Optimal control of antagonistic muscle stiffness during voluntary movements

This paper presents a study on the control of antagonist muscle stiffness during single-joint arm movements by optimal control theory with a minimal effort criterion. A hierarchical model is developed based on the physiology of the neuromuscular control system and the equilibrium point hypothesis. For point-to-point movements, the model provides predictions on (1) movement trajectory, (2) equilibrium trajectory, (3) muscle control inputs, and (4) antagonist muscle stiffness, as well as other variables. We compared these model predictions to the behavior observed in normal human subjects. The optimal movements capture the major invariant characteristics of voluntary movements, such as a sigmoidal movement trajectory with a bell-shaped velocity profile, an N-shaped equilibrium trajectory, a triphasic burst pattern of muscle control inputs, and a dynamically modulated joint stiffness. The joint stiffness is found to increase in the middle of the movement as a consequence of the triphasic muscle activities. We have also investigated the effects of changes in model parameters on movement control. We found that the movement kinematics and muscle control inputs are strongly influenced by the upper bound of the descending excitation signal that activates motoneuron pools in the spinal cord. Furthermore, a class of movements with scaled velocity profiles can be achieved by tuning the amplitude and duration of this excitation signal. These model predictions agree with a wide body of experimental data obtained from normal human subjects. The results suggest that the control of fast arm movements involves explicit planning for both the equilibrium trajectory and joint stiffness, and that the minimal effort criterion best characterizes the objective of movement planning and control.     

Velocity-Based Planning of Rapid Elbow Movements Expands the Control Scheme of the Equilibrium Point Hypothesis

According to the equilibrium point hypothesis of voluntary motor control, control action of muscles is not explicitly computed, but rather arises as a consequence of interaction between moving equilibrium position, current kinematics and stiffness of the joint. This approach is attractive as it obviates the need to explicitly specify the forces controlling limb movements. However, many debatable aspects of this hypothesis remain in the manner of specification of the equilibrium point trajectory and muscle activation (or its stiffness), which elicits a restoring force toward the planned equilibrium trajectory. In this study, we expanded the framework of this hypothesis by assuming that the control system uses the velocity measure as the origin of subordinate variables scaling descending commands. The velocity command is translated into muscle control inputs by second order pattern generators, which yield reciprocal command and coactivation commands, and create alternating activation of the antagonistic muscles during movement and coactivation in the post-movement phase, respectively. The velocity command is also integrated to give a position command specifying a moving equilibrium point. This model is purely kinematics-dependent, since the descending commands needed to modulate the visco-elasticity of muscles are implicitly given by simple parametric specifications of the velocity command alone. The simulated movements of fast elbow single-joint movements corresponded well with measured data performed over a wide range of movement distances, in terms of both muscle excitations and kinematics. Our proposal on a synthesis for the equilibrium point approach and velocity command, may offer some insights into the control scheme of the single-joint arm movements.     

A model for learning human reaching movements

 Reaching movement is a fast movement towards a given target. The main characteristics of such a movement are straight path and a bell-shaped speed profile. In this work a mathematical model for the control of the human arm during ballistic reaching movements is presented. The model of the arm contains a 2 degrees of freedom planar manipulator, and a Hill-type, non-linear mechanical model of six muscles. The arm model is taken from the literature with minor changes. The nervous system is modeled as an adjustable pattern generator that creates the control signals to the muscles. The control signals in this model are rectangular pulses activated at various amplitudes and timings, that are determined according to the given target. These amplitudes and timings are the parameters that should be related to each target and initial conditions in the workspace. The model of the nervous system consists of an artificial neural net that maps any given target to the parameter space of the pattern generator. In order to train this net, the nervous system model includes a sensitivity model that transforms the error from the arm end-point coordinates to the parameter coordinates. The error is assessed only at the termination of the movement from knowledge of the results. The role of the non-linearity in the muscle model and the performance of the learning scheme are analysed, illustrated in simulations and discussed. The results of the present study demonstrate the central nervous system’s (CNS) ability to generate typical reaching movements with a simple feedforward controller that controls only the timing and amplitude of rectangular excitation pulses to the muscles and adjusts these parameters based on knowledge of the results. In this scheme, which is based on the adjustment of only a few parameters instead of the whole trajectory, the dimension of the control problem is reduced significantly. It is shown that the non-linear properties of the muscles are essential to achieve this simple control. This conclusion agrees with the general concept that motor control is the result of an interaction between the nervous system and the musculoskeletal dynamics. Received : 21 May 1996 / Accepted in revised form : 10 June 1997     

