit into the Methods.

- Lines 121-122: This sentence is awkward to read.

- line 125: there is a typo

- line 198: Rather than “recursive” network, I suggest to use a more common expression of recurrently connected or recurrent neural network.

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6. 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.

Do you want your identity to be public for this peer review? If you choose “no”, your identity will remain anonymous but your review may still be made public.

For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

Figure resubmission:

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. Registration is free. 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 PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. If there are other versions of figure files still present in your submission file inventory at resubmission, please replace them with the PACE-processed versions.

Reproducibility:

To enhance the reproducibility of your results, we recommend that authors of applicable studies deposit 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

PCSY-D-24-00137R1

Neuronal networks quantified as vector fields

PLOS Complex Systems

Dear Dr. Jörntell,

Thank you for submitting your manuscript to PLOS Complex Systems. After careful consideration, we feel that it has merit but does not fully meet PLOS Complex Systems's publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript within 30 days Apr 26 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at complexsystems@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pcsy/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

* A rebuttal letter that responds to each point raised by the editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. This file does not need to include responses to any formatting updates and technical items listed in the 'Journal Requirements' section below.

* A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

* An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, competing interests statement, or data availability statement, please make these updates within the submission form at the time of resubmission. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

We look forward to receiving your revised manuscript.

Kind regards,

Juan Gonzalo Barajas-Ramirez

Academic Editor

PLOS Complex Systems

Additional Editor Comments (if provided):

AE: The reviewers are in-disagreement as to their recommendations for the revised version of the manuscript. Although a significant improvement is recognized there are points that require additional corrections. In particular the comments of Reviewers 3 and 4 need to be addressed. The authors are encourage to prepare an additional revision where a larger discussion and recognition of limitations in their approach.

Reviewer 3:

I thank the authors for their thorough revision of the manuscript - the manuscript is now substantially improved. I have a further minor suggestion for an additional comment of the current paper with respect to previous work on across-neuron interactions measured by correlations. Since the manuscript is rather abstract and presents a new idea, I find it important that its relation to most relevant existing approaches is as clear as possible.

In neuroscience, it is common to assess the interactions across neurons with the measure of pair-wise correlation of neuronal activity (Cohen and Kohn, Nat. Neurosci. 2011; Averbeck et al. Nat. Rev. Neurosci. 2006; Panzeri et al. Nat. Rev Neurosci. 2022). Noise correlations between pairs of neurons were shown to have non-random structure (Koren et al. Cell Reports 2020), which could be reflected by constructing vector fields from neuronal activity. Can vector fields use correlations as activity-defining constituents instead of synaptic weights? Correlations are a functional measure and can change from one context to another, while synaptic weights are much closer to a “ground truth” quantity that determines across-neuron interactions in any context. However, synaptic weights are still today very difficult to assess experimentally, while assessing correlations is straightforward. Can authors comment on the relation of their model to the related research on correlations of neural activity?

Reviewer 4:

Dear Authors,

Thank you for sharing this thoughtfully worded paper. In particular I found the introduction to have a greater than average level of nuance, and your assertion that some hypothesis-driven framework is necessary for optimal extraction of meaningful signals from populations is interesting (I suppose this would correspond to the strong hypothesis of neural dimensionality reduction). There's an inter-related cloud of topics your paper touches, of dynamics, task and behavior representations, structure-function maps and internal vs human-explainable representations of neuronal activity. I'm sure you're aware of the work from Byron Yu, as well as younger researchers like Alex Cayco-Gajic in this area. I think part of what makes their work particularly compelling is how it's tied very closely to neurophysiology. I can see that you're trying to do that in your work, but I think showing practical experimental relevance is going to be important to all but some (hypothetical) work of such mathematical elegance that such concerns are thrown to the wind. This point also pertains to how your future work might extend to spiking neurons, additional canonical neuronal models, particular experimental paradigms. In the spirit that all models are wrong, what can make some of them useful, it may be helpful to consider which particular aspect of neural coding you're going to focus on in future work - is it more analytical, will it be trying to recapitulate dimensionally from human EEG, or single unit recordings in a particular area of mouse brain - many many options to consider, but focusing in to a particular paradigm may be helpful, even in the service of ultimately developing more general models/representations.

[Note: HTML markup is below. Please do not edit.]

