DOI: 10.5061/dryad.2280gb5p6
title: Bending moments in swimming fish

description: 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 3D 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 3D 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.
description: <strong>For a detailed description, see our PLOS Biology
 paper (some parts copied from this paper) and its S1 Text supplement,
 including references. </strong>
 <strong>Video analysis of swimming fish
 larvae.</strong> We used three batches of 50 zebrafish larvae from
 3–12 days post fertilisation (dpf). We recorded free-swimming larvae at
 2000 fr/s with three synchronised high-speed video cameras (raw video
 recording available upon request from the corresponding author). We
 reconstructed the swimming kinematics from the high-speed videos, using
 previous published in-house developed automated 3D tracking software in
 MATLAB (R2013a, The Mathworks)(doi:10.1371/journal). For every time point
 of the video sequence, the best fit for the larva's 3D position,
 orientation and body curvature was calculated. These parameters served to
 calculate the position of the motion of the larva's central axis and
 its outer surface. <strong>Near-periodic
 swimming. </strong>We calculated phase-averaged quantities for an
 individual swimming sequence to study the generated bending moments and
 powers. We selected (subset of a) sequence that was near-periodic. For
 every possible subset of a swimming sequence, we calculated the sum of
 absolute difference with a time shifted version of the curvature, similar
 to an autocorrelation. We then calculated extrema in this function – if
 extrema are detected, their maximum value determines the “periodicity” of
 the sequence. We then selected the longest possible subsequence that has a
 periodicity value higher than a threshold of 35 for a swimming sequence of
 a 3 dpf fish. We divided this sequence in half-phases based on peaks in
 the body angle. Body curvature, bending moment, fluid power, kinetic
 power, and resultant power were then phase-averaged based on these
 subdivisions. <strong>Analysis of aperiodic
 swimming.</strong> We divided each swimming bout in half tail-beats,
 for each of which we determined a mean swimming speed and acceleration. We
 mirrored all half-beats towards the left side of the fish such that all
 extracted half-beats were in the same direction. This allowed us to
 systematically analyse aperiodic motion. We used lateral bending moment
 patterns to divide the swimming motion into half tail-beats—we defined the
 start of each half tail-beat as the moment at which the bending moment at
 0.5ℓ (0.5 body length) crossed the zero line. Because some of these zero
 crossings are related to noise, we evaluated every possible permutation of
 zero crossings per sequence on several criteria with a custom MATLAB
 (R2018b, The Mathworks) program (see our PLOS Biology paper for details).
 Out of 113 swimming sequences, we selected 398 half-beats with this
 procedure, and calculated the duration, mean speed, and peak bending
 moment. Mean acceleration was calculated as the difference in mean speed
 (i.e. velocity magnitude) between the following and current half-beat and
 dividing by the time difference between the tail-beat mid-points. Because
 we could not calculate mean acceleration for the last half-beat in each
 sequence, 285 half-beats remained for which we computed all
 quantities. <strong>Calculation of fluid forces
 on the fish.</strong> We used computational fluid dynamics (CFD) to
 solve the full Navier-Stokes equations numerically. This results in an
 accurate representation of the fluid force distributions, as all flow
 scales can be represented numerically. We developed an add-on to the
 open-source immersed boundary method implementation IBAMR to calculate
 fluid force distributions around swimming fish. To validate this method
 and assess its accuracy when calculating internal forces and moments, we
 used a second, experimentally validated solver (details of the methodology
 are described in the PLOS Biology paper and its supplement).
 <strong>Calculation of bending moments (see S1 Text of our
 PLOS Biology for details). </strong>To calculate bending moments, we
 represented the fish by its central axis only. Effects of muscles, spine,
 and other tissues were combined for every transversal slice along this
 axis. This simplification allowed us to describe the fish as a non-linear,
 one-dimensional beam in two-dimensional space. We derived the equations of
 motion for this beam in an accelerating and rotating coordinate system
 attached to the fish’s head. We modelled the fish as a beam with varying
 cross-sections, undergoing arbitrarily large deformation. Plane
 cross-sections are assumed to remain plane and perpendicular to the
 neutral line (no shear deformation), but axial deformation is allowed. We
 assumed that the fish deforms in a single plane (with an arbitrary 3D
 position and orientation). We used a two dimensional beam model to
 represent the deformation of the fish in this plane. Because we attached
 the non-inertial reference frame to the head of the fish, the model
 contains both translation and rotational accelerations with respect to the
 inertial reference frame. We took these accelerations into account using
 fictitious forces. In summary, we modelled the fish as a beam undergoing
 large bending deformations in two dimensions. We validated the bending
 moment computations with reference data based on a known external force
 distribution and internal moment distribution. We also computed the
 resultant fluid dynamic power on the beam from the derived fluid-dynamic
 forces, as well as the kinetic energy of the beam.
 <strong>Further descriptions can be found in the readme
 files for each of the supplied data sets.</strong>