Roles of dynamic and stationary components of a motor command in establishing the equilibrium state at simplest targeted movements

We studied in humans interrelations between the kinematic characteristics of targeted movements of the arm and current levels of EMG of the muscles providing these movements; the movements were relatively slow, and the attained joint angle was held for a time. The EMG level was considered a correlate of the level of integral motor commands (efferent activity of the respective motoneuronal pools). Application of low-amplitude non-inertial loadings, directed against the forces developed by one or another muscle group, allowed us to provide realization of targeted movements exclusively by the activity of this muscle group, without Involvement of the antagonists. It was demonstrated that the target equilibrium joint angle is reached synchronously with the dynamic phase of EMG activity, before the latter reaches a stationary level. The structure of the dynamic EMG phase itself is complex; in most cases it is split into several components. The dependence between the joint angle and amplitude of the EMG stationary phase is rather complex and variable, and usually it is difficult to predict the characteristics of this phase based on simple biomechanical considerations. There are proofs that at the performance of the studied movements and maintaining a target position there are some components in the mechanical muscle activity, which are not controlled by the motor commands. Thus, the stationary level of a motor command represents only one of several factors responsible for attaining and maintaining a target equilibrium position. Establishing this position is provided, first of all, by interaction of dynamic components of the motor commands to different muscles. Our results show that the attempts to interpret the processes of control of targeted movements on the basis of modifications of the equilibrium point hypothesis are inadequate; these data are in better compliance with the concept of impulse-temporal control; at their interpretation it is also necessary to take more thoroughly into account nonlinear properties of the muscle reactions.     

Equilibrium point control of a monkey arm simulator by a fast learning tree structured artificial neural network

A planar 17 muscle model of the monkey's arm based on realistic biomechanical measurements was simulated on a Symbolics Lisp Machine. The simulator implements the equilibrium point hypothesis for the control of arm movements. Given initial and final desired positions, it generates a minimum-jerk desired trajectory of the hand and uses the backdriving algorithm to determine an appropriate sequence of motor commands to the muscles (Flash 1987; Mussa-Ivaldi et al. 1991; Dornay 1991b). These motor commands specify a temporal sequence of stable (attractive) equilibrium positions which lead to the desired hand movement. A strong disadvantage of the simulator is that it has no memory of previous computations. Determining the desired trajectory using the minimum-jerk model is instantaneous, but the laborious backdriving algorithm is slow, and can take up to one hour for some trajectories. The complexity of the required computations makes it a poor model for biological motor control. We propose a computationally simpler and more biologically plausible method for control which achieves the benefits of the backdriving algorithm. A fast learning, tree-structured network (Sanger 1991c) was trained to remember the knowledge obtained by the backdriving algorithm. The neural network learned the nonlinear mapping from a 2-dimensional cartesian planar hand position {x, y} to a 17-dimensional motor command space {u ₁, ..., u ₁₇}. Learning 20 training trajectories, each composed of 26 sample points {{x y{,{u ₁, ..., u ₁₇} took only 20 min on a Sun-4 Spare workstation. After the learning stage, new, untrained test trajectories as well as the original trajectories of the hand were given to the neural network as input. The network calculated the required motor commands for these movements. The resulting movements were close to the desired ones for both the training and test cases.     

