Detection of Anticipatory Dynamics between a Pair of Zebrafish

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the article “Detection of Anticipatory Dynamics Between a Pair of Zebra fish”, the authors present an experiment on zebra fish, and find that the fish show anticipatory dynamics in some cases. This means that there is a leading fish and a lead fish, and the lead fish anticipates the trajectory of the leading fish. This paper is very interesting, as it describes a behavioral outcome by means of physics, with a mathematical model. Such a finding is very rare and pushes the envelope of the description of biological systems.

Before publication, I would consider the following.

The conundrum that some measures of information direction cannot capture the case of AS has been theoretically investigated in [1]. I believe in their simulations, and maybe in their experiment, too, the authors are getting into a regime with too coarse resolution, in which causality detectors give wrong results. This could be further investigated or at least stated in the discussion.

 

 This is the abstract of [1]:

 

 “We report that transfer entropy estimates obtained from low-resolution and/or small data sets show net

 

information flow away from a purely anticipatory element whereas transfer entropy calculated using exact

distributions show the flow towards it. This means that for real-world data sets anticipatory elements can

appear to be strongly driving the network dynamics even when there is no possibility of such an influence.”

 [1] D. W. Hahs, and S. D. Pethel: "Distinguishing anticipation from causality: anticipatory bias in the estimation of information flow," Phys Rev Lett, vol. 107, pp. 128701, 2011.

 

 Ethical concerns: The animal research protocol number seems to be missing (“xxx”)

Author Response

 Thanks for the reivewer to point out this important reference for us. We have performed further simulation and the discussions of Hahs'work is included in the new section of "Limitations" as:

\subsection{Issues of Entropy based Causality}
It is well known that one must be cautious when dealing with entropy; especially when it is used to deal with causality \cite{zenil2017low}. The counter-intuitive anticipatory dynamics under consideration here might even make matter worse. For example, in a simulation study of using TE to determine the direction of causality in coupled logistic maps with anticipatory properties, Hahs et al \cite{hahs2011distinguishing} found that the direction of causality in this relatively simple case is dependent on the data resolution being used. Correct direction of causality can be found only when high enough data resolution is used to compute the transfer entropy. In fact, we have also checked that, for our simulation model above, the results could be quite different from that shown in Figure~\ref{TE_hist} if a higher data resolution is used in the computation of TE. 

The two lines in Figure~\ref{TE_hist} will crossover at a certain history length when the data resolution is 6 bit or higher; suggesting that the direction of information flow is reversed. However, we find that this crossover can be removed by using longer simulated time (larger sample size). It is not clear if this phenomenon originates from the bias of sample size mentioned above or it is related to the effect mentioned in Ref\cite{hahs2011distinguishing}. Nevertheless, these results from the NGD model simulation clearly demonstrated that TE based causality must be taken with care. Unlike numerical simulations in which data resolution can be increased relatively easily, our experimental data resolution is fixed by the setup. The use of computed TE based causality in our experiments might not be conclusive.  Of course, our assumption that the "driver" fish will go through the door first in the experiment also needs to be further tested.

Reviewer 2 Report

Comments and Suggestions for Authors

 

The topic is very interesting. Also, the paper has some novelty. I have the following comments before consideration for publication.

-        Include more references and expand your literature review. The literature review on this specific application of proposed model is sufficient, however the literature review on similar models in other applications is not enough, for example for applications of entropy in PHM, include this related work: https://www.mdpi.com/1999-4893/15/11/393 and for Zebrafish, include this related work: https://www.sciencedirect.com/science/article/pii/S0149763422001683.

-        Create a table and summarize all the notations (parameters, variables, etc.).

 

Comments on the Quality of English Language

Minor edit is needed.

Author Response

Original comments are listed first and followed by our response:

The topic is very interesting. Also, the paper has some novelty. I have the following comments before consideration for publication.

