Resonance tuning in a neuro-musculo-skeletal model of the forearm

In rhythmic movements, humans activate their muscles in a robust and energy efficient way. These activation patterns are oscillatory and seem to originate from neural networks in the spinal cord, called central pattern generators (CPGs). Evidence for the existence of CPGs was found for instance in lampreys, cats and rats. There are indications that CPGs exist in humans as well, but this is not proven yet. Energy efficiency is achieved by resonance tuning: the central nervous system is able to tune into the resonance frequency of the limb, which is determined by the local reflex gains. The goal of this study is to investigate if the existence of a CPG in the human spine can explain the resonance tuning behavior, observed in human rhythmic limb movement. A neuro-musculo-skeletal model of the forearm is proposed, in which a CPG is organized in parallel to the local reflexloop. The afferent and efferent connections to the CPG are based on clues about the organization of the CPG, found in literature. The model is kept as simple as possible (i.e., lumped muscle models, groups of neurons are lumped into half-centers, simple reflex model), but incorporates enough of the essential dynamics to explain behavior—such as resonance tuning—in a qualitative way. Resonance tuning is achieved above, at and below the endogenous frequency of the CPG in a highly non-linear neuro- musculo-skeletal model. Afferent feedback of muscle lengthening to the CPG is necessary to accomplish resonance tuning above the endogenous frequency of the CPG, while feedback of muscle velocity is necessary to compensate for the phase lag, caused by the time delay in the loop coupling the limb to the CPG. This afferent feedback of muscle lengthening and velocity represents the Ia and II fibers, which—according to literature—is the input to the CPG. An internal process of the CPG, which integrates the delayed muscle lengthening and feeds it to the half-center model, provides resonance tuning below the endogenous frequency. Increased co-contraction makes higher movement frequencies possible. This agrees with studies of rhythmic forearm movements, which have shown that co-contraction increases with movement frequency. Robustness against force perturbations originates mainly from the CPG and the local reflex loop. The CPG delivers an increasing part of the necessary muscle activation for increasing perturbation size. As far as we know, the proposed neuro-musculo-skeletal model is the first that explains the observed resonance tuning in human rhythmic limb movement.     

A comparison of resonance tuning with positive versus negative sensory feedback

We used a computational model of rhythmic movement to analyze how the connectivity of sensory feedback affects the tuning of a closed-loop neuromechanical system to the mechanical resonant frequency (ω_r). Our model includes a Matsuoka half-center oscillator for a central pattern generator (CPG) and a linear, one-degree-of-freedom system for a mechanical component. Using both an open-loop frequency response analysis and closed-loop simulations, we compared resonance tuning with four different feedback configurations as the mechanical resonant frequency, feedback gain, and mechanical damping varied. The feedback configurations consisted of two negative and two positive feedback connectivity schemes. We found that with negative feedback, resonance tuning predominantly occurred when ω_r was higher than the CPG’s endogenous frequency (ω_CPG). In contrast, with the two positive feedback configurations, resonance tuning only occurred if ω_r was lower than ω_CPG. Moreover, the differences in resonance tuning between the two positive (negative) feedback configurations increased with increasing feedback gain and with decreasing mechanical damping. Our results indicate that resonance tuning can be achieved with positive feedback. Furthermore, we have shown that the feedback configuration affects the parameter space over which the endogenous frequency of the CPG or resonant frequency the mechanical dynamics dominates the frequency of a rhythmic movement.     

Kinematics of lower limbs during walking are emulated by springy walking model with a compliantly connected,off-centered curvy foot

The dynamics of the center of mass (CoM) during walking and running at various gait conditions are well described by the mechanics of a simple passive spring loaded inverted pendulum (SLIP). Due to its simplicity, however, the current form of the SLIP model is limited at providing any further information about multi-segmental lower limbs that generate oscillatory CoM behaviors and their corresponding ground reaction forces. Considering that the dynamics of the CoM are simply achieved by mass-spring mechanics, we wondered whether any of the multi-joint motions could be demonstrated by simple mechanics. In this study, we expand a SLIP model of human locomotion with an off-centered curvy foot connected to the leg by a springy segment that emulates the asymmetric kinematics and kinetics of the ankle joint. The passive dynamics of the proposed expansion of the SLIP model demonstrated the empirical data of ground reaction forces, center of mass trajectories, ankle joint kinematics and corresponding ankle joint torque at various gait speeds. From the mechanically simulated trajectories of the ankle joint and CoM, the motion of lower-limb segments, such as thigh and shank angles, could be estimated from inverse kinematics. The estimation of lower limb kinematics showed a qualitative match with empirical data of walking at various speeds. The representability of passive compliant mechanics for the kinetics of the CoM and ankle joint and lower limb joint kinematics implies that the coordination of multi-joint lower limbs during gait can be understood with a mechanical framework.     

