Based on the idea of synergy to explore the building blocks of movements, this study focused on the muscle space for reaching movements by human upper limbs on a horizontal plane to estimate the relationship among muscle synergies, equilibrium-point (EP) trajectories, and endpoint stiffness in two ways: (1) a novel estimation method that analyzes electromyographic signals under the concept of agonistantagonist (A-A) muscle pairs and (2) a conventional estimation method that uses mechanical perturbations. The experimental results suggest that (1) muscle activities of reaching movements by human upper limbs are represented by only three functional muscle synergies; (2) each muscle synergy balances the coactivations of A-A muscle pairs; (3) two of the muscle synergies are invariant bases that form an EP trajectory described in polar coordinates centered on a shoulder joint, where one is a composite unit for radial movement and the other is for angular movement; and (4) the third muscle synergy is the invariant basis for additional adjustment of the endpoint stiffness and has some influence on the direction and size of the endpoint stiffness ellipse.

Investigation of neural representation of movement planning has attracted the attention of neuroscientists, as it may reveal the sensorimotor transformation essential to motor control. The analysis of muscle synergies based on the activity of agonist–antagonist (AA) muscle pairs may provide insight into such transformations, especially for a reference frame in the muscle space. In this study, we examined the AA concept using the following explanatory variables: the AA ratio, which is related to the equilibrium-joint angle, and the AA sum, which is associated with joint stiffness. We formulated muscle synergies as a function of AA sums, positing that muscle synergies are composite units of mechanical impedance. The AA concept can be regarded as another form of the equilibrium-point (EP) hypothesis, and it can be extended to the concept of EP-based synergies. We introduce, here, a novel tool for analyzing the neurological and motor functions underlying human movements and review some initial insights from our results about the relationships between muscle synergies, endpoint stiffness, and virtual trajectories (time series of EP). Our results suggest that (1) muscle synergies reflect an invariant balance in the co-activation of AA muscle pairs; (2) each synergy represents the basis for the radial, tangential, and null movements of the virtual trajectory in the polar coordinates centered on the specific joint at the base of the body; and (3) the alteration of muscle synergies (for example, due to spasticity or rigidity following neurological injury) results in significant distortion of endpoint stiffness and concomitant virtual trajectories. These results indicate that muscle synergies (i.e., the balance of muscle mechanical impedance) are essential for motor control.

This paper proposes a novel method for assessment of muscle imbalance based on muscle synergy hypothesis and equilibrium point (EP) hypothesis of motor control. We explain in detail the method for extracting muscle synergies under the concept of agonist-antagonist (AA) muscle pairs and for estimating EP trajectories and endpoint stiffness of human upper limbs in a horizontal plane using an electromyogram. The results of applying this method to the reaching movement of one normal subject and one hemiplegic subject suggest that (1) muscle synergies (the balance among coactivation of AA muscle pairs), particularly the synergies that contributes to the angular directional kinematics of EP and the limb stiffness, are quite different between the normal subject and the hemiplegic subject; (2) the concomitant EP trajectory is also different between the normal and hemiplegic subjects, corresponding to the difference of muscle synergies; and (3) the endpoint (hand) stiffness ellipse of the hemiplegic subject becomes more elongated and orientation of the major axis rotates clockwise more than that of the normal subject. The level of motor impairment would be expected to be assessed from a comparison of these differences of muscle synergies, EP trajectories, and endpoint stiffness among normal and pathological subjects using the method.

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Stroke rehabilitation requires intralimb coordination to achieve natural movement after recovery. Focusing on mechanical impedance by the coactivation of agonist muscles, we performed two experiments to assess the intralimb coordination of a post-stroke subject using two indices of the endpoint stiffness and muscle synergies. The results of the first experiment showed that the endpoint stiffness of a post-stroke subject during posture maintenance estimated from muscle synergy analysis resembled that estimated from the mechanical perturbation method. Based on the validity of proposed muscle synergy analysis shown in the first experiment, the results of the second experiment revealed that muscle activities of both the post-stroke and healthy subjects are composed of three muscle synergies in the circle-tracing task. These muscle synergies were invariant despite being determined from time-variant muscle activities; muscle synergies of the post-stroke subject before rehabilitation were different from those of the healthy subject. In addition, the muscle synergies of the post-stroke subject after rehabilitation resembled those of the healthy subject. It is assumed that the post-stroke subject regained appropriate muscle synergies (i.e., the balance of mechanical impedance) after rehabilitation. This study tested the feasibility for practical uses in the assessment, diagnosis, and interventions for stroke rehabilitation using two indices of muscle synergies and endpoint stiffness.