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Introduction

Most fish swim by rhythmically passing neural waves of muscle activation from head to tail, alternating left and right. This yields travelling waves of local muscle shortening, which in turn produce travelling waves of body curvature. These mechanical waves interact with the water, developing reactive thrust that pushes the animal forward. Breder [1] divided this type of swimming into two classes, depending on the proportion of the body undergoing undulations. In the anguilliform mode, as exhibited by, e.g. lampreys and eels, most or all of the body is flexible and participates in the propulsive movement. In carangiform swimming, as exhibited by, e.g. mackerel, the amplitude of lateral motion is concentrated near the tail. See [2] for an overview of animal locomotion, and [3]–[5] for vertebrate swimming in particular.

At any point on the body, rhythmic cycles of muscle activation alternate with silence, causing cycles of muscle shortening and lengthening (see Figure 1A). However, in all species which have been studied [8] except the leopard shark [9], delays between the onsets of activation and of shortening increase along the body from head to tail (see Figure 1C), i.e., the wave of shortening travels more slowly than the wave of activation. In consequence, near the tail the greater portion of the activation phase occurs during muscle lengthening, giving rise to negative work during part of the cycle. There are a number of possible functions assigned to this change in timing (e.g., providing stiffness as the tail moves laterally through the water, thereby contributing to power transmission, or tuning the resonant body frequency to match tailbeats [10]), but the mechanism or mechanisms responsible for it are not known [11]. In this paper, we throw light on this phenomenon.

Figure 1. Relative timing of activation and movement.

(A) Passage toward the tail of the waves of activation (EMG) and curvature. Speed of activation wave (gradient of dotted line), 1.0 body lengths/cycle. Solid lines show curvature toward left side; speed of mechanical wave (gradient of solid line), 0.72 body lengths/cycle. Arrows indicate time periods over which the muscle on the left side is lengthening and shortening. Abscissa, time (cycles); ordinate, position on body. (B) Lamprey outline from above. Arrows show electrode placement on left side only; dashed lines show active region on each side at approx 0.6 cycles in panel A; (C) Phase delay (fraction of cycle) from onset of activation to onset of shortening (time of maximum convex curvature), plotted against position on body. Data replotted from [6],[7].

doi:10.1371/journal.pcbi.1000157.g001

Previous computational models of anguilliform swimming have incorporated the known timing of muscle activation within a mechanical representation of the body and water [12],[13], resulting in a travelling mechanical wave. In [13] no phase delay was seen between the waves of activation and curvature, and in [12], none was reported. However, both models assumed specific scalings of muscle density with body location, and that muscle force was simply proportional to activation. In reality, the force developed by activated muscle takes time to develop. Furthermore, because of the changing relative timing of activation and curvature, the patterns of muscle length and velocity vary significantly along the body length. This results in changing patterns in the developed muscle force, and such variation is further complicated by the body taper.

In the present study we investigate this phenomenon by incorporating a revised version of a kinetic muscle force model, originally due to Williams et al. [14], in the continuum mechanical model for anguilliform swimming of [13]. The resulting integrated neuromechanical system models the swimmer as an elastic rod with time-dependent preferred curvature arising from interactions of muscles with the body configuration. The model's modular structure—coupled sets of differential equations—allows us to selectively “lesion” it to probe the sources of its collective behavior. We find that the wave speed difference results primarily from the body's tapered geometry and passive viscoelastic damping, and that it does not require prioprioceptive sensory feedback. Depending on force density, the nonlinear dependence of force on muscle length and shortening velocity can also contribute to the wave speed difference, although it is not necessary for it. In a preliminary study, however, we find that length and velocity dependence can reduce the mechanical work output during swimming. When further coupled with a central pattern generator and motoneurons, this integrated muscle-body-enviroment model will also allow us to examine proprioceptive feedback, cf. [15].

This paper is organized as follows. In the methods section we review the equations of motion of the actuated rod and the fluid loading model. We show that the discretized rod equations are equivalent to equations describing a chain of interconnected links. This allows us to relate torques at the joints, and the forces responsible for them, to the preferred curvature and elastic properties of the rod. The model for muscle forces is developed in the penultimate subsection and in the final subsection we combine the muscle and body models to produce an integrated computational model. Simulations of the model are presented in Results and a discussion ensues in the concluding section, in which some larger implications of the work are noted.