Fibre operating lengths of human lower limb muscles during walking

Muscles actuate movement by generating forces. The forces generated by muscles are highly dependent on their fibre lengths, yet it is difficult to measure the lengths over which muscle fibres operate during movement. We combined experimental measurements of joint angles and muscle activation patterns during walking with a musculoskeletal model that captures the relationships between muscle fibre lengths, joint angles and muscle activations for muscles of the lower limb. We used this musculoskeletal model to produce a simulation of muscle-tendon dynamics during walking and calculated fibre operating lengths (i.e. the length of muscle fibres relative to their optimal fibre length) for 17 lower limb muscles. Our results indicate that when musculotendon compliance is low, the muscle fibre operating length is determined predominantly by the joint angles and muscle moment arms. If musculotendon compliance is high, muscle fibre operating length is more dependent on activation level and force-length-velocity effects. We found that muscles operate on multiple limbs of the force-length curve (i.e. ascending, plateau and descending limbs) during the gait cycle, but are active within a smaller portion of their total operating range.     

Energetics and passive dynamics of the ankle in downhill walking

This study investigated the energetics of the human ankle during the stance phase of downhill walking with the goal of modeling ankle behavior with a passive spring and damper mechanism. Kinematic and kinetic data were collected on eight male participants while walking down a ramp with inclination varying from 0° to 8°. The ankle joint moment in the sagittal plane was calculated using inverse dynamics. Mechanical energy injected or dissipated at the ankle joint was computed by integrating the power across the duration of the stance phase. The net mechanical energy of the ankle was approximately zero for level walking and monotonically decreased (i.e., became increasingly negative) during downhill walking as the slope decreased. The indication is that the behavior of the ankle is energetically passive during downhill walking, playing a key role in dissipating energy from one step to the next. A passive mechanical model consisting of a pin joint coupled with a revolute spring and damper was fit to the ankle torque and its parameters were estimated for each downhill slope using linear regression. The passive model demonstrated good agreement with actual ankle dynamics as indicated by low root-mean-square error values. These results indicate the stance phase behavior of the human ankle during downhill walking may be effectively duplicated by a passive mechanism with appropriately selected spring and damping characteristics.     

Virtual slope control of a forward dynamic bipedal walker

Active joint torques are the primary source of power and control in dynamic walking motion. However the amplitude, rate, timing and phasic behavior of the joint torques necessary to achieve a natural and stable performance are difficult to establish. The goal of this study was to demonstrate the feasibility and stable behavior of an actively controlled bipedal walking simulation wherein the natural system dynamics were preserved by an active, nonlinear, state-feedback controller patterned after passive downhill walking. A two degree-of-freedom, forward-dynamic simulation was implemented with active joint torques applied at the hip joints and stance leg ankle. Kinematic trajectories produced by the active walker were similar to passive dynamic walking with active joint torques influenced by prescribed walking velocity. The control resulted in stable steady-state gait patterns, i.e. eigenvalue magnitudes of the stride function were less than one. The controller coefficient analogous to the virtual slope was modified to successfully control average walking velocity. Furture developments are necessary to expand the range of walking velocities.     

