How the bending kinematics of swimming lampreys build negative pressure fields for suction thrust

Form and function of anguilliform swimming

Anguilliform swimmers are long and narrow animals that propel themselves by undulating their bodies. Observations in nature and recent investigations suggest that anguilliform swimming is highly efficient. However, understanding the underlying reasons for the efficiency of this type of locomotion requires interdisciplinary studies spanning from biology to hydrodynamics. Regrettably, these different fields are rarely discussed together, which hinders our ability to understand the repeated evolution of this swimming mode in vertebrates. This review compiles the current knowledge of the anatomical features that drive anguilliform swimming, compares the resulting kinematics across a wide range of anguilliform swimmers, and describes the resulting hydrodynamic interactions using data from both in vivo experiments and computational studies.

Fish can use coordinated fin motions to recapture their own vortex wake energy

During swimming, many fishes use pectoral fins for propulsion and, in the process, move substantial amounts of water rearward. However, the effect that this upstream wake has on the caudal fin remains largely unexplored. By coordinating motions of the caudal fin with the pectoral fins, fishes have the potential to create constructive flow interactions which may act to partially recapture the upstream energy lost in the pectoral fin wake. Using experimentally derived velocity and pressure fields for the silver mojarra (Eucinostomus argenteus), we show that pectoral-caudal fin (PCF) coordination enables the circulation and interception of pectoral fin wake vortices by the caudal fin. This acts to transfer energy to the caudal fin and enhance its hydrodynamic efficiency at swimming speeds where this behaviour occurs. We also find that mojarras commonly use PCF coordination in nature. The results offer new insights into the evolutionary drivers and behavioural plasticity of fish swimming as well as for developing more capable bioinspired underwater vehicles.

The hydrodynamics and kinematics of the appendicularian tail underpin peristaltic pumping

Planktonic organisms feed while suspended in water using various hydrodynamic pumping strategies. Appendicularians are a unique group of plankton that use their tail to pump water over mucous mesh filters to concentrate food particles. As ubiquitous and often abundant members of planktonic ecosystems, they play a major role in oceanic food webs. Yet, we lack a complete understanding of the fluid flow that underpins their filtration. Using high-speed, high-resolution video and micro particle image velocimetry, we describe the kinematics and hydrodynamics of the tail in Oikopleura dioica in filtering and free-swimming postures. We show that sinusoidal waves of the tail generate peristaltic pumping within the tail chamber with fluid moving parallel to the tail when filtering. We find that the tail contacts attachment points along the tail chamber during each beat cycle, serving to seal the tail chamber and drive pumping. When we tested how the pump performs across environmentally relevant temperatures, we found that the amplitude of the tail was invariant but tail beat frequency increased threefold across three temperature treatments (5°C, 15°C and 25°C). Investigation into this unique pumping mechanism gives insight into the ecological success of appendicularians and provides inspiration for novel pump designs.

Identification of the trade-off between speed and efficiency in undulatory swimming using a bio-inspired robot

Anguilliform swimmers, like eels or lampreys, are highly efficient swimmers. Key to understanding their performances is the relationship between the body's kinematics and resulting swimming speed and efficiency. But, we cannot prescribe kinematics to living fish, and it is challenging to measure their power consumption. Here, we characterise the swimming speed and cost of transport of a free-swimming undulatory bio-inspired robot as we vary its kinematic parameters, including joint amplitude, body wavelength, and frequency. We identify a trade-off between speed and efficiency. Speed, in terms of stride length, increases for increasing maximum tail angle, described by the newly proposed specific tail amplitude and reaches a maximum value around the specific tail amplitude of unity. Efficiency, in terms of the cost of transport, is affected by the whole-body motion. Cost of transport decreases for increasing travelling wave-like kinematics, and lower specific tail amplitudes. Our results suggest that live eels tend to choose efficiency over speed and provide insights into the key characteristics affecting undulatory swimming performance.

Measuring the 3D wake of swimming snakes (Natrix tessellata) using volumetric particle image velocimetry

We describe a method for measuring the 3D vortical structures produced by an anguilliform swimmer using volumetric velocimetry. The wake of freely swimming dice snakes (Natrix tessellata) was quantified, revealing the creation of multiple vortices along the body of the snake due to its undulation. The 3D structure of the vortices generally consisted of paired vortex tubes, some of which were linked together to form a hairpin structure. The observations match predictions from computational fluid dynamic studies of other anguilliform swimmers. Quantitative measurements allowed us to study vortex circulation and size, and global kinetic energy of the flow, which varied with swimming speed, vortex topology and individual characteristics. Our findings provide a baseline for comparing wake structures of snakes with different morphologies and ecologies and investigating the energetic efficiency of anguilliform swimming.

