Synergies in the space of control variables within the equilibrium-point hypothesis

Abstract

We use an approach rooted in the recent theory of synergies to analyze possible co-variation between two hypothetical control variables involved in finger force production based on the equilibrium-point (EP) hypothesis. These control variables are the referent coordinate (R) and apparent stiffness (C) of the finger. We tested a hypothesis that inter-trial co-variation in the {R; C} space during repeated, accurate force production trials stabilizes the fingertip force. This was expected to correspond to a relatively low amount of inter-trial variability affecting force and a high amount of variability keeping the force unchanged. We used the "inverse piano" apparatus to apply small and smooth positional perturbations to fingers during force production tasks. Across trials, R and C showed strong co-variation with the data points lying close to a hyperbolic curve. Hyperbolic regressions accounted for over 99% of the variance in the {R; C} space. Another analysis was conducted by randomizing the original {R; C} data sets and creating surrogate data sets that were then used to compute predicted force values. The surrogate sets always showed much higher force variance compared to the actual data, thus reinforcing the conclusion that finger force control was organized in the {R; C} space, as predicted by the EP hypothesis, and involved co-variation in that space stabilizing total force.

Keywords: apparent stiffness; equilibrium-point hypothesis; finger force; isometric; synergy; uncontrolled manifold hypothesis.

Figure 1

Model for finger force generation. Finger force generated by the fingertip in proportion to the difference between the nervous-system-defined referent coordinate (R_FT) and the fingertip actual configuration (X_FT). In Panels (A) and (B), the lab-fixed coordinate frame is conveniently located at the actual fingertip position, and the distance coordinate is measured positive upwards. Panels A and B depict the initial configuration and the configuration at the end of the upward perturbation of the sensor, respectively. Panel (C) depicts the sensor displacement vs fingertip force (absolute value) relation. The slope provides an estimate of the apparent fingertip stiffness (C_FT), and the force-axis intercept provides the estimate of R_FT.

Figure 2

The experimental setup. Panel (A) shows the subject with the fingers of the test hand resting on four force sensors. The laser measures the sensor displacement, and visual feedback of the produced force is presented on a computer screen placed in front of the subject. Panel (B) shows the visual feedback. The target force is presented as a solid horizontal line. The subject’s output force is displayed as a continuous trace moving from the left to the right for the first 5 s of the trial. Then the feedback disappears.

Figure 3

Typical subject response and computation of the referent variables. Panel (A) shows the temporal evolution of the normalized fingertip force (absolute value). Panel (B) plots the sensor displacement against time. Visual feedback is removed at the 5-s mark. The sensor was lifted and lowered in each trial. However, only the initial portion when the sensor height increased was used for further analysis. This portion is indicated by shaded rectangles in Panels (A) and (B). Panel (C) plots the fingertip force against the sensor displacement during the rising phase of the sensor displacement. It also depicts the computation of the referent fingertip position R_FT and the fingertip apparent stiffness C_FT.

Figure 4

Generation of surrogate data. Panel (A) shows a set of twenty {R_FT; C_FT} pairs for one finger condition (data is pooled across the hands) as the solid dots that align with the hyperbolic curve. The surrogate data {R_FTsur; C_FTsur} obtained by permuting R_FT and C_FT are depicted as crosses. Panel (B) depicts the actual forces obtained from the original data set as solid dots, and the surrogate forces obtained from the permuted data as crosses. The standard deviation (StDev) in the original forces is lower than that in the forces generated using the surrogate data (f_sur).

Figure 5

Referent variables for a random subject for all finger conditions are depicted as solid dots for the left and right hands in Panels (A) and (B), respectively. The solid bands represent the spread of the data for each condition. The bands are displaced along the horizontal axis for clarity.

Figure 6

The referent fingertip position R_FT is plotted against the apparent stiffness C_FT for each finger condition. Data is poled across all subjects. The data for nine subjects are displaced horizontally along the R_FT axis to illustrate the spread for each subject. The dashed lines are linearly-displaced replicates of the hyperbola for ideal performance (the solid curve): R_FT × C_FT = −0.25.

Figure 7

The across-subject mean ± SE of the mean apparent stiffness C_FT and the referent fingertip R_FT coordinate for all finger conditions and both hands are shown in Panel (A) and Panel (B), respectively.

Figure 8

Possible across-trials changes in {R_FT; C_FT} that yield the same force variance. Panel (A) depicts finger force (f) plotted against the fingertip coordinate (X). Force variability over repeated trials results from co-variation of R_FT and C_FT. These data are spread along the hyperbolic UCM depicted in Panel (B). Panel (C) depicts two hypothetical behaviors that yield the same force variance. In one example (dashed inclined lines), R_FT is constant (square point on the X axis), and the force variability stems from the variability in C_FT. In the other example (solid inclined lines), the slope (C_FT) is constant and force variability stems from R_FT variability. These data appear as vertical and horizontal lines, respectively, in Panel (D).

Figure 9

(A): The control of a single joint can be described with two variables (thresholds of the tonic stretch reflex, λ) for the agonist (λ_AG) and antagonist (λ_ANT) muscles. The overall joint behavior will be defined by its torque-angle characteristic (the solid line), which is the algebraic sum of the muscle characteristics (solid curves). Its location is defined by the r-command (the mid-point between the two λs), while its slope is defined by the c-command (the spatial range where both muscles are activated). (B): Force production by the fingertip may be viewed as an interaction between groups of muscles: generalized flexors (G-flexor) and generalized extensors (G-extensor). (C): Neural commands change the spatial ranges of activation of the G-flexor and G-extensor muscles, which may be viewed as changing the mid-point (R) and the range (C) where both muscles are active on the force-coordinate plane. (D): Changing the R and C commands results in a shift of the X intercept (R_FT) and slope (C_FT) of the fingertip force-coordinate characteristic. A given task may be accomplished by different combinations {R; C} resulting in different combinations {R_FT; C_FT} shown by different force-coordinate lines.