Kinematics.

X-ray videos were undistorted, calibrated and digitized using XMALab 1.5.4 [87], which also computed rigid body motions for the bones where possible (i.e., when a bone had three beads implanted in it). In addition, the tip of the ungual of digit III from both limbs was digitized to grossly capture movement at the metatarsophalangeal (MTP) joints. The limited experimental data collected, in terms of both number of trials and number of beads, did not permit an assessment of tracking accuracy or precision [86] for the current dataset. However, in a separate cadaveric study of tinamous (using different individuals and bead placements) a much larger dataset of bone motions was obtained using the same XROMM system as above. Across the 49 trials in this dataset there were 358 pairwise co-osseous intermarker distances recorded over a total of 149,795 frames; the known distances between these markers (measured in the cadaveric CT scans) was used to determine the accuracy of the experimental setup and digitization procedure. In terms of accuracy, there was a mean relative error of 0.1259% (mean absolute error of 0.08146 mm), and in terms of precision, there was a mean standard deviation of 0.06839 mm. Although it remains uncertain how strictly these results translate to the kinematic data of the present study, they at least give a qualitative indication of acceptable levels of accuracy and precision.

From the digitized kinematic data and recorded forceplate data, two trials were identified to be the focus of the inverse simulations on the basis of cleanness of the forceplate data and completeness of kinematic data. Moreover, these trials spanned practically the entire range of speeds that was able to be elicited from the birds–a slow walk (DDT09, 0.39 m/s or a relative speed [90] of approximately 0.32, duty factor 0.71, overground) and a grounded run (DDT04, 1.39 m/s or a relative speed of approximately 1.15, duty factor 0.57, treadmill)–and they therefore capture the full range of variation in locomotor behaviour that was elicited in the birds.

Bone motions in these focal trials were reconstructed from the digitized bead trajectories and rigid body motion output using a combination of markerless and maker-based rotoscoping [85] in Maya 2018 (Autodesk, San Rafael, USA). This involved first generating the geometries of the bones, as well as the location of the XROMM beads in them, which was accomplished via a combination of manual and automatic segmentation of the carcass CT scans using Mimics 20.0 (Materialise NV, Leuven, Belgium), followed by mesh refinement in 3-Matic 12.0 (Materialise NV, Leuven, Belgium). Rotoscoped bone motions in the global (laboratory) coordinate system were then expressed in terms of physiologically meaningful joint movements using a series of joint coordinate systems, following the approach and convention of Kambic et al. [91]. Briefly, geometric primitives (e.g., spheres, cylinders) were fitted to the articular surfaces of bones using MATLAB code previously described by Bishop et al. [92], and used to mathematically derive anatomical coordinate systems for the proximal and distal ends of each bone. These were then united to produce a joint coordinate system for each joint, using the ‘jointAxes’ script in the XROMM_MayaTools package (available at www.xromm.org), which computed joint translations and rotations. The fibula and tibia were not permitted any relative motion between them, rather moving as a single entity. This study follows the previously used convention of expressing joint rotations as three intrinsic Euler rotations, measured as flexion–extension, followed by abduction–adduction, followed by long-axis rotation [91,93]. Due to the small size of the calibrated X-ray volume in relation to the size of the birds, the pelvis was not fully visible in both X-ray videos throughout the entirety of the trials. As such, its position and orientation were estimated from the reconstructed positions of the femora, wherein the centres of the femoral heads were used as ‘virtual points’ to help constrain the pelvis’s disposition by assuming that these corresponded as closely as possible to the centres of the acetabulae [cf. 91,94].

An additional consequence of the small size of the calibrated X-ray volume was that neither of the focal trials’ data comprised an unbroken kinematic sequence covering a full stride cycle. Missing sections of the kinematic sequences were filled in by making a composite sequence from corresponding sections recorded elsewhere, using instances of foot touchdown and liftoff (i.e., stance and swing) as points by which to bring different strides to a common timescale. The approach inherently involves a level of subjectivity, but the overarching philosophy was to synthesize a complete stride cycle that as faithfully as possible reflected the kinematics of the focal limb or stride involved. A hierarchy of importance was followed in terms of how missing data were sourced. If possible, gaps in the focal limb’s kinematics were first filled in using kinematics from the same limb in an earlier or later stride. Then, gaps were filled using kinematics from the contralateral limb in the same trial; this assumed bilateral symmetry in limb movements, which for the trials involved was supported by a comparison of duty factors and phase offsets between left and right limbs [but see 95]. Lastly, gaps were filled in using kinematics from the focal limb, but from a separate trial (or trials) of as similar speed as possible. For the treadmill trial, which encompassed three strides, a missing section of data for a given joint was filled in using data for that joint in a different stride (via cubic spline interpolation in MATLAB), preferably the same joint but otherwise that of the contralateral limb. The overground trial encompassed less than a single stride within the field of view of the X-ray cameras, and missing sections of data for one joint were filled in using data from the other joint, again using cubic spline interpolation. However, some small gaps remained in the kinematic sequence, and these were filled in using data from a separate treadmill trial involving the same bird using a similar slow walk (~0.45 m/s). The completed kinematic sequences were then filtered with a fourth-order, zero-lag, low-pass Butterworth filter; for the walking trial a cutoff frequency of 20 Hz was used, whereas a cutoff frequency of 30 Hz was used for the running trial. Lastly, for the running trial the tread speed was added to the forward translational coordinate of the pelvis to emulate overground running. Ultimately, slightly more than one whole stride was reconstructed for both walking and running. The approach used to derive whole-stride kinematics is admittedly not ideal, and implicitly assumes that kinematics obtained from overground and treadmill setups are equivalent; the observations of Jacobson and Hollyday [96] suggest that this is broadly the case, but more detailed studies may detect consistent finer-level differences [e.g., 97,98]. Nevertheless, the resulting kinematic sequences were qualitatively highly comparable to those previously reported for other small- to medium-sized ground birds, including Eudromia elegans, moving at similar relative speeds [26,95,99].

Kinetics.

As the running trial occurred on a treadmill, no ground reaction data were collected; see below for how this was estimated and used in the simulations. The walking trial fortuitously involved each foot landing cleanly and wholly on adjacent forceplates, such that complete force and moment profiles were recorded. The raw signal from the forceplates was calibrated, baselined and filtered using a custom script in MATLAB (v9.5; MathWorks, Natick, USA), which computed forces, moments and the instantaneous centre of pressure; filtering used a fourth-order, zero-lag, low-pass Butterworth filter, with a cutoff frequency of 15 Hz. A lower cutoff frequency was used here compared to the kinematic data due to differences in the quality of the two data sets; for example, the forceplate data tended to contain more noise, and so a more aggressive filtering was chosen to conservatively keep only the major aspects of data variation. The cutoff frequencies used in the present study are similar to those used in previous studies of avian terrestrial locomotion [28]. However, in future studies that use greater sample sizes than used here, and use higher-quality experimental data, a single consistent cutoff frequency would likely be desirable [100]. Reconstructing the complete GRF and GRM profiles through time for the whole stride involved reconstructing GRFs and GRMs for the period of double stance at the beginning and end of the trials, periods for which forces and moments of only one foot were recorded. This was achieved by duplicating the GRF and GRM profile for the foot involved and shifting it forward or backward in time as appropriate so as to correspond with instances of foot touchdown and liftoff (estimated using equation S4 of [28]). The centre of pressure of these unmeasured footfalls was estimated from the position of the relevant foot involved, using the reconstructed kinematic sequence derived above.

