A compact representation of drawing movements with sequences of parabolic primitives

Abstract

Some studies suggest that complex arm movements in humans and monkeys may optimize several objective functions, while others claim that arm movements satisfy geometric constraints and are composed of elementary components. However, the ability to unify different constraints has remained an open question. The criterion for a maximally smooth (minimizing jerk) motion is satisfied for parabolic trajectories having constant equi-affine speed, which thus comply with the geometric constraint known as the two-thirds power law. Here we empirically test the hypothesis that parabolic segments provide a compact representation of spontaneous drawing movements. Monkey scribblings performed during a period of practice were recorded. Practiced hand paths could be approximated well by relatively long parabolic segments. Following practice, the orientations and spatial locations of the fitted parabolic segments could be drawn from only 2-4 clusters, and there was less discrepancy between the fitted parabolic segments and the executed paths. This enabled us to show that well-practiced spontaneous scribbling movements can be represented as sequences ("words") of a small number of elementary parabolic primitives ("letters"). A movement primitive can be defined as a movement entity that cannot be intentionally stopped before its completion. We found that in a well-trained monkey a movement was usually decelerated after receiving a reward, but it stopped only after the completion of a sequence composed of several parabolic segments. Piece-wise parabolic segments can be generated by applying affine geometric transformations to a single parabolic template. Thus, complex movements might be constructed by applying sequences of suitable geometric transformations to a few templates. Our findings therefore suggest that the motor system aims at achieving more parsimonious internal representations through practice, that parabolas serve as geometric primitives and that non-Euclidean variables are employed in internal movement representations (due to the special role of parabolas in equi-affine geometry).

Figure 1. The behavioral procedure used with monkey U.

Shown are the grid of possible targets (monkey O had an equivalent grid of circles) and an example of a scribbling movement produced by the monkey. The grey hexagons indicate the single currently active targets. Both the trajectory and the grid were invisible to the monkey. The only visual feedback was produced by the cursor (circle), which indicated the online hand position. A. The monkey's hand is near the target. B. As soon as the monkey's hand entered the target, a beep was heard, the monkey received a little orange juice reinforcement, and another target was randomly selected.

Figure 2. Segments of motion and rest.

Three movement segments and four intervals of rest.

Figure 3. A parabola in the canonical coordinate system and fitted parabolic strokes.

A. Parabola is shown using the canonical coordinate system. The orientation of the normal at the point of maximal curvature is 270° and the focal parameter p = 1. B. An example of a pattern of monkey drawing that emerged after several practice sessions and could be well approximated by three parabolic pieces with different orientations. C. These parabolic strokes were fitted to monkey drawings. Different strokes have similar orientations and are grouped according to their focal parameter .

Figure 4. Demonstration of behavior at different stages of practice.

A. Paths drawn by monkeys O and U at the beginning of the practice period (left), during the 16^th practice session (middle), and path drawn by monkey U during the period of well-trained performance. The dotted segments in each plot have the same duration of 1.98 sec. Although slow and jerky in the beginning, with practice the movements became smoother, faster and more regular. B. Dwell distributions for the end-point position, same sessions as in A. Depicted are the frequencies of visits weighted by tangential velocity; that is, visiting the same location once with a tangential velocity equal to 450 mm/s has the same contribution as visiting 3 times with a tangential velocity equal to 150 mm/s. These weighted frequencies indicate that monkey movements become more stereotypical after a period of practice. C. Average tangential velocities of the end-point, same sessions as in A.

Figure 5. Emerging parabolic clusters and dimensionality reduction.

A. Typical histograms for the fitted parabolic segments. In the one-dimensional histogram (left), the segments were tabulated according to their orientation. In the color histogram, they were tabulated in bins identified by two values: the orientation and the focal parameter of the parabola. B. Location of the vertex and orientation of the parabola for every 10^th parabolic segment for the recording sessions in (A). Locations of the vertices of the similarly oriented parabolas are also clustered. The clusters are marked by ellipses and the mean orientations of the parabolas within each cluster are indicated by arrows.