Methods

An Integrated Model and Its Computational Realization

We model the swimmer's body as an isotropic, inextensible, unshearable, viscoelastic rod that obeys a linear constitutive relation and is subject to hydrodynamic body forces. We assume that passive material properties such as density and bending stiffness remain constant in time, but allow them to vary along the rod. We endow the rod with a time-dependent preferred curvature in the form of a traveling wave, representing muscular activations. We adopt the conventions of [16],[17], and use an elliptical cross section to compute hydrodynamic reaction forces, although we restrict to planar motions, since lampreys and eels in “normal” steady swimming flex their bodies primarily in the horizontal plane [18],[19]. The calcium kinetics and muscle force model, which produces the preferred curvature, is described in the penultimate subsection and the integrated model is summarized in the final subsection of this section. The material of the first three subsections below is drawn from [13], to which the reader should refer for further detail, and where the numerical method and validation tests are also described.

A Continuum Description of the Actuated Rod

The independent variable sЄ[0,l] denotes arc-length along the rod, and a configuration of the rod is given at each time t by the space curve s ↦ r(s,t) = (x(s,t),y(s,t)) describing its centerline in the inertial (x,y)-plane. Derivatives with respect to s and t will be denoted by subscripts. The inextensibility condition |∂r/∂s| = 1, can be written in terms of the angle φ between the tangent to the curve t = ∂r/∂s and the inertial x-axis:
(1)
see Figure 2. The normal to r is then given by n = (−sin φ, cos φ). Each element of the rod is subject to contact forces f = (f,g), a contact moment M, and body forces W = (W_x,W_y) per unit length, vector components again being referred to the inertial frame. The contact forces and moment are those exerted on the region (s,s+ds) by [0,s), which maintain the inextensibility constraint, and the body forces arise from interactions with the fluid environment.

Figure 2. A viscoelastic rod with elliptical cross-sections and variable semi-axes undergoes bending motions in the (x,y) inertial coordinate plane; s denotes arclength along the body centerline.

(Figure modified from Figure 1 in [13]).

doi:10.1371/journal.pcbi.1000157.g002

Balance of linear and angular momenta yields the equations of motion (cf. [17],[20]):
(2)

(3)

(4)
where ρ is the volumetric material density and A and I the cross-sectional area and moment of inertia of the rod. For an elliptical cross-section with semi-axes a and b, as in Figure 2, A = πab and the moment of inertia for motions in the (x,y)-plane is . We assume that ρ is constant, but allow A = A(s), I = I(s) to vary (both remaining strictly positive); specifically, we will study a tapered elliptical cross section based on lamprey body geometry.

In [13] the activation of the rod was determined by an externally-specified function κ(s,t), representing its intrinsic or preferred curvature. The muscle model developed later in this section effectively replaces κ with a function that depends on neural activation and the local curvature and its rate of change, but we retain the usual linear constitutive relation [20] so that moments are proportional to departures from preferred curvature:
(5)
Here E>0 and δ≥0 are the Young's modulus and viscoelastic damping coefficient and the flexural rigidity EI, with SI units N m², determines the overall stiffness. The equations of motion (Equations 2–4), the constraints (Equation 1), and the constitutive relation (Equation 5), along with specified body forces and suitable boundary and initial conditions, form a closed system of evolution equations. Natural boundary conditions for free swimming are that contact forces and moments vanish at the head and tail: M = f = g = 0 at s = 0,l.

Approximation of Hydrodynamic Reaction Forces

In swimming the local body forces are due to hydrodynamic reactions that depend on the global velocity field of the fluid relative to the body. To avoid the complexity and computational expense of solving coupled rod and Navier-Stokes equations, we adopt the model of G. I. Taylor [21] in which W(s,t) depends only on the local relative velocity. This approximation accurately predicts forces on a straight rod in steady flow, but fails to capture unsteady effects including vortex shedding, which are undoubtedly important in swimming propulsion [22],[23]. We believe that it suffices as a first approximation for the present purpose, since we are mainly concerned with the interaction of muscle forces and configuration dynamics. Unlike the Kirchhoff and Lighthill theories [24],[25], we neglect added mass effects. See [13] for further discussion.