Hip actuations can be used to control bifurcations and chaos in a passive dynamic walking model

We explored how hip joint actuation can be used to control locomotive bifurcations and chaos in a passive dynamic walking model that negotiated a slightly sloped surface (gamma<0.019 rad). With no hip actuation, our passive dynamic walking model was capable of producing a chaotic locomotive pattern when the ramp angle was 0.01839 rad相似文献   

Resonant frequencies of arms and legs identify different walking patterns

The present study is aimed at investigating changes in the coordination of arm and leg movements in young healthy subjects. It was hypothesized that with changes in walking velocity there is a change in frequency and phase coupling between the arms and the legs. In addition, it was hypothesized that the preferred frequencies of the different coordination patterns can be predicted on the basis of the resonant frequencies of arms and legs with a simple pendulum model. The kinematics of arms and legs during treadmill walking in seven healthy subjects were recorded with accelerometers in the sagittal plane at a wide range of different velocities (i.e., 0.3-1. 3m/s). Power spectral analyses revealed a statistically significant change in the frequency relation between arms and legs, i.e., within the velocity range 0.3-0.7m/s arm movement frequencies were dominantly synchronized with the step frequency, whereas from 0.8m/s onwards arm frequencies were locked onto stride frequency. Significant effects of walking speed on mean relative phase between leg and arm movements were found. All limb pairs showed a significantly more stable coordination pattern from 0.8 to 1.0m/s onwards. Results from the pendulum modelling demonstrated that for most subjects at low-velocity preferred movement frequencies of the arms are predicted by the resonant frequencies of individual arms (about 0.98Hz), whereas at higher velocities these are predicted on the basis of the resonant frequencies of the individual legs (about 0.85Hz). The results support the above-mentioned hypotheses, and suggest that different patterns of coordination, as shown by changes in frequency coupling and phase relations, can exist within the human walking mode.     

A passive dynamic walking robot that has a deterministic nonlinear gait

There is a growing body of evidence that the step-to-step variations present in human walking are related to the biomechanics of the locomotive system. However, we still have limited understanding of what biomechanical variables influence the observed nonlinear gait variations. It is necessary to develop reliable models that closely resemble the nonlinear gait dynamics in order to advance our knowledge in this scientific field. Previously, Goswami et al. [1998. A study of the passive gait of a compass-like biped robot: symmetry and chaos. International Journal of Robotic Research 17(12)] and Garcia et al. [1998. The simplest walking model: stability, complexity, and scaling. Journal of Biomechanical Engineering 120(2), 281-288] have demonstrated that passive dynamic walking computer models can exhibit a cascade of bifurcations in their gait pattern that lead to a deterministic nonlinear gait pattern. These computer models suggest that the intrinsic mechanical dynamics may be at least partially responsible for the deterministic nonlinear gait pattern; however, this has not been shown for a physical walking robot. Here we use the largest Laypunov exponent and a surrogation analysis method to confirm and extend Garcia et al.'s and Goswami et al.'s original results to a physical passive dynamic walking robot. Experimental outcomes from our walking robot further support the notion that the deterministic nonlinear step-to-step variations present in gait may be partly governed by the intrinsic mechanical dynamics of the locomotive system. Furthermore the nonlinear analysis techniques used in this investigation offer novel methods for quantifying the nature of the step-to-step variations found in human and robotic gait.     

Kinematic control of walking

The planar law of inter-segmental co-ordination we described may emerge from the coupling of neural oscillators between each other and with limb mechanical oscillators. Muscle contraction intervenes at variable times to re-excite the intrinsic oscillations of the system when energy is lost. The hypothesis that a law of coordinative control results from a minimal active tuning of the passive inertial and viscoelastic coupling among limb segments is congruent with the idea that movement has evolved according to minimum energy criteria (1, 8). It is known that multi-segment motion of mammals locomotion is controlled by a network of coupled oscillators (CPGs, see 18, 33, 37). Flexible combination of unit oscillators gives rise to different forms of locomotion. Inter-oscillator coupling can be modified by changing the synaptic strength (or polarity) of the relative spinal connections. As a result, unit oscillators can be coupled in phase, out of phase, or with a variable phase, giving rise to different behaviors, such as speed increments or reversal of gait direction (from forward to backward). Supra-spinal centers may drive or modulate functional sets of coordinating interneurons to generate different walking modes (or gaits). Although it is often assumed that CPGs control patterns of muscle activity, an equally plausible hypothesis is that they control patterns of limb segment motion instead (22). According to this kinematic view, each unit oscillator would directly control a limb segment, alternately generating forward and backward oscillations of the segment. Inter-segmental coordination would be achieved by coupling unit oscillators with a variable phase. Inter-segmental kinematic phase plays the role of global control variable previously postulated for the network of central oscillators. In fact, inter-segmental phase shifts systematically with increasing speed both in man (4) and cat (38). Because this phase-shift is correlated with the net mechanical power output over a gait cycle (3, 4), phase control could be used for limiting the overall energy expenditure with increasing speed (22). Adaptation to different walking conditions, such as changes in body posture, body weight unloading and backward walk, also involves inter-segmental phase tuning, as does the maturation of limb kinematics in toddlers.     