Skeletal and joint definitions.

A high-fidelity 3-D musculoskeletal model of the pelvis and hindlimbs was developed for use in OpenSim 3.3 [101,102] (Fig 2 and see S1 Model File). It was based principally on individual DDT09, with data on muscle lines of action and segment mass properties drawn from two additional individuals (Table 1). The skeletal geometry and joint definitions of the model were identical to those used above in deriving locomotor kinematics, wherein a bilaterally symmetrical, rigged hierarchical marionette [85] was created in Maya and then transcribed to the OpenSim modelling environment as per the workflow of Bishop et al. [92]. Digits II–IV were modelled together as a single segment. For computational simplicity, joints in the OpenSim model were permitted rotation only (no translations), and moreover the MTP joint was permitted rotation about the flexion–extension axis only. Therefore, each limb comprised 10 degrees of freedom (DOFs), with three each at the hip, knee and ankle, in addition to six describing the location and orientation of the pelvis and trunk segment in the global coordinate system. In the XROMM joint convention followed here, the neutral pose of the joint coordinate systems (i.e., where all joint translations and rotations are zero) involves the bones collapsed into a highly non-physiological concertina (Fig 2B; see also [91]). The neutral pose of the musculoskeletal model was set differently to this by incorporating a fixed offset in all flexion–extension axes (hip extension +90°, knee extension +180°, ankle extension +180°, MTP angle + 90°), such that the limb was vertically straightened with the digits segment parallel to the ground plane (Fig 2A). By not accounting for joint translations in the model, this inevitably resulted in a degree of interpenetration between adjoining bones. As such, fixed translational offsets were added to definitions of the knee (5 mm), ankle (4 mm) and MTP joints (3.5 mm) in order to avoid interpenetration. The displacements were applied only along the proximodistal axis of the ‘child’ body (e.g., the tarsometatarsus for the ankle joint), and the magnitude of displacements was based on the joint translations measured in the in vivo kinematic sequences.

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Fig 2. Musculoskeletal model of the tinamou.

(A) Rigid body mechanics component of the model, shown in the neutral pose used in this study; the green and black sphere denotes the whole-body centre of mass in this pose. Scale bar refers to this panel. (B) Skeleton and joints shown collapsed in the XROMM neutral pose [91], which differs from the neutral pose of the musculoskeletal model by offsets of 90° or 180° about the flexion–extension axes. In both A and B, anatomical coordinate systems used to define joint coordinate systems are illustrated; blue (z-axis) corresponds to flexion–extension, green (y-axis) corresponds to abduction–adduction and red (x-axis) corresponds to long-axis rotation. (C, D) The 36 muscle–tendon actuators for the right leg, shown for an arbitrary standing posture in lateral (C) and anterior (D) views. (E–G) Close-ups on different parts of the model to show the actuator paths used to model muscle–tendon units, along with various wrapping surfaces (blue and purple geometries) used to help constrain these paths. (H) A single contact sphere was applied to the digits (terminal) segment of each limb, with radius and location as shown.

https://doi.org/10.1371/journal.pcbi.1008843.g002

Segment mass properties. To derive the mass properties (mass, centre of mass [COM] location and inertial tensor) of each segment in the musculoskeletal model, the intact carcass of a separate tinamou individual (DDT07) was CT scanned (Toshiba Aquilion ONE, 120 kV peak tube voltage, 75 mAs exposure, 500 ms exposure time, 0.25 mm slice thickness, 0.427 mm pixel resolution) in a roughly ‘neutral’ and symmetric standing position with the wings folded and the neck and head directed cranially and straight out from the trunk. It was then disarticulated to produce the major segments, which were massed and then re-scanned (same machine settings as above, but with 0.407 mm pixel resolution) to digitally measure segment volume and in turn calculate segment-specific densities (reported in S1 Table). The geometries of flesh and bones in the intact carcass CT scans were segmented using a combination of manual and automated techniques in Mimics, and the flesh segments were scaled and reoriented to the bones of the DDT09 Maya marionette via affine transformation, using the point registration tool in CloudCompare 2.5.4 (http://www.cloudcompare.org/). Two flesh geometries (trunk + wings, neck + head) were modelled for the ‘trunk’ body in the model, and flesh segments of the left hindlimb were mirrored for the right hindlimb, to maintain bilateral symmetry. The scaled flesh geometries were then used to compute segment mass properties as per [103,104], incorporating segment-specific densities, as part of the above process of transcribing the Maya model to the OpenSim environment. To ensure bilateral symmetry in the model, the mediolateral position of the trunk body COM was set to lie perfectly on the body midline.

Muscle–tendon unit paths.

The paths of each muscle–tendon unit (MTU) were reconstructed in OpenSim with reference to anatomical dissection of DDT09, supplemented with observations made of other tinamou dissections and comparison to Suzuki et al. [105]. These paths were constrained to follow realistic lines of action using a combination of via points and cylindrical, ellipsoidal and toroidal wrapping surfaces (Fig 2E–2G, [106–108]), and only the minimum number of these required to achieve realistic paths were used for each muscle. To provide additional clarification, the 3-D geometries of muscles in the right hindlimb of a separate specimen (DDT03) were obtained through iodine staining [109] (Lugol’s solution, 4% iodine with 10% neutral-buffered formalin, stained for 3.5 months with solution refreshed every two to three weeks) and micro-CT scanning (Nikon XTEK XTH 225 ST, 190 kV peak tube voltage, 245 mA tube current, 1,000 ms exposure time, 0.0907 mm isotropic voxel resolution), followed by manual and automated segmentation in Amira 2019.2 (ThermoFisher Scientific Inc., Waltham, USA). This separated the muscle bellies of almost all muscles that crossed the hip and knee joints. The geometries of these bellies were then imported into previously published MATLAB code [48] to determine lines of action, taken as the path passing through the centroid of 50 slices running transversely across the 3-D muscle volume. The bone geometries and muscle paths were linearly scaled to the size of DDT09 and imported into Maya, wherein the marionette of DDT09 was rotoscoped to match the in situ posture of the DDT03 cadaver. Once the bones were aligned between the two birds, the muscle paths of DDT03 were transferred to the bones of DDT09 and subsequently followed these bones through the process of OpenSim model generation. Thus, the muscle paths in their correct spatial relationships to the bones were directly visible in the OpenSim graphical interface, enabling more detailed refinement of MTU paths. Once the left limb’s MTU paths were completely specified, the whole limb (bones, muscles, joints and mass properties) was mirrored about the sagittal plane to produce a perfectly symmetric model.