-        Include more references and expand your literature review. The literature review on this specific application of proposed model is sufficient, however the literature review on similar models in other applications is not enough, for example for applications of entropy in PHM, include this related work: https://www.mdpi.com/1999-4893/15/11/393 (namdari2022lithium)

Our response>  The following sentences are added in the discussion to include the references cited by the reviewer: “Most likely these news tools are also information/entropy based as it is being used in a wide varieties of applications. A recent example  is the use of permutation entropy of the voltage sequences in a rechargeable batter together with machine learning for the prediction of the next-cycle battery capacity \cite{namdari2022lithium}.”

and for Zebrafish, include this related work: https://www.sciencedirect.com/science/article/pii/{bS0149763422001683.

 

Our response>The following sentences has been added to address this point: "Finally, we would like to note that the method of analysis presented here can be further tested by using zebra fish with known altered behavior; for example, fish with neurodegenerative disease\cite{bashirzade2022modeling}. Presumably, the dynamics of interaction between these fish might be quite different because of their impaired information processing capability."

-        Create a table and summarize all the notations (parameters, variables, etc.).

Our response> A list of symbols used in the paper has been added in the “Abbreviations and Symbols” section

Reviewer 3 Report

Comments and Suggestions for Authors

The study tests the hypothesis of the presence of anticipatory dynamics in a pair of zebrafish.

The study is interesting, but the authors should be more cautions in interpretating markers derived from their tools in terms of causality.

 

1)      Anticipatory responses do not provide any indication about causality because causal mechanisms might exhibit anticipatory responses of the output with respect to the input, for example cerebral autoregulation (see F. Gelpi et al, Auton Neurosci: Basic Clin, 237, 102920, 2022), and anticipatory dynamics might suggest a causal relationship even in the opposite time direction (i.e., from the output to input). Please clarify throughout the manuscript.

2)      Description of anticipatory dynamics needs the definition of the driver and target signals. Please pay attention that this definition is given and unambiguous in any parts of the manuscript.

3)      Anticipatory dynamics is usually detected via the transfer function phase and a relationship between phase markers and Granger causality indexes has been established (se e.g. A. Porta et al, Am J Physiol, 300, R378-R386, 2011). The authors should compute phase and compare the results with those derived from Granger causality tools.

4)      The presence of closed-loop interactions could make the definition of anticipatory response much more complex. For example, in the case of cardiorespiratory coupling a positive phase relationship from respiration to heart rate suggested the possible action of the heart on the respiratory system (see A. Porta et al, Phil Trans R Soc A, 371, 20120161, 2013).

5)      Simulations should be more complex to become really useful. Closed-loop relationship should be included between Z and S as it is likely to occur in a pair of interacting zebrafish.

6)      Computation of phase should be extended to the simulated data as well.

Comments on the Quality of English Language

The quality of English is sufficient

Author Response

Comments of the reviwer will be list first and followed by our response:

1) Anticipatory responses do not provide any indication about causality because causal mechanisms might exhibit anticipatory responses of the output with respect to the input, for example cerebral autoregulation (see F. Gelpi et al, Auton Neurosci: Basic Clin, 237, 102920, 2022), and anticipatory dynamics might suggest a causal relationship even in the opposite time direction (i.e., from the output to input). Please clarify throughout the manuscript.

Our Response> We agree with the reviewer that “anticipatory dynamics might suggest a causal relationship even in the opposite time direction”. There might be other definitions of anticipatory dynamics (AD). But this is exactly what we meant by AD in our manuscript. That is: information flows from the input to the output but the time course of output is ahead of that of the input. Our experiment was setup to look for the existence of this definition of AD. We will make this definition of AD more explicit in the manuscript. The reference cited by the reviewer is added in the revised manuscript when we discuss the case of a closed-loop control system.

2) Description of anticipatory dynamics needs the definition of the driver and target signals. Please pay attention that this definition is given and unambiguous in any parts of the manuscript.

Our Response> Thanks for the suggestion and we have added the following sentence to make this point clear: "The time series $S(t)$ is sometimes labelled as the master or driver signal and as slave or target signal for $Z(t)$." in the simulation section.

