Force transmission via axial tendons in undulating fish: a dynamic analysis

Sonomicrometrics of in vivo axial strain of muscle has shown that the swimming fish body bends like a homogenous, continuous beam in all species except tuna. This simple beam-like behavior is surprising because the underlying tendon structure, muscle structure and behavior are complex. Given this incongruence, our goal was to understand the mechanical role of various myoseptal tendons. We modeled a pumpkinseed sunfish, Lepomis gibbosus, using experimentally-derived physical and mechanical attributes, swimming from rest with steady muscle activity. Axially oriented muscle-tendons, transverse and axial myoseptal tendons, as suggested by current morphological knowledge, interacted to replicate the force and moment distribution. Dynamic stiffness and damping associated with muscle activation, realistic muscle force generation, and force distribution following tendon geometry were incorporated. The vertebral column consisted of 11 rigid vertebrae connected by joints that restricted bending to the lateral plane and endowed the body with its passive viscoelasticity. In reaction to the acceleration of the body in an inviscid fluid and its internal transmission of moment via the vertebral column, the model predicted the kinematic response. Varying only tendon geometry and stiffness, four different simulations were run. Simulations with only intrasegmental tendons produced unstable axial and lateral tail forces and body motions. Only the simulation that included both intra- and intersegmental tendons, muscle-enhanced segment stiffness, and a stiffened caudal joint produced stable and large lateral and axial forces at the tail. Thus this model predicts that axial tendons function within a myomere to (1) convert axial force to moment (moment transduction), (2) transmit axial forces between adjacent myosepta (segment coupling), and, intersegmentally, to (3) distribute axial forces (force entrainment), and (4) stiffen joints in bending (flexural stiffening). The fact that all four functions are needed to produce the most realistic swimming motions suggests that axial tendons are essential to the simple beam-like behavior of fish.     

The Importance of Body Stiffness in Undulatory Propulsion

During steady swimming in fish, the dynamic form taken by theaxial undulatory wave may depend on the bending stiffness ofthe body. Previous studies have suggested the hypothesis thatfish use their muscles to modulate body stiffness. In orderto expand the theoretical and experimental tools available fortesting this hypothesis, we explored the relationship betweenbody stiffness, muscle activity, and undulatory waveform inthe mechanical context of dynamically bending beams. We proposethat fish minimize the mechanical cost of bending by increasingtheir body stiffness, which would allow them to tune their body'snatural frequency to match the tailbeat frequency at a givenswimming speed. A review of the literature reveals that theform of the undulatory wave, as measured by propulsive wavelength,is highly variable within species, a result which calls intoquestion the use of propulsive wavelength as a species-specificindicator of swimming mode. At the same time, the smallest wavelengthwithin a species is inversely proportional to the number ofvertebrae across taxa (r² = 0.21). In order to determine ifintact fish bodies are capable of increasing bending stiffness,we introduce a method for stimulating muscle in the body ofa dead fish while it is being cyclically bent at physiologicalfrequencies. The bending moment (N m) and angular displacement(radians) are measured during dynamic bending with and withoutmuscle stimulation. Initial results from these whole body workloops demonstrate that largemouth bass possess the capabilityto increase body stiffness by using their muscles to generatenegative mechanical work.     

Locomotion of Gymnarchus Niloticus： Experiment and Kinematics

In addition to forward undulatory swimming, Gymnarchus niloticus can swim via undulations of the dorsal fin while the body axis remains straight; furthermore, it swims forward and backward in a similar way, which indicates that the undulation of the dorsal fin can simultaneously provide bidirectional propulsive and maneuvering forces with the help of the tail fin. A high-resolution Charge-Coupled Device （CCD） imaging camera system is used to record kinematics of steady swimming as well as maneuvering in G. niloticus. Based on experimental data, this paper discusses the kinematics （cruising speed, wave speed, cycle frequency, amplitude, lateral displacement） of forward as well as backward swimming and maneuvering. During forward swimming, the propulsive force is generated mainly by undulations of the dorsal fin while the body axis remains straight. The kinematic parameters （wave speed, wavelength, cycle frequency, amplitude） have statistically significant correlations with cruising speed. In addition, the yaw at the head is minimal during steady swimming. From experimental data, the maximal lateral displacement of head is not more than 1% of the body length, while the maximal lateral displacement of the whole body is not more than 5% of the body length. Another important feature is that G. niloticus swims backwards using an undulatory mechanism that resembles the forward undulatory swimming mechanism. In backward swimming, the increase of lateral displacement of the head is comparatively significant; the amplitude profiles of the propulsive wave along the dorsal fin are significantly different from those in forward swimming. When G. niloticus does fast maneuvering, its body is first bent into either a C shape or an S shape, then it is rapidly unwound in a travelling wave fashion. It rarely maneuvers without the help of the tail fin and body bending.     