Passive bipedal walking with phasic muscle contraction

The existence of self-organizing walking patterns is often considered the result of a mechanical system interacting with the environment and a (neural) oscillating unit. The pattern generators might be thought of as an indispensable component for the existence of limit cycle behavior. This paper shows that this is not a necessity for the existence of a self-organizing bipedal walking pattern. Stable walking cycles emerge from a simple passive bipedal structure, with an energy source inevitably present to sustain the oscillation. In this work the energy source is chosen to be phasic muscle contraction. A two-dimensional model is composed of two legs and a hip mass, symbolizing the trunk. The stance leg stiffness is generated by two muscles. The hip stiffness is generated by four muscles. Muscle activation is caused by two reflex-like trigger signals, without feedback control. Human equivalent model parameters such as geometry and mass distribution were assumed. With return map analysis, the model is analyzed on periodic behavior. Stable walking cycles were found and could be manipulated during walking by varying the muscle or reflex parameters, forcing the oscillation to converge to a new attractor. Received: 5 November 1998 / Accepted in revised form: 26 March 1999     

Contributions of muscles and passive dynamics to swing initiation over a range of walking speeds

Stiff-knee gait is a common walking problem in cerebral palsy characterized by insufficient knee flexion during swing. To identify factors that may limit knee flexion in swing, it is necessary to understand how unimpaired subjects successfully coordinate muscles and passive dynamics (gravity and velocity-related forces) to accelerate the knee into flexion during double support, a critical phase just prior to swing that establishes the conditions for achieving sufficient knee flexion during swing. It is also necessary to understand how contributions to swing initiation change with walking speed, since patients with stiff-knee gait often walk slowly. We analyzed muscle-driven dynamic simulations of eight unimpaired subjects walking at four speeds to quantify the contributions of muscles, gravity, and velocity-related forces (i.e. Coriolis and centrifugal forces) to preswing knee flexion acceleration during double support at each speed. Analysis of the simulations revealed contributions from muscles and passive dynamics varied systematically with walking speed. Preswing knee flexion acceleration was achieved primarily by hip flexor muscles on the preswing leg with assistance from biceps femoris short head. Hip flexors on the preswing leg were primarily responsible for the increase in preswing knee flexion acceleration during double support with faster walking speed. The hip extensors and abductors on the contralateral leg and velocity-related forces opposed preswing knee flexion acceleration during double support.     

A model of cerebrocerebello-spinomuscular interaction in the sagittal control of human walking

A computationally developed model of human upright balance control (Jo and Massaquoi on Biol cybern 91:188–202, 2004) has been enhanced to describe biped walking in the sagittal plane. The model incorporates (a) non-linear muscle mechanics having activation level -dependent impedance, (b) scheduled cerebrocerebellar interaction for control of center of mass position and trunk pitch angle, (c) rectangular pulse-like feedforward commands from a brainstem/ spinal pattern generator, and (d) segmental reflex modulation of muscular synergies to refine inter-joint coordination. The model can stand when muscles around the ankle are coactivated. When trigger signals activate, the model transitions from standing still to walking at 1.5 m/s. Simulated natural walking displays none of seven pathological gait features. The model can simulate different walking speeds by tuning the amplitude and frequency in spinal pattern generator. The walking is stable against forward and backward pushes of up to 70 and 75 N, respectively, and with sudden changes in trunk mass of up to 18%. The sensitivity of the model to changes in neural parameters and the predicted behavioral results of simulated neural system lesions are examined. The deficit gait simulations may be useful to support the functional and anatomical correspondences of the model. The model demonstrates that basic human-like walking can be achieved by a hierarchical structure of stabilized-long loop feedback and synergy-mediated feedforward controls. In particular, internal models of body dynamics are not required.     