Taylor models the force on a rod of radius a due to perpendicular flow of fluid of density ρ_f and dynamic viscosity μ with speed v as
(6)
where the drag coefficient C_N varies between 0.9 and 1.1 for Reynolds numbers 20<R<10⁵, and C_T is closely approximated by in the range 10<R<10⁵, cf. Figure 1 of [21]. Drag forces for smooth oblique cylinders can be decomposed into normal and tangential components in terms of the normal and tangential velocities v_⊥ and v_∥ at (s,t) as:
(7)
and the body forces are given by
(8)
where n and t denote the normal and tangential unit vectors to the rod's centerline at s.

In calculating W, we consider only the height 2a of the rod, assuming that fluid reaction forces are equal to those on a cylinder of radius a, although the constant C_N does change slightly for elliptical rods. Further, we set C_N = 1, since Reynolds numbers for lampreys and eels lie well within the range 20<Re<10⁵; for example, in their work on the eel Anguilla rostrata, Tytell and Lauder cite Re = 60,000 based on body length l = 20 cm for a specimen swimming at 1.4l/s. [22], and speeds reported in [23] range from 0.5 to 2 body lengths per second. In terms of Taylor's body-diameter-based Reynolds number, this corresponds to R≈2000–8000.

Discretization of the Actuated Rod

We discretize the rod equations with spatial step size h = l/N in the arclength variable s, letting x_i(t) = x(ih,t), i = 0, …, N, and similarly for the other field variables y_i,φ_i and parameters A_i,I_i: see Figure 3. The inextensibility constraints in Equation 1 are approximated by
(9)
and Equations 2–4 are approximated by the ordinary differential equations (ODEs):
(10)

(11)

(12)
where m_i = ρA_ih and J_i = ρI_ih. The constitutive relation in Equation 5 becomes:
(13)
The force and moment free boundary conditions M = f = g = 0 at s = 0,l become:
(14)

Figure 3. Representation of the swimmer as a chain of interconnected links.

doi:10.1371/journal.pcbi.1000157.g003

The finite-difference discretization of Equations 10–13 is closely related to representions of the body as a planar chain of rigid links subject to forces and moments. In modeling lamprey Bowtell and Williams [26],[27] take a chain of N massless rigid rods each of length h, with mass m_i at each pivot and at both free ends. The pivots are actuated by passive springs, dashpots, and active force generators. Ekeberg [12],[28] adopts a similar configuration but in place of time-dependent force generators, the spring constants vary with time, and instead of point masses at the pivots, the center of mass of each link is placed at its midpoint. Here we adopt the mass distribution of [12], and include active muscle elements, to be described in succeeding subsections, in the force-generating components. The configuration of the ith link is described by its midpoint (x_i,y_i) and the angle φ_i between its centerline and the inertial basis vector ê_x (Figure 3). Equaions 9 then express the constraint that links remain connected at the joints. Letting (f_i,g_i) and M_i denote the components of contact force and the torque at the joint connecting link i to link i+1 and (hW_xi,hW_yi) be the body force acting on the midpoint of link i (Figure 4a), balances of linear and angular momenta yield Equations 10–12 above with mass m_i = ρA_ih and moment of inertia of the ith link. The discrepancy between the discretized rod equations and the equations for the chain of N pivoted rods thus consists only in the terms in the moments of inertia, and the two models coincide in the limit h → 0. We employ the exact formula above for the moments of inertia J_i in all the calculations below, although the approximation J_i = ρhI_i yields results (not shown) that are nearly identical, even for quite large values of h≈1.