Computational methods for the analysis of swimming biomechanics

A series of computer programs is presented which enables the analysis of fish body shape and mass distributions, spine positions, spine curvatures and coordinates for the centre of mass. Data are derived from silhouette outlines of swimming fish, white muscle strains during swimming, white muscle force-time development functions for body bending cycles, muscle force and power production along the whole fish body and hydrodynamic efficiencies for fast-start swimming behaviours.     

Nonlinear muscles, passive viscoelasticity and body taper conspire to create neuromechanical phase lags in anguilliform swimmers

Locomotion provides superb examples of cooperation among neuromuscular systems, environmental reaction forces, and sensory feedback. As part of a program to understand the neuromechanics of locomotion, here we construct a model of anguilliform (eel-like) swimming in slender fishes. Building on a continuum mechanical representation of the body as an viscoelastic rod, actuated by a traveling wave of preferred curvature and subject to hydrodynamic reaction forces, we incorporate a new version of a calcium release and muscle force model, fitted to data from the lamprey Ichthyomyzon unicuspis, that interactively generates the curvature wave. We use the model to investigate the source of the difference in speeds observed between electromyographic waves of muscle activation and mechanical waves of body curvature, concluding that it is due to a combination of passive viscoelastic and geometric properties of the body and active muscle properties. Moreover, we find that nonlinear force dependence on muscle length and shortening velocity may reduce the work done by the swimming muscles in steady swimming.     

Fish swimming stride by stride: speed limits and endurance

Summary Steadily swimming fish show a species-specific stride length and tail tip amplitude. These are constant over the entire speed range if expressed as a fraction of the body length. The speed of a fish equals the stride length times the tail beat frequency. We describe how maximum tail beat frequencies, and hence maximum swimming speeds, are related to temperature and body length.Maximum sustained swimming speeds, endurance during swimming at higher speeds, and maximum burst velocities of 27 species are compared. The rate of decline of endurance with increasing speed is either gradual or steep, with only a few cases in between Steady swimmers show the steepest decline.The published effects of temperature on endurance are not consistent.The effect of body size on the endurance curve could be investigated for two species. The maximum sustained speed decreases with increasing length, and the slope of the endurance curves steepens with increasing length with the same factor in both species. The maximum burst speed is 10 Ls^-1 on average.     

Power isn't everything: muscle function and energetic costs during steady swimming in Atlantic cod (Gadus morhua)

Power produced by red myotomal muscles of fish during cruise swimming appears seldom maximized, so we sought to investigate whether economy may impact or dominate muscle function. We measured cost of transport (COT) using oxygen consumption and the strain trajectories and electromyographic activity of red muscle measured at anterior (ANT) and posterior (POST) locations while Atlantic cod (Gadus morhua) swam steadily at speeds between 0.3 and 1.0 body lengths (BL) s(-1). We then measured the power produced by isolated segments of red muscle when activated either as in the swimming cod or such that maximal net power was produced. Patterns of activation during swimming were not optimal for power output and were highly variable between tail beats, particularly at the ANT location and at slow swim speeds. Muscle strain amplitude did not increase until swimming speed reached 0.9 (ANT) versus 0.5 (POST) BL s(-1). These limited power to only 53% (ANT) and 71% (POST) of maximum at slower swim speeds and to 70%-80% of maximum at high swim speeds. COT (resting metabolism subtracted) was minimal at the slowest swim speed, surprisingly, where power was most impaired by activation and strain. Thus, production of powered forces for maneuverability/stability appeared to greatly impact red muscle function during cruise swimming in cod, particularly at slow speeds and in ANT muscle.     

The role of mechanical resonance in the neural control of swimming in fishes

The bodies of many fishes are flexible, elastic structures; if you bend them, they spring back. Therefore, they should have a resonant frequency: a bending frequency at which the output amplitude is maximized for a particular input. Previous groups have hypothesized that swimming at this resonant frequency could maximize efficiency, and that a neural circuit called the central pattern generator might be able to entrain to a mechanical resonance. However, fishes swim in water, which may potentially damp out many resonant effects. Additionally, their bodies are elongated, which means that bending can occur in complicated ways along the length of the body. We review previous studies of the mechanical properties of fish bodies, and then present new data that demonstrate complex bending properties of elongated fish bodies. Resonant peaks in amplitude exist, but there may be many of them depending on the body wavelength. Additionally, they may not correspond to the maximum swimming speed. Next, we describe experiments using a closed-loop preparation of the lamprey, in which a preparation of the spinal cord is linked to a real-time simulation of the muscle and body properties, allowing us to examine resonance entrainment as we vary the simulated resonant frequency. We find that resonance entrainment does occur, but is rare. Gain had a significant, though weak, effect, and a nonlinear muscle model produced resonance entrainment more often than a linear filter. We speculate that resonance may not be a critical effect for efficient swimming in elongate, anguilliform swimmers, though it may be more important for stiffer carangiform and thunniform fishes.     

