A Compact Representation of Drawing Movements with Sequences of Parabolic Primitives

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).

A b a y x A y x Behavioral evidence for parabolic primitives which are constrained by the condition 1 det A , are called equi-affine transformations.
These include a linear part defined by 3 independent terms appearing in the matrix A; where the fourth term depends on the other 3, provided that 1 det A . It also includes translation components a and b . The variables x and y are coordinates of a point located along a planar curve, and * x and * y are the coordinates of the same point following the equi-affine transformation. The condition on the determinant means that the area contained within any closed curve is preserved under equi-affine transformations; these transformations are therefore also called area-preserving. The set of Euclidian transformations consists of rigid rotations and translations that preserve Euclidian distance and curvature, and constitutes a particular subset of equi-affine transformations. An equi-affine transformation can be applied to all the points along a curve. The Euclidian length of a curve and its Euclidian curvature are modified accordingly. However the equi-affine arc-length and the equi-affine curvature, defined below, remain the same.
For a twice differentiable planar trajectory described by the vector function , the equi-affine velocity, which is equal to the time derivative of the equi-affine arc-length , is invariant under equi-affine transformations and is expressed as follows [32,33]: Here and elsewhere, a dot above a symbol denotes a time derivative and boldfaced symbols signify vector quantities.
It equals the cubic root of the area of the parallelogram defined by the vectors of the movement velocity and acceleration and is invariant under equi-affine (area-preserving) where t corresponds to the movement duration during which an equi-affine distance (mm 2/3 ) of the path is drawn. Given that for an equi-affine transformation 1 det A (see equation 1 above), the condition for an equi-affine parameterization is: Behavioral evidence for parabolic primitives with the proportionality coefficient obtained from equations (S2) and (S3) being expressed as The coefficient is the equi-affine curvature of a curve [32,33], which is another equi-affine invariant. Curves having the same equi-affine curvature can be aligned by using some equi-affine transformation [32]; therefore equi-affine curvature can be used for curve classification in equi-affine geometry.
Curves having a constant equi-affine curvature are the conics (the ellipse, parabola, and hyperbola). Parabolas have zero equi-affine curvature. Therefore any parabolic stroke can be uniquely associated with any other parabolic stroke, whenever the two strokes have the same equi-affine length [27,30]. For this reason any parabolic segment can be obtained from an arbitrary parabolic template by an affine transformation (composed of a unique uniform spatial scaling -adjusting the equi-affine length -and a unique equiaffine transformation). The equi-affine curvature of an ellipse is a positive constant defined only by its enclosed area A : The constrained minimum-jerk model predicts maximally smooth movements in the jerk sense for a given movement path [4]; more details are provided in the next section.
Predictions of the constrained minimum-jerk model and the two-thirds power law model for a given path are generally different. They are equal only for curves satisfying the following equation [25,27]: The optimization procedure described below was aimed to find a prediction of the constrained minimum-jerk model or, alternatively, a speed profile t s that minimizes the cost (S5). The optimal trajectory is constrained by a prescribed path s y s x , and the total duration T of a movement segment. No constraints were imposed on the velocity or acceleration at the segment boundaries as we seek for predictions along movement segments (by definition, the monkeys' hand velocity was always non-zero along the movement segments, see Methods).
A classical minimum-jerk trajectory for moving from rest to rest passing through a single via-point (3 points constrain the task instead of the entire path) can be very well approximated by a parabola, though it is not an exact parabola [25,27]. The constrained Behavioral evidence for parabolic primitives minimum-jerk model predicts movements with constant equi-affine speed, that is satisfying the two-thirds power-law model, for parabolic paths.
Given a sequence of recorded samples along the path of a movement segment, Finally, the jerk cost (S5) can be approximated with the squared norm of J: sum(J .* J).
The time increments were adjusted during the process of cost minimization implemented with the Matlab (Mathworks) function "lsqnonlin" from the Optimization toolbox. The function was run with the large-scale algorithm provided by the toolbox.
The initial guess for the time increments was taken from the recorded trajectory: i / (Recording frequency). A non-trivial time-warping relationship between the actual and predicted trajectories was needed to align the predicted and measured time courses and is defined as follows: We show below that the predicted trajectories differ from the actual trajectories.
Estimates of the fit to the constrained minimum-jerk model and to the two-thirds power law Now we explain formula (S7) in more detail. Let us use the following notation: , we conclude that the total maximal possible difference between the actual and predicted time-courses ( w ), integrated over the movement duration is equal to half the area of the squared time interval T: . This maximal possible value is set as 100% and is used in the normalization of the deviation between the time-courses of the recorded and predicted trajectories. The term t is cancelled by the same term which is used in the summation (for integration, 1 2 ) in the numerator.
Therefore, the normalization factor is 2 / 1 T N .
The two-thirds power law establishes a relationship between geometric and temporal properties of hand trajectories. It assumes that the tangential velocity s for producing a given path and the Euclidian curvature c of that path are related via a piecewise constant gain factor K : 3 / 1 c K s [18]. The gain factor K in the above expression is equal to the equi-affine velocity of the movement trajectory, namely: K y x y x 3 . Therefore, movements obeying the two-thirds power law have Behavioral evidence for parabolic primitives piece-wise constant equi-affine velocity [22][23][24]; this property is invariant under equiaffine transformations of the trajectory.
The constancy of the ratio is used to test whether a given trajectory complies with the two-thirds power law. Whenever the ratio is constant, the angle between the two multidimensional vectors is zero. Therefore, for each movement segment the constancy of the ratio is estimated by the angle between the multidimensional vectors and t : t , a for the actual trajectories; t , p for the predicted trajectories ; with t t t arccos , .
The smaller the angle in (S8), the closer the equi-affine speed is to being constant for a given movement segment and therefore the better the fit to the two-thirds power law.

Regularization of
Numerical calculation of the equi-affine parameters is sensitive to noise in the original data because high-order derivatives are used. Therefore, these parameters occasionally show large fluctuations between adjacent samples. We introduce a regularity criterion for the magnitude of the increments of the equi-affine arc-length between adjacent samples on a movement path. The criterion is based on the proximity of the neighboring values of . As is illustrated in Figure S2, a block of data is considered Behavioral evidence for parabolic primitives regular whenever it contains a sufficient number (at least 5) of consecutive that are close enough to their neighbors (0.075 mm 2/3 or less). The data analysis, which involved equi-affine speed, was performed on those parts of movement segments satisfying the regularity conditions for .

Example: parameters analyzed for a single movement segment
Equi-affine geometry, the two-thirds power law and the constrained minimum-jerk model were used to mathematically infer that parabolas are candidate movement primitives. We next describe the properties of the equi-affine curvature of the monkey scribbling movements and how well these movements fit the two-thirds power law and the constrained minimum-jerk model. First, we use one movement segment scribbled by monkey O to describe in detail the parameters involved in the data analysis presented below.
The segment path ( Figure S3A) is smooth and consists of several repetitions of the same piece-wise parabolic pattern. Figure S3B shows the actual and predicted timecourses versus path samples. Their difference defines the time-warping needed for the recorded trajectory to obey the constrained minimum-jerk model. Figure S3C shows the speed profiles of the actual and predicted trajectories. For the actual trajectories, the sampling interval is proportional to time due to the constant recording frequency. For the predicted trajectories, however, the time taken to pass between the pairs of consecutive samples is not constant. Hence, we plotted a single profile for the actual speed and two profiles for the predicted speed: one was plotted as