A tale of two fish tails: does a forked tail really perform better than a truncate tail when cruising?

Many fishes use their tail as the main thrust producer during swimming. This fin's diversity in shape and size influences its physical interactions with water as well as its ecological functions. Two distinct tail morphologies are common in bony fishes: flat, truncate tails which are best suited for fast accelerations via drag forces, and forked tails that promote economical, fast cruising by generating lift-based thrust. This assumption is based primarily on studies of the lunate caudal fin of Scombrids (i.e. tuna, mackerel), which is comparatively stiff and exhibits an airfoil-type cross-section. However, this is not representative of the more commonly observed and taxonomically widespread flexible forked tail, yet similar assumptions about economical cruising are widely accepted. Here, we present the first comparative experimental study of forked versus truncate tail shape and compare the fluid mechanical properties and energetics of two common nearshore fish species. We examined the hypothesis that forked tails provide a hydrodynamic advantage over truncate tails at typical cruising speeds. Using experimentally derived pressure fields, we show that the forked tail produces thrust via acceleration reaction forces like the truncate tail during cruising but at increased energetic costs. This reduced efficiency corresponds to differences in the performance of the two tail geometries and body kinematics to maintain similar overall thrust outputs. Our results offer insights into the benefits and tradeoffs of two common fish tail morphologies and shed light on the functional morphology of fish swimming to guide the development of bio-inspired underwater technologies.

Fish-inspired segment models for undulatory steady swimming

Many aquatic animals swim by undulatory body movements and understanding the diversity of these movements could unlock the potential for designing better underwater robots. Here, we analyzed the steady swimming kinematics of a diverse group of fish species to investigate whether their undulatory movements can be represented using a series of interconnected multi-segment models, and if so, to identify the key factors driving the segment configuration of the models. Our results show that the steady swimming kinematics of fishes can be described successfully using parsimonious models, 83% of which had fewer than five segments. In these models, the anterior segments were significantly longer than the posterior segments, and there was a direct link between segment configuration and swimming kinematics, body shape, and Reynolds number. The models representing eel-like fishes with elongated bodies and fishes swimming at high Reynolds numbers had more segments and less segment length variability along the body than the models representing other fishes. These fishes recruited their anterior bodies to a greater extent, initiating the undulatory wave more anteriorly. Two shape parameters, related to axial and overall body thickness, predicted segment configuration with moderate to high success rate. We found that head morphology was a good predictor of its segment length. While there was a large variation in head segments, the length of tail segments was similar across all models. Given that fishes exhibited variable caudal fin shapes, the consistency of tail segments could be a result of an evolutionary constraint tuned for high propulsive efficiency. The bio-inspired multi-segment models presented in this study highlight the key bending points along the body and can be used to decide on the placement of actuators in fish-inspired robots, to model hydrodynamic forces in theoretical and computational studies, or for predicting muscle activation patterns during swimming.

Swimming kinematics and performance of spinal transected lampreys with different levels of axon regeneration

Axon regeneration is critical for restoring neural function after spinal cord injury. This has prompted a series of studies on the neural and functional recovery of lampreys after spinal cord transection. Despite this, there are still many basic questions remaining about how much functional recovery depends on axon regeneration. Our goal was to examine how swimming performance is related to degree of axon regeneration in lampreys recovering from spinal cord transection by quantifying the relationship between swimming performance and percent axon regeneration of transected lampreys after 11 weeks of recovery. We found that while swimming speeds varied, they did not relate to percent axon regeneration. In fact, swimming speeds were highly variable within individuals, meaning that most individuals could swim at both moderate and slow speeds, regardless of percent axon regeneration. However, none of the transected individuals were able to swim as fast as the control lampreys. To swim fast, control lampreys generated high amplitude body waves with long wavelengths. Transected lampreys generated body waves with lower amplitude and shorter wavelengths than controls, and to compensate, transected lampreys increased their wave frequencies to swim faster. As a result, transected lampreys had significantly higher frequencies than control lampreys at comparable swimming velocities. These data suggest that the control lampreys swam more efficiently than transected lampreys. In conclusion, there appears to be a minimal recovery threshold in terms of percent axon regeneration required for lampreys to be capable of swimming; however, there also seems to be a limit to how much they can behaviorally recover.

