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.     

Spinal Mechanisms in the Control of Lamprey Swimming

SYNOPSIS. The lamprey, an anguilliform fish, swims using lateralundulatory movement; a transverse wave passes backward, fromhead to tail, the amplitude of the wave increasing as it movestailward. The wave of muscle activity producing this movementtravels down the body faster than the mechanical wave. The wayin which certain features of anguilliform movement contributeto its efficiency have been described. The neural activity underlyingswimming is characterized by: 1) rhythmical alternation betweenthe two sides of a single segment; 2) a burst duration thatremains a constant proportion of the cycle time and is independentof the cycle frequency; 3) rostrocaudal phase lag that is constantand also independent of the cycle frequency. Local circuitsin the lamprey spinal cord can generate this locomotory patternin the absence of sensory feedback following activation of excitatoryamino acid receptors; the pattern is centrally generated. Ithas been hypothesized that the spinal central pattern generatorfor locomotion consists of a series of segmental burst generatorscoupled together by an intersegmental coordinating system. Theintersegmental coordinating system functions to keep the frequenciesof the oscillators along the cord constant and to provide theappropriate rostrocaudal phase lag. Mechanosensitive units withinthe spinal cord are sensitive to movement of the spinal cord\notochordand movement of the spinal cord/notochord can entrain the burstpattern. Entrainment occurs through movement-related feedbackonto neurons at the local level. The possible roles this movement-relatedfeedback plays during locomotion are discussed.     

Flexibility in the patterning and control of axial locomotor networks in lamprey

In lower vertebrates, locomotor burst generators for axial muscles generally produce unitary bursts that alternate between the two sides of the body. In lamprey, a lower vertebrate, locomotor activity in the axial ventral roots of the isolated spinal cord can exhibit flexibility in the timings of bursts to dorsally-located myotomal muscle fibers versus ventrally-located myotomal muscle fibers. These episodes of decreased synchrony can occur spontaneously, especially in the rostral spinal cord where the propagating body waves of swimming originate. Application of serotonin, an endogenous spinal neurotransmitter known to presynaptically inhibit excitatory synapses in lamprey, can promote decreased synchrony of dorsal-ventral bursting. These observations suggest the possible existence of dorsal and ventral locomotor networks with modifiable coupling strength between them. Intracellular recordings of motoneurons during locomotor activity provide some support for this model. Pairs of motoneurons innervating myotomal muscle fibers of similar ipsilateral dorsoventral location tend to have higher correlations of fast synaptic activity during fictive locomotion than do pairs of motoneurons innervating myotomes of different ipsilateral dorsoventral locations, suggesting their control by different populations of premotor interneurons. Further, these different motoneuron pools receive different patterns of excitatory and inhibitory inputs from individual reticulospinal neurons, conveyed in part by different sets of premotor interneurons. Perhaps, then, the locomotor network of the lamprey is not simply a unitary burst generator on each side of the spinal cord that activates all ipsilateral body muscles simultaneously. Instead, the burst generator on each side may comprise at least two coupled burst generators, one controlling motoneurons innervating dorsal body muscles and one controlling motoneurons innervating ventral body muscles. The coupling strength between these two ipsilateral burst generators may be modifiable and weakening when greater swimming maneuverability is required. Variable coupling of intrasegmental burst generators in the lamprey may be a precursor to the variable coupling of burst generators observed in the control of locomotion in the joints of limbed vertebrates.     

Muscle energy transfer during a given wave-like movement of human body segments

The paper deals with a movement of two voluntary segments fixed in a joint and connected by a muscle in a multi-segment biomechanical system of human body. The muscle model is a four-element mechanical system. The mechanical movement energy brought into the "segments-muscle" system from the segments preceding the next ones is studied. The movement in which the total multi-segment system of the human body participates is described by the wave equation. Conditions concerning applying active muscle efforts and correlating velocities of muscle ends movement which provide the maximal value of transferred energy have been found. It is shown that the use of "artificial muscles" type devices promotes activization of energy transfer processes between segments.     

Rate theory models for ion transport through rigid pores. I. Time-dependent analysis in the case of vanishing interactions

In this first of a series of papers concerning the theoretical analysis of rate theory models for ion transport through rigid pores, the case of vanishing interactions is investigated. "Rigidity" means that ions crossing membranes through pores see a fixed structure of the pores, not changing in time. A single pore is considered to be a sequence of (n + 1) activation barriers separated by n energy minima. The explicit analytical treatment is restricted to pores with regular internal barrier structure, including the nonequilibrium situation of an applied electric field. In this case the connection with continuum diffusion models is demonstrated by performing in the limit n leads to infinity (n = number of binding sites within the pores) the transition to continuum. Thus, from diffusion equations describing a discrete number of jumps, the corresponding diffusion-like partial differential equations and boundary conditions are generated. For regular pores, from the time dependent solutions of the discrete equations, the corresponding solutions of the continuum equations are explicitly generated. The time-dependent relaxation behaviour of the discrete model is in good agreement with the continuum model if one assumes more than two binding sites in the pores.     

