Experimental–numerical method for calculating bending moments in swimming fish shows that fish larvae control undulatory swimming with simple actuation

Most fish swim with body undulations that result from fluid–structure interactions between the fish’s internal tissues and the surrounding water. Gaining insight into these complex fluid–structure interactions is essential to understand how fish swim. To this end, we developed a dedicated experimental–numerical inverse dynamics approach to calculate the lateral bending moment distributions for a large-amplitude undulatory swimmer that moves freely in three-dimensional space. We combined automated motion tracking from multiple synchronised high-speed video sequences, computation of fluid dynamic stresses on the swimmer’s body from computational fluid dynamics, and bending moment calculations using these stresses as input for a novel beam model of the body. The bending moment, which represent the system’s net actuation, varies over time and along the fish’s central axis due to muscle actions, passive tissues, inertia, and fluid dynamics. Our three-dimensional analysis of 113 swimming events of zebrafish larvae ranging in age from 3 to 12 days after fertilisation shows that these bending moment patterns are not only relatively simple but also strikingly similar throughout early development and from fast starts to periodic swimming. This suggests that fish larvae may produce and adjust swimming movements relatively simply, yet effectively, while restructuring their neuromuscular control system throughout their rapid development.


1.
Is the hypothesis as expressed worthwhile? Or is it actually almost circular? The hypothesis (L.14) can be summarized as: a simple brain results in simple actuation. Is this not a (sorry) nobrainer? If it is the brain that initiates activation, is it not part of the definition of a simple brain that it should only be capable of simple activation?

Reply 2.2
The question that we intend to solve is not necessarily whether the larvae use a simple actuation pattern. Instead, we wondered how these larvae, despite their simple brains, can control their swimming speed and acceleration, and how they can maintain swimming control throughout their rapid development.
We rephrased the abstract and introduction to more accurately reflect the intent of the article-focusing more on the adjusting of swimming speed and changes throughout development, rather than the (simple) production of swimming motion. In addition, we now put more focus on the methodological innovations presented, considering the "methods and resources" status of the article.

Reply 2.3
We agree with the reviewer that although some of our results might seem intuitive-which they are not necessarily-it is worthwhile to confirm them with evidence. In addition, we do not only show that swimming action is similar at different speeds and accelerations, but also remains strikingly similar throughout development. This is less intuitive, because with increasing size, the fluid characteristics change (expressed by Re) and throughout development the neuro-muscular system is rapidly being reorganized. Despite these changes, the swimming actuations remain very similar, illustrating the robustness of the locomotor system.
In the revised manuscript, we rephrased the introduction of our question to more clearly demonstrate what we are answering, and why this is interesting. In addition, to address comments by another reviewer and the editor, we changed the focus of the article more towards the methodology without downplaying the interesting biological results.  Figure 3 to be a demonstration that the relationships between force, work, power, and between hydrodynamic and whole-body… all sort of relate intuitively.

Reply 2.5
To clarify our intent and explain the purpose of our custom-derived metrics, we extended the explanation in the results (lines 162-164). Furthermore, we extended the discussion on the physical meaning of the vigour (lines 324-335). For more details about why we use vigour and effort instead of power, see our reply to Comment 1.2.

Comment 2.6
There appears to be the implication of an adaptive slant... and this does not feel justified. (L.22 allows function during development).

Reply 2.6
We changed the sentence (line 26-28) to avoid any ambiguities or possible misinterpretation of what we intend to say.

Comment 2.7
The suggestion that complex physics would be (initially) thought to require a sophisticate control system (L. 41) probably overstates matters. Most biologists should be familiar with complex physics occurring with very simple (or zero) control.

Reply 2.7
We rephrased this section to be more accurate.

Comment 2.8
To what extent is the lack of curvature towards the tail tip a consequence of the shape reconstruction?
I am not sure how this could be dealt with neatly… but I am suspicious that a 90 degree bend in the last 1% might get smoothed out, whereas the same angle bend at 50% would make for an obviously right-angle fish, and would persist. I don't think this affects the story of the paper, but if it is an inevitable consequence of methodology and not a reliable measurement, this should be noted.

Reply 2.8
Indeed, strong curvatures toward the tail get smoothed out, as discussed in our article that describes the three-dimensional tracking method [2]. However, even a 90° bend in the last 1% moving at the tail beat frequency would make little difference to the fluid mechanics compared to a 90° bend over a length region centred at 50%. The region where curvature is reconstructed most accurately is also the region where curvature changes influence the solution the most. We now mention this in the discussion (lines 292-295).

Reply 3.1
We thank the reviewer for the thorough review of the paper and the positive assessment of its quality. In the point-by-point reply below, we address the four major comments as well as the minor comments.