Trajectory formation based on physiological characteristics of skeletal muscles

Human arm trajectories in natural unrestricted reaching movements were studied. They have particular properties such that a hand path is a rather simple straight or curved line, and a tangential velocity profile of hand is bell-shaped. Also these properties are invariant, independent of movement duration and hand-held load. In this study, trajectory formation is investigated on the basis of physiological characteristics of skeletal muscles, and a criterion prescribed by a derivative of isometric muscle torque is proposed. Subsequently, optimal trajectories are formulated under various conditions of movement to account for a planning strategy of human arm trajectories. In addition to such a theoretical approach, human arm trajectories are experimentally observed by a measuring system which provides a visual sensor and a target tracking device, enabling totally unrestricted movements. Then, optimal trajectories are quantitatively evaluated in comparison with experimental data in which essential properties of human arm trajectories are demonstrated. These results support the idea that human arm trajectories are planned in order to minimize the proposed criterion which is determined from physiological aspects. Finally, the physiological advantages of human arm trajectories are discussed with regard to the analysis of observed and optimal trajectories. Received: 2 December 1997 / Accepted in revised form: 20 March 1998     

On the possibility of linear modelling the human arm neuromuscular apparatus

It has been widely claimed that linear models of the neuromuscular apparatus give very inaccurate approximations of human arm reaching movements. The present paper examines this claim by quantifying the contributions of the various non-linear effects of muscle force generation on the accuracy of linear approximation. We performed computer simulations of a model of a two-joint arm with six monarticular and biarticular muscles. The global actions of individual muscles resulted in a linear dependence of the joint torques on the joint angles and angular velocities, despite the great non-linearity of the muscle properties. The effect of time delay in force generation is much more important for model accuracy than all the non-linear effects, while ignoring this time delay in linear approximation results in large errors. Thus, the viscosity coefficients are rather underestimated and some of them can even be paradoxically estimated to be negative. Similarly, our computation showed that ignoring the time delay resulted in large errors in the estimation of the hand equilibrium trajectory. This could explain why experimentally estimated hand equilibrium trajectories may be complex, even during a simple reaching movement. The hand equilibrium trajectory estimated by a linear model becomes simple when the time delay is taken into account, and it is close to that actually used in the non-linear model. The results therefore provide a theoretical basis for estimating the hand equilibrium trajectory during arm reaching movements and hence for estimating the time course of the motor control signals associated with this trajectory, as set out in the equilibrium point hypothesis. Received: 17 February 1999 / Accepted in revised form: 22 October 1999     

Motor commands for fast point-to-point arm movements are customized for small changes in inertial load

For repeated point-to-point arm movements it is often assumed that motor commands are customized in a trial-to-trial manner, based on previous endpoint error. To test this assumption, we perturbed movement execution without affecting the endpoint error by using a modest manipulation of inertia. Participants made point-to-point elbow flexion and extension movements in the horizontal plane, under the instruction to move as fast as possible from one target area to another. In selected trials the moment of inertia of the lower arm was increased or decreased by 25%. First, we found that an unexpected increase or decrease of inertia did not affect the open loop controlled part of the movement path (and thus endpoint error was not affected). Second, we found that when the increased or decreased inertia was presented repeatedly, after 5-11 trials motor commands were customized: the first 100ms of agonistic muscle activity in the smoothed and rectified electromyographic signal of agonistic muscles was higher for the high inertia compared to the low inertia. We conclude that endpoint error is not the only parameter that is used to evaluate if motor commands lead to movements as planned.     

Novel peripheral motor neurons in the posterior tentacles of the snail responsible for local tentacle movements

Three flexor muscles of the posterior tentacles of the snail Helix pomatia have recently been described. Here, we identify their local motor neurons by following the retrograde transport of neurobiotin injected into these muscles. The mostly unipolar motor neurons (15–35 µm) are confined to the tentacle digits and send motor axons to the M2 and M3 muscles. Electron microscopy revealed small dark neurons (5–7 µm diameter) and light neurons with 12–18 (T1 type) and 18–30 µm diameters (T2 type) in the digits. The diameters of the neurobiotin-labeled neurons corresponded to the T1 type light neurons. The neuronal processes of T1 type motor neurons arborize extensively in the neuropil area of the digits and receive synaptic inputs from local neuronal elements involved in peripheral olfactory information processing. These findings support the existence of a peripheral stimulus–response pathway, consisting of olfactory stimulus—local motor neuron—motor response components, to generate local lateral movements of the tentacle tip (“quiver”). In addition, physiological results showed that each flexor muscle receives distinct central motor commands via different peritentacular nerves and common central motor commands via tentacle digits, respectively. The distal axonal segments of the common pathway can receive inputs from local interneurons in the digits modulating the motor axon activity peripherally without soma excitation. These elements constitute a local microcircuit consisting of olfactory stimulus—distal segments of central motor axons—motor response components, to induce patterned contraction movements of the tentacle. The two local microcircuits described above provide a comprehensive neuroanatomical basis of tentacle movements without the involvement of the CNS.     