3) Anticipatory dynamics is usually detected via the transfer function phase and a relationship between phase markers and Granger causality indexes has been established (se e.g. A. Porta et al, Am J Physiol, 300, R378-R386, 2011).(porta2011causal) The authors should compute phase and compare the results with those derived from Granger causality tools.

Our Response> We agree with the reviewer that anticipatory dynamics can be detected by the phase of a transfer function in the frequency domain. In fact, the negative group delay model used to generate anticipatory dynamic data in our manuscript is based on the phase of the transfer function (see Voss[2]). In the paper cited by the reviewer (Porta et al), the Granger causality indices were an extension of the idea of the PRE paper by Palus et al (cited as Ref 24 in Porta’s paper). In Palus’ PRE paper, they were considering a system of weakly interacting phase oscillators. Therefore, the time course of the phases of the oscillators are used in the computation of directional indices. However, in our case of two interacting fish, we do not have any good argument to think of them as two weakly coupled oscillators. We could have computed the corresponding phases from the Hilbert transform of the trajectory signals, but they have no physical meanings because there are no obvious repeating patterns in the trajectories. In fact, in Porta’s paper, no phase data were used. In this sense, our methodology is similar to those of Porta’s work. We felt that the precision of our data was not good enough to distinguish different types of interactions as in Porta’s work. In the revised manuscript, we have added a few sentences to alert the readers about this problem.

4)  The presence of closed-loop interactions could make the definition of anticipatory response much more complex. For example, in the case of cardiorespiratory coupling a positive phase relationship from respiration to heart rate suggested the possible action of the heart on the respiratory system (see A. Porta et al, Phil Trans R Soc A, 371, 20120161, 2013).(porta2013cardiovascular)

Our response> Thanks for pointing this out. We agree that the presence of closed-loop interactions could make the definition of anticipatory response much more complex as in the case of cardiorespiratory coupling as mentioned by the reviewer. In this closed-loop case, the heart rate the respiration rate of an individual are controlled to maintain a suitable state for the daily activities of the individual. Therefore, the overall control mechanism can be quite complex. However, in the case of two interacting fish, it is not clearly known what is being controlled. Therefore, there might not be any closed-loop. Of course, there is a possibility that a bi-directional anticipatory dynamic can take place between the two fish. However, with the narrow definition of AD in our manuscript, if the net information flow in close to zero or there is no obvious leader in the system (in statistical sense), our method will not be able to detect it. Therefore, it is not easy to apply our method to detect bi-directional interaction. We have stated clearer the restriction of our AD detection method in the new section of Limitation as:

\subsection{Problem of bi-directional coupling}

It should be stated again that our definition of anticipatory dynamics is quite narrow. For example, if there are bi-directional anticipatory interactions between the two fish with almost equal strength, the resultant cross-TLMI will peak very close to zero time lag. In such a case, there will be no obvious leader in the two resultant time series and our method will reach the wrong conclusion that there is no AD. Therefore, our method of detecting AD probably cannot be used to detect causality in a closed-loop control system such as the coupling between heart rate and respiration rate \cite{porta2013cardiovascular} or auto-regulation of blood pressure \cite{gelpi2022dynamic}. In a closed-loop control system, different variables of the system are coupled directly or indirectly in such a way to maintain the system in some desirable state. Therefore, the causal effects of different variables in such a system are investigated because these causal relations might reveal the underlying control mechanism. Methods such as Granger indices \cite{porta2011causal} have been used to infer the types of interaction between different variables in the controlled system. However, such an approach is probably not suitable for our investigation of the interaction between two fish. It is not clear if their trajectories can be regarded as the result of some kind of control. We could have also extended our NGD simulation model to have a bi-directional coupling. But as mentioned above, our detection method would probably fail.

5) Simulations should be more complex to become really useful. Closed-loop relationship should be included between Z and S as it is likely to occur in a pair of interacting zebrafish.