Summary: Lampreys that have recovered from having their spinal cords transected do not fully regain swimming abilities and are not able to swim as efficiently as non-transected lampreys.

Biorobotic insights into neuromechanical coordination of undulatory swimming

Skin sensors on an eel-like robot couple external hydrodynamic pressure with internal neural patterns for robust swimming.

Anguilliform Swimming Performance of an Eel-Inspired Soft Robot

In this article, we propose a soft eel robot design using soft pneumatic actuators that mimic eel muscles. Four pairs of soft actuators are used to construct the eel robot body. Pulse signals with suitable shifting phases are utilized to control delivery of compressed air to the actuators in sequence to create a sinusoidal wave from head to tail of the robot body. A model of hydrodynamic forces acting on an anguilliform swimmer when moving in fluid was built to estimate the thrust force generated by the robot at different tail beat frequencies. Experimental data revealed that the generated thrust force was positively correlated with the beat frequency. Measured data showed that swimming efficiency depended on both generated thrust force and body posture in situ. At the beat frequency of 1.25 Hz, and air pressure at three segments from head to tail of 65, 50, and 30 kPa, respectively, the eel robot body showed the best cost of transport (COT) of 19.21 with velocity of 10.5 cm/s (or 0.198 body length per second [BL/s]), compared to the other's values of operation frequency and air pressure. We also found that control shifting phase strongly affects the swimming speed and COT. The robot body reached the highest velocity at around 19 cm/s (0.36 BL/s) with the COT of 10.72. Obtained result in this research would contribute to development of soft elongated swimming robot and enhance the knowledge on swimming performance of both robot and natural eels.

Metachronal motion across scales: current challenges and future directions

Metachronal motion is used across a wide range of organisms for a diverse set of functions. However, despite its ubiquity, analysis of this behavior has been difficult to generalize across systems. Here we provide an overview of known commonalities and differences between systems that use metachrony to generate fluid flow. We also discuss strategies for standardizing terminology and defining future investigative directions that are analogous to other established subfields of biomechanics. Lastly, we outline key challenges that are common to many metachronal systems, opportunities that have arisen due to the advent of new technology (both experimental and computational), and next steps for community development and collaboration across the nascent network of metachronal researchers.

Automatic segmentation of fish midlines for optimizing robot design

While fish use continuous and flexible bodies to propel themselves, fish robots are often made from interconnected segments. How many segments do robots need to represent fish movements accurately? We propose a new method to automatically determine parsimonious robot models from actual fish data. We first identify key bending points (i.e., joint positions) along the body and then study the concerted movement of the segments so that the difference between actual fish and modelled bending kinematics is minimized. To demonstrate the utility of our method, we analyse the steady swimming kinematics of 10 morphologically distinct fish species. Broadly classified as sub-carangiform (e.g., rainbow trout) and carangiform (e.g., crevalle jack) swimmers, these species exhibit variations in the way they undulate when traditional parameters (including head and tail beat amplitudes, body wavelength and maximum curvature along the body) are considered. We show that five segments are sufficient to describe the kinematics with at least 99% accuracy. For optimal performance, segments should progressively get shorter towards the tail. We also show that locations where bending moments are applied vary among species, possibly because of differences in morphology. More specifically, we find that wider fish have shorter head segments. We discover that once bending points are factored in, the kinematics differences observed in these species collapse into a single undulatory pattern. The amplitude and timing of how body segments move entirely depend on their respective joint positions along the body. Head and body segments are also coupled in a timely manner, which depends on the position of the most anterior joint. Our findings provide a mechanistic understanding of how morphology relates to kinematics and highlight the importance of head control, which is often overlooked in current robot designs.

