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)

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

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

Movement variability in stroke patients and controls performing two upper limb functional tasks: a new assessment methodology

Background: In the evaluation of upper limb impairment post stroke there remains a gap between detailed kinematic analyses with expensive motion capturing systems and common clinical assessment tests. In particular, although many clinical tests evaluate the performance of functional tasks, metrics to characterise upper limb kinematics are generally not applicable to such tasks and very limited in scope. This paper reports on a novel, user-friendly methodology that allows for the assessment of both signal magnitude and timing variability in upper limb movement trajectories during functional task performance. In order to demonstrate the technique, we report on a study in which the variability in timing and signal magnitude of data collected during the performance of two functional tasks is compared between a group of subjects with stroke and a group of individually matched control subjects. Methods: We employ dynamic time warping for curve registration to quantify two aspects of movement variability: 1) variability of the timing of the accelerometer signals' characteristics and 2) variability of the signals' magnitude. Six stroke patients and six matched controls performed several trials of a unilateral ('drinking') and a bilateral ('moving a plate') functional task on two different days, approximately 1 month apart. Group differences for the two variability metrics were investigated on both days. Results: For 'drinking from a glass' significant group differences were obtained on both days for the timing variability of the acceleration signals' characteristics (p = 0.002 and p = 0.008 for test and retest, respectively); all stroke patients showed increased signal timing variability as compared to their corresponding control subject. 'Moving a plate' provided less distinct group differences. Conclusion: This initial application establishes that movement variability metrics, as determined by our methodology, appear different in stroke patients as compared to matched controls during unilateral task performance ('drinking'). Use of a user-friendly, inexpensive accelerometer makes this methodology feasible for routine clinical evaluations. We are encouraged to perform larger studies to further investigate the metrics' usefulness when quantifying levels of impairment

A pilot study evaluating use of a computer-assisted neurorehabilitation platform for upper-extremity stroke assessment

<p>Abstract</p> <p>Background</p> <p>There is a need to develop cost-effective, sensitive stroke assessment instruments. One approach is examining kinematic measures derived from goal-directed tasks, which can potentially be sensitive to the subtle changes in the stroke rehabilitation process. This paper presents the findings from a pilot study that uses a computer-assisted neurorehabilitation platform, interfaced with a conventional force-reflecting joystick, to examine the assessment capability of the system by various types of goal-directed tasks.</p> <p>Methods</p> <p>Both stroke subjects with hemiparesis and able-bodied subjects used the force-reflecting joystick to complete a suite of goal-directed tasks under various task settings. Kinematic metrics, developed for specific types of goal-directed tasks, were used to assess various aspects of upper-extremity motor performance across subjects.</p> <p>Results</p> <p>A number of metrics based on kinematic performance were able to differentiate subjects with different impairment levels, with metrics associated with accuracy, steadiness and speed consistency showing the best capability. Significant differences were also shown on these metrics between various force field settings.</p> <p>Conclusion</p> <p>The results support the potential of using UniTherapy software with a conventional joystick system as an upper-extremity assessment instrument. We demonstrated the ability of using various types of goal-directed tasks to distinguish between subjects with different impairment levels. In addition, we were able to show that different force fields have a significant effect on the performance across subjects with different impairment levels in the trajectory tracking task. These results provide motivation for studies with a larger sample size that can more completely span the impairment space, and can use insights presented here to refine considerations of various task settings so as to generalize and extend our conclusions.</p