1.
Turning. The most confusing part of this paper is how it does not seem to address turning. The calculated bending moments seem to be left-right symmetric, or at least that is the implication of Fig. 2 and Fig. 4A

Reply 3.1
The reviewer points out an important aspect of our analysis. In our study, we focus on the effect of tail beat kinematics on linear speed and accelerations, and ignore turning dynamics.
We chose to do this because the left-right asymmetry of turning is rather more subtle than linear speed and acceleration, and preliminary analyses showed that our current dataset was simply too small and variable in behaviour for systematically analysing the turn dynamics.
The reviewer is correct that many of the analysed tail beats were of turning manoeuvres, and therefore we developed a method that takes this into account: we analysed the motions of the fish per half tail-beat, rather than per full tail-beat. By mirroring all left-half tail-beats, we could analyse the complete dataset without the need to assume that the bending moments are left--right symmetric. This method allowed us to analyse the overall swimming motion in terms of speed and net acceleration, while removing the effects of asymmetries. We rephrased the section "Subdividing motion" to more clearly explain this (lines 446-448).
Although turning dynamics is outside the scope of the current manuscript, we aim to address this in a future study. In the revised manuscript, we therefore now mention turning behaviour as a potential future application of this method. We feel that the current description of the method along with the analysis of linear swimming dynamics meets the goals of a Methods and Resources paper. I think the most novel aspect of the study is the analysis of acceleration and speed, as shown in Fig. 5. This paper is also listed as a "Methods and Resource" article, but the authors do not appear to describe any way to get the code to apply these methods to other cases. If the goal of the study is to provide a method, more detail should be given in the main text on the method itself, and the authors should perhaps try applying it to another fish species.

Reply 3.3
We reframed our study to put more focus on the methodological aspect. We therefore expanded the materials and methods section to provide more detail, which was first present only in the Supporting Information. In addition, we rephrased our aims in the introduction and adapted the discussion accordingly. See also our reply to Comments 2.2 and 2.3 for more details. Applying our developed methodology to another fish species would require a whole set of new experiments which is impossible given time, cost and labour constraints. Nobody has so far captured the required 3D motion data with sufficient accuracy for a different species.
The parameter. The authors introduce two new parameters to quantify swimming: "effort", the ratio of bending moment and half-tail-beat duration, and "vigour", which they define as where is mass, is linear swimming velocity, is linear acceleration, and is a parameter found by optimizing a linear fit between effort and vigour. In Fig. 5

Reply 3.15
We addressed the reviewer's comment by expanded this section considerably (section "Calculating bending moments" in the Methods).

Comment 3.16
11. Fig. 4. It would be helpful to provide some frequency distributions for the data. Do fish modulate half-beat duration more often, or peak bending moment?

Reply 3.16
We thank the reviewer for the interesting suggestion; we added two extra panels to Fig 5 showing the frequency distribution of peak bending moment and half-beat duration for low, medium, and high effort. We explained these panels in lines 259-266 and discussed the change in relative contribution of half-beat duration and peak bending moment with increasing effort in lines 386-390.

Reply 3.17
We thank the reviewer for the feedback on the visualisation of our data set. In making this figure, we tried several permutations of the axes, and concluded that having the "input" variables on the axes and the "output" colour-coded showed the clearest result. In addition, Fig. 3C and D show the data in this fashion, against effort on the x-axis. The new information in Fig. 5 is the subdivision in peak bending moment and half-beat duration, which we concluded to be the main input parameters based on the analysis of Fig. 4. However, we added two additional panels showing the frequency distribution of the peak bending moment and the half-beat duration (see Reply 3.16).

Reply 3.18
We moved this section to the main text and added an explanation of the CFL-number (lines 490-491).

Comment 3.19
14. Supplemental section 3.1. If I understand the analysis correctly, this is a highly underconstrained optimization problem, which means that multiple optima are possible. How did the authors select a particular optimum?

Reply 3.19
For several test cases, we tried to initialise optimisation from any particular frame from the previous frame, from a distribution of zeros, and from random distributions. All these different initialisations converged to the same solution-the problem is relatively insensitive to the initial conditions. In addition, our validation of the method reproduces the reference internal forces and moments almost perfectly from the same information as the real dataset: motion and external force. This provides further confidence in the reliability of the method.
We now mention this in the revised manuscript (lines 560-561).

Comment 3.20
15. Supplemental Fig. 5C, D. The difference between the reference and IBAMR solutions seems fairly substantial. Please justify further.

Reply 3.20
We expanded the explanation of the differences. (section 6.2 in S1 Text).