An elastic rod model for anguilliform swimming

We develop a model for anguilliform (eel-like) swimming as an elastic rod actuated via time-dependent intrinsic curvature and subject to hydrodynamic drag forces, the latter as proposed by Taylor (in Proc Roy Proc Lond A 214:158–183, 1952). We employ a eometrically exact theory and discretize the resulting nonlinear partial differential evolution both to perform numerical simulations, and to compare with previous models consisting of chains of rigid links or masses connected by springs, dampers, and prescribed force generators representing muscles. We show that muscle activations driven by motoneuronal spike trains via calcium dynamics produce intrinsic curvatures corresponding to near-sinusoidal body shapes in longitudinally-uniform rods, but that passive elasticity causes Taylor’s assumption of prescribed shape to fail, leading to time-periodic motions and lower speeds than those predicted Taylor (in Proc Roy Proc Lond A 214:158–183, 1952). We investigate the effects of bending stiffness, body geometry, and activation patterns on swimming speed, turning behavior, and acceleration to steady swimming. We show that laterally-uniform activation yields stable straight swimming and laterally differential activation levels lead to stable turns, and we argue that tapered bodies with reduced caudal (tail-end) activation (to produce uniform intrinsic curvature) swim faster than ones with uniform activation.     

The role of the notochard in forward and reverse swimming and burrowing in the amphioxus Branchiostoma lanceolatum

Analyses of high speed cinefilm have shown that amphioxus swims either forward or backward with undulatory movement generated at the leading end, the wave of displacement passing along the body with increasing amplitude. The leading end, whether this is "head" or tail, is evidently more rigid than the trailing end, flexibility at each end changing with reversal in direction of swimming. It is suggested that control of the amplitude of the waves of displacement in different regions of the body in swimming is a function of the notochord, contraction of the muscular notochordal plates increasing its stiffness. Connections between the central nervous system and the notochordal plates via the notochordal pits are already known to exist.
As exposure to light invariably induces swimming in dark–adapted animals, it seems probable that the eyes function in initiating movement. The rate of increase in number and size of the eye cups during larval and adult growth and their pattern of distribution in the nerve cord are given. In the adult the eye cups occur predominantly in the anterior and posterior regions of the body. This may be of significance in providing the stimulus for changes in flexibility of these regions in swimming.
High speed cinefilm has also shown that amphioxus can burrow "head" or tail-first and move through sand in a forward or a reverse direction. It is suggested that rapid reversal of direction is of greater importance in movement through sand than in swimming.     

A Forced Damped Oscillation Framework for Undulatory Swimming Provides New Insights into How Propulsion Arises in Active and Passive Swimming

A fundamental issue in locomotion is to understand how muscle forcing produces apparently complex deformation kinematics leading to movement of animals like undulatory swimmers. The question of whether complicated muscle forcing is required to create the observed deformation kinematics is central to the understanding of how animals control movement. In this work, a forced damped oscillation framework is applied to a chain-link model for undulatory swimming to understand how forcing leads to deformation and movement. A unified understanding of swimming, caused by muscle contractions (“active” swimming) or by forces imparted by the surrounding fluid (“passive” swimming), is obtained. We show that the forcing triggers the first few deformation modes of the body, which in turn cause the translational motion. We show that relatively simple forcing patterns can trigger seemingly complex deformation kinematics that lead to movement. For given muscle activation, the forcing frequency relative to the natural frequency of the damped oscillator is important for the emergent deformation characteristics of the body. The proposed approach also leads to a qualitative understanding of optimal deformation kinematics for fast swimming. These results, based on a chain-link model of swimming, are confirmed by fully resolved computational fluid dynamics (CFD) simulations. Prior results from the literature on the optimal value of stiffness for maximum speed are explained.     