A total of 36 MTU actuators representing 34 separate muscles–all the important muscles of the hindlimb insofar as locomotion is concerned–were included in each limb of the musculoskeletal model (Fig 2C and 2D and Table 2; see also S1 Fig). As with Hudson et al. [110], but in contrast to Suzuki et al. [105], a caudofemoralis pars caudalis was not observed and so was not modelled. Either this muscle was genuinely lacking in the current study’s specimen, or it was so small as to be indistinct from surrounding muscles (principally, the caudofemoralis pars pelvica); in the latter scenario its (very small) mass and force-generating capacity would have been incorporated into that of surrounding muscles. A similar situation applies to the fibularis brevis, which was not observed in the present study and so not modelled. On account of their broad area of origin on the ilium, the iliotibialis lateralis pars postacetabularis (ILPO) and iliotrochantericus caudalis (ITCa) were modelled with two MTUs. Two very small crural muscles, the popliteus and the plantaris, were not modeled here as their contributions to locomotion were deemed minimal enough to be safely ignored; moreover, the popliteus spans the tibiotarsus and fibula, which were immobile relative to one another in the model. The small intrinsic muscles of the pes (e.g., abductors, extensores proprii, lumbricales) were also ignored, largely due to the modelling simplification of treating all digits as a single segment. In a similar fashion to previous avian models [49,111], both parts of the flexor cruris lateralis, partes pelvica (FCLP) and accessoria (FCLA), were modelled, following the same line of action proximally towards the pelvis but diverging distally towards the tibiotarsus and femur, respectively.

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Table 2. Muscle–tendon units included in the musculoskeletal model.

The original architectural parameters as derived from anatomical dissection are listed (values for fibre length and pennation angle are means of multiple measurements), and tuned fibre and slack lengths are given in parentheses. The mass and maximal isometric force for the ILPO and ITC was split evenly between two actuators.

https://doi.org/10.1371/journal.pcbi.1008843.t002

Muscle–tendon unit parameterization.

Muscle architectural data for DDT09 (Table 2) were measured during dissection following standard protocols [e.g., 111,112], and included muscle belly mass, resting fibre length and pennation angle (α_o). The measurements obtained were found to be comparable to that measured in a second set of tinamous as part of a separate study (A.R. Cuff et al., in prep.). Architectural data were then used to estimate each muscle’s maximal isometric force as (1) where m is muscle belly mass, σ is the maximum stress developed in the fibres and ρ is muscle tissue density; this assumes that the measured resting fibre length was equivalent to ℓ_o in subsequent use of a Hill-type model of contraction dynamics [5]. The values used for σ and ρ were taken as standard for vertebrate skeletal muscle: 300,000 N/m² [45,64,74,113] and 1060 kg/m³ [111,114], respectively. Note that the calculated F_max does not yet factor in the effect of α_o, for this is taken into account in the geometric underpinnings of the Hill-type formulation used in the simulations outlined below; that is, α_o is explicitly accounted for in the maintenance of constant muscle thickness [5,7,50,115]. The values thus obtained for F_max, ℓ_o and α_o were assigned directly to the corresponding MTUs of the model. An exception was the ILPO and ITCa, since these were modelled with two MTUs; in both cases, mass and F_max obtained for these muscles was distributed equally between their respective MTUs, but ℓ_o and α_o remained unaltered from the original measurements. In addition to architectural measurements of muscle bellies, the lengths and masses of the corresponding tendons were also measured, for estimation of tendon stiffness in the inverse and static simulations (below). Following assignment of architectural properties and definition of MTU paths, tendon slack length (L_S) for each MTU was estimated using the approach of Manal and Buchanan [116]. This parameter denotes the length at which the in-series tendon is slack (i.e., resting length), which in practice is difficult to determine empirically due to the complicating effect of internal tendon or aponeuroses. The calculation of tendon slack length was implemented assuming that muscle fibres had the capability to range in length from 0.5–1.5× ℓ_o across the limb’s entire range of motion. Whilst the underlying model of muscle contraction used in the procedure of Manal and Buchanan [116] is different to that used in the inverse and static simulations, it should be noted that this was only used to obtain a first approximation of L_S, since this was subject to modification in the inverse simulations (below). Maximal fibre contraction velocity was set to 10× ℓ_o/second.

Residual reduction algorithm.

The above dynamic inconsistency was rectified by first applying OpenSim’s residual reduction algorithm to the walking trial data, using a two-pass process [101,117]. In the first pass, the algorithm was run with high tracking weights on the input kinematics and was allowed to modify the mass and COM location for the most massive segment in the body (trunk) to improve the match between kinematics and kinetics. This resulted in the trunk COM being shifted 5 mm caudally and 5 mm ventrally in the pelvis coordinate system (equivalent to ~6% of glenoacetabular distance), and the total body mass being increased by 2.1%, which was applied equally to all body segments; this scaling factor was also applied to the inertial tensor of each body segment as well, but COM location remained unaltered (effectively, density was increased, but density distribution remained the same). In the second pass, the adjusted model was then used with the original kinematic and kinetic data, with low tracking weights on the kinematics to allow the algorithm to modify the kinematics so as to reduce dynamic inconsistency (i.e., the magnitude of force and moment residuals at the ground–pelvis joint). Despite this procedure, the resulting residuals remained unacceptably high, sometimes exceeding double the threshold suggested by Hicks et al. [118] (particularly for moments), necessitating additional modification to the input data of the model.

Tracking simulations.