Our Response> The simulation was not intended to be realistic or useful for describing the interaction between a pair of zebrafish. The negative group delay simulation in our manuscript was used to demonstrate that even in this simple case of anticipatory dynamics, the common methods of TE and Granger causality test failed to detect the correct direction of causality. Therefore, even these tools failed to detect anticipatory dynamics in the experiments, it is not clear that no anticipatory dynamics were present in our data. To make this point clearer, we have added the sentence: "With this simple example, it should be clear that even the methods of TE and GC failed to detect anticipatory dynamics in our experimental data, it is not conclusive that no anticipatory dynamics is present in our experiments." at the end of the simulation section.

6) Computation of phase should be extended to the simulated data as well.

Our Response> Please see our response to point 1) above.

Note that all the references cited by thre reviewer are all included in the revised version of the manuscript

Reviewer 4 Report

Comments and Suggestions for Authors

Please see attachment.

Comments for author File: Comments.pdf

Author Response

Report 4)

Comments are in the pdf file

Our Response> We are sorry that our use of reference of Voss[3] work in the introduction of the manuscript might be miss-leading. Ref.3 is the case of anticipatory synchronization in which two identical systems are synchronized by a delayed feedback drive from the master to the slave system. We agree with the reviewer that “Anticipatory dynamics in two autonomous systems will violate causality.” In Ref.3, when the two identical systems are synchronized, there is no transfer of information. They can be synchronized because they have identical intrinsic dynamics and the driving term (information flow) will disappear after synchronization. Therefore, there is no question of causality in this case because of no information flow.

However, this is not the case we were considering in our manuscript. In the simulation of the Voss model in our manuscript, we are using the Voss’s model published in 2006 (Ref[2] in our manuscript); with the title: Signal prediction by anticipatory relaxation dynamics. In this work, the two systems under consideration are different. In fact, the intrinsic dynamics of the slave system is just pure relaxation. The slave system can be anticipatory of the master system because of the delayed feedback drive and the predictable properties of the master system. There is no synchronization between the master and slave and therefore the driving term (information flow) will not be zero all the time. In this case, the causal relation between the two systems is clear. We have made this point clear in the “correlation and causation” under the new Limitations section in the revised manuscript as:

\subsection{Correlation and Causation}

One common mistake in the study of causality is the confusion between correlation and causality. In Ref\cite{voss2000anticipating}, where anticipatory synchronization is considered, there is a master and a slave with identical intrinsic dynamics (before coupling). These two systems will synchronize with the slave system ahead of the master system. It would be easy to conclude that the slave system is the cause of the master system because its time course is ahead of that from the master. However, in such a case, there is no causal relation between them because when the two systems are synchronized, there is no information flow between them. In fact, the coupling term vanishes when the two systems are synchronized. In such a case, these two systems are perfectly correlated but there is no casual relation between them.

 

However, in the case of Ref\cite{Voss2016}, the NGD model (Eqn(7)) in our manuscript, $S(t)$ and $Z(t)$ are not only correlated but there is also a causal relation between them. Note that the intrinsic dynamics of $Z(t)$ is purely relaxation without the coupling term. It is the coupling term delayed feedback, $k(S(t)-Z(t-t_d))$, which uses $S(t)$ as the drive for the system $Z(t)$. The system $Z(t)$ can produce anticipatory output of $S(t)$ because of the negative group delay properties of the system $Z(t)$ as well as the predictability of signal $S(t)$ by using a lowpass filter on a random signal. Had $S(t)$ been totally random, there will be no anticipatory output from $Z(t)$. For this NGD model, the causal direction is clearly from $S(t)$ to $Z(t)$.

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript has been substantially improved. The authors replied satisfactorily to all my issues and took into account the suggestions given. I have no additional comments. 

Comments on the Quality of English Language

It is fine

Author Response

Thanks for the report.