Fish without Tail Fins-Exploring the Function of Tail Morphology of the First Vertebrates

We use a series of hydrodynamic experiments on abstracted models to explore whether primitive vertebrates may have swum under various conditions without a clearly-differentiated tail fin. Cambrian vertebrates had post-anal stubby tails, some had single dorsal and ventral fins, but none had yet evolved a clearly differentiated caudal fin typical of post-Cambrian fishes, and must have relied on their long and flexible laterally-compressed bodies for locomotion, i.e., by bending their bodies side-to-side in order to propagate waves from head to tail. We approach this problem experimentally based on an abstracted model of Metaspriggina walcotti from the 506-million-year old Burgess Shale by using oscillating thin flexible plates while varying the tail fin geometry from rectangular to uniform, and finally to a no tail-fin condition. Despite a missing tail fin, this study supports the observation that the abstracted Metaspriggina model can generate a strong propulsive force in cruise conditions, both away from, and near the sea bed (in ground effect). However, when the abstracted Metaspriggina model moves in ground effect, a weaker performance is observed, indicating that Metaspriggina may not necessarily have been optimized for swimming near the sea bed. When considering acceleration from rest, we find that the Metaspriggina model's performance is not significantly different from other morphological models (abstracted truncate tail and abstracted heterocercal tail). Statistical analysis shows that morphological parameters, swimming modes, and ground effect all play significant roles in thrust performance. While the exact relationships of Cambrian vertebrates are still debated, as agnathans, they share some general characteristics with modern cyclostomes, in particular an elongate body akin to lampreys. Lampreys, as anguilliform swimmers, are considered to be some of the most efficient swimmers using a particular type of suction thrust induced by the traveling body wave as it travels from head to tail. Our current experiments suggest that Metaspriggina's ability in acceleration from rest, through possibly a similar type of suction thrust, which is defined as the ability to generate low pressure on upstream facing sections of the body, might have evolved early in response to increasing predator pressure during the Cambrian Explosion.

The most efficient metazoan swimmer creates a 'virtual wall' to enhance performance

It has been well documented that animals (and machines) swimming or flying near a solid boundary get a boost in performance. This ground effect is often modelled as an interaction between a mirrored pair of vortices represented by a true vortex and an opposite sign 'virtual vortex' on the other side of the wall. However, most animals do not swim near solid surfaces and thus near body vortex-vortex interactions in open-water swimmers have been poorly investigated. In this study, we examine the most energetically efficient metazoan swimmer known to date, the jellyfish Aurelia aurita, to elucidate the role that vortex interactions can play in animals that swim away from solid boundaries. We used high-speed video tracking, laser-based digital particle image velocimetry (dPIV) and an algorithm for extracting pressure fields from flow velocity vectors to quantify swimming performance and the effect of near body vortex-vortex interactions. Here, we show that a vortex ring (stopping vortex), created underneath the animal during the previous swim cycle, is critical for increasing propulsive performance. This well-positioned stopping vortex acts in the same way as a virtual vortex during wall-effect performance enhancement, by helping converge fluid at the underside of the propulsive surface and generating significantly higher pressures which result in greater thrust. These findings advocate that jellyfish can generate a wall-effect boost in open water by creating what amounts to a 'virtual wall' between two real, opposite sign vortex rings. This explains the significant propulsive advantage jellyfish possess over other metazoans and represents important implications for bio-engineered propulsion systems.

The role of suction thrust in the metachronal paddles of swimming invertebrates

An abundance of swimming animals have converged upon a common swimming strategy using multiple propulsors coordinated as metachronal waves. The shared kinematics suggest that even morphologically and systematically diverse animals use similar fluid dynamic relationships to generate swimming thrust. We quantified the kinematics and hydrodynamics of a diverse group of small swimming animals who use multiple propulsors, e.g. limbs or ctenes, which move with antiplectic metachronal waves to generate thrust. Here we show that even at these relatively small scales the bending movements of limbs and ctenes conform to the patterns observed for much larger swimming animals. We show that, like other swimming animals, the propulsors of these metachronal swimmers rely on generating negative pressure along their surfaces to generate forward thrust (i.e., suction thrust). Relying on negative pressure, as opposed to high pushing pressure, facilitates metachronal waves and enables these swimmers to exploit readily produced hydrodynamic structures. Understanding the role of negative pressure fields in metachronal swimmers may provide clues about the hydrodynamic traits shared by swimming and flying animals.