To achieve an acceptable level of dynamic consistency between the model, kinematics and kinetics, a tracking simulation of walking was performed using a direct collocation approach [119,120]. Here, the rigid body dynamics component of the musculoskeletal model (with modified segment mass properties derived from the residual reduction step above) was transcribed to an algorithmically differentiable C++ source file, and called as an external function within a custom MATLAB script that formulated the tracking simulation as an optimal control problem (see S1 Code). The model was driven by torque actuators acting at each joint, and was constrained to move at the same average speed as in the trial, whilst tracking as closely as possible the recorded kinematics and GRFs: (2) for L torque actuator activations a_torque (= 20 in the current model), N pairwise deviations between model coordinates in the simulation q_sim and the kinematic data q_exp (= 26 in the current model), K pairwise deviations between each GRF component for the right and left feet in the simulation R_sim and kinetic data R_exp (= 6 in the current model), and N model coordinate accelerations. Due to differences between the contact model used (involving GRMs in all three axes; see below) and experimental data (only the vertical component of the GRM is non-zero), the simulation was not encouraged to track the experimentally recorded GRMs. The values for activation and acceleration weighting terms were low (w₁ = w₄ = 0.1) whilst the weightings on the tracking terms were set to high values (w₂ = 200, w₃ = 10) to ensure that the solution maintained high fidelity to the original experimental data. The torque actuators were modelled with excitation–activation dynamics (basic first-order differential equation, approximating time delay) to produce a smoother solution for joint moments. To simulate GRFs and GRMs in the problem, a single contact sphere was applied at the digits segment (Fig 2H), and foot–ground contact was modeled using a smoothed implementation of OpenSim’s Hunt–Crossley formulation [119–121], with contact stiffness set to 250,000 N/m. It should be noted that both the GRFs and GRMs applied to the digits segments were derived strictly from the kinematics of the digits and their attached contact spheres. Unlike the residual reduction algorithm, the tracking simulation approach allows for both kinematics and kinetics to be altered in the course of the simulation. By enforcing Newtonian rigid body dynamics as a path constraint in the optimal solution, this guaranteed that at the converged solution, the output kinematics and kinetics were dynamically consistent with each other and the model (within a specified tolerance, here 10⁻⁴). The optimal control problem was discretized across 200 evenly spaced mesh intervals using CasADi 3.4.5 [122], and solved using the solver IPOPT 3.12.3 [123]. The resulting optimal solution involved generally only minor changes to the kinematics, GRFs and (vertical) GRM compared to the original experimental data (Table 3; see also S2 Fig). Greater modification occurred to the MTP angle, which is not surprising given the simplified representation of the distal pes (a single DOF, rather than 12 or more) and its contact with the ground (a single large contact sphere) in the model. It was deemed acceptable for the purposes of the current study that realism in the distal pes was sacrificed for greater kinematic and kinetic realism in the remainder of the limb. Modification to the GRFs was on average less than 10% of body weight, with peak deviations of 14.7%, 28.5% and 7.5% of body weight in the x, y and z directions, respectively. Deviation between the vertical GRM in the simulation and as measured experimentally (the ‘free vertical moment’) was on average less than 2% of the product of body weight and standing hip height, with a peak deviation of 7.8% of the product of body weight and standing hip height. It should be recognized that in the present simulation framework there exists an interdependency of GRFs, GRMs and kinematics; for example, the centre of pressure of the GRF (which in turn affects the computed GRMs as expressed in a local or global reference system) is dictated by the kinematics of the body segment to which the contact sphere is appended, and displacement of the centre of pressure will therefore affect the achievement of force and moment balance (i.e., dynamic consistency).

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Table 3. Comparison of the results of the tracking simulations in relation to the original experimental data used as input.

This is reported as root mean squared error for kinematics and kinetics; error is reported in degrees for rotational kinematics, millimetres for translational kinematics, Newtons for GRFs and Newton-metres for GRMs. Note that only the vertical component of the GRM is considered, as horizontal components in reality are practically zero (yet non-zero components can exist in the simulation, rendering any comparison approximate only). For entries pertaining to the limbs, error is reported as left limb/right limb.

https://doi.org/10.1371/journal.pcbi.1008843.t003

A tracking simulation was also performed for the running trial, so as to generate ground reactions that were dynamically consistent with the experimentally derived kinematics, and to improve digit placement with respect to the ground plane during the stance phase. The same torque-driven optimal control problem formulation used for the walking simulation was also used here. However, in lieu of experimentally derived kinetic data, the inputs used for the GRF in the x (anteroposterior) and y (vertical) directions were estimated using the BIRDS model of Bishop et al. [28], given the known values for speed and duty factor. The x and y components of the simulated GRFs were then encouraged to track these estimates in the simulation; the z (mediolateral) component was allowed to freely vary so as to allow dynamic consistency. Given the highly simplified representation of foot–ground interaction in the current model, tracking just the (less than ideal) kinematic data alone would have resulted in unrealistic GRF profiles. Forcing the simulation to track a theoretically expected GRF (which is based on prior empirical observations) in addition to the kinematic data, kept the simulation better grounded in biological realism. One further difference from the walking simulation was that a lower contact stiffness (150,000 N/m) was found to produce markedly better results in terms of tracking error; this is consistent with the empirical observation of greater limb compliance at faster speeds in a variety of species [124,125]. As with the walking trial, the resulting optimal solution involved generally minor changes to the kinematics compared to the original data, again except for MTP angle (Table 3). In addition to producing modified locomotor kinematics (and kinetics), the above procedure also computed the external joint moments as would be calculated via an inverse dynamics routine (i.e., the moments incurred about each joint due to gravity and inertia). These formed the final inputs used in the inverse simulation below.

Muscle redundancy solver modifications.

In the approach of De Groote et al. [7], the force–length curve of the in-series tendon is in part parameterized by dimensionless tendon stiffness, calculated as (3) where E is Young’s modulus for tendon, A_T is the cross-sectional area of the tendon and F_max is the maximal isometric force for the muscle. Note that in this calculation F_max takes muscle pennation into account, in contrast to Eq 1 above, because the muscle is here reduced down to a force along a line of action in series with the tendon (i.e., fibre kinematics are irrelevant). In the human model of De Groote et al. [7], k_T was nominally set to 35 [cf. 5]. However, anatomical measurements of all MTUs for the tinamou, including tendon length and mass, indicated that k_T was in general markedly higher, assuming typical values of 1.2 GPa and 1,120 kg/m³ for tendon modulus and density, respectively [126]. Values for k_T ranged from 23 for some digital flexors to well over 1,000 for some hip muscles with short, thick tendons, with a median value of 114. As such, a first-pass estimate of k_T = 100 was used for all MTUs in the current study. The markedly higher stiffness inferred for many of the tendons in the tinamou, in comparison to humans, is consistent with prior empirical studies showing that smaller animals tend to have lower ratios of muscle physiological cross-sectional area compared to tendon cross-sectional area [e.g., 127,128–130]. Activation and deactivation time constants (in the muscle excitation–activation dynamics) were also altered from the default human model values of 0.015s and 0.06 s, to 0.007s and 0.027s, respectively. Although no published data exist to support these specific values, it was considered likely that both constants would scale with body size; smaller animals need to have less electromechanical delay due to being adapted for higher stride frequencies, and have a shorter distance for electrical signal to travel from motor neuron to muscle fibres [131,132]. Here it was assumed that the time constants would scale proportional to body mass^1/6 [29], and were scaled from the 64.8 kg human model to the 0.545 kg tinamou model in accordance with this. All other parameters regarding muscle excitation–activation and activation–contraction dynamics used the default values as per De Groote et al. [7].

