Hydrodynamic model of fish orientation in a channel flow

  1. Maurizio Porfiri
  2. Peng Zhang
  3. Sean D Peterson

  • Curated by eLife

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    Evaluation Summary:

    The authors address the problem of fish orienting against the mean flow when deprived of visual cues. They study a simple model of swimming dipole and argue that in the absence of flow-sensing feedback, fluid-structure coupling alone is sufficient to generate upstream orienting behavior, above a given flow speed. A comparison with the experimental literature on fish behavior is attempted.

    (This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. The reviewers remained anonymous to the authors.)

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Abstract

For over a century, scientists have sought to understand how fish orient against an incoming flow, even without visual and flow cues. Here, we elucidate a potential hydrodynamic mechanism of rheotaxis through the study of the bidirectional coupling between fish and the surrounding fluid. By modeling a fish as a vortex dipole in an infinite channel with an imposed background flow, we establish a planar dynamical system for the cross-stream coordinate and orientation. The system dynamics captures the existence of a critical flow speed for fish to successfully orient while performing cross-stream, periodic sweeping movements. Model predictions are examined in the context of experimental observations in the literature on the rheotactic behavior of fish deprived of visual and lateral line cues. The crucial role of bidirectional hydrodynamic interactions unveiled by this model points at an overlooked limitation of existing experimental paradigms to study rheotaxis in the laboratory.

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  1. Version published to 10.7554/elife.75225 on eLife
  2. eLife

    Author Response:

    Reviewer #1:

    The authors argue that in the absence of flow-sensing feedback, fluid-structure coupling alone is sufficient to generate upstream orienting behaviors of fish. If true, this would be an interesting phenomenon of moderately wide interest.

    The strengths of this paper are:

    1. A needed consideration of coupled interactions in fluids that can potentially augment or replace flow-sensitive feedback behaviors.
    2. A simplified mathematical model that reveals an interesting passive hydrodynamic mechanism of rheotaxis that exists only above critical flow speeds
    3. The authors do a respectable job combing the literature for lateral line studies.

    We are thankful to the Reviewer for the constructive feedback.

    The weaknesses of this paper are:

    1. The discrepancy between what can be supported by the biological literature and the simplification of the model is large. One has the impression that the authors are fitting a round peg in a square hole rather than uncovering a realistic mechanism for behavior in the absence of sensory information. Part of this is not the authors' fault, it is the lack of relevant experiments (e.g. inconclusive or indirect) in the biological literature.

    We thank the Reviewer for the comments on the significance of our literature review. We have added to the Discussion section a suggested experimental protocol that could be pursued to verify our theoretical predictions, based on robotic fish (see paragraphs five and six of the Discussion section).

    1. The authors' claims are not justified by their data. They acknowledge shortcomings of their model as a departure from real animals, neglecting elasticity and inertia of the fish and added mass effects. Water is known for its non-linear properties, and yet their model assumes a linear hydrodynamic feedback system.

    We are thankful to the Reviewer for the comment, which prompted us to clarify the assumptions of the model throughout the paper. In our model, the response of the fish to flow perturbations are generally nonlinear, whereby they are the composition of two effects: i) passive advection and ii) lateral line feedback. Passive advection is nonlinear, as shown in Eq. (8) of the revised manuscript; fish modify their turn rate in response to their current position and heading in a nonlinear way. The lateral line feedback is linear as a result of the simplifying assumption of a parabolic flow superposed on a uniform flow and the choice of a linear relationship between circulation and turn rate. Such a choice bears no consequences on the local stability analysis. (See the added paragraph after Equation (22).)

    1. Biological relevance is lacking. Real fish don't orient in the absence of all sensory inputs, and yet the model does not account for vision, balance and touch.

    We agree with the Reviewer that real fish may rely on several sensory cues for rheotaxis. With this study, we introduce an alternative, passive pathway by which fish may orient against the flow. Such a pathway has never been explored before. In light of the majority of the literature being inconclusive with regards to our proposed pathway, we have de-emphasized the biological relevance by relocating these sections to Appendix 3. We have largely rewritten the Discussion section to clearly identify the contribution of this work and better place it in the context of a multisensory framework of real fish to obtain rheotaxis. Furthermore, we have added to the Discussion section (paragraphs five and six) an experimental protocol utilizing robotic fish that could be used to validate our model findings.