Estimating fish swimming metrics and metabolic rates with accelerometers: the influence of sampling frequency

Accelerometry is growing in popularity for remotely measuring fish swimming metrics, but appropriate sampling frequencies for accurately measuring these metrics are not well studied. This research examined the influence of sampling frequency (1–25 Hz) with tri‐axial accelerometer biologgers on estimates of overall dynamic body acceleration (ODBA), tail‐beat frequency, swimming speed and metabolic rate of bonefish Albula vulpes in a swim‐tunnel respirometer and free‐swimming in a wetland mesocosm. In the swim tunnel, sampling frequencies of ≥ 5 Hz were sufficient to establish strong relationships between ODBA, swimming speed and metabolic rate. However, in free‐swimming bonefish, estimates of metabolic rate were more variable below 10 Hz. Sampling frequencies should be at least twice the maximum tail‐beat frequency to estimate this metric effectively, which is generally higher than those required to estimate ODBA, swimming speed and metabolic rate. While optimal sampling frequency probably varies among species due to tail‐beat frequency and swimming style, this study provides a reference point with a medium body‐sized sub‐carangiform teleost fish, enabling researchers to measure these metrics effectively and maximize study duration.     

A Flexible Fin with Bio-Inspired Stiffness Profile and Geometry

Biological evidence suggests that fish use mostly anterior muscles for steady swimming while the caudal part of the body is passive and,acting as a carrier of energy,transfers the momentum to the surrounding water.Inspired by those findings we hypothesize that certain swimming patterns can be achieved without copying the distributed actuation mechanism of fish but rather using a single actuator at the anterior part to create the travelling wave.To test the hypothesis a pitching flexible fin made of silicone rubber and silicone foam was designed by copying the stiffness distribution profile and geometry of a rainbow trout.The kinematics of the fin was compared to that of a steadily swimming trout.Fin's propulsive wave length and tail-beat amplitude were determined while it was actuated by a single servo motor.Results showed that the propulsive wave length and tail-beat amplitude of a steadily swimming 50 cm rainbow trout was achieved with our biomimetic fin while stimulated using certain actuation parameters (frequency 2.31 Hz and amplitude 6.6 degrees).The study concluded that fish-like swimming can be achieved by mimicking the stiffness and geometry of a rainbow trout and disregarding the details of the actuation mechanism.     

The timing of the electromyogram in the lateral myotomes of mackerel and saithe at different swimming speeds

Electromyogram (EMG) signals from two points at about 40% L and 65% L ( L = length) in the left latera1 muscle of mackerel ( Scomber scombrus L.) L = 28–33 cm a nd saithe ( Pollachius virens L.) L = 42–50 cm were recorded synchronously with films of steady straight swimming motions. In both species, the duration of EMG activity at both electrodes, remains a constant proportion of the tail cycle period Tat all the tail beat frequencies between 1–8 and 13 Hz. In mackerel and saithe respectively: onset of EMG activity at the front was 74% T and 77% T before the left-most tail blade position and fronl EMG-onset occurred 15% T and 18% T before rear onset. The duration of the EMG burst is longer at the front position (41% T and 47% T ) than at the rear (25% T and 27% T ), At all swimming speeds the wave of electrical activation of the muscle travelled between the two electrodes 25% L apart at a velocity between 1.5 and 1.6 L T ⁻¹. Frequencies of spikes within the burst of EMG activity rose from 30–40 Hz at 2 T s⁻¹ to 50–80 Hz at 8 T s⁻¹. In muscle at 40%L EMG-onset happens at phase 30° just after muscle strain at this point reaches its resting length while lengthening (360°). At 65% L EMG-onset occurs earlier in the strain cycle-350° just before the muscle reaches it resting length while lengthening (360°). This could represent within the length of the fish, a phase shift of up to 90° in the EMG-onset in relation to the muscle strain cycle. These timings are discussed in relation to optimized work output and a single instant of maximum bending moment all along the left side of the body.     

Muscle Dynamics in Fish During Steady Swimming

SYNOPSIS. Recent research in fish locomotion has been dominatedby an interest in the dynamic mechanical properties of the swimmingmusculature. Prior observations have indicated that waves ofmuscle activation travel along the body of an undulating fishfaster than the resulting waves of muscular contraction, suggestingthat the phase relation between the muscle strain cycle andits activation must vary along the body. Since this phase relationis critical in determining how the muscle performs in cycliccontractions, the possibility has emerged that dynamic musclefunction may change with axial position in swimming fish. Quantificationof muscle contractile properties in cyclic contractions relieson in vitro experiments using strain and activation data collectedin vivo. In this paper we discuss the relation between theseparameters and body kinematics. Using videoradiographic datafrom swimming mackerel we demonstrate that red muscle straincan be accurately predicted from midline curvature but not fromlateral displacement. Electromyographic recordings show neuronalactivation patterns that are consistent with red muscle performingnet positive work at all axial positions. The relatively constantcross-section of red muscle along much of the body suggeststhat positive power for swimming is generated fairly uniformlyalong the length of the fish.     