The original Muscle Redundancy Solver code was further modified to solve for both walking and running trials together, whilst simultaneously also allowing MTU architectural parameters to be modified from their original values. The measurement of muscle architectural parameters during dissection inevitably involves a level of error, which may be magnified by simplified representation of an entire muscle effectively as a single fibre in Hill-type models [5]. It was therefore considered appropriate to allow some parameters to be adjusted from their original values such that the MTUs were better ‘tuned’ at executing the observed behaviours [22,133–136]. This helps reduce the discrepancy between model capabilities and measured performance, reducing reliance on reserve actuators (see also below), and also produces more physiologically plausible MTU behaviour. Here, ℓ_o and L_S were tuned, with a single new set of parameters being derived that worked well for both walking and running trials. F_max was kept at its original value for all MTUs; for a given physiological cross-sectional area, this implicitly assumes that the number of fibres varies in inverse proportion to ℓ_o. By tuning for different gaits simultaneously, this reduced the chance of the MTUs becoming too well adapted for the execution of any single behaviour and possibly less effective in the execution of other behaviours. As in real life, an MTU’s parameters will reflect a tradeoff in requirements for the effective execution of different behaviours throughout daily activity. Ideally, experimental data from a wider range of behaviours could be used to obtain even more ‘generically adapted’ MTU parameters, such as maximal speed running, sit-to-stand or jumping manoeuvres; unfortunately, suitable data were unavailable for the present study.

Problem formulation and solving.

From the supplied joint kinematics, MTU length (l_MT) and moment arms throughout the trials were computed for each MTU using the MuscleAnalysis tool in OpenSim. The resulting data, and the external joint moments, were first filtered using a fourth-order low-pass Butterworth filter with a cutoff frequency of 20 Hz, prior to being used in the inverse simulations. This second pass of filtering was applied to remove additional noise or artifact that had collectively accumulated throughout the data processing steps, and kept the focus of the simulations and analyses on the major patterns in the data (especially considering only a single trial each of walking and running were studied here). As tinamou movement was quite bilaterally symmetrical, MTU excitations were only solved for the left limb, which encompassed complete and contiguous stance and swing phases in both trials. In addition to the 36 MTUs, all ten DOFs in the left hindlimb were actuated by torque or ‘reserve’ actuators. These reserve actuators account for modelling simplifications or errors that collectively make it difficult, if not impossible, for the MTUs alone to completely balance external joint moments [17,118]. The maximal moment-generating capacity of these actuators was generally set to be a low value (relative to peak external joint moment), to encourage the use of MTUs in countering joint moments and driving movement. However, for abduction–adduction and long-axis rotation at both the knee and ankle, these maximal capacities were set to higher values to achieve more realistic results (see Results); this is consistent with the muscles that actuate these DOFs being positioned so as to act largely in the flexion–extension plane of the joints. Given that previous in vivo kinematic studies have demonstrated considerable mobility of the avian knee and ankle in abduction–adduction or long-axis rotation [91,137], it was felt that including these additional DOFs in the simulations could be more insightful than excluding them a priori.

The optimal control problem was posed thus: (4) subject to muscle excitation–activation and activation–contraction dynamics, muscle–tendon equilibrium within each MTU, state continuity across successive mesh intervals throughout each trial (i.e., collocation defect constraints are zero), and net moment balance between MTU forces, reserve actuators and external joint moments (see S1 Code). The problem was discretized across 100 evenly spaced mesh intervals (using third order Radau collocation) for both trials in CasADi and solved using IPOPT, and the initial guess supplied for the state variables was seeded with the results obtained from a static optimization solution. Note that the Muscle Redundancy Solver simply solves for muscle states and controls given provided input experimental data, without imposing constraints on the initial or final conditions of the system; thus, non-cyclical experimental data will result in non-cyclical muscle results.

The objective function aimed to minimize six terms:

1. The sum of squared muscle activations (a_muscle) across all M muscles for both trials, integrated across the trial; activations could range between 0 (‘fully off’) and 1 (‘fully on’). The corresponding weighting used for this was w₁ = 2.
2. The sum of squared reserve activations (a_reserve) across all L DOFs in the limb for both trials, integrated across the trials. In a similar fashion to the muscles, reserve activations could range between -1 and 1 (allowing torques of either sense to be produced), with the absolute magnitude of supplied torque computed by scaling activations by pre-defined maximum external joint moments. This term was weighted heavily relative to the previous term (w₂ = 500), penalizing the use of reserve actuators unless absolutely necessary.
3. The sum of squared time derivatives of muscle activations (da/dt) and MTU forces (dF_T/dt) across all M muscles for both trials, integrated across the trials. The inclusion of this term is to improve numerical conditioning of the nonlinear program, avoiding situations for which these implicit controls are not uniquely defined by optimality conditions [120]. The corresponding weighting used for this term was w₃ = 1; it was found that lower values (as used in the human simulations of [7]) resulted in qualitatively similar excitation patterns, but which were markedly less smooth over time.
4. The sum of squared passive muscle fibre forces (F_passive) across all M muscles for both trials, integrated across the trials. The corresponding weighting used for this term was w₄ = 100. A high value here helps penalize the optimizer from ‘cheating’ and tuning ℓ_o such that a given MTU operates largely or wholly on the descending limb of its active force–length curve, thereby using passive forces to help counter joint moments with no associated activation cost [138]. Whilst this term may artefactually decrease or constrain the recovered operating ranges of some muscle fibres, it was found that preliminary simulations that did not include this term resulted in a few muscles optimizing their operating ranges at very high (and unrealistic) values of ℓ*. The weighting value ultimately used was chosen as a compromise that avoided these unrealistic results but minimized potential side-effects on other muscles.
5. The sum of squared deviations in ℓ_o tuning factors (p_ℓ) from one; encouraging the tuned fibre lengths to differ only as much as is needed from the original values in the musculoskeletal model. The corresponding weighting used for this was w₅ = 2.
6. The sum of squared deviations in L_S tuning factors (p_L) from one; encouraging the tuned slack lengths to differ only as much as is needed from the original values in the musculoskeletal model. The corresponding weighting used for this was w₆ = 1; a lower weighting is used compare to that for fibre lengths as greater confidence was placed a priori in the original values for fibre length, since these were directly measured whereas slack lengths were estimated. It was found that using higher values of w₅ in relation to w₆ did not have any important effect on the resulting excitation patterns.

Although the specific (absolute) values of the weighting terms were somewhat arbitrarily chosen (for both Eqs 2 and 4), their relative magnitudes reflected the relative importance of the corresponding terms in each objective function as per the study’s objectives. A systematic exploration of the consequences of using differing weighting terms was beyond the scope of the present investigation, but could be useful in future comparative studies of larger datasets, especially across species.

Premise.

Following the execution of inverse simulations, the tuned musculoskeletal model was subject to four sets of static simulations. Only a subset of all possible limb postures was used in vivo by the tinamous in the locomotor experiments, and so these additional simulations helped explore the full range of possible fibre length variation across the entirety of limb poses. Moreover, in concert with the results of the inverse simulations, a key aim of the static simulations was to explicitly test the assumption outlined in the Introduction, that ℓ* will vary from 0.5–1.5 across the entire range of limb motion. Deconstructing this assumption, it was hypothesized that the length change an MTU experiences over the entire range of limb motion is equal to ℓ_o [66,73,74]; this in turn implies that ℓ* will vary from approximately 0.5–1.5. Additionally, it was posited that by assuming a fixed moment arm, l_MT change could therefore be computed directly from total joint range of motion. By exploring the full range of limb poses in the tinamou model, the static simulations therefore sought to elicit the greatest range of physiologically possible muscle lengths, providing a more comprehensive test of the aforementioned assumption than could be achieved with the in vivo locomotor simulations alone.