Reviewer 4 Report

Comments and Suggestions for Authors

You remark that Voss's paper of 2016 supports your idea, (I do not find his paper of 2006) but as I said before, Voss's anticipatory relaxation dynamics cannot be applied to movement of two zebrafishes in a tank.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

English in the text is fair.  In the response to reviewer, there are mistypes like casual-> causal. 

Author Response

The comments of the reviewer are listed first and followed by our response.

Reviewer>The authors added references that study anticipatory elements and causality based on Voss’s argument (Hahs et al.) and information transfers based on Schreiber’s argument (Barnett et al.) both are well written papers. However, I do not think that these papers supports results of the paper contin characteristic information transfer between two zebrafishes

Our Response> The papers cited above are used to illustrate that it is not trivial to use transfer entropy to detect causality between time series; especially when there is anticipatory dynamics (AD). They are not used to support our claim of detection of AD. Our claim of detection of AD in the paper is based on experimental observations: the leading fish in the channel becomes the follower when going through the door. Of course, this is also based on the assumption that the roles of the fish do not change during the experiment.  We have also alerted the readers about this in the revised manuscript.  From an experimental point of view, we have no doubt that there is information exchange between the fish pair because a) their trajectories are always close together and b) trajectory of a single fish is quite different from those of a fish pair. Their correlated motions are the result of information exchanges between them.

Reviewer>For discussing chaos and entropy transfer, trajectory of two zebrafishes among about 20 fishes may not be enough. Authors might found two intimate zebrafish pairs.

Our Response> We agree that it is better to have more fish pairs if we want to establish the percentage of fish pair exhibiting anticipatory behavior. However, our goal is the detection of anticipatory dynamics a real animal system. We are satisfied that at least some pairs fulfil our requirement for positive detection.

Reviewer>Relations between the trajectories of two zebrafish in a tank and entropy measure of Lithium-Ion Battery prognostics (Nandari et al.) are obscure

Our Response> We have in mind the machine learning part of the reference; not about the battery.  We speculate that if there are new methods of detecting anticipatory dynamics, it might be based on methods of entropy/information and machine learning. Since this is only our speculation, we are happy to remove this reference in the revised manuscript.

Round 3

Reviewer 4 Report

Comments and Suggestions for Authors

Please see my comment to the editor.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Please see my comment to the editor.

Author Response

Our response to the comments of reviewer 4 (round 3).

The reviewer’s comment will be listed first and followed by our response.

1. The authors added a comment that the simple Negative Group Delay model cannot be used to understand the interaction between the fish pair. It is OK. But the authors’ claim that Anticipatory Dynamics was experimentally detected is not true.

Our response> We do not agree with the reviewer. With our definition of AD and our experimental assumptions, we have detected some fish pair interacting with AD. Our claim is not based on any references. Our claim is based on our experimental design of using the passing order of the fish through the door to assign the roles of information source and receiver to the two fish and the assumption that the roles of the fish do not change during the experiment. Therefore, if the temporal order of the receiver is ahead of the source, AD is said to be detected. Of course, one might challenge our method of assignment of source and receiver and our assumption of the constancy of the roles of the fish during experiment. We have also alerted the readers about these caveats in the manuscript.

2. The authors speak about leader and follower citing [23]. But in [23], anticipation of moving stimuli by the retina of one salamander and that of one rabbit are discussed and there are no leader or follower.

Our response> We do not agree with the reviewer and we think there are some confusions here. The reviewer must have in mind Reference 4 instead of reference 23. In reference 4, a classic paper by Berry et al, responses of retinae from salamanders and rabbits were used to study the anticipatory dynamics of a retina in response to a moving stimulation. In reference 4, the responses of the retina from salamander and rabbit show that their spatial responses to a moving bar are all shifted towards the moving direction of the moving bar when the contrast of the moving is high enough (see Figure 3 of Ref 4). This means that the retina is responding ahead of the moving bar stimulation. In other words, detecting the arrival of the moving bar before it actually arrives. Here the “leader” is the retina and there is no follower because the retina (receiver) is producing anticipatory response to the moving bar (source). It is anticipatory because the direction of information flow is unambiguously from the moving bar (source) to the retina (receiver) but the response of the receiver is ahead of the source. This the unusual aspect of AD; namely the temporal leader is the information receiver. We will have the retina as a follower only when the contrast is low or the moving speed of the bar is very slow.