Alternative migration and host parasitism strategies and their long-term stability in river lampreys from the River Endrick, Scotland

The stability of a discrete body size dimorphism of sexually mature river lamprey Lampetra fluviatilis from the River Endrick, Scotland, was examined over a 21 year period. Stable isotope analysis was used to test the hypothesis that the two size forms comprise individuals with differing migration and parasitic foraging strategies. Maturing river lamprey and the brook lamprey Lampetra planeri were trapped over 3 months each year in the periods 1983–1984 and 2004–2005. Brook lamprey catches and catches of both species combined showed no significant trend in catch rate with time. The catch rate of small body size river lamprey declined between 1983–1984 and 2004–2005 (although the difference did not reach statistical significance; P = 0·055). In contrast, there was a significant increase in the catch rate of the large body size river lamprey and as a consequence, a significant change in the relative proportion of each of the two river lamprey morphs over the study period. Analysis of the stable isotopes of C and N in muscle tissue showed that brook lamprey tissue derived its carbon from a freshwater source and had a δ¹³C more consistent with that of the River Endrick than with Loch Lomond. δ¹⁵N values for this species showed it to be feeding at the base of the food chain, consistent with filter feeding as an ammocoete. The large body size and the small body size river lamprey adults differed substantially in their δ¹³C values, with the small body size δ¹³C signature indicative of a freshwater carbon source and the large body size morph of a marine source. The small body size morph had a δ¹³C signature that was consistent with that of Loch Lomond powan Coregonus lavaretus suggesting that they share a common carbon source. The large body size morph was clearly feeding at a higher trophic level than the small body size morph. A single small body size river lamprey individual with typical morphology for that group, however, had C and N signatures that clustered with those of the large body size morphs. This individual had either migrated to sea to forage, as is typical for the species, or had been feeding on an anadromous fish with a strong marine C signature in fresh water. It is concluded that the body size dimorphism is indicative of a differential migration and foraging strategy in the parasitic phase of the life cycle of river lamprey at this site.     

Shifts in a single muscle's control potential of body dynamics are determined by mechanical feedback

Muscles are multi-functional structures that interface neural and mechanical systems. Muscle work depends on a large multi-dimensional space of stimulus (neural) and strain (mechanical) parameters. In our companion paper, we rewrote activation to individual muscles in intact, behaving cockroaches (Blaberus discoidalis L.), revealing a specific muscle's potential to control body dynamics in different behaviours. Here, we use those results to provide the biologically relevant parameters for in situ work measurements. We test four hypotheses about how muscle function changes to provide mechanisms for the observed control responses. Under isometric conditions, a graded increase in muscle stress underlies its linear actuation during standing behaviours. Despite typically absorbing energy, this muscle can recruit two separate periods of positive work when controlling running. This functional change arises from mechanical feedback filtering a linear increase in neural activation into nonlinear work output. Changing activation phase again led to positive work recruitment, but at different times, consistent with the muscle's ability to also produce a turn. Changes in muscle work required considering the natural sequence of strides and separating swing and stance contributions of work. Both in vivo control potentials and in situ work loops were necessary to discover the neuromechanical coupling enabling control.     

A continuous dynamic beam model for swimming fish

When a fish swims in water, muscle contraction, controlled by the nervous system, interacts with the body tissues and the surrounding fluid to yield the observed movement pattern of the body. A continuous dynamic beam model describing the bending moment balance on the body for such an interaction during swimming has been established. In the model a linear visco-elastic assumption is made for the passive behaviour of internal tissues, skin and backbone, and the unsteady fluid force acting on the swimming body is calculated by the 3D waving plate theory. The body bending moment distribution due to the various components, in isolation and acting together, is analysed. The analysis is based on the saithe (Pollachius virens), a carangiform swimmer. The fluid reaction needs a bending moment of increasing amplitude towards the tail and near-standing wave behaviour on the rear-half of the body. The inertial movement of the fish results from a wave of bending moment with increasing amplitude along the body and a higher propagation speed than that of body bending. In particular, the fluid reaction, mainly designed for propulsion, can provide a considerable force to balance the local momentum change of the body and thereby reduce the power required from the muscle. The wave of passive visco-elastic bending moment, with an amplitude distribution peaking a little before the mid-point of the fish, travels with a speed close to that of body bending. The calculated muscle bending moment from the whole dynamic system has a wave speed almost the same as that observed for EMG-onset and a starting instant close to that of muscle activation, suggesting a consistent matching between the muscle activation pattern and the dynamic response of the system in steady swimming. A faster wave of muscle activation, with a variable phase relation between the strain and activation cycle, appears to be designed to fit the fluid reaction and, to a lesser extent, the body inertia, and is limited by the passive internal tissues. Higher active stress is required from caudal muscle, as predicted from experimental studies on fish muscle. In general, the active force development by muscle does not coincide with the propulsive force generation on the tail. The stiffer backbone may play a role in transmitting force and deformation to maintain and adjust the movement of the body and tail in water.     

