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)

“Biological Geometry Perception”: Visual Discrimination of Eccentricity Is Related to Individual Motor Preferences

In the continuum between a stroke and a circle including all possible ellipses, some eccentricities seem more “biologically preferred” than others by the motor system, probably because they imply less demanding coordination patterns. Based on the idea that biological motion perception relies on knowledge of the laws that govern the motor system, we investigated whether motorically preferential and non-preferential eccentricities are visually discriminated differently. In contrast with previous studies that were interested in the effect of kinematic/time features of movements on their visual perception, we focused on geometric/spatial features, and therefore used a static visual display.In a dual-task paradigm, participants visually discriminated 13 static ellipses of various eccentricities while performing a finger-thumb opposition sequence with either the dominant or the non-dominant hand. Our assumption was that because the movements used to trace ellipses are strongly lateralized, a motor task performed with the dominant hand should affect the simultaneous visual discrimination more strongly. We found that visual discrimination was not affected when the motor task was performed by the non-dominant hand. Conversely, it was impaired when the motor task was performed with the dominant hand, but only for the ellipses that we defined as preferred by the motor system, based on an assessment of individual preferences during an independent graphomotor task.Visual discrimination of ellipses depends on the state of the motor neural networks controlling the dominant hand, but only when their eccentricity is “biologically preferred”. Importantly, this effect emerges on the basis of a static display, suggesting that what we call “biological geometry”, i.e., geometric features resulting from preferential movements is relevant information for the visual processing of bidimensional shapes

Affine differential geometry analysis of human arm movements

Humans interact with their environment through sensory information and motor actions. These interactions may be understood via the underlying geometry of both perception and action. While the motor space is typically considered by default to be Euclidean, persistent behavioral observations point to a different underlying geometric structure. These observed regularities include the “two-thirds power law” which connects path curvature with velocity, and “local isochrony” which prescribes the relation between movement time and its extent. Starting with these empirical observations, we have developed a mathematical framework based on differential geometry, Lie group theory and Cartan’s moving frame method for the analysis of human hand trajectories. We also use this method to identify possible motion primitives, i.e., elementary building blocks from which more complicated movements are constructed. We show that a natural geometric description of continuous repetitive hand trajectories is not Euclidean but equi-affine. Specifically, equi-affine velocity is piecewise constant along movement segments, and movement execution time for a given segment is proportional to its equi-affine arc-length. Using this mathematical framework, we then analyze experimentally recorded drawing movements. To examine movement segmentation and classification, the two fundamental equi-affine differential invariants—equi-affine arc-length and curvature are calculated for the recorded movements. We also discuss the possible role of conic sections, i.e., curves with constant equi-affine curvature, as motor primitives and focus in more detail on parabolas, the equi-affine geodesics. Finally, we explore possible schemes for the internal neural coding of motor commands by showing that the equi-affine framework is compatible with the common model of population coding of the hand velocity vector when combined with a simple assumption on its dynamics. We then discuss several alternative explanations for the role that the equi-affine metric may play in internal representations of motion perception and production

Altered Perceptual Sensitivity to Kinematic Invariants in Parkinson's Disease

Ample evidence exists for coupling between action and perception in neurologically healthy individuals, yet the precise nature of the internal representations shared between these domains remains unclear. One experimentally derived view is that the invariant properties and constraints characterizing movement generation are also manifested during motion perception. One prominent motor invariant is the “two-third power law,” describing the strong relation between the kinematics of motion and the geometrical features of the path followed by the hand during planar drawing movements. The two-thirds power law not only characterizes various movement generation tasks but also seems to constrain visual perception of motion. The present study aimed to assess whether motor invariants, such as the two thirds power law also constrain motion perception in patients with Parkinson's disease (PD). Patients with PD and age-matched controls were asked to observe the movement of a light spot rotating on an elliptical path and to modify its velocity until it appeared to move most uniformly. As in previous reports controls tended to choose those movements close to obeying the two-thirds power law as most uniform. Patients with PD displayed a more variable behavior, choosing on average, movements closer but not equal to a constant velocity. Our results thus demonstrate impairments in how the two-thirds power law constrains motion perception in patients with PD, where this relationship between velocity and curvature appears to be preserved but scaled down. Recent hypotheses on the role of the basal ganglia in motor timing may explain these irregularities. Alternatively, these impairments in perception of movement may reflect similar deficits in motor production

Movement Timing and Invariance Arise from Several Geometries

Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartan's moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limaçons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations

An eight-week mindfulness-based stress reduction (MBSR) workshop increases regulatory choice flexibility

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Neural representations of kinematic laws of motion: Evidence for action-perception coupling

Behavioral and modeling studies have established that curved and drawing human hand movements obey the 2/3 power law, which dictates a strong coupling between movement curvature and velocity. Human motion perception seems to reflect this constraint. The functional MRI study reported here demonstrates that the brain's response to this law of motion is much stronger and more widespread than to other types of motion. Compliance with this law is reflected in the activation of a large network of brain areas subserving motor production, visual motion processing, and action observation functions. Hence, these results strongly support the notion of similar neural coding for motion perception and production. These findings suggest that cortical motion representations are optimally tuned to the kinematic and geometrical invariants characterizing biological actions