Similar movements are associated with drastically different muscle contraction velocities

We investigated how kinematic redundancy interacts with the neurophysiological control mechanisms required for smooth and accurate, rapid limb movements. Biomechanically speaking, tendon excursions are over-determined because the rotation of few joints determines the lengths and velocities of many muscles. But how different are the muscle velocity profiles induced by various, equally valid hand trajectories? We used an 18-muscle sagittal-plane arm model to calculate 100,000 feasible shoulder, elbow, and wrist joint rotations that produced valid basketball free throws with different hand trajectories, but identical initial and final hand positions and velocities. We found large differences in the eccentric and concentric muscle velocity profiles across many trajectories; even among similar trajectories. These differences have important consequences to their neural control because each trajectory will require unique, time-sensitive reflex modulation strategies. As Sherrington mentioned a century ago, failure to appropriately silence the stretch reflex of any one eccentrically contracting muscle will disrupt movement. Thus, trajectories that produce faster or more variable eccentric contractions will require more precise timing of reflex modulation across motoneuron pools; resulting in higher sensitivity to time delays, muscle mechanics, excitation/contraction dynamics, noise, errors and perturbations. By combining fundamental concepts of biomechanics and neuroscience, we propose that kinematic and muscle redundancy are, in fact, severely limited by the need to regulate reflex mechanisms in a task-specific and time-critical way. This in turn has important consequences to the learning and execution of accurate, smooth and repeatable movements—and to the rehabilitation of everyday limb movements in developmental and neurological conditions, and stroke.     

Using voluntary motor commands to inhibit involuntary arm movements

A hallmark of voluntary motor control is the ability to stop an ongoing movement. Is voluntary motor inhibition a general neural mechanism that can be focused on any movement, including involuntary movements, or is it mere termination of a positive voluntary motor command? The involuntary arm lift, or ‘floating arm trick’, is a distinctive long-lasting reflex of the deltoid muscle. We investigated how a voluntary motor network inhibits this form of involuntary motor control. Transcranial magnetic stimulation of the motor cortex during the floating arm trick produced a silent period in the reflexively contracting deltoid muscle, followed by a rebound of muscle activity. This pattern suggests a persistent generator of involuntary motor commands. Instructions to bring the arm down voluntarily reduced activity of deltoid muscle. When this voluntary effort was withdrawn, the involuntary arm lift resumed. Further, voluntary motor inhibition produced a strange illusion of physical resistance to bringing the arm down, as if ongoing involuntarily generated commands were located in a ‘sensory blind-spot’, inaccessible to conscious perception. Our results suggest that voluntary motor inhibition may be a specific neural function, distinct from absence of positive voluntary motor commands.     

Correlations in state space can cause sub-optimal adaptation of optimal feedback control models

Control of our movements is apparently facilitated by an adaptive internal model in the cerebellum. It was long thought that this internal model implemented an adaptive inverse model and generated motor commands, but recently many reject that idea in favor of a forward model hypothesis. In theory, the forward model predicts upcoming state during reaching movements so the motor cortex can generate appropriate motor commands. Recent computational models of this process rely on the optimal feedback control (OFC) framework of control theory. OFC is a powerful tool for describing motor control, it does not describe adaptation. Some assume that adaptation of the forward model alone could explain motor adaptation, but this is widely understood to be overly simplistic. However, an adaptive optimal controller is difficult to implement. A reasonable alternative is to allow forward model adaptation to ‘re-tune’ the controller. Our simulations show that, as expected, forward model adaptation alone does not produce optimal trajectories during reaching movements perturbed by force fields. However, they also show that re-optimizing the controller from the forward model can be sub-optimal. This is because, in a system with state correlations or redundancies, accurate prediction requires different information than optimal control. We find that adding noise to the movements that matches noise found in human data is enough to overcome this problem. However, since the state space for control of real movements is far more complex than in our simple simulations, the effects of correlations on re-adaptation of the controller from the forward model cannot be overlooked.     