Shortening velocity and power output of skinned muscle fibers from mammals having a 25,000-fold range of body mass

The shortening velocities of single, skinned, fast and slow skeletal muscle fibers were measured at 5-6 degrees C in five animal species having a 25,000-fold range of body size (mouse, rat, rabbit, sheep, and cow). While fiber diameter and isometric force showed no dependence on animal body size, maximum shortening velocity in both fast and slow fibers and maximum power output in fast fibers were found to vary with the -1/8 power of body size. Maximum power output in slow fibers showed a slightly greater (-1/5 power) dependence on body size. The isometric force produced by the fibers was correlated (r = 0.74) inversely with fiber diameter. For all sizes of animal the average maximum velocity was 1.7 times faster in fast fibers than in slow fibers. The large difference in mechanical properties found between fibers from large and small animals suggests that properties of the contractile proteins vary in a systematic manner with the body size. These size-dependent changes can be used to study the correlations of structure and function of these proteins. Experimental results also suggest that the different metabolic rates observed in different sizes of animals could be accounted for, at least in part, by the difference in the properties of the contractile proteins.     

Mechanical demand and multijoint control during landing depend on orientation of the body segments relative to the reaction force.

The purpose of this study was to determine how diverse momentum conditions and anatomical orientation at contact influences mechanical loading and multijoint control of the reaction force during landings. Male collegiate gymnasts (n=6) performed competition style landings (n=3) of drop jumps, front saltos, and back saltos from a platform (0.72 m) onto landing mats (0.12 m). Kinematics (200 fps), reaction forces (800 Hz) and muscle activation patterns (surface EMG, 1600 Hz) of seven lower extremity muscles were collected simultaneously. Between-task differences in segment orientation relative to the reaction force contributed to significant between-task differences in knee and hip net joint moments (NJM) during the impact phase. During the stabilization phase, ankle, knee, and hip NJMs acted to control joint flexion. Between-task differences in muscle activation patterns indicated that gymnasts scaled biarticular muscle activation to accommodate for between-task differences in NJM after contact. Activation of muscles on both sides of the joint suggests that impedance like control was used to stabilize the joints and satisfy the mechanical demand imposed on the lower extremity. Between-subject differences in the set of muscles used to control total body center of mass (TBCM) trajectory and achieve lower extremity NJMs suggests that control of multijoint movements involving impact needs to incorporate mechanical objectives at both the total body and local level. The functional consequences of such a control structure may prove to be an asset to gymnasts, particularly when required to perform a variety of landing tasks under a variety of environmental constraints.     

Capturing the Dynamics of a Hybrid Multiscale Cancer Model with a Continuum Model

Cancer is a complex disease involving processes at spatial scales from subcellular, like cell signalling, to tissue scale, such as vascular network formation. A number of multiscale models have been developed to study the dynamics that emerge from the coupling between the intracellular, cellular and tissue scales. Here, we develop a continuum partial differential equation model to capture the dynamics of a particular multiscale model (a hybrid cellular automaton with discrete cells, diffusible factors and an explicit vascular network). The purpose is to test under which circumstances such a continuum model gives equivalent predictions to the original multiscale model, in the knowledge that the system details are known, and differences in model results can be explained in terms of model features (rather than unknown experimental confounding factors). The continuum model qualitatively replicates the dynamics from the multiscale model, with certain discrepancies observed owing to the differences in the modelling of certain processes. The continuum model admits travelling wave solutions for normal tissue growth and tumour invasion, with similar behaviour observed in the multiscale model. However, the continuum model enables us to analyse the spatially homogeneous steady states of the system, and hence to analyse these waves in more detail. We show that the tumour microenvironmental effects from the multiscale model mean that tumour invasion exhibits a so-called pushed wave when the carrying capacity for tumour cell proliferation is less than the total cell density at the tumour wave front. These pushed waves of tumour invasion propagate by triggering apoptosis of normal cells at the wave front. Otherwise, numerical evidence suggests that the wave speed can be predicted from linear analysis about the normal tissue steady state.     