Reviewers' Comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #3: (No Response)

Reviewer #4: (No Response)

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2. Does this manuscript meet PLOS Complex Systems's publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Yes

Reviewer #3: Yes

Reviewer #4: No

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #3: Yes

Reviewer #4: No

**********

4. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data 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—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #3: Yes

Reviewer #4: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS Complex Systems does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #3: Yes

Reviewer #4: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: (No Response)

Reviewer #3: I thank the authors for their thorough revision of the manuscript - the manuscript is now substantially improved. I have a further minor suggestion for an additional comment of the current paper with respect to previous work on across-neuron interactions measured by correlations. Since the manuscript is rather abstract and presents a new idea, I find it important that its relation to most relevant existing approaches is as clear as possible.

In neuroscience, it is common to assess the interactions across neurons with the measure of pair-wise correlation of neuronal activity (Cohen and Kohn, Nat. Neurosci. 2011; Averbeck et al. Nat. Rev. Neurosci. 2006; Panzeri et al. Nat. Rev Neurosci. 2022). Noise correlations between pairs of neurons were shown to have non-random structure (Koren et al. Cell Reports 2020), which could be reflected by constructing vector fields from neuronal activity. Can vector fields use correlations as activity-defining constituents instead of synaptic weights? Correlations are a functional measure and can change from one context to another, while synaptic weights are much closer to a “ground truth” quantity that determines across-neuron interactions in any context. However, synaptic weights are still today very difficult to assess experimentally, while assessing correlations is straightforward. Can authors comment on the relation of their model to the related research on correlations of neural activity?

Reviewer #4: Dear Authors,

Thank you for sharing this thoughtfully worded paper.  In particular I found the introduction to have a greater than average level of nuance, and your assertion that some hypothesis-driven framework is necessary for optimal extraction of meaningful signals from populations is interesting (I suppose this would correspond to the strong hypothesis of neural dimensionality reduction).  There's an inter-related cloud of topics your paper touches, of dynamics, task and behavior representations, structure-function maps and internal vs human-explainable representations of neuronal activity.  I'm sure you're aware of the work from Byron Yu, as well as younger researchers like Alex Cayco-Gajic in this area.  I think part of what makes their work particularly compelling is how it's tied very closely to neurophysiology.  I can see that you're trying to do that in your work, but I think showing practical experimental relevance is going to be important to all but some (hypothetical) work of such mathematical elegance that such concerns are thrown to the wind.  This point also pertains to how your future work might extend to spiking neurons, additional canonical neuronal models, particular experimental paradigms.  In the spirit that all models are wrong, what can make some of them useful, it may be helpful to consider which particular aspect of neural coding you're going to focus on in future work - is it more analytical, will it be trying to recapitulate dimensionally from human EEG, or single unit recordings in a particular area of mouse brain - many many options to consider, but focusing in to a particular paradigm may be helpful, even in the service of ultimately developing more general models/representations.

**********

7. 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.

Do you want your identity to be public for this peer review? If you choose “no”, your identity will remain anonymous but your review may still be made public.

For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #3: No

Reviewer #4: No

**********

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

Figure resubmission:

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. Registration is free. 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 PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. If there are other versions of figure files still present in your submission file inventory at resubmission, please replace them with the PACE-processed versions.

Reproducibility:

To enhance the reproducibility of your results, we recommend that authors of applicable studies deposit 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

Neuronal networks quantified as vector fields

PCSY-D-24-00137R2

Dear Dr. Jörntell,

We are pleased to inform you that your manuscript 'Neuronal networks quantified as vector fields' has been provisionally accepted for publication in PLOS Complex Systems.

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

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.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they'll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact complexsystems@plos.org.

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

Best regards,

Juan Gonzalo Barajas-Ramirez

Academic Editor

PLOS Complex Systems

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

The reviewers are satisfied with the revised version.

Reviewer Comments (if any, and for reference):

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #3: All comments have been addressed

**********

2. Does this manuscript meet PLOS Complex Systems's publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Yes

Reviewer #3: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #3: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data 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—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #3: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS Complex Systems does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #3: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: All comments have been addressed

Reviewer #3: Authors have addressed all my concerns.

**********

7. 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.

Do you want your identity to be public for this peer review? If you choose “no”, your identity will remain anonymous but your review may still be made public.

For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #3: No

**********