Figure 6. Properties of the paths and fitted parabolic strokes at different stages of practice, demonstration.

Histograms of the parameters of the fitted parabolic strokes for recording sessions taken from different practice periods of both monkeys. A. Equi-affine lengths of the path strokes fitted with parabolic strokes. B. Equi-affine lengths of the fitted parabolic strokes. Practice makes the measures from A and B more similar. C. Discrepancy between the path strokes and the parabolic strokes fitted to them. Practice decreases the discrepancy. D. Euclidian lengths of the fitted parabolic strokes.

Figure 7. Properties of the paths and fitted parabolic strokes during practice and examples of drawing patterns.

A–C. First column: monkey O from the beginning of practice. Second column: monkey U from the beginning of practice. Third column: well-trained behavior of monkey U. A. Mean values of the equi-affine length of the path strokes fitted with parabolas (continuous line), mean values of the equi-affine length of the fitted parabolic strokes (triangles). B. Parabolic discrepancy, an estimate of the deviation of the fitted paths from piece-wise parabolicity. C. Mean values of the Euclidian length of the path strokes fitted with parabolas. D. Different drawing patterns corresponding to the well-trained behavior of monkey U. D1, D2. Correspond to typical patterns described by ordered sequences of parabolic strokes taken from the different identified clusters. The examples depicted in D3–D8 are relatively rare. D3–D5. Different patterns with reversals in movement direction. D6–D8. Movement patterns with “irregular” parabolic strokes, i.e., some of the fitted parabolic strokes fell outside the directionally identified clusters.

Figure 8. Rewarded and non-rewarded trajectories.

A. Selection of areas with high density of rewarded locations. (Upper) Locations at which the monkey received a reward. (Lower) Same locations as in the upper plot, different colors correspond to different targets. PCA ellipses designate areas with a high density of rewarded locations. B. Three movement patterns that cross ellipses #18 and #16 without being rewarded there. For 2 trajectories, red and purple, the monkey completes the primitive sequence (1–2) and continues to scribble further via parabolic element 3 without stopping. For the green trajectory, the monkey completes the primitive sequence as well and decelerates after its completion. If the monkey is rewarded at target 18, it completes parabolic segments 1 and 2 and then nearly stops (see Figure 8C). If the monkey is not rewarded at target 18, and gets a reward at target 16, it completes parabolic segment 2, decelerates and nearly stops after that (see Figure 8D). In general, after obtaining a reward at targets 18 and 16 (at the beginning of parabolic strokes 1 or 2) the monkey decelerates and nearly stops after completing the sequence (1–2) (the data from all 17 recording sessions are summarized in Figure 9A). C, D. After receiving a reward inside ellipses 16, 18 during an ongoing movement the monkey tended to decelerate and even stop moving but only after completing a movement sequence composed of several parabolic segments. The set of rewarded locations is marked by a blue ellipse. The monkey scribbled counter clockwise. C. An example of the completed (before stopping) particular cycle that consisted of two parabolic segments (oriented downward and rightward). D. The last parabolic element of the cycle mostly corresponded to the parabolic strokes whose orientations ranged from 0° to 100°. The trajectories which were not rewarded inside the ellipse were more variable than the rewarded ones, which may have resulted from the composition of an ongoing movement sequence with another movement element.

Figure 9. Differences between the rewarded and non-rewarded trajectories, statistical estimates.

Differences between rewarded and non-rewarded trajectories. A. Minimal values of the average tangential velocities of the rewarded and non-rewarded trajectories for the sequence (1–2) which is a higher level primitive (target #18, as in Figure 8C) and for the last parabolic element (target #16, as in Figure 8D). B. Corresponds to parabolic strokes fitted to the trajectories constituting the last parabolic element (ellipses corresponding to target #16, as in Figure 8D). The orientations of 15.8% of the non-rewarded and 5.23% of the rewarded segments were outside the orientation interval [0, 70] degrees. This difference in variability was statistically significant (see text).