Implementation.

A total of 5,000 random limb postures were generated across the model’s full range of motion. For the sake of simplicity and tractable analysis of the results, the knee and ankle joint were reduced to a single DOF each, with abduction–adduction and long-axis rotation fixed at constant angles, equal to the mean angles across the stride for both the walking and running trials’ kinematics. This simplification also obviated the need for reserve actuators, which were strongly required in the inverse simulations (see also Results), keeping the focus directed towards those DOFs for which muscles are primarily responsible for actuation (and therefore those DOFs most relevant to the assumption being tested). Thus, the 5,000 random limb postures resulted in a 6 DOF joint space. In defining range of motion limits, ranges for the knee ([-145°, -10°]), ankle ([-135°, -0°]) and MTP joint ([0°, 180°]) were set based on what is physiologically realistic, such as the knee not being permitted full extension due to soft tissue restrictions observed in intact and defleshed carcasses. For the 3-DOF hip, flexion–extension was nominally allowed full flexion through to full vertical extension (i.e., [-90°, 0°]), which is more extensive than that possible in vivo, as observed in cadaveric manipulations due to the limiting effects of intrinsic soft tissues of the hip [139], as well as far-field influences such as the ribcage. Ranges for hip abduction–adduction ([-5°, 30°]) and long-axis rotation ([-15°, 30°]) were determined from bone-on-bone collisions with the other two DOFs set at 0°. This simplified approach did not accurately or completely capture the true nature of joint mobility in the hip (for which an automated method like that of [139] would be required), but it was deemed acceptable here for it almost certainly overestimated the total mobility of the joint [93]. It therefore enabled a conservative assessment of fibre length changes: the total scope of possible fibre length changes in reality would be less than what would be achieved here. Values for l_MT and moment arms were then determined for the 5,000 test postures using the MuscleAnalysis tool in OpenSim.

For each test posture, four basic, static simulations were performed. In the first two, all 36 MTUs in the limb were considered together and an optimization problem was formulated, which sought to either minimize (simulation 1) or maximize (simulation 2) muscle activations, (5) subject to maintaining static equilibrium throughout the leg, (6) and maintaining muscle–tendon equilibrium within each MTU; r_m,k is the moment arm of muscle m about DOF k. Gravity was ignored here as it is otherwise very difficult to standardize the results against limb posture to enable meaningful comparison; and as the simulations were static, there were no external joint moments due to segment inertia. That is, only the geometric, skeletal aspect of the system was considered here. In the third and fourth simulations each MTU was considered sequentially in isolation, whereby an optimization problem was formulated in which activation was set to zero (simulation 3) or one (simulation 4) and then the MTU was solved for muscle–tendon equilibrium.

In all four simulations, MTU behaviour was modelled using the same formulation as in the inverse simulations, and the optimization problem in each was formulated in CasADi and solved with IPOPT. Simulations 2–4 arguably did not follow any physiologically relevant principle, but were included so as to circumscribe as great a range as possible of muscle shortening and lengthening, beyond what occurred in vivo in the locomotor experiments. For example, in considering a single muscle at a time, simulations 3 or 4 may elicit a greater range of fibre lengths than simulations 1 or 2 alone. Regardless, the ranges of ℓ* possible in the real animal will be less than what was achieved here, and therefore the simulations lent themselves towards a conservative testing of the above assumption. In combination with the inverse simulations of in vivo locomotor behaviour, the static simulations may be seen as progressively simplified analyses of the tinamou hindlimb musculoskeletal system, focusing increasingly more on the muscles themselves (insofar as they relate to skeletal anatomy) in isolation of other complicating factors, namely dynamics, gravitational and inertial loading and muscle co-contraction.

Results filtering and analysis.

Prior to analysis and interpretation, the results of the above simulations first required filtering, to remove poses that produced justifiably ‘bad’ results. Three such instances of this occurred:

1. Poses in which it was impossible to achieve muscle–tendon equilibrium within all MTUs and simultaneously achieve moment balance across all the joints, regardless of activation. The underlying cause of this was very high ℓ* for at least one MTU, producing extremely high passive fibre forces. Only simulations 1 and 2 were affected by this.
2. Poses where l_MT < L_S for a given MTU, in which case muscle–tendon equilibrium cannot be achieved regardless of activation and no solution therefore exists. Only simulations 3 and 4 were affected by this.
3. Poses where unrealistically high values of l_MT were achieved for the flexores cruris lateralis (FLCA and FLCP) due to the poor wrapping behaviour for these muscles’ actuator paths at extreme hip flexion (coupled with low hip abduction). This poor behaviour occurred in poses well exceeding those observed in vivo during locomotion or in situ during cadaveric manipulations (due to soft tissue limits on hip mobility); it was hence a consequence of the model performing poorly in a non-physiological situation that it was never intended to represent, due to the use of highly inclusive range of motion limits defined above. These instances could be easily identified from a histogram of l_MT for the FCLA and by visual inspection of the musculoskeletal model in OpenSim; it was a priori set that poses with l_MT of the FCLA exceeding 80 mm were considered inviable. This affected all four simulations.

Results for the viable poses were then examined with respect to the above assumption, via three comparisons. Firstly, the gross range of ℓ* and l_MT achieved for each MTU across the simulations was computed, to test if this met the expectation of 0.5–1.5× ℓ_o. Secondly, the range of ℓ* and l_MT achieved was compared muscle-by-muscle. Thirdly, as it was posited that l_MT change could be computed from total joint range of motion by assuming a fixed moment arm, this implies that ℓ* should vary approximately in proportion to (linearly with) joint angle(s). As such, a hyperplane was fitted to plots of ℓ* and joint angle(s) using linear algebra: (7) where C is a η × 1 vector of coefficients describing the hyperplane (of the form c₁x₁ + c₂x₂ +…+ c_η), X⁺ is the η × o Moore-Penrose pseudoinverse of an augmented matrix formed from the joint angle data and a column of ones, and y is a o × 1 vector of the ℓ* data, for o observations over η – 1 joint angles. The resulting hyperplane is the minimum norm least-squares fit for the data; for comparisons involving a single joint angle, η = 2 and the hyperplane is a line, and for comparisons involving two joint angles, η = 3 and the hyperplane is a regular plane. Following the derivation of the hyperplane, the coefficient of determination (r²), root mean squared error (RMSE) and maximum absolute error (ε_max) were computed for the fit versus the original data. For each MTU, hyperplane fitting was applied twice: considering the results from simulations 1 and 2 together (analyses of all muscles holistically), and considering the results from simulations 3 and 4 together (analyses of each muscle individually). Hyperplane fitting, and the goodness of the resulting fit, therefore helped assess the scale of the effect of variable moment arms, compliant tendons and pennation angles on an otherwise linear relation between ℓ*and joint angle(s).