In the case of two fish, we follow this usual definition of anticipatory response. If motions of the two fish are correlated but not synchronized, there must be a leader (deduced from the temporal correlation or time lag mutual information between the two time series of their trajectories). If information flows from this leader to the other fish, then the other fish is just a follower. This is the trivial case. However, if information flows from the other fish (source) to the leader (receiver), then the leader is producing anticipatory response to the other fish and there is no follower. In other words, the temporal leader is the information receiver. This is similar to the case of a retina (leader) which is producing response ahead of the moving bar.

In our opinion, the most critical task in the detection of anticipatory response is the determination of the direction of information flow. In the case of a retina, the direction of information flow in obvious but not in the case of two fish when only their trajectories (two time series) are known. As stated in the manuscript, it is not trivial to detect direction of information flow between two given time series by using common tools such as TE or GC. In fact, in our work, our claim of detection of anticipatory dynamics between two fish is not based on these tools but rather based on an experimental design to detect which fish is the information source. We stated clearly in the manuscript that this method needs to be further tested.

3. Physics should predict chaotic behaviors or controlled behaviors of subsystems of a system. Voss [3] discussed anticipatory chaotic synchronization of a coupled harmonic oscillator system. Not a system of two fishes.

Our response> We agree totally with the reviewer. That is why we added this discussion in the \subsection{Correlation and Causation}. There was a typo in the first sentence in this section. It is now corrected. Please read it again.

4. Voss [2] discussed anticipatory relaxation dynamics of a complex signal processing system with relaxation.

Our response> We have stressed in the manuscript that the Voss model is not used in the manuscript to understand the anticipatory dynamics of the fish. But rather, it is used to simply to generate anticipatory data which is used to test the validity of the common tools of TE and GC for causality.

5. I asked relevance of the article with [22]. Authors erased the reference.

Our response> We have given reasons to cite Ref 22 and its removal. We cited it because it was our speculation that the use of methods of entropy/information together with machine learning might provide a new tool for the detection of information flow. We removed it because we felt that our speculation might be far-fetched.

6. Authors searched trajectories of zebrafish pairs and claimed Antipatory Dynamics was observed and referred various references. But no reference supports authors’ claim. After I read the text carefully, I found not understandable sentences like ”causality between to time series”.

 

Our response> We are sorry that there was a typo error. The “to” should be “two”.

As explain above, our definition of anticipatory dynamics (AD) in fish pair is based on a usual definition of AD used in Ref 4 (Berry et al). However, our claim of detection of AD is not based on any references. Our claim is based on our experimental design of using the passing order of the fish through the door to assign the roles of information source and receiver to the two fish and the assumption that the roles of the fish do not change during the experiment.

Round 4

Reviewer 4 Report

Comments and Suggestions for Authors

I don’t think the article is suitable for publication. The reason is written in the attached comment.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

/

Author Response

Our response to the reviewer’s commented dated November 26, 2023

We will list the comments of the reviewer first in bold italics and followed by our comments.

The authors added a comment on the negative group delay (NGD) that was introduced by Voss in [2] for understanding the Anticipatry Relaxation Dynamics (ARD). The authors claim that they detected Anticipatory Dynamics of fish pairs from experimental observations. It is not true. Dynamics of fish pairs are governed by classical dynamics, but Voss’ argument apply for quantum mechanical systems in which the time delay and frequency response are related. There are no corresponding ingredients in the dynamics of zebrafishes.