Jellyfish and Fish Solve the Challenges of Turning Dynamics Similarly to Achieve High Maneuverability

Turning maneuvers by aquatic animals are essential for fundamental life functions such as finding food or mates while avoiding predation. However, turning requires resolution of a fundamental dilemma based in rotational mechanics: the force powering a turn (torque) is favored by an expanded body configuration that maximizes lever arm length, yet minimizing the resistance to a turn (the moment of inertia) is favored by a contracted body configuration. How do animals balance these opposing demands? Here, we directly measure instantaneous forces along the bodies of two animal models—the radially symmetric Aurelia aurita jellyfish, and the bilaterally symmetric Danio rerio zebrafish—to evaluate their turning dynamics. Both began turns with a small, rapid shift in body kinematics that preceded major axial rotation. Although small in absolute magnitude, the high fluid accelerations achieved by these initial motions generated powerful pressure gradients that maximized torque at the start of a turn. This pattern allows these animals to initially maximize torque production before major body curvature changes. Both animals then subsequently minimized the moment of inertia, and hence resistance to axial rotation, by body bending. This sequential solution provides insight into the advantages of re-arranging mass by bending during routine swimming turns.

Airfoil-like mechanics generate thrust on the anterior body of swimming fishes

The anterior body of many fishes is shaped like an airfoil turned on its side. With an oscillating angle to the swimming direction, such an airfoil experiences negative pressure due to both its shape and pitching movements. This negative pressure acts as thrust forces on the anterior body. Here, we apply a high-resolution, pressure-based approach to describe how two fishes, bluegill sunfish (Lepomis macrochirus Rafinesque) and brook trout (Salvelinus fontinalis Mitchill), swimming in the carangiform mode, the most common fish swimming mode, generate thrust on their anterior bodies using leading-edge suction mechanics, much like an airfoil. These mechanics contrast with those previously reported in lampreys-anguilliform swimmers-which produce thrust with negative pressure but do so through undulatory mechanics. The thrust produced on the anterior bodies of these carangiform swimmers through negative pressure comprises 28% of the total thrust produced over the body and caudal fin, substantially decreasing the net drag on the anterior body. On the posterior region, subtle differences in body shape and kinematics allow trout to produce more thrust than bluegill, suggesting that they may swim more effectively. Despite the large phylogenetic distance between these species, and differences near the tail, the pressure profiles around the anterior body are similar. We suggest that such airfoil-like mechanics are highly efficient, because they require very little movement and therefore relatively little active muscular energy, and may be used by a wide range of fishes since many species have appropriately shaped bodies.

PTPσ Knockdown in Lampreys Impairs Reticulospinal Axon Regeneration and Neuronal Survival After Spinal Cord Injury

Traumatic spinal cord injury (SCI) results in persistent functional deficits due to the lack of axon regeneration within the mammalian CNS. After SCI, chondroitin sulfate proteoglycans (CSPGs) inhibit axon regrowth via putative interactions with the LAR-family protein tyrosine phosphatases, PTPσ and LAR, localized on the injured axon tips. Unlike mammals, the sea lamprey, Petromyzon marinus, robustly recovers locomotion after complete spinal cord transection (TX). Behavioral recovery is accompanied by heterogeneous yet predictable anatomical regeneration of the lamprey's reticulospinal (RS) system. The identified RS neurons can be categorized as "good" or "bad" regenerators based on the likelihood that their axons will regenerate. Those neurons that fail to regenerate their axons undergo a delayed form of caspase-mediated cell death. Previously, this lab reported that lamprey PTPσ mRNA is selectively expressed in "bad regenerator" RS neurons, preceding SCI-induced caspase activation. Consequently, we hypothesized that PTPσ deletion would reduce retrograde cell death and promote axon regeneration. Using antisense morpholino oligomers (MOs), we knocked down PTPσ expression after TX and assessed the effects on axon regeneration, caspase activation, intracellular signaling, and behavioral recovery. Unexpectedly, PTPσ knockdown significantly impaired RS axon regeneration at 10 weeks post-TX, primarily due to reduced long-term neuron survival. Interestingly, cell loss was not preceded by an increase in caspase or p53 activation. Behavioral recovery was largely unaffected, although PTPσ knockdowns showed mild deficits in the recovery of swimming distance and latency to immobility during open field swim assays. Although the mechanism underlying the cell death following TX and PTPσ knockdown remains unknown, this study suggests that PTPσ is not a net negative regulator of long tract axon regeneration in lampreys.

