Do muscle synergies reduce the dimensionality of behavior?

Abstract

The muscle synergy hypothesis is an archetype of the notion of Dimensionality Reduction (DR) occurring in the central nervous system due to modular organization. Toward validating this hypothesis, it is important to understand if muscle synergies can reduce the state-space dimensionality while maintaining task control. In this paper we present a scheme for investigating this reduction utilizing the temporal muscle synergy formulation. Our approach is based on the observation that constraining the control input to a weighted combination of temporal muscle synergies also constrains the dynamic behavior of a system in a trajectory-specific manner. We compute this constrained reformulation of system dynamics and then use the method of system balancing for quantifying the DR; we term this approach as Trajectory Specific Dimensionality Analysis (TSDA). We then investigate the consequence of minimization of the dimensionality for a given task. These methods are tested in simulations on a linear (tethered mass) and a non-linear (compliant kinematic chain) system. Dimensionality of various reaching trajectories is compared when using idealized temporal synergies. We show that as a consequence of this Minimum Dimensional Control (MDC) model, smooth straight-line Cartesian trajectories with bell-shaped velocity profiles emerged as the optima for the reaching task. We also investigated the effect on dimensionality due to adding via-points to a trajectory. The results indicate that a trajectory and synergy basis specific DR of behavior results from muscle synergy control. The implications of these results for the synergy hypothesis, optimal motor control, motor development, and robotics are discussed.

Keywords: Hankel singular values; dimensionality reduction; modular motor control; muscle synergies; optimal motor control; system balancing.

Figure 1

Conceptual schematic of the proposed methods and control model. First a simple feedforward control and learning scheme using temporal muscle synergies Ψ(t) and full-dimensional system dynamics is used to convert Cartesian task requirements C obtained from behavior goals into the necessary synergy weight matrix Ŵ. A trajectory (and synergy basis) specific constrained reformulation is then obtained and the procedure of system balancing is used to reduce the dimensionality: the proposed Trajectory Specific Dimensionality Analysis (TSDA). The Hankel Singular Value measure computed through system balancing is developed into a performance index for minimization in the Minimum Dimensional Control (MDC) model. The resulting reduced dimensional model can instead be used within the synergy control learning in the control and learning scheme (on top) to speed up learning and adaptation in a task-specific manner.

Figure 2

Physical systems employed for demonstrating the TSDA. (A) Tethered mass (linear): motion of the mass is constrained to a 2D plane. The mass is anchored to the origin by weak passive forces and actuator forces are applied in two orthogonal directions. (B) Two-link planar compliant kinematic chain (non-linear): end-point motion is constrained to a 2D surface. Passive joint stiffness and damping effects are present and joint torques are used to actuate the system. The state-space descriptions of these systems have identical input (2), state (4), and output (2) dimensionality.

Figure 3

Trajectory Specific Dimensionality Analysis (TSDA) used to compare four benchmark trajectories. (A) The task is to reach position (0.5, 0.5) in 3 s tracing each of the four trajectories [T₁, … T₄]. Two kinds of temporal synergies are tested: (B) Fourier basis (order 4), and (C) Legendre polynomial basis (order 4) actuating the tethered mass system.

Figure 4

Trajectory Specific Dimensionality Analysis (TSDA) for comparing the Fourier and Legendre polynomial basis temporal synergies actuating the tethered mass system, tracing the benchmark trajectories [T₁, … T₄]. The synergy training is carried out using least-squares and full-dimensional inverse dynamics—The obtained weight matrices for the four trajectories are represented as Hinton diagrams (ellipse size is the magnitude, a dark region denotes positive weight and white region denotes a negative weight) for the (A) Fourier basis of size 2 × 9, and (B) Legendre polynomials of size 2 × 5. The corresponding cumulative normalized HSV magnitudes for (C) Fourier, and (D) Legendre polynomial basis synergies with the threshold t_r = 0.975 represented in both cases by the solid black line. The DR was computed as _fourier = [1, 3, 2, 3], and _legendre = [1, 3, 3, 3]. The straight line trajectory has the minimum dimensionality for both of these synergy bases.

