Remarks on the Mathematical Modeling of Gene and Neuronal Networks by Ordinary Differential Equations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

No. Well written and constructed paper.

Author Response

Thanks to Reviewer 1

Reviewer 2 Report

Comments and Suggestions for Authors

Greetings!
See review in attached .pdf file.

Comments for author File: Comments.pdf

Author Response

Thanks to Reviewer 2. For responses to the reviewer, see the attached .docx file.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

The work is very interesting from both a mathematical and an applied point of view. 

The types of W matrices for the case of three-dimensional systems are selected in a rather cross-sectional way. This choice allowed not only the analysis of the dynamics of the system at given values of the W matrix, but also allowed the observation of different types of system behaviour, including Hopf bifurcations. 

Author Response

Thanks to Reviewer 3.

Reviewer 4 Report

Comments and Suggestions for Authors

1. The abstract needs to rewritten. This is the first time I saw someone mentioned in the abstract the number of citations.

2. The language needs improved.

3. This manuscript needs a complete rewriting. What is the question being aksed here? Why is it important? What did you find and how did you do it? None of these are clear to me in the current version. The readers won’t feel interested just because you chose an arbitrary system and plotted their nullclines and attractors.

Comments on the Quality of English Language

Needs rewritten.

Author Response

Thanks to Reviewer 4.  See answers to the reviewer in the attached .docx file.

Author Response File: Author Response.docx

Reviewer 5 Report

Comments and Suggestions for Authors

The paper refers to a mathematical model which describes an autonomous neural network. The authors followed the well-known procedures and did computations specific for the studies of dynamical systems: invariant set, nullclines, critical and focus points, linearization of the system around these points, bifurcations and periodic orbits. The cases of networks with 2, 3 and n nodes are considered. There is nothing special from a mathematical point of view. Since the reported results are practically no connected with neural networks, there is nothing special from this point of view either. Moreover, the paper has gaps, both in conception and presentation, gaps that make it difficult to follow and impossible to publish in its current form. Here are some examples:

-      The authors chose to study a network described by system (4). It's just one model (not the only one as we could understand from the text). It is not clear how the paper is related to other published studies related to the same model and what it brings as novelty.

-      The structuring of the material does not follow a certain logic, generating mix-ups of information about 3D, 2D and n-D networks, and unnecessary repetitions of relationships, such as (7) and (9). It would be much more natural to first have a section establishing a general theoretical framework, showing, for the generic system (4), or (even better) for an n-node network, how to determine the invariant set, the critical and focal points, the bifurcations and different types of orbits. Examples may follow, grouped into one or more sections.

-      The results are structured as Propositions and Corollaries, some of which are actually simple, evident statements (e.g. Proposition 2.2), but most with inconsistent proofs.

-      Example 2.1 consists in a figure that, without any explanations, cannot be understood. No explanations are given related to how and why the parameters were chosen in the other examples, too.  What means, for example, negative coefficients in W?

-      The authors use the same captures for Figures 2, 3 and 4, without mentioning that they corresponds to different choices of W. Moreover, the lack of any interpretation makes the graphs practically meaningless for ANN.

-      The findings the authors reported in the paper are without any interpretation related to ANN. There are nice comments in Introduction and in section 7, but they are general, not related to the findings at all.

-      The paper is sloppily written, with many typos and mistakes in English. See, for example, relations (3) and (6), or the sentence appearing in the Introduction "the illustrative example are".

Comments on the Quality of English Language

The paper has to be improved.

Author Response

Thanks to Reviewer 5. For responses to the reviewer, see the attached .docx file.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Accepting authors' responses.

Author Response

Thanks to Reviewer 2.

Reviewer 4 Report

Comments and Suggestions for Authors

This paper considers a 3D dynamical system, often used as components in ANNs, and analyzed the fixed points, limit cycles and bifurcations under various conditions. My major issue with this paper is not about the results, but the motivation and writing of this paper.

See my comments below:

1. First of all, when people use the term “Artificial Neural Networks” (ANNs), they usually use them in the sense of machine learning models, where the most common types of ANNs are feedforward neural networks (FNNs), recurrent neural networks (RNNs), convolutional neural networks (CNNs), etc. The ANNs that are being discussed in the manuscript are most related to the recurrent neural networks since the authors consider time evolution of the system using ODEs. However, all types of ANNs are usually trained on some kind of tasks (like image recognition etc) and they usually have hundreds or thousands of units. But neither is true for the current manuscript. No task or training is involved and the model only has 3 dimensions. For sure, there is nothing wrong to study a low-dimensional system because the bifurcations of the low-dimensional system may give us insight into the dynamics in higher-dimensional ones. But I wouldn’t motivate the entire paper only from the perspective of ANNs (of course, modeling of networks using dynamical system are ANNs in broad), because it is kind of misleading and not consistent with the common usage of this term. And it’s too broad. It’s good to start broad in the Introduction but the authors also need to narrow it down quickly.

2. Moreover, the title of this manuscript is “… mathematical modeling of gene and neuronal networks …”, but nothing about the genetic/neuronal networks were discussed in the Introduction. For example, the authors studies different dynamical regimes of the system, like Regulatory matrices with zero diagonal elements, focus-type fixed points, Inhibition-activation etc. So the questions are what do they mean for genetic/neuronal networks and why understanding them is important? This can be a way to narrow down the Introduction and highlight the motivation and significance of this paper. By analyzing the bifurcation of the system, we can gain insight into the some of the properties of genetic/neuronal networks in terms of their function and computation. As I said in my previous review, choosing a system and analyzing its bifurcations and limits sets is not interesting per se. But these bifurcations and dynamics must provide mechanisms for some physical processes (as in this case, genetic/neuronal networks) and understanding the mechanisms is why it’s important. But that part is unclear in the current version. Some good example papers with a similar flavor are as follows:

   Kraynyukova, N., & Tchumatchenko, T. (2018). Stabilized supralinear network can give rise to bistable, oscillatory, and persistent activity. Proceedings of the National Academy of Sciences, 115(13), 3464-3469.

   Ahmadian, Y., & Miller, K. D. (2021). What is the dynamical regime of cerebral cortex?. Neuron, 109(21), 3373-3391.

   …

3. Section 8 about the control is interesting. My questions are: what makes theta2 special and important in this case? What would be theta in higher dimensions? What can be the biological mechanism that change theta2 for genes and neurons?

Comments on the Quality of English Language

English is good.

Author Response

The authors thank the referee for carefully reading the article and making valuable suggestions leading to several improvements.  See answers to the reviewer in the attached .docx file.

Author Response File: Author Response.docx

Reviewer 5 Report

Comments and Suggestions for Authors

The paper was improved and can be now published after minor revisions.

1) The authors have to correct the first two equations from (6) where x_n appears.

2) The claim from Introduction "we study properties of the mathematical model of a three-dimensional ANN" is not true as long as in section 6 for example a n-dimensional network is considered. I suggest a reformulation of the form: "We mainly study properties of the mathematical model of a three-dimensional ANN, but part of our results will refer to two-dimensional or, more generally, to n-dimensional networks".

3) Authors should check for errors in newly inserted paragraphs (highlighted in yellow). For example, the English wording of the last paragraph of the Conclusions should be revised, and reference (4) should probably be (43).

Comments on the Quality of English Language

The English is OK, but, as already mentioned, the authors need to revise some wording in the new paragraphs.

Author Response

Thanks to Reviewer 5.  See answers to the reviewer in the attached .docx file.

Author Response File: Author Response.docx