    1. No discussion or interpretation of neural feedback (reafference and motor copy re Bell and Bodznick) that could alter the interpretation of their results in the context of the literature.

    We thank the Reviewer for the comment, which has prompted us to improve on the manuscript in two ways: i) as discussed above, the biological claims have been de-emphasized, and ii) the hypotheses of the model have been more clearly articulated throughout the manuscript. In particular, with respect to the latter point, our model considers only the mean flow so that it practically averages fish locomotion in time and responds solely to the circulation of the background flow (that is, an infinite signal-to-noise ratio from the neural feedback perspective). We have added to the Discussion section (second-to-last paragraph) text highlighting the need for research at the interface of fluid mechanics and neuroscience to hone a multisensory framework combining active and passive mechanisms that can support rheotaxis.

    1. Justification of results based on few biological papers that have their own shortcomings.

    We thank the Reviewer for the comment. We have moved the literature review to Appendix 3 in the revised manuscript and better articulated a pathway for future validation in the Discussion section (paragraphs five and six).

    1. The relevance and importance of the finding is exaggerated.

    We have softened the claims in the manuscript and further emphasized the realm of applicability of the model and its contribution in with respect to passive rheotaxis.

    Reviewer #2:

    The paper describes a dipole model of fish swimming. The model is very much based on existing work. But more importantly the model entails several parameters and is validated in rather qualitative terms. I would suggest comparisons of this 2D model with 2D viscous simulations that should be easy to produce. At the moment there are far too many parameters that are evaluated in rather qualitative terms. There is no sensitivity analysis of any form to warrant reassurance as to the validity of the results. In turn the results have only some qualitative value.

    We thank the Reviewer for the insightful comments which prompted us to undertake a thorough validation of the dipole model through numerical solution of the two-dimensional Navier-Stokes equations. (See the new Numerical Validation of the Dipole Model subsection, with additional details in Appendix 2.)

    Reviewer #3:

    This manuscript proposes a hydrodynamic model of a fish in a channel flow. This work is based on important assumptions (dipole model, potential flow, parabolic channel flow) that lead to a simple dynamical system. This dynamical system is only stable when the incoming flow is over a threshold value. This result is compared with experimental data of the literature.

    Although the problem addressed is interesting, the assumptions of the model are not justified and probably not appropriate in the context studied. First, the dipole model used to model the flow generated by the fish does not seem appropriate to study small animals that swim with bursts. Second, the channel flow is a superposition of a constant velocity and a parabolic profile, which is not what is expected at moderate Re. Finally, the feedback mechanism based on vorticity does not seem plausible and, since vorticity is a linear function of the cross-stream coordinate in a parabolic flow, it is not distinguishable from visual feedback.

    We thank the Reviewer for the comment. The Reynolds numbers for experimental studies of rheotaxis reported in the literature span a broad range, from 10 to 10,000. At the low end of the range, a parabolic velocity profile is expected. At the higher end of the spectrum, the velocity profile is expected to be turbulent, resembling a top hat (plug flow). In both cases, near the channel centerline there will be a degree of shear flow, offering some bias to the animal by which it can appraise the flow environment and distinguish downstream from upstream. This is the only information that we utilize in our model and its local stability analysis to demonstrate the existence of a critical flow speed above which fish will perform rheotaxis. In the revised manuscript, we have clarified that other flow profiles could be considered, and the results remain valid provided the flow retains non-zero vorticity. (See added text in the Results section after Equation (4) and in the Methods and Materials section after Equation (22).)

    In the same spirit, retaining nonlinear dependencies for the relationship between circulation and feedback will have no effect on the linear stability analysis. As a result, while we are aware that a linear feedback by the lateral line could be overly simplistic, it will capture the leading order physics that is needed to elucidate the stability of rheotaxis. We have further clarified that all sensory modalities other than the lateral line are excluded from our present model, including vision. (See added text in the last paragraph of the Introduction and in the Methods and Materials section after Equation (22).)

    The other issue is the comparison with experimental data. The model predicts a channel flow threshold, that is necessary to have a stable point. But this is not the only prediction, it also predicts the dynamics around this point, for instance. The authors choose to only compare the threshold and their comparison presented as tables is mostly inconclusive.