A review on directional information in neural signals for brain-machine interfaces

Brain-machine interfaces (BMIs) can be characterized by the technique used to measure brain activity and by the way different brain signals are translated into commands that control an effector. We give an overview of different approaches and focus on a particular BMI approach: the movement of an artificial effector (e.g. arm prosthesis to the right) by those motor cortical signals that control the equivalent movement of a corresponding body part (e.g. arm movement to the right). This approach has been successfully applied in monkeys and humans by accurately extracting parameters of movements from the spiking activity of multiple single-units. Here, we review recent findings showing that analog neuronal population signals, ranging from intracortical local field potentials over epicortical ECoG to non-invasive EEG and MEG, can also be used to decode movement direction and continuous movement trajectories. Therefore, these signals might provide additional or alternative control for this BMI approach, with possible advantages due to reduced invasiveness.     

Motor control of voluntary arm movements

The motor control of pointing and reaching-to-grasp movements was investigated using two different approaches (kinematic and modelling) in order to establish whether the type of control varies according to modifications of arm kinematics. Kinematic analysis of arm movements was performed on subjects' hand trajectories directed to large and small stimuli located at two different distances. The subjects were required either to grasp and to point to each stimulus. The kinematics of the subsequent movement, during which subject's hand came back to the starting position, were also studied. For both movements, kinematic analysis was performed on hand linear trajectories as well as on joint angular trajectories of shoulder and elbow. The second approach consisted in the parametric identification of the black box (ARMAX) model of the controller driving the arm movement. Such controller is hypothesized to work for the correct execution of the motor act. The order of the controller ARMAX model was analyzed with respect to the different experimental conditions (distal task, stimulus size and distance). Results from kinematic analysis showed that target distance and size influenced kinematic parameters both of angular and linear displacements. Nevertheless, the structure of the motor program was found to remain constant with distane and distal task, while it varied with precision requirements due to stimulus size. The estimated model order of the controller confirmed the invariance of the control law with regard to movement amplitude, whereas it was sensitive to target size.     

Influence of Vibration on Motor Responses in the Upper Arm Muscles under Stationary Conditions and during Voluntary and Evoked Arm Movements under Limb Unloading Conditions

Locomotion of mammals, including humans, is based on the rhythmic activity of spinal cord circuitries. The functioning of these circuitries depends on multimodal afferent information and on supraspinal influences from the motor cortex. Using the method of transcranial magnetic stimulation (TMS) of arm muscle areas in the motor cortex, we studied the motor evoked potentials (MEP) in the upper arm muscles in stationary conditions and during voluntary and vibration-evoked arm movements. The study included 13 healthy subjects under arm and leg unloading conditions. In the first series of experiments, with motionless limbs, the effect of vibration of left upper arm muscles on motor responses in these muscles was evaluated. In the second series of experiments, MEP were compared in the same muscles during voluntary and rhythmic movements generated by left arm m. triceps brachii vibration (the right arm was stationary). Motionless left arm vibration led to an increase in MEP values in both vibrated muscle and in most of the non-vibrated muscles. For most target muscles, MEP was greater with voluntary arm movements than with vibration-evoked movements. At the same time, a similar MEP modulation in the cycle of arm movements was observed in the same upper arm muscles during both types of arm movements. TMS of the motor cortex significantly potentiated arm movements generated by vibration, but its effect on voluntary movements was weaker. These results indicate significant differences in the degree of motor cortex involvement in voluntary and evoked arm movements. We suppose that evoked arm movements are largely due to spinal rather than central mechanisms of generation of rhythmic movements.     

