Timing of continuous motor imagery: the two-thirds power law originates in trajectory planning

Abstract

The two-thirds power law, v = γκ(-1/3), expresses a robust local relationship between the geometrical and temporal aspects of human movement, represented by curvature κ and speed v, with a piecewise constant γ. This law is equivalent to moving at a constant equi-affine speed and thus constitutes an important example of motor invariance. Whether this kinematic regularity reflects central planning or peripheral biomechanical effects has been strongly debated. Motor imagery, i.e., forming mental images of a motor action, allows unique access to the temporal structure of motor planning. Earlier studies have shown that imagined discrete movements obey Fitts's law and their durations are well correlated with those of actual movements. Hence, it is natural to examine whether the temporal properties of continuous imagined movements comply with the two-thirds power law. A novel experimental paradigm for recording sparse imagery data from a continuous cyclic tracing task was developed. Using the likelihood ratio test, we concluded that for most subjects the distributions of the marked positions describing the imagined trajectory were significantly better explained by the two-thirds power law than by a constant Euclidean speed or by two other power law models. With nonlinear regression, the β parameter values in a generalized power law, v = γκ(-β), were inferred from the marked position records. This resulted in highly variable yet mostly positive β values. Our results imply that imagined trajectories do follow the two-thirds power law. Our findings therefore support the conclusion that the coupling between velocity and curvature originates in centrally represented motion planning.

Keywords: equi-affine geometry; motor control; motor imagery; trajectory planning; two-thirds power law.

Fig. 1.

Experimental procedure. A: each subject performed the experimental procedure using 1 of the 3 templates shown for 4 sessions, each of 30 trials. B: each trial was composed of a complex task involving observation, imagery (including key pressing each time the image passed through the red mark and memorization of imagery location at the beep), marking (the memorized location) on the tablet, and rest (see Imagine, stop, and report—a paradigm for recording imagined motion). Eyes were closed only during the imagery part. C: each trial yielded 1 measurement of each of the 5 parameters (durations t, T, T_p and matching arc lengths s, S, S). Using these parameters we calculated the normalized durations (t̃ and t̃_p) matched by normalized arc lengths (s̃) (see Normalization and correction of recorded imagery data).

Fig. 2.

Single-subject profiles and distributions. Normalized segment length and duration profiles (t̃, s̃) for imagery motion (Obs), along with derived distributions of each subject's s̃ data (Obs Density) for 6 different subjects (S1–S6), 2 subjects for each of the 3 shapes. Also shown are predicted distributions for movement according to the two-thirds power law (model β = ⅓) and for movement with a constant Euclidean speed (model β = 0), assuming a uniform distribution for t̃.

Fig. 3.

Group distributions of s̃. A resampled distribution of s̃ values (Obs Density), such that t̃ distribution was uniform for grouped data for imagery and drawing of MEM subjects. Also shown are predicted distributions for movement according to the two-thirds power law (model β = ⅓) and for movement with a constant Euclidean speed (model β = 0) assuming a uniform distribution of t̃. The distributions of t̃ and s̃ were resampled such that the t̃ distribution became uniform. This eliminated the random effect of the t̃ distribution from the presented s̃ distribution. These distributions were compared to the predicted distributions of s̃ for the 2 power laws with the assumption that t̃ was uniformly distributed. This resampling, i.e., the process of randomly selecting 1 data point [a pair (t̃, s̃) with t̃ within the bin], was repeated 10,000 times for each bin of the t̃ variable, and the resulting s̃ values were collected in a histogram for each template.

Fig. 4.

Comparison of likelihood values of the 4 power laws: values of the log likelihoods of the model, mean and SE for all subjects of the MEM experiment for each of the 4 power laws β = ⅔, ⅓, 0, −⅓. The two-thirds power law, β = ⅓, had higher log likelihood values than the other power laws (P ≤ 0.05, see Distribution of subjects' marked locations).

Fig. 5.

Nonlinear regression results for β: for each of the 3 templates and for each subject, the results of the β values derived by the nonlinear regression process along with confidence intervals for P = 0.05, for imagery (top) and drawing (bottom) data. Blue, MEM subjects; black, NO MEM subjects. Shown for comparison are the constant β values of the two-thirds power law (β = ⅓, dot-dashed pink line) and of movement with a constant Euclidean speed (β = 0, red line).

Fig. 6.

Comparison of imagery and drawing β values. Comparison of the best β value for each subject for imagery and drawing (see Fig. 5) in the MEM condition, extracted with the nonlinear regression procedure. No correlation between the imagery and drawing β values is apparent.