Figure 4. Forces and moments acting on link i (A), bending moments are determined by muscles on both sides of the body modeled by springs and dashpots with additional active elements (B), and forces and moments associated with a single joint (C).

doi:10.1371/journal.pcbi.1000157.g004

As shown in section 4.3.4 of [13], for the large segment numbers typical of eels and lampreys, the behaviors of the discrete and continuum models are very close. Additionally, the discretization reveals how activation determines preferred curvature κ(s,t) and affects bending stiffness EI of the continuum model. As in [26], the joint connecting each pair of links of length h is actuated by a pair of spring-dashpot-actuators in parallel, with spring constant ν and damping coefficient γ, anchored to arms of length w that project normally from the links' midpoints (Figure 4b). These arms represent myosepta, the connective tissue layers to which the muscle fibres connect. The linear springs and dashpots represent passive tissue viscoelasticity, and the actuators generate prescribed contractile muscle forces f_Li and f_Ri on the right and left sides of the body respectively. Suppressing the dependence on i and denoting the relative extensions and of the spring-dashpot-actuators as Δ_R and Δ_L (Figure 4c), the total forces on the right and left sides may be written
(15)
Since the relative extensions are dimensionless, stiffness ν and damping γ have the units N and N s. respectively. The springs are in tension (and hence generating contractile forces) when Δ_R, Δ_L>0. The forces are applied at a distance w from the centerline of the rod, so elementary trignometry gives:
(16)
where ψ_i = φ_i+1−φ_i is the angle between neighboring links and . Finally, computing the moment arms L_R,L_L to the joint along normals from the lines AB and CD on which the forces act (Figure 4c):
(17)
we find that, for small angles ψ_i, the resulting torque at joint i is given by
(18)

Comparing the linearized moment in Equation 18 in the limit h → 0 with the discretized constitutive relation in Equation 13 we see that the link and discretized rod models coincide if the stiffness EI_i, intrinsic curvature κ_i and viscoelastic damping δ are interpreted as follows:
(19)

We propose that the stiffness ν and damping γ are proportional to cross-sectional area A(s). Thus we set
(20)
so that the stiffness and damping have units N/m² and N s./m² respectively. To approximate a uniform distribution of the muscle, we set w = b/2, where b is the half-width of the body. Equations 19 now become
(21)
In particular, using I = πab³/4 we can write Young's modulus in terms of the spring stiffness as . One of the questions we address is the influence of force density as a function of arclength. We take up this question after a discussion of force generation in muscle fibers.

Muscle Activation and Force Generation

Recordings such as those of [29] show that waves of motoneuronal activity consisting of bursts of closely-spaced action potentials (APs), separated by near-silent interburst periods, travel the length of the lamprey spinal cord (see Figure 1A and 1B). The waves are generated spontaneously by a distributed central pattern generator (CPG) within the spinal cord [30], which has been modelled as a chain of coupled oscillators [31]–[33]. The waves are in antiphase contralaterally and maintain approximately constant duty cycles (burst/cycle period ratios) and segment-to-segment ipsilateral phase lags, regardless of overall frequency. This activity pattern is transmitted via nerves that enter the myotomes through the ventral roots [34], producing muscle activation with similar phasing, evident in electromyograms (EMGs) [7]. Each myotome corresponds to a segment of the spinal cord.

Bundles of myofibrils make up the muscle fibres within the myotomes. The AP bursts cause calcium release from the sarcoplasmatic reticulum (SR) that surrounds the myofibrils and is encircled by T-tubuli at repeated intervals. The resulting muscle contraction occurs in three phases. (i) A motoneuronal AP arrives at the neuromuscular junction, producing an AP at the motor end plate which spreads along the surface and T-tubular membranes of the muscle fiber. (ii) This depolarization opens gates in the SR and releases Ca²⁺ ions into the muscle protein filaments. (iii) Ca²⁺ causes conformational changes in the thick filaments which form cross-bridges to the thin filaments; a subsequent conformational change then develops a force tending to slide the thin filaments over the thick ones [35], shortening the muscle (unless overcome by opposing force via the muscle attachments). This is followed by resequestering of Ca²⁺ by the SR, resulting in relaxation of the muscle. The force developed during muscle activation is dependent upon both the length of the muscle and the velocity of its shortening [36]. Traditionally, shortening is taken as positive, but here we use the opposite convention, referring to the time derivative of muscle length as velocity, which is negative for shortening.

