Exploring the high-dimensional structure of muscle redundancy via subject-specific and generic musculoskeletal models

Abstract

Subject-specific and generic musculoskeletal models are the computational instantiation of hypotheses, and stochastic techniques help explore their validity. We present two such examples to explore the hypothesis of muscle redundancy. The first addresses the effect of anatomical variability on static force capabilities for three individual cat hindlimbs, each with seven kinematic degrees of freedom (DoFs) and 31 muscles. We present novel methods to characterize the structure of the 31-dimensional set of feasible muscle activations for static force production in every 3-D direction. We find that task requirements strongly define the set of feasible muscle activations and limb forces, with few differences comparing individual vs. species-average results. Moreover, muscle activity is not smoothly distributed across 3-D directions. The second example explores parameter uncertainty during a flying disc throwing motion by using a generic human arm with five DoFs and 17 muscles to predict muscle fiber velocities. We show that the measured joint kinematics fully constrain the eccentric and concentric fiber velocities of all muscles via their moment arms. Thus muscle activation for limb movements is likely not redundant: there is little, if any, latitude in synchronizing alpha-gamma motoneuron excitation-inhibition for muscles to adhere to the time-critical fiber velocities dictated by joint kinematics. Importantly, several muscles inevitably exhibit fiber velocities higher than thought tenable, even for conservative throwing speeds. These techniques and results, respectively, enable and compel us to continue to revise the classical notion of muscle redundancy for increasingly more realistic models and tasks.

Keywords: Computational models; Monte Carlo simulation; Muscle redundancy; Stochastic modeling.

Figure 1

Bone lengths, joint axes of rotation, and moment arm matrix for the species average cat hindlimb model, in cm. Positive values are shown in red and negative values in blue, as per the right-hand-rule.

Figure 2

Left: The polygon of the 2-D feasible force set in the sagittal plane. The color-coded vectormapping of radial lines indicate the magnitude of the maximal feasible force along that direction, then vectormapped onto the perimeter of the circle surrounding the FFS. The very thin lines emanating from the origin are the lines of action of each of the 31 muscles. Center: the polyhedron of the 3-D FFS, again with the vectormapping of force magnitude values onto a circle in the sagittal plane. Right: The color-coded vectormapping onto the surface of a sphere indicating the maximal feasible force in every direction in 3-D. Note the FFS is rather flat on the sagittal plane, but elongated towards the posterior direction. All data are for the cat called Birdy in [2]. For 3-D views see [[[add Elsevier URL]]].

Figure 3

Top: Vectormap of the average of maximal feasible force across all sampled output vectors in three feline hindlimbs. Bottom: A vectormap displaying regions of the feasible force space that have higher standard deviation across three cat hindlimbs. Force in Newtons represented by color scale.

Figure 4

Structure of the feasible activation set for three muscles. The large vectormaps on the far Right show their unique activation level for maximal force output in every 3-D direction. Because multiple activation levels can produce submaximal forces, the small vectormaps to the Left show the lower and upper bounds of those feasible activation levels for force magnitudes (a) gradually increasing from 50% of maximal in every 3-D

Figure 5

Moment arm values for human arm model. The moment arms from the 17 muscles considered in this model and their associations with the five DoFs are illustrated, in cm. The moment arms are grouped by DoF and are shown below the associated joint. Positive values are shown in red and negative values in blue.

Figure 6

Top view of the 3-D human arm model. This figure illustrates the initiation of forward motion through follow-through of the flying disc throw. The reference posture is shown in black and the release point in the throw is shown in red. The interpolated joint angles for the 45 postures describing this motion, obtained from [1], are shown in the bottom panel.

Figure 7

Normalized instantaneous fiber velocities during the throw for the nominal model. Top: The muscles are listed on the y-axis and the 45 postures making up the throw are shown on the x-axis. Excessive muscle velocities are shown in red (shortening) and blue (lengthening). Bottom: The same data are illustrated with individual traces for each muscle that show the fiber velocity. Muscles controlling the shoulder, elbow, and wrist are illustrated in blue, red, and green, respectively. Instantaneous fiber velocity is given on the y-axis and the postures during the throw are on the x-axis. Regions of the traces outside of the horizontal dashed lines indicate excessive muscle velocities. In both figures, the release point of the throw is indicated with a vertical dashed line.