Effect of body roll amplitude and arm rotation speed on propulsion of arm amputee swimmers

Only a limited amount of research has gone into evaluating the contribution made by the upper arm to the propulsion of elite swimmers with an amputation at elbow level. With assistance of computational fluid dynamics (CFD) modelling, the swimming technique of competitive arm amputee swimmers can be assessed through numerical simulations which test the effect of various parameters on the effectiveness of the swimming propulsion.This numerical study investigates the effect of body roll amplitude and of upper arm rotation speed on the propulsion of an arm amputee swimmer, at different mean swimming speeds. Various test cases are simulated resulting in a thorough analysis of the complex body/fluid interaction with a detailed quantitative assessment of the effect of the variation of each parameter on the arm propulsion. It is found that a body roll movement with an amplitude of 45° enhances greatly the propulsive contribution from the upper arm with an increase of about 70% in the propulsive force compared to the no roll condition. An increase in the angular velocity of the upper arm also leads to a concomitant increase in the propulsive forces produced by the arm.Such results have direct implications for competitive arm amputee front crawl swimmers and for those who coach them. One important message that emerges in this present work is that there exists, for any given swimming speed, a minimum angular velocity at which the upper arm must be rotated to generate effective propulsion. Below this velocity, the upper arm will experience a net resistive drag force which adversely affects swimming performance.     

Kinematics of swimming garter snakes (Thamnophis sirtalis)

We investigate the kinematics of swimming garter snakes (Thamnophis sirtalis) using a novel nonlinear regression-based digitization method to establish quantitative statistical support for non-constant wavelengths in the undulatory pattern exhibited by swimming snakes. We find that in swimming snakes, the growth of the amplitude of the propulsive wave head-to-tail is strongly correlated (p < 0.005) with the head-to-tail growth in the wavelength. We investigate correlations between kinematic parameters and steady swimming speed, and find a very strong positive correlation between swimming speed and undulation frequency. We furthermore find a statistically well-supported positive correlation between swimming speed and both the initial amplitude of the propulsive wave at the head and the degree of amplitude growth from head to tail.     

Locomotion of free-swimming ghost knifefish: anal fin kinematics during four behaviors

The maneuverability demonstrated by the weakly electric ghost knifefish (Apteronotus albifrons) is a result of its highly flexible ribbon-like anal fin, which extends nearly three-quarters the length of its body and is composed of approximately 150 individual fin rays. To understand how movement of the anal fin controls locomotion we examined kinematics of the whole fin, as well as selected individual fin rays, during four locomotor behaviors executed by free-swimming ghost knifefish: forward swimming, backward swimming, heave (vertical) motion, and hovering. We used high-speed video (1000 fps) to examine the motion of the entire anal fin and we measured the three-dimensional curvature of four adjacent fin rays in the middle of the fin during each behavior to determine how individual fin rays bend along their length during swimming. Canonical discriminant analysis separated all four behaviors on anal fin kinematic variables and showed that forward and backward swimming behaviors contrasted the most: forward behaviors exhibited a large anterior wavelength and posterior amplitude while during backward locomotion the anal fin exhibited both a large posterior wavelength and anterior amplitude. Heave and hover behaviors were defined by similar kinematic variables; however, for each variable, the mean values for heave motions were generally greater than for hovering. Individual fin rays in the middle of the anal fin curved substantially along their length during swimming, and the magnitude of this curvature was nearly twice the previously measured maximum curvature for ray-finned fish fin rays during locomotion. Fin rays were often curved into the direction of motion, indicating active control of fin ray curvature, and not just passive bending in response to fluid loading.     

Kinematics of swimming garter snakes (Thamnophis sirtalis).

We investigate the kinematics of swimming garter snakes (Thamnophis sirtalis) using a novel nonlinear regression-based digitization method to establish quantitative statistical support for non-constant wavelengths in the undulatory pattern exhibited by swimming snakes. We find that in swimming snakes, the growth of the amplitude of the propulsive wave head-to-tail is strongly correlated (p < 0.005) with the head-to-tail growth in the wavelength. We investigate correlations between kinematic parameters and steady swimming speed, and find a very strong positive correlation between swimming speed and undulation frequency. We furthermore find a statistically well-supported positive correlation between swimming speed and both the initial amplitude of the propulsive wave at the head and the degree of amplitude growth from head to tail.