Multi-scale modeling of a wound-healing cell migration assay

A continuum model and a discrete model are developed to capture the population-scale and cell-scale behavior in a wound-healing cell migration assay created from a scrape wound in a confluent cell monolayer. During wound closure, the cell population forms a sustained traveling wave, with close contact between cells behind the wavefront. Cells exhibit contact inhibition of migration and contact-limited proliferation. The continuum model includes the two dominant mechanisms and characteristics of cell migration and proliferation, using a cell diffusivity function that decreases with cell density and a logistic proliferative growth term. The discrete model arises naturally from the continuum model. Individual cells are simulated as continuous-time random walkers with nearest-neighbor transitions, together with a birth/death process. The migration and proliferation parameters are determined by analysing individual mice 3T3 fibroblast cell trajectories obtained during the development of a confluent cell monolayer and in a wound healing assay. The population-scale model successfully predicts the shape and speed of the traveling wave, while the discrete model is also successful in capturing the contact inhibition of migration effects.     

A model of flagellar movement based on cooperative dynamics of dynein-tubulin cross-bridges

A theoretical model based on molecular mechanisms of both dynein cross-bridges and radial spokes is used to study bend propagation by eukaryotic flagella. Though nine outer doublets are arranged within an axoneme, a simplified model with four doublets is constructed on the assumption that cross-bridges between two of the four doublets are opposed to those between the other two, corresponding to the geometric array of cross-bridges on the 6-9 and the 1-4 doublets in the axoneme. We also assume that external viscosity is zero, whereas internal viscosity is non-zero in order to reduce numerical complexity. For demonstrating flagellar movement, computer simulations are available by dividing a long flagellum into many straight segments. Considering the fact that dynein cross-bridge spacing is almost equal to attachment site spacing, we may use a localized cross-bridge distribution along attachment sites in each straight segment. Dynamics of cross-bridges are determined by a three-state model, and effects of radial spokes are represented by a periodic mechanical potential whose periodicity is considered to be a stroke distance of the radial spoke. First of all, we examine the model of a short segment to know basic properties of the system. Changing parameters relating to "activation" of cross-bridges, our model demonstrates various phenomena; for example "excitable properties with threshold phenomena" and "limit cycle oscillation". Here, "activation" and "inactivation" (i.e. switching mechanisms) between a pair of oppositely-directed cross-bridges are essential for generation of excitable or oscillatory properties. Next, the model for a flagellar segment is incorporated into a flagellum with a whole length to show bending movement. When excitable properties of cross-bridges, not oscillatory properties, are provided along the length of the flagellum and elastic links between filaments are presented at the base, then our model can demonstrate self-organization of bending waves as well as wave propagation without special feedback control by the curvature of the flagellum. Here, "cooperative interaction" between adjacent short segments, based on "cooperative dynamics" of cross-bridges, is important for wave propagation.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Simulation analysis of muscle activity changes with altered body orientations during pedaling

Testing hypotheses related to the effect of gravitational orientation on neural control mechanisms is difficult for most locomotor tasks, like walking, because body orientation with respect to gravity affects both sensorimotor control and task mechanics. To examine the mechanical effect of body orientation independently from changes in workload and posture, Brown et al. (J. Biomech. 29 p. 1349, 1996) studied pedaling at altered body orientations. They found that subjects pedaling at different orientations changed needlessly their muscle excitations, putatively to preserve body-upright pedaling kinematics. We tested the feasibility of this hypothesis using simulations based on a three biomechanical-function pair organization for control of lower limb muscles (limb extension/flexion pair, extension/flexion transition pair, and foot plantarflexion/dorsiflexion pair), where each pair consists of alternating agonistic/antagonistic muscles. Adjustment of only three parameters, one to scale the muscle excitations of each pair, was sufficient to preserve pedaling kinematics to altered body orientation. Because these adjustments produced changes in muscle excitation and net joint moments similar to those observed in pedaling subjects, the hypothesis is supported. Moreover, the effectiveness of a decoupled gain adjustment procedure where each parameter was adjusted by error in only one aspect of the pedaling trajectory during each iteration (i.e., cadence adjusted the Ext/Flex parameter; peak-to-peak variation in crank velocity over the cycle adjusted the transition parameter; average ankle angle over the cycle adjusted the foot parameter) further supports the distinct function of each muscle pair.     