Activity of upper limb muscles during human walking

The EMG activity of upper limb muscles during human gait has rarely been studied previously. It was examined in 20 normal volunteers in four conditions: walking on a treadmill (1) with unrestrained natural arm swing (Normal), (2) while volitionally holding the arms still (Held), (3) with the arms immobilized (Bound), and (4) with the arms swinging in phase with the ipsilateral legs, i.e. opposite-to-normal phasing (Anti-Normal). Normal arm swing involved weak rhythmical lengthening and shortening contractions of arm and shoulder muscles. Phasic muscle activity was needed to keep the unrestricted arms still during walking (Held), indicating a passive component of arm swing. An active component, possibly programmed centrally, existed as well, because some EMG signals persisted when the arms were immobilized during walking (Bound). Anti-Normal gait involved stronger EMG activity than Normal walking and was uneconomical. The present results indicate that normal arm swing has both passive and active components.     

Dynamic stability of superior vs. inferior body segments in individuals with transtibial amputation walking in destabilizing environments

Interestingly, young and highly active people with lower limb amputation appear to maintain a similar trunk and upper body stability during walking as able-bodied individuals. Understanding the mechanisms underlying how this stability is achieved after lower-leg amputation is important to improve training regimens for improving walking function in these patients. This study quantified how superior (i.e., head, trunk, and pelvis) and inferior (i.e., thigh, shank, and feet) segments of the body respond to continuous visual or mechanical perturbations during walking. Nine persons with transtibial amputation (TTA) and 12 able-bodied controls (AB) walked on a 2 m×3 m treadmill in a Computer Assisted Rehabilitation Environment (CAREN). Subjects were perturbed by continuous pseudo-random mediolateral movements of either the treadmill platform or the visual scene. TTA maintained a similar local and orbital stability in their superior body segments as AB throughout both perturbation types. However, for their inferior body segments, TTA subjects exhibited greater dynamic instability during perturbed walking. In TTA subjects, these increases in instability were even more pronounced in their prosthetic limb compared to their intact leg. These findings demonstrate that persons with unilateral lower leg amputation maintain upper body stability in spite of increased dynamic instability in their impaired lower leg. Thus, transtibial amputation does significantly impair sensorimotor function, leading to substantially altered dynamic movements of their lower limb segments. However, otherwise relatively healthy patients with unilateral transtibial amputation appear to retain sufficient remaining sensorimotor function in their proximal and contralateral limbs to adequately compensate for their impairment.     

An upper-body can improve the stability and efficiency of passive dynamic walking

The compass-gait walker proposed by McGeer can walk down a shallow slope with a self-stabilizing gait that requires no actuation or control. However, as the slope goes to zero so does the walking speed, and dynamic gait stability is only possible over a very narrow range of slopes. Gomes and Ruina have results demonstrating that by adding a torso to the compass-gait walker, it can walk passively on level-ground with a non-infinitesimal constant average speed. However, the gait involves exaggerated joint movements, and for energetic reasons horizontal passive dynamic walking cannot be stable. We show in this research that in addition to collision-free walking, adding a torso improves stability and walking speed when walking downhill. Furthermore, adding arms to the torso results in a collision-free periodic gait with natural-looking torso and limb movements. Overall, in contrast to the suggestions that active control may be needed to balance an upper-body on legs, it turns out that the upper and lower bodies can be integrated to improve the stability, efficiency and speed of a passive dynamic walker.     

Energetics of actively powered locomotion using the simplest walking model

We modified an irreducibly simple model of passive dynamic walking to walk on level ground, and used it to study the energetics of walking and the preferred relationship between speed and step length in humans. Powered walking was explored using an impulse applied at toe-off immediately before heel strike, and a torque applied on the stance leg. Although both methods can supply energy through mechanical work on the center of mass, the toe-off impulse is four times less costly because it decreases the collision loss at heel strike. We also studied the use of a hip torque on the swing leg that tunes its frequency but adds no propulsive energy to gait. This spring-like actuation can further reduce the collision loss at heel strike, improving walking energetics. An idealized model yields a set of simple power laws relating the toe-off impulses and effective spring constant to the speed and step length of the corresponding gait. Simulations incorporating nonlinear equations of motion and more realistic inertial parameters show that these power laws apply to more complex models as well.     