Surprising simplicities and syntheses in limbless self-propulsion in sand

Animals moving on and in fluids and solids move their bodies in diverse ways to generate propulsion and lift forces. In fluids, animals can wiggle, stroke, paddle or slap, whereas on hard frictional terrain, animals largely engage their appendages with the substrate to avoid slip. Granular substrates, such as desert sand, can display complex responses to animal interactions. This complexity has led to locomotor strategies that make use of fluid-like or solid-like features of this substrate, or combinations of the two. Here, we use examples from our work to demonstrate the diverse array of methods used and insights gained in the study of both surface and subsurface limbless locomotion in these habitats. Counterintuitively, these seemingly complex granular environments offer certain experimental, theoretical, robotic and computational advantages for studying terrestrial movement, with the potential for providing broad insights into morphology and locomotor control in fluids and solids, including neuromechanical control templates and morphological and behavioral evolution. In particular, granular media provide an excellent testbed for a locomotion framework called geometric mechanics, which was introduced by particle physicists and control engineers in the last century, and which allows quantitative analysis of alternative locomotor patterns and morphology to test for control templates, optimality and evolutionary alternatives. Thus, we posit that insights gained from movement in granular environments can be translated into principles that have broader applications across taxa, habitats and movement patterns, including those at microscopic scales.

Thrust generation during steady swimming and acceleration from rest in anguilliform swimmers

Escape swimming is a crucial behavior by which undulatory swimmers evade potential threats. The hydrodynamics of escape swimming have not been well studied, particularly for anguilliform swimmers, such as the sea lamprey Petromyzon marinus. For this study, we compared the kinematics and hydrodynamics of larval sea lampreys with those of lampreys accelerating from rest during escape swimming. We used experimentally derived velocity fields to calculate pressure fields and distributions of thrust and drag along the body. Lampreys initiated acceleration from rest with the formation of a high-amplitude body bend at approximately one-quarter body length posterior to the head. This deep body bend produced two high-pressure regions from which the majority of thrust for acceleration was derived. In contrast, steady swimming was characterized by shallower body bends and negative-pressure-derived thrust, which was strongest near the tail. The distinct mechanisms used for steady swimming and acceleration from rest may reflect the differing demands of the two behaviors. High-pressure-based mechanisms, such as the one used for acceleration from rest, could also be important for low-speed maneuvering during which drag-based turning mechanisms are less effective. The design of swimming robots may benefit from the incorporation of such insights from unsteady swimming.

Multichannel stroboscopic videography (MSV): a technique for visualizing multiple channels for behavioral measurements

Biologists commonly visualize different features of an organism using distinct sources of illumination. Such multichannel imaging has largely not been applied to behavioral studies because of the challenges posed by a moving subject. We address this challenge with the technique of multichannel stroboscopic videography (MSV), which synchronizes multiple strobe lights with video exposures of a single camera. We illustrate the utility of this approach with kinematic measurements of a walking cockroach (Gromphadorhina portentosa) and calculations of the pressure field around a swimming fish (Danio rerio). In both, transmitted illumination generated high-contrast images of the animal's body in one channel. Other sources of illumination were used to visualize the points of contact for the feet of the cockroach and the water flow around the fish in separate channels. MSV provides an enhanced potential for high-throughput experimentation and the capacity to integrate changes in physiological or environmental conditions in freely-behaving animals.

Accelerating fishes increase propulsive efficiency by modulating vortex ring geometry

Swimming animals need to generate propulsive force to overcome drag, regardless of whether they swim steadily or accelerate forward. While locomotion strategies for steady swimming are well characterized, far less is known about acceleration. Animals exhibit many different ways to swim steadily, but we show here that this behavioral diversity collapses into a single swimming pattern during acceleration regardless of the body size, morphology, and ecology of the animal. We draw on the fields of biomechanics, fluid dynamics, and robotics to demonstrate that there is a fundamental difference between steady swimming and forward acceleration. We provide empirical evidence that the tail of accelerating fishes can increase propulsive efficiency by enhancing thrust through the alteration of vortex ring geometry. Our study provides insight into how propulsion can be altered without increasing vortex ring size and represents a fundamental departure from our current understanding of the hydrodynamic mechanisms of acceleration. Our findings reveal a unifying hydrodynamic principle that is likely conserved in all aquatic, undulatory vertebrates.