The inactivation principle: mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements

An important question in the literature focusing on motor control is to determine which laws drive biological limb movements. This question has prompted numerous investigations analyzing arm movements in both humans and monkeys. Many theories assume that among all possible movements the one actually performed satisfies an optimality criterion. In the framework of optimal control theory, a first approach is to choose a cost function and test whether the proposed model fits with experimental data. A second approach (generally considered as the more difficult) is to infer the cost function from behavioral data. The cost proposed here includes a term called the absolute work of forces, reflecting the mechanical energy expenditure. Contrary to most investigations studying optimality principles of arm movements, this model has the particularity of using a cost function that is not smooth. First, a mathematical theory related to both direct and inverse optimal control approaches is presented. The first theoretical result is the Inactivation Principle, according to which minimizing a term similar to the absolute work implies simultaneous inactivation of agonistic and antagonistic muscles acting on a single joint, near the time of peak velocity. The second theoretical result is that, conversely, the presence of non-smoothness in the cost function is a necessary condition for the existence of such inactivation. Second, during an experimental study, participants were asked to perform fast vertical arm movements with one, two, and three degrees of freedom. Observed trajectories, velocity profiles, and final postures were accurately simulated by the model. In accordance, electromyographic signals showed brief simultaneous inactivation of opposing muscles during movements. Thus, assuming that human movements are optimal with respect to a certain integral cost, the minimization of an absolute-work-like cost is supported by experimental observations. Such types of optimality criteria may be applied to a large range of biological movements.     

A control model of human tongue movements in speech

Tongue movements during speech production have been investigated by means of a simple yet realistic biomechanical model, based on a finite elements modeling of soft tissues, in the framework of the equilibrium point hypothesis (-model) of motor control. In particular, the model has been applied to the estimation of the “central” control commands issued to the muscles, for a data set of mid-sagittal digitized tracings of vocal tract shape, r ecorded by means of low-intensity X-ray cineradiographies during speech. In spite of the highly non-linear mapping between the shape of the oral cavity and its acoustic consequences, the organization of control commands preserves the peculiar spatial organization of vowel phonemes in acoustic space. A factor analysis of control commands, which have been decomposed into independent or “orthogonal” muscle groups, has shown that, in spite of the great mobility of the tongue and the highly complex arrangement of tongue muscles, its movements can be explained in terms of the activation of a small number of independent muscle groups, each corresponding to an elementary or “primitive” movement. These results are consistent with the hypothesis that the tongue is controlled by a small number of independent “articulators”, for which a precise biomechanical substrate is provided. The influence of the effect of jaw and hyoid movements on tongue equilibrium has also bee n evaluated, suggesting that the bony structures cannot be considered as a moving frame of reference, but, indeed, there may be a substantial interaction between them and the tongue, that may only be accounted for by a “global” model. The reported results also define a simple control model for the tongue and, in analogy with similar modelling studies, they suggest that, because of the peculiar geometrical arrangement of tongue muscles, the central nervous system (CNS) may not need a de tailed representation of tongue mechanics but rather may make use of a relatively small number of muscle synergies, that are invariant over the whole space of tongue configurations. Received: 27 August 1996 / Accepted in revised form: 25 February 1997     

Interlimb interactions during cyclic in-phase and antiphase movements of arms and legs and their dependence on afferent influences

Coordinated arm and leg movements imply neural interactions between the rhythmic generators of the upper and lower extremities. In ten healthy subjects in the lying position, activity of the muscles of the upper and lower extremities was recorded during separate and joint cyclic movements of the arms and legs with different phase relationships between the movements of the limbs and under various conditions of the motor task. Antiphase active arm movements were characterized by higher muscle activity than during the inphase mode. The muscle activity during passive arm movements imposed by the experimentalist was significantly lower than muscle activity during passive arm movements imposed by the other arm. When loading one arm, the muscle activity in the other, passively moving, arm increased independently from the synergy of arm movements. During a motor task implementing joint antiphase movements of both upper and lower extremities, compared to a motor task implementing their joint in-phase movements, we observed a significant increase in activity in the biceps brahii muscle, the tibialis anterior muscle, and the biceps femoris muscle. Loading of arms in these motor tasks has been accompanied by increased activity in some leg muscles. An increase in the frequency of rhythmic movements resulted in a significant growth of the muscle activity of the arms and legs during their cooperative movements with a greater rate of rise in the flexor muscle activity of the arms and legs during joint antiphase movements. Thus, both the spatial organization of movements and the type of afferent influences are significant factors of interlimb interactions, which, in turn, determine the type of neural interconnections that are involved in movement regulation.