Tuning the cochlea: wave-mediated positive feedback between cells

Frequency analysis by the mammalian cochlea is traditionally thought to occur via a hydrodynamically coupled ‘travelling wave’ along the basilar membrane. A persistent difficulty with this picture is how sharp tuning can emerge. This paper proposes, and models, a supplementary or alternative mechanism: it supposes that the cochlea analyses sound by setting up standing waves between parallel rows of outer hair cells. In this scheme, multiple cells mutually interact through positive feedback of wave-borne energy. Analytical modelling and numerical evaluation presented here demonstrate that this can provide narrow-band frequency analysis. Graded cochlear tuning will then rely on the distance between rows becoming greater as distance from the base increases (as exhibited by the actual cochlea) and on the wave’s phase velocity becoming slower. In effect, tuning is now a case of varying the feedback delay between the rows, and a prime candidate for a wave exhibiting suitably graded phase velocity—a short-wavelength ‘squirting wave’—is identified and used in the modelling. In this way, resonance between rows could supply both amplification and high Q, characteristics underlying the ‘cochlear amplifier’—the device whose action has long been evident to auditory science but whose anatomical basis and mode of operation are still obscure.     

The kinematic consequences of invariant dynamics in children 6-18 years of age

The effect of limb dynamics on trajectory formation is unclear. The natural frequency of a limb is the major factor in its dynamics. It has previously been shown with an indirect measurement method that the natural frequency of body segments is invariant during human growth from the age of 6 to 18. The aim of our study was to determine, using a direct measurement method, whether human growth affects: (1) lower limb dynamics (i.e. the natural frequency of the lower leg) and (2) the maximum velocities of the knee during selected motor tasks. In 20 non-disabled children, 6-18 years of age, measurements were taken of the natural frequency of the lower leg (including the foot), and the maximum velocities of knee flexion and extension during voluntary movement (MVV) and at initial and terminal swing phases of self-paced walking (WAL). The velocities were also estimated using a dynamic model and the results were compared to the measured velocities with a paired t-test. Correlations among the frequencies, velocities, and body height (an indicator of growth) were calculated. The natural frequency of the lower leg (mean+/-standard deviation, omega(0)=6.58+/-0.54s(-1)), maximum velocities of knee extension and flexion during voluntary movement (MVV(e)=10.1+/-1.8rads(-1) and MVV(f)=7.8+/-1.3rads(-1), respectively), and maximum velocities of knee flexion and extension during the swing phase of walking (WAL(f)=5.4+/-0.6rads(-1) and WAL(e)=6.3+/-0.87rads(-1), respectively) were each found to be independent of body height. The MVV measured velocities were 22% larger and WAL(f) measured velocities were 25% smaller than the velocities predicted from the dynamic model (p<0.05). The study found that a segment's dynamic properties, as well as selected kinematics, may be considered invariant with human growth.     

Contributions of muscles to terminal-swing knee motions vary with walking speed

Many children with cerebral palsy walk with diminished knee extension during terminal swing, at speeds much slower than unimpaired children. Treatment of these gait abnormalities is challenging because the factors that extend the knee during normal walking, over a range of speeds, are not well understood. This study analyzed a series of three-dimensional, muscle-driven dynamic simulations to determine whether the relative contributions of individual muscles and other factors to angular motions of the swing-limb knee vary with walking speed. Simulations were developed that reproduced the measured gait dynamics of seven unimpaired children walking at self-selected, fast, slow, and very slow speeds (7 subjects×4 speeds=28 simulations). In mid-swing, muscles on the stance limb made the largest net contribution to extension of the swing-limb knee at all speeds examined. The stance-limb hip abductors, in particular, accelerated the pelvis upward, inducing reaction forces at the swing-limb hip that powerfully extended the knee. Velocity-related forces (i.e., Coriolis and centrifugal forces) also contributed to knee extension in mid-swing, though these contributions were diminished at slower speeds. In terminal swing, the hip flexors and other muscles on the swing-limb decelerated knee extension at the subjects’ self-selected, slow, and very slow speeds, but had only a minimal net effect on knee motions at the fastest speeds. Muscles on the stance limb helped brake knee extension at the subjects’ fastest speeds, but induced a net knee extension acceleration at the slowest speeds. These data—which show that the contributions of muscular and velocity-related forces to terminal-swing knee motions vary systematically with walking speed—emphasize the need for speed-matched control subjects when attempting to determine the causes of a patient's abnormal gait.