Figure 5

Minimum dimensional control computed on the tethered mass for reaching position (0.5, 0.5) from the origin—two kinds of synergies (Legendre basis of order 6 and Fourier basis of order 4) and three desired time spans (t_d = [0.8, 1.0, 1.2]) analyzed. Trajectory of mass traces a sigmoid for all time spans and for both kinds of synergies. Trajectories (A) are similar to the minimum jerk (MJ) criterion for the Legendre polynomial basis and minimum acceleration (MA) for the Fourier basis case; (B) The corresponding bell-shaped velocity profiles. Weights corresponding to minimum dimension (C) for both Legendre polynomial and Fourier basis synergies linearly increase with movement duration across both inputs.

Figure 6

Dimensionality analysis of via-point tasks. (A) A set of via-points were specified on a circle of radius 0.353 m centered on the target (0.5, 0.5) for a reaching movement from the initial position (origin); (B) Polar plot of the variation in the dimensionality performance index against orientation of via-point with respect to origin for the two kinds of tested synergies composed of Legendre polynomial (blue) and Fourier bases (red). The minimum value of 0 is located exactly along the straight line linking origin and target for both kinds of synergies.

Figure 7

Generalization of the minimum dimensional control for reaching tasks in the tethered mass system. Optimization was initialized with a trajectory passing through a via-point offset by 0.075 m from the straight line connecting origin and target ϕ_d = (0.4, 0.4); Gradual convergence to Cartesian straight-lines with bell-shaped time-normalized velocity profiles seen during intermediate stages of the optimization (shades of gray) in the, (A) Cartesian endpoint trajectories, and (B) position and velocity traces of endpoints. (C) Change in J(_T) cost with each iteration of optimization, and (D) Hinton diagram of the initial and optimal weights and the corresponding normalized Hankel singular values.

Figure 8

Trajectory Specific Dimensionality Analysis (TSDA) computed on Legendre basis synergies (order 7) actuating the compliant kinematic chain system, the task was to reach position P_d = (0.5, 0.2) in a time span of 2.5 s, the initial condition was a nearly fully extended kinematic chain. (A) Four benchmark trajectories [T₁, … T₄] traced by the mass under synergy control—synergy weights were computed from via-points using a least-squares approach; (B) Hinton diagram of the weight matrix (ellipse size is the magnitude, a dark region denotes positive weight and white region denotes a negative weight). (C) The normalized empirical HSV magnitudes for the non-linear reformulated composite systems for each trajectory. For a threshold magnitude choice of t_r = 0.935, represented by the solid black line, the DR was computed as = [1, 2, 2, 2]. The straight line trajectory T₁ has minimum dimensionality as measured by the HSV magnitudes.

Figure 9

Minimum dimensional control on the kinematic chain for reaching various positions using Legendre basis synergy (order 7). Minimum dimensional trajectories were obtained for targets (0.7, 0.2), (0.6, 0.4), (0.6, 0.2), and (0.5, 0.1) in a duration of 2.5, (0.6, 0.0) in 3.5 s and (0.4, 0.2) in 4 s, respectively. (A) Near straight lines seen in the Cartesian endpoint trajectories. (B) Trajectory of endpoint is sigmoidal, and (C) time-normalized velocity profiles show slightly skewed bell shapes. The peaks of the velocity profiles, however, are close match to the minimum acceleration (MA) criterion result.

Figure 10

Dimensionality analysis for via-point tasks. (A) A set of cartesian via-points were specified on a circle of radius 0.216 m centered on the target (0.4, 0.2) for a reaching movement from the initial position; (B) Polar plot of the variation in the dimensionality performance index against orientation of via-point with respect to origin. The minimum index of 11.21 is located at the orientation of 286.15° corresponding to the via-point at (0.61, 0.14) which is very close to the straight line linking origin and target.

Figure 11

Generalization of the minimum dimensional control in reaching in the compliant kinematic chain system. Optimization was initialized with a trajectory passing through a via-point offset by 0.04 m from the straight line connecting origin and target (0.4, 0.4); Gradual convergence to Cartesian straight-lines with bell-shaped time-normalized velocity profiles seen during intermediate stages of the optimization (shades of gray) in the, (A) cartesian endpoint trajectories, and (B) position and velocity traces of endpoints. (C) Change in J(_T) cost with each iteration of optimization, and (D) Hinton diagram of the initial and optimal weights and the corresponding normalized Hankel singular values.