Tuning factors.

Almost all tuning factors in the optimal solution exceeded one (Fig 3), indicating that most MTUs needed both ℓ_o and L_S to be increased from their original values (Table 2). Whereas ℓ_o was increased on average by 11% (range -1.1 to 49.3%), L_S was increased on average by 10.5% (range -3.1 to 33.3%). Therefore, many MTUs only required a level of tuning well within the scope of variation due to either measurement error (considering that a whole muscle is essentially represented as a single fibre) or genuine intra- or inter-individual variation. Moreover, the two most drastically altered MTUs can probably be explained in large part as a consequence of the simplified approach used to represent their paths in the model. Firstly, the FMTL (ℓ_o tuned by 32.4%, or 9.1 mm) crosses the knee joint over the lateral condyle region, and its involvement with the common aponeurosis of the knee extensors (‘patellar tendon’) is complex. It would not be surprising if the MTU path as modelled here deviates significantly from the ‘true’ path in vivo over at least part of the knee’s range of motion, introducing error into muscle kinematics during movement, and thereby necessitating greater tuning in the simulation. Secondly, the posterior part of the ITCa (ITCap; ℓ_o tuned by 49.3%, or 9.1 mm) is one of two MTUs that was used to model a highly pennate muscle, and it is therefore possible that considerable tuning occurred here to compensate for this modelling simplification. It is worth noting that the anterior part (ITCaa) also had a high tuning factor for ℓ_o (+23.0%), and had the highest tuning factor for L_S (+33.3%).

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Fig 3. Tuning factors used in the optimal solution for the inverse simulation.

These factors are multiplied against the original assigned values for optimal fibre length (ℓ_o) and tendon slack length (L_S) to derive the tuned parameters for each MTU.

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Reserve contributions.

For both walking and running trials, reserve actuator contributions almost always remained less (often much less) than 10% of the external joint moment at any given instant in the stance phase, as well as during much of the swing phase, for the following DOFS: hip extension, abduction and rotation, knee extension, ankle extension and MTP angle (Fig 4). This indicated that the muscles provided the vast majority of the effort required to stabilize and propel the limb. Given that the original (dissection-derived) values for muscle F_max were used in these simulations, this result is encouraging, as even in the running trial the muscles were acceptably sufficient in sustaining the prescribed locomotor dynamics. The reserve contributions for the abduction–adduction and long-axis rotation DOFs of the knee and ankle were almost always markedly higher than 10% of the external joint moment, which at least in part stems from assigning a higher maximal moment-generating capacity to these actuators. In preliminary runs of the inverse simulations, it was found that whilst using lower maximal capacities led to lower reserve contributions to external joint moment, it also resulted in higher excitations for muscles crossing the DOFs involved. The excitation patterns obtained for several of these muscles (particularly the ankle flexors and extensors) were grossly inconsistent with previously published empirical data as well as a priori expectations based on anatomy, and were therefore considered implausible. These results suggest an apparent necessity for increased use of non-muscular (i.e., passive) forces in controlling these particular DOFs (see below for further discussion).

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Fig 4. Reserve actuator contributions to external joint moments.

This is shown for walking (A) and running (B) trials. Grey regions denote the stance phase, white regions denote the swing phase. Note the strong contributions to external joint moment made by reserves acting about the abduction–adduction and long-axis rotation DOFs of the knee and ankle joints. Also illustrated are the kinematics of the left hindlimb (reversed for visualization) at various parts of the stride cycle.

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Muscle recruitment.

The time histories of muscle excitations recovered for the walking trial are presented in Fig 5, and those for the running trial in Fig 6. Excitation patterns are also visualized in the context of kinematics and kinetics in S1 and S2 Movies. There were no pronounced changes in muscle recruitment (timing or duration with respect to the stance and swing phases) between walking and running, except for generally greater excitations in running, corresponding to greater force production in this more strenuous gait. Exemplar comparisons of simulated excitations with previously published electromyographic data for walking and running birds are presented in Fig 7; as previous studies rarely quantified the magnitude of the recorded electrical signal (either in absolute terms, or in relation to a maximum voluntary contraction), comparisons with the literature are necessarily qualitative only. The recovered patterns of recruitment for many muscles were grossly similar to that recorded experimentally in birds [9,27,33,41,55,58,88,96,140], including for tinamous at similar speeds to those simulated here [88].

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Fig 5. Time histories of muscle excitation and fibre length change for each MTU in the walking simulation.

Excitations (from zero recruitment to possible maximal recruitment) are plotted above normalized fibre lengths for each MTU, the latter of which are colour coded according to where on the active force–length curve fibres are operating: purple = steep ascending limb, blue = shallow ascending limb, green = plateau, orange = descending limb (divisions approximately correspond to those of Arnold and Delp [15]). Excitation profiles obtained in the simulation are reflected about the abscissa so as to visually emulate the appearance of experimental EMG signals. Grey regions denote the stance phase, white regions denote the swing phase. See Table 2 for muscle abbreviations.

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Fig 6. Time histories of muscle excitation and fibre length change for each MTU in the running simulation.

Formatting is the same as for Fig 5.

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Fig 7. Comparison of exemplar outputs for excitation and fibre length change in the running simulation against previous experimental data (electromyography and sonomicrometry), for validation of the modelling and simulation approach.

For each muscle, experimental data are shown in blue, and simulation outputs are shown in red; in some studies sonomicrometry data were reported in absolute terms, whereas in others it was reported in relative terms. See Table 2 for muscle abbreviations. Fibre length data do not exist for the AMB or ILFB. Experimental results were digitized from the original studies and scaled according to swing and stance phase durations. Sources of experimental data are as follows: ILPOp, guineafowl running at 2.5 m/s [9]; FCLP and FCLA, guineafowl running at 1.5 m/s [38]; GL, guineafowl running at 1.3 m/s [36]; GM, guineafowl running at 2 m/s [141]; FP4, guineafowl running at 1.3 m/s [36]; AMB and ILFB, guineafowl running at 1 m/s [27].