To describe the forces f_R(t) and f_L(t) in Equations 15, 18, and 19, we adapt the model developed by Williams et al., who carried out experiments on portions of single myotomes of lamprey muscle [14]. Intermittent tetanic stimulation was applied during isometric and constant-velocity movements, and analysis and modelling of the resulting force trajectories were used to predict the trajectories recorded during applied sinusoidal movement. Experimental data are reproduced in Figure 5 below (for details of experimental protocol, see [14]). We follow a modified form of the simple kinetic model used in that study, including calcium ions, SR sites and contractile filaments (CF). The rates at which calcium ions are bound and released approximately follows the principle of mass action (see Figure 6). For example, the rate of binding of calcium ions to the CF is proportional to the product of concentrations of free calcium ions and unbound filaments, with rate constant k₃. The resulting equations for the kinetics of the calcium, sarcoplasmic reticulum sites and bound filaments are as follows:
(22)

(23)

(24)

(25)

(26)
where brackets denote concentrations of the relevant quantity. When the muscle is activated, k₁>0 and k₂ = 0; in the absence of activation k₁ = 0 and k₂>0.

Figure 5. Least squares fit of model to isometric data at three muscle lengths (left), and sinusoidal forcing data with predictions from isometric data fit (right).

Tetanic stimuli applied periodically for 0.36 s, while length of preparation varies sinusoidally with various phase offsets φ. The sine waves show the length of the preparation as a function of time. Solid lines, simulation; dashed lines, data.

doi:10.1371/journal.pcbi.1000157.g005

Figure 6. The model of calcium kinetics.

c, free calcium ion; s, unbound SR calcium-binding sites; cs, calcium-bound SR sites; f, unbound contractile filament calcium-binding sites; cf, calcium-bound filament sites; k₁–k₄, rate constants of binding and release.

doi:10.1371/journal.pcbi.1000157.g006

We assume that the total number of calcium ions, SR binding sites and filament binding sites per liter remain constant so that [cs]+[c]+[cf] = C_T, [cs]+[s] = S_T, and [cf]+[f] = F_T. This allows us to reduce the five Equations 22–26 to a system of two in [c] and [cf]. We further scale by the number of filament sites F_T, writing Caf = [cf/F_T], Ca = [c]/F_T and introducing the new constants C = C_T/F_T and S = S_T/F_T. Since the number of bound filament sites cannot exceed F_T, Caf≤1, Ca≤C, and Caf = 1 when all of the filaments are bound. Although appropriate values for C and S are not known, general knowledge of skeletal muscle indicates that C is large enough for the filament binding sites to be saturated during tetanic stimulation and that S is large enough to reduce free calcium to a negligible amount during rest. We obtain similar data fits over a range of values for these constants, so we arbitrarily set C = 2 and S = 6. Thus twice as much calcium is available than is necessary to bind all of the filaments and thrice as many binding sites are available in the SR than are required to bind all the calcium.

Following Hill [37], each myotome is modeled as a contractile element (CE) in series with an elastic element (SE). (The Hill model includes a second elastic element in parallel [38], but for our purposes this can be included in the linear spring of Figure 4b.) Because they are in series, the CE and SE experience equal forces at steady-state. We begin by describing them separately, as a force P exerted by the SE, and a force P_c developed by the active element CE.

The SE is modelled as a linear spring and hence P is proportional to the length l_s of this element minus its resting length l_s0: P = μ_s(l_s−l_s0). This force is never negative. The total length L of the segment is the sum of l_s and the length l_c of the contractile element. The length and velocity v_c = l ˙_c of the contractile element are therefore given in terms of the length and velocity V = L ˙ of the segment and the force P as follows:
(27)

(28)

We assume that the the force P_c exerted by the contractile element can be described by independent multiplicative factors of its length l_c and velocity v_c,
(29)
where the constant P₀ is the force exerted in isometric tetanic contraction (Caf = 1) at the optimum length l_c0. The functions λ(l_c) and α(v_c) are estimated from force measurements (described below), from which we obtain a piecewise linear function for α and a quadratic for λ:
(30)

(31)
We additionally restrict these functions such that 0≤α(v_c)≤α_max and 0≤λ(l_c)≤1. The fact that α_p>α_m>0 (see Table 1) reflects the ability of muscle fibers to exert progressively greater forces during lengthening than in shortening.