    We opted to compare the flow speed with the threshold speed, Uc, due to the availability of flow speed data in the experimental studies, which can be precisely quantified and easily varied in most experiments (see Appendix 3 Table 1). While in principle other dynamic phenomena, such as the frequency of cross-stream sweeping could be compared, experiments where these data are available are rare. We acknowledge that most data in the literature only provide inconclusive evidence. In the revised manuscript, we have moved the literature review to Appendix 3 and amended the Discussion section accordingly.

  3. eLife

    Evaluation Summary:

    The authors address the problem of fish orienting against the mean flow when deprived of visual cues. They study a simple model of swimming dipole and argue that in the absence of flow-sensing feedback, fluid-structure coupling alone is sufficient to generate upstream orienting behavior, above a given flow speed. A comparison with the experimental literature on fish behavior is attempted.

    (This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. The reviewers remained anonymous to the authors.)

  4. eLife

    Reviewer #1 (Public Review):

    The authors argue that in the absence of flow-sensing feedback, fluid-structure coupling alone is sufficient to generate upstream orienting behaviors of fish. If true, this would be an interesting phenomenon of moderately wide interest.

    The strengths of this paper are:

    1. A needed consideration of coupled interactions in fluids that can potentially augment or replace flow-sensitive feedback behaviors.
    2. A simplified mathematical model that reveals an interesting passive hydrodynamic mechanism of rheotaxis that exists only above critical flow speeds
    3. The authors do a respectable job combing the literature for lateral line studies.

    The weaknesses of this paper are:

    1. The discrepancy between what can be supported by the biological literature and the simplification of the model is large. One has the impression that the authors are fitting a round peg in a square hole rather than uncovering a realistic mechanism for behavior in the absence of sensory information. Part of this is not the authors' fault, it is the lack of relevant experiments (e.g. inconclusive or indirect) in the biological literature.
    2. The authors' claims are not justified by their data. They acknowledge shortcomings of their model as a departure from real animals, neglecting elasticity and inertia of the fish and added mass effects. Water is known for its non-linear properties, and yet their model assumes a linear hydrodynamic feedback system.
    3. Biological relevance is lacking. Real fish don't orient in the absence of all sensory inputs, and yet the model does not account for vision, balance and touch.
    4. No discussion or interpretation of neural feedback (reafference and motor copy re Bell and Bodznick) that could alter the interpretation of their results in the context of the literature.
    5. Justification of results based on few biological papers that have their own shortcomings.
    6. The relevance and importance of the finding is exaggerated.
  5. eLife

    Reviewer #2 (Public Review):

    The paper describes a dipole model of fish swimming. The model is very much based on existing work. But more importantly the model entails several parameters and is validated in rather qualitative terms. I would suggest comparisons of this 2D model with 2D viscous simulations that should be easy to produce. At the moment there are far too many parameters that are evaluated in rather qualitative terms. There is no sensitivity analysis of any form to warrant reassurance as to the validity of the results. In turn the results have only some qualitative value.

  6. eLife

    Reviewer #3 (Public Review):

    This manuscript proposes a hydrodynamic model of a fish in a channel flow. This work is based on important assumptions (dipole model, potential flow, parabolic channel flow) that lead to a simple dynamical system. This dynamical system is only stable when the incoming flow is over a threshold value. This result is compared with experimental data of the literature.

    Although the problem addressed is interesting, the assumptions of the model are not justified and probably not appropriate in the context studied. First, the dipole model used to model the flow generated by the fish does not seem appropriate to study small animals that swim with bursts. Second, the channel flow is a superposition of a constant velocity and a parabolic profile, which is not what is expected at moderate Re. Finally, the feedback mechanism based on vorticity does not seem plausible and, since vorticity is a linear function of the cross-stream coordinate in a parabolic flow, it is not distinguishable from visual feedback.

    The other issue is the comparison with experimental data. The model predicts a channel flow threshold, that is necessary to have a stable point. But this is not the only prediction, it also predicts the dynamics around this point, for instance. The authors choose to only compare the threshold and their comparison presented as tables is mostly inconclusive.

  7. Version published to 10.1101/2021.11.11.468193v2 on bioRxiv
  8. Version published to 10.1101/2021.11.11.468193v1 on bioRxiv