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Muscles recruited exclusively or mostly during the stance phase included the ILPO, FMTL, FMTI, FMTM, FCLP, FLCA, ITCa, CFP, PIFML, GM, FL and the digital flexors. Muscles recruited exclusively or mostly during the swing phase included the IC, ILFB, OM and EDL. Several muscles showed small bouts of excitation during both swing and stance phases, the timing of which in some cases was not entirely consistent with published experimental data; one example (AMB) is illustrated in Fig 7. The results for a few specific muscles are worth noting in more detail. Firstly, the GL is a large, biarticular ankle extensor that has been frequently studied in birds; across all species studied to date, it is always recruited in late swing and into most of the stance phase [88]. The simulations for both walking and running recovered a consistent excitation pattern for this muscle that matches experimental observations well (Fig 7). Secondly, the FCLA was recruited exclusively in stance phase, consistent with published experimental data [27,33,38]. This result is encouraging, because the MTU geometry in the model was distinctly different from the actual course of the muscle; in reality, the muscle originates from the belly of another muscle (FCLP) rather than the pelvis, a topology that cannot be easily incorporated in standard musculoskeletal modelling approaches. Lastly, the excitation pattern for the ILFB, with recruitment occurring almost exclusively in swing phase, was in marked contrast to published experimental data for various bird species (including tinamou), which shows it to predominantly be a stance phase muscle (Fig 7, [27,33,88,96,140]), although Marsh et al. [33] recorded activity around the stance–swing transition for anterior fibres of this muscle in guineafowl. This anomaly may suggest an incomplete description of the muscle’s functional anatomy and physiological role, in either the musculoskeletal model or the OCP formulation (see also below).

There are several muscles whose recruitment during avian terrestrial locomotion remains to be experimentally investigated; combined with the results of ostrich simulations by Rankin et al. [17], the present study therefore provides important insight into the behaviour of such muscles. The FCM in the present study was negligibly recruited in the present simulations (both walking and running), whereas it was heavily recruited in the ostrich simulations, particularly for running. This muscle is fairly large and pennate in ostriches (~0.18% of body mass [111]) but is small and parallel-fibred in the tinamou (~0.05% of body mass; Table 2), potentially explaining the difference in recovered recruitment patterns. Alternatively, such differences may reflect other genuine differences between the two species, for instance due to differing moment arms or the more than 100-fold difference in body size. In contrast to this discrepancy, the EDL and most digital flexors had similar recovered excitation patterns between the two studies; the EDL was active mostly in the swing phase, whilst the digital flexors mostly or exclusively in the stance phase, and both of these results are strongly consistent with expectations based on gross anatomy.

Fibre operating ranges.

The time histories of fibre length changes recovered for the walking trial are presented in Fig 5, and those for the running trial in Fig 6. The patterns of change, in terms of timing of increases, decreases or stasis of fibre length with respect to stance and swing phases remained largely consistent for almost all muscles across walking and running. However, the net fibre length excursion across the stride for given muscle tended to be greater in running. Exemplar comparisons of simulated fibre length changes with previously published sonomicrometry data for walking and running birds are presented in Fig 7. However, unlike electromyography, these data have previously been collected for only a few muscles in the avian hindlimb, limiting the extent to which simulation results can be validated here.

All muscle fibres operated between 0.5 ≤ ℓ* ≤ 1.21 in walking and running, with some muscles using a greater portion of this range than others. The vast majority of muscles’ fibres spent most of the stride cycle on the ascending limb or the plateau of the active force–length curve. There were a few muscles whose fibres tended to operate on the descending limb for most or all of the stride cycle, which may be a consequence of the optimization combined with the initial parameters derived from dissection. Alternatively, this result may indicate that these muscles’ fibres are more tuned (adapted) for other behaviours not studied here. By comparing the time histories of excitation and fibre length change, it is evident that some muscles produced force whilst fibres were undergoing contraction (concentric force production; e.g., FCLP, PIFML), others were actively producing force whilst fibres were lengthening (eccentric force production; e.g., TCf), others remained nearly isometric during force production (e.g., ITCa), and others exhibited a combination of these. This diversity of behaviours is consistent with a diversity of functional roles inferred for ostrich hindlimb muscles by Rankin et al. [17], although a detailed analysis of force–length behaviour (e.g., work loops) in the tinamou is beyond the scope of the present study.

In relation to previously recorded fascicle strain data, the time histories of fibre length changes for previously studied muscles is generally concordant with published data. In the ILPO, fibre length change across the stride was more pronounced during running than walking, and exhibited a near-monotonic decrease during the stance phase (Fig 7); this is largely consistent with published data for guineafowl, except for a small increase in length at the beginning of stance [9,41,58]. Moreover, ℓ* remained on the ascending limb or plateau of the active force–length curve, also consistent with empirical observations [9,41,58]. In the FCLP, ℓ* oscillated around the peak of the active force–length curve; it decreased monotonically during stance phase and continued to decrease into early swing, before increasing for the remainder of swing phase (Fig 7), a pattern experimentally observed in guineafowl [38]. Similarly, fibre length of the FCLA decreased almost monotonically throughout much of stance and increased over the swing phase (Fig 7), also consistent with experimental observations. In running, ℓ* of the FCLP oscillated evenly about the peak of the active force–length curve, comparable to the results of Ellerby and Marsh [38], but in walking ℓ* tended to occur more on the ascending limb. As with the excitation pattern recovered for the FCLA, this concordance between simulation and empirical observations is encouraging given the simplified representation of its MTU in the model.

The avian gastrocnemius has been extensively studied experimentally, particularly the lateral (GL) and medial (GM) heads. In the GL, fibres shortened over the whole stance phase and lengthened in the first half of swing, before shortening again in late swing (Fig 7), a pattern largely consistent with previously published data, although modest variation across species and studies is known [36,55,78,141–145]. In the simulations ℓ* operated almost exclusively on the ascending limb for both walking and running, whereas published data for turkey GL indicates an operating range that includes part of the descending limb [78,143]; however, it is not certain if those studies’ measures of resting fascicle length is truly representative of optimal fascicle length. Published data for guineafowl [141] and mallard ducks [142] suggest that mean ℓ* also lies on the ascending limb, consistent with simulation results, but an uncertain measure of resting fascicle length was used by those studies, too. In the GM, fibre lengths underwent a small decrease across the stance phase, except for a slight increase around midstance (inverted U shape, more prominent in running), and they increased and then decreased over the course of the swing phase (Fig 7). This pattern, including the inverted U shape around midstance, is consistent with published data, especially for more proximal fibres in the muscle [41,141,145]. In contrast to the GL and GM, the gastrocnemius pars intermedia (GI) displayed a pattern of fibre length change that is not entirely consistent with limited published data for guineafowl [38], but this may in part be due to anatomical topologies (osseous and soft tissue) involving the GI and surrounding muscles that differ between guineafowl and the tinamou as modelled here [38].

More distally, only two additional avian hindlimb muscles have been experimentally investigated. In the FL, ℓ* remained fairly constant over the stride in walking, operating on the ascending limb of the active force–length curve; however, in running, ℓ* showed an increase-then-decrease pattern during the swing phase, oscillating from ascending to descending limbs, a pattern somewhat consistent with published data [78,143]. In the FP4 (a digital flexor), fibres increased in length over the stance phase followed by a lengthening-then-shortening pattern in swing; this pattern was more exaggerated in running, and is partially consistent with data reported for guineafowl [36,144].