Table 1. Muscle model parameters used in simulations.

doi:10.1371/journal.pcbi.1000157.t001

Table 2. List of major symbols.

doi:10.1371/journal.pcbi.1000157.t002

If we set P_c = P, the calculation suffers from instability, and in reality the stretch of the SE due to activation of the CE is not instantaneous. We therefore model the transfer of force from the CE to the SE by simple linear kinetics:
(32)

Combining Equations 22–32 and using the three conserved quantities C_T, S_T, and F_T, we obtain three ODEs for the concentrations of free calcium, bound calcium and the force exerted by the preparation:
(33)

(34)

(35)

The parameters of the model are determined from analysis of the data of [14], as follows. μ_s and l_s0 are determined from quick-release experiments [37]. The maximum values of force P₀ in the three isometric experiments (Figure 5) are used in Equations 27, 29, and 31 to determine the values of λ₂ and l_c0. The results of constant-velocity ramp experiments are then used with Equations 27–31 and the parameters λ₂ and l_c0 to determine α_m and α_p. The limiting value of α_max was not determined in [14], so α_max is taken from results in dogfish [39]. In practice, results vary little over a range of values for α_max.

We set the time constant k₅ = 100 s⁻¹, so that P_c closely tracks P. The remaining time constants k₁, k₂, k₃, and k₄ are found by fitting force trajectories from the experimental data, using the least-squares curve-fitting facilities in the software XPPAUT devised by G. Bard Ermentrout and available at http://www.pitt.edu/phase/.

The parameters k₁–k₄ are fit in two different ways. The isometric fit follows the approach in [14] by using only data from the isometric experiments at the three lengths L = 2.7 and 2.7±0.125. The main aim of [14] was to show that a model based on isometric and constant-velocity experiments could be used to approximately predict forces that occur during swimming, even though it excludes known properties such as the observation that the length-tension and force-velocity relationships change during muscle activation and relaxation [36]. Such secondary features cause discrepancies between the predictions and the data seen in the sinusoidal traces of Figure 5, but the model nonetheless produces forces during sinusoidal movement that capture the overall behavior well.

The present study demands our best estimate of force development during swimming, and for this reason we have made a second, dynamic fit of the time constants k₁–k₄ based not on isometric data but on muscle force data during sinusoidal movement at 1 Hz. To best match swimming behavior, we chose the experiment with a delay of 0.1 from onset of stimulation to onset of shortening (cf. Figure 1), and as the upper panels of Figure 7 show, the resulting force trace is much closer to the data than the fit to isometric data. The discrepancy between the isometric data and the prediction using these parameters is primarily in the repolarisation phase (Figure 7, lower panel), reflecting the model's inadequacy during this phase of the force trajectory. Values for both fits, along with the other muscle parameters, are given in Table 1. The most striking difference is in the rate constant k₂ (uptake of free Ca²⁺ by the SR), which doubles. Using this, the dynamic fit captures the rapid force decay seen in the sinusoidal data at low phase delays.

Figure 7. Fit of model to sinusoidal forcing data.

The upper left panel is the same as that in Figure 5 for phase offset φ = 0.1; the upper right panel shows the model behavior under the same conditions but with rate parameters k₁–k₄ fit to the sinusoidal data. Lower panel shows the model behaviour under isometric conditions with these parameters. Solid lines, simulation; dashed lines, data.

doi:10.1371/journal.pcbi.1000157.g007

Sinusoidal forcing data were only available at 1 Hz [14] and in most of the simulations described below we retain this frequency, but we also briefly investigate swimming behavior at 2 Hz. The muscle parameters are listed in Table 1. It is worth noting that neither set of time constants is unique: in both cases it was possible to find more than one set of time constants that gave a good fit, by starting from different initial guesses. The primary goal of this study is not to discover accurate parameters, but to find a good prediction of muscle behaviour for use in our neuromechanical model.

The Integrated Model

Muscle dynamics is incorporated into the discretized rod model as follows. The forces P_Ri and P_Li generated by the right and left myotomes associated with the ith link ar