Caenorhabditis elegans body mechanics are regulated by body wall muscle tone

Body mechanics in the nematode Caenorhabditis elegans are central to both mechanosensation and locomotion. Previous work revealed that the mechanics of the outer shell, rather than internal hydrostatic pressure, dominates stiffness. This shell is comprised of the cuticle and the body wall muscles, either of which could contribute to the body mechanics. Here, we tested the hypothesis that the muscles are an important contributor by modulating muscle tone using optogenetic and pharmacological tools, and measuring animal stiffness using piezoresistive microcantilevers. As a proxy for muscle tone, we measured changes in animal length under the same treatments. We found that treatments that induce muscle contraction generally resulted in body shortening and stiffening. Conversely, methods to relax the muscles more modestly increased length and decreased stiffness. The results support the idea that body wall muscle activation contributes significantly to and can modulate C. elegans body mechanics. Modulation of body stiffness would enable nematodes to tune locomotion or swimming gaits and may have implications in touch sensation.     

A computational model of visually guided locomotion in lamprey

This study addresses mechanisms for the generation and selection of visual behaviors in anamniotes. To demonstrate the function of these mechanisms, we have constructed an experimental platform where a simulated animal swims around in a virtual environment containing visually detectable objects. The simulated animal moves as a result of simulated mechanical forces between the water and its body. The undulations of the body are generated by contraction of simulated muscles attached to realistic body components. Muscles are driven by simulated motoneurons within networks of central pattern generators. Reticulospinal neurons, which drive the spinal pattern generators, are in turn driven directly and indirectly by visuomotor centers in the brainstem. The neural networks representing visuomotor centers receive sensory input from a simplified retina. The model also includes major components of the basal ganglia, as these are hypothesized to be key components in behavior selection. We have hypothesized that sensorimotor transformation in tectum and pretectum transforms the place-coded retinal information into rate-coded turning commands in the reticulospinal neurons via a recruitment network mimicking the layered structure of tectal areas. Via engagement of the basal ganglia, the system proves to be capable of selecting among several possible responses, even if exposed to conflicting stimuli. The anatomically based structure of the control system makes it possible to disconnect different neural components, yielding concrete predictions of how animals with corresponding lesions would behave. The model confirms that the neural networks identified in the lamprey are capable of responding appropriately to simple, multiple, and conflicting stimuli.     

Feeding Motor Patterns in Anurans: Insights from Biomechanical Modeling

During feeding in anurans, the mouth opens while the tongue,which is attached to the mandible at the front of the mouth,rotates forward. Due to the relative simplicity of its anatomyand the complexity of its motion, tongue protraction in frogspresents an ideal system for exploring the neural control ofmultijoint movements. In this study, we used a forward dynamic,rigid body model with four segments and two muscles to investigateopen loop control of tongue protraction in the Australian white-lippedtree frog, Litoria caerulea. Model parameters include the massdistribution, initial position and initial angular velocityof each segment and the anatomy and physiology of each muscle.Model variables include the level of muscle activation at eachtime step and impulsive torques to open and close the mouth.The model gives X,Y coordinates of each segment and joint anglesat each time step as output. The model was tested using scaled,normalized EMG signals and impulsive joint torques to predictthe paths of the lower jaw tip and tongue tip. Predicted pathswere compared to experimentally observed paths using Pearsonproduct-moment correlation coefficients. Simulations demonstratethat the genioglossus muscles likely play a minor role, if any,in determining the trajectory of the tongue in most anurans.Most of the force for tongue protraction comes from angularmomentum transferred to the tongue by the opening jaws. In anurans,tongue protraction is dynamically stable and will occur as longas the musculoskeletal elements are in the correct initial position.     

A single muscle's multifunctional control potential of body dynamics for postural control and running

A neuromechanical approach to control requires understanding how mechanics alters the potential of neural feedback to control body dynamics. Here, we rewrite activation of individual motor units of a behaving animal to mimic the effects of neural feedback without concomitant changes in other muscles. We target a putative control muscle in the cockroach, Blaberus discoidalis (L.), and simultaneously capture limb and body dynamics through high-speed videography and a micro-accelerometer backpack. We test four neuromechanical control hypotheses. We supported the hypothesis that mechanics linearly translates neural feedback into accelerations and rotations during static postural control. However, during running, the same neural feedback produced a nonlinear acceleration control potential restricted to the vertical plane. Using this, we reject the hypothesis from previous work that this muscle acts primarily to absorb energy from the body. The conversion of the control potential is paralleled by nonlinear changes in limb kinematics, supporting the hypothesis that significant mechanical feedback filters the graded neural feedback for running control. Finally, we insert the same neural feedback signal but at different phases in the dynamics. In this context, mechanical feedback enables turning by changing the timing and direction of the accelerations produced by the graded neural feedback.