Our response> We do not understand these statements because Voss model is not related to quantum mechanics. We have checked that in the paper of Voss (Ref.2), the word “quantum” never appeared. In fact, in Ref.2, Voss demonstrated that his model can be used to anticipate the beating of a human heart by using real human cardiac data. In our opinion, it is not easy to treat the contraction dynamics of a heart with quantum mechanics. The reviewer might have mistaken Ref.2 with some of the application of Liang’s T method (also used in our work) to detect direction of information flow. Recently, some people are using Liang’s method in quantum systems. (See for example: https://arxiv.org/abs/2201.00197#:~:text=Liang%20information%20flow%20is%20a,other%20nodes%20of%20the%20network.

However, even for Liang’s method, it was successfully used to understand the causal structure between CO2 and global temperature which are classical systems. (Scientific Reports | 6:21691 | DOI: 10.1038/srep21691)

End of our response>

To discuss Granger Causality and Transfer Entropy, it is necessary to check whether the system can be simulated by Gaussian random variables of large numbers as pointed out by Barnett, Barrett and Seth (Phys. Rev. Lett 103,238701 (2009)).

Our response> We do not agree that “To discuss Granger Causality and Transfer Entropy, it is necessary to check whether the system can be simulated by Gaussian random variables of large numbers”. The most important finding of the quoted paper is that TC and TE are the same if the data are Gaussian random variables. There is no a priori requirement that the data should be Gaussian when one discuss TE and TC. In fact, the method of TE is well-suited for analyzing non-Gaussian data. Unlike some traditional statistical methods that might rely on assumptions of normality, transfer entropy is based on information theory and does not require data to follow a Gaussian distribution.

The question of whether our number of data points are large enough to use TC and TE, we have already addressed this point in our manuscript by pointing out the problem of bias in the computation of entropies from limited number of experimental data points. We avoided this problem by using a relatively small number of states in computation of TE. For the case of using Voss’s model to generate anticipatory data to test TE, we have stated clearly in the caption of Figure 11 that: for 300s of simulated data, our TE results will not change when the time step is 0.01s or 0.001s. That means: our number of data point is large enough.

End of our response>

The authors uses the MATLAB package MVGC and the python package PyInform to analyze the time series classical data of zebrafishs’ trajectory. They claim that the direction of information flow (DIF) for anticipatory dynamics. The statement is based on their assumption that among the pair of zebrafihes, there is a leader and a follower the roles can be interchanged in the time series. Hahs and Pethel [18] considered DIF of systems either coupled or posessing common exogeneous input. The two zebrafish system do not follow the condition that DIF can be defined.

Our response> There is a leader and follower in a zebra fish pair during their interaction is not an assumption. It is the conclusion from the measurement of time lag mutual information (TLMI) because the peak the TLMI (Figure 5a) is not at zero-time lag; indicating that one fish is leading the other. Of course, this conclusion is correct only in statistical sense. We also warned the readers that their roles might change over time. However, overall, one fish is leading the other one.

As for the point related to the whether DIF can be defined for a pair of fish in the sense of Hahs et al [18]. It can be seen clearly from our data (Figure 3 and Figure 4) that the two trajectories of the fish are coupled because they are always close to each other so that they give a non-zero TLMI. If they were not coupled, the two trajectories would have been totally un-correlated and there will be no shared mutual information and the measured TLMI would have been just be zero. Therefore, we do not agree with the statement that: “The two zebrafish system do not follow the condition that DIF can be defined.”

End of our resonse>

Authors admitted that ”causality between to time series” is a typo of ”causality between two time series”, but the revised version of line 389, it is not corrected. 

Our response> We do no understand this because we just downloaded our submitted version from the website of Entropy and checked that the “to” had been corrected to ”two” which is marked in red. The reviewer must have used a different version.

End of our response>

There remain mistakes in English and expressions.

• Before eq. (3) {vi−1, vi−2 as → {vi−1, vi−2, · · · } as
• After Figure 4. two fish → two fishes
• Same errors in the following lines.
• In subsection 3.5 MVGC → The Multivariate Granger Causality (MVGC

Our response> Thanks for pointing out the typo error. We have corrected these error in the revised manuscript. But we prefer to use "fish" instead of "fishes" as it is commonly used for the plural of fish.

End of our response>