On the similarities between the perception and production of elliptical trajectories

Role of the Cerebellum in the Construction of Functional and Geometrical Spaces

The perceptual and motor systems appear to have a set of movement primitives that exhibit certain geometric and kinematic invariances. Complex patterns and mental representations can be produced by (re)combining some simple motor elements in various ways using basic operations, transformations, and respecting a set of laws referred to as kinematic laws of motion. For example, point-to-point hand movements are characterized by straight hand paths with single-peaked-bell-shaped velocity profiles, whereas hand speed profiles for curved trajectories are often irregular and more variable, with speed valleys and inflections extrema occurring at the peak curvature. Curvature and speed are generically related by the 2/3 power law. Mathematically, such laws can be deduced from a combination of Euclidean, affine, and equi-affine geometries, whose neural correlates have been partially detected in various brain areas including the cerebellum and the basal ganglia. The cerebellum has been found to play an important role in the control of coordination, balance, posture, and timing over the past years. It is also assumed that the cerebellum computes forward internal models in relationship with specific cortical and subcortical brain regions but its motor relationship with the perceptual space is unclear. A renewed interest in the geometrical and spatial role of the cerebellum may enable a better understanding of its specific contribution to the action-perception loop and behavior's adaptation. In this sense, we complete this overview with an innovative theoretical framework that describes a possible implementation and selection by the cerebellum of geometries adhering to different mathematical laws.

Motor Control Adherence to the Two-thirds Power Law Differs in Autistic Development

Autistic individuals often exhibit motor atypicalities, which may relate to difficulties in social communication. This study utilized a smart tablet activity to computationally characterize motor control by testing adherence to the two-thirds power law (2/3 PL), which captures a systematic covariation between velocity and curvature in motor execution and governs many forms of human movement. Children aged 4-8 years old participated in this study, including 24 autistic children and 33 typically developing children. Participants drew and traced ellipses on an iPad. We extracted data from finger movements on the screen, and computed adherence to the 2/3 PL and other kinematic metrics. Measures of cognitive and motor functioning were also collected. In comparison to the typically developing group, the autistic group demonstrated greater velocity modulation between curved and straight sections of movement, increased levels of acceleration and jerk, and greater intra- and inter-individual variability across several kinematic variables. Further, significant motor control development was observed in typically developing children, but not in those with autism. This study is the first to examine motor control adherence to the 2/3 PL in autistic children, revealing overall diminished motor control. Less smooth, more varied movement and an indication of developmental stasis in autistic children were observed. This study offers a novel tool for computational characterization of the autism motor signature in children's development, demonstrating how smart tablet technology enables accessible assessment of children's motor performance in an objective, quantifiable and scalable manner.

Representational momentum of biological motion in full-body, point-light and single-dot displays

Observing the actions of others triggers, in our brain, an internal and automatic simulation of its unfolding in time. Here, we investigated whether the instantaneous internal representation of an observed action is modulated by the point of view under which an action is observed and the stimulus type. To this end, we motion captured the elliptical arm movement of a human actor and used these trajectories to animate a photorealistic avatar, a point-light stimulus or a single dot rendered either from an egocentric or an allocentric point of view. Crucially, the underlying physical characteristics of the movement were the same in all conditions. In a representational momentum paradigm, we then asked subjects to report the perceived last position of an observed movement at the moment in which the stimulus was randomly stopped. In all conditions, subjects tended to misremember the last configuration of the observed stimulus as being further forward than the veridical last showed position. This misrepresentation was however significantly smaller for full-body stimuli compared to point-light and single dot displays and it was not modulated by the point of view. It was also smaller when first-person full body stimuli were compared with a stimulus consisting of a solid shape moving with the same physical motion. We interpret these findings as evidence that full-body stimuli elicit a simulation process that is closer to the instantaneous veridical configuration of the observed movements while impoverished displays (both point-light and single-dot) elicit a prediction that is further forward in time. This simulation process seems to be independent from the point of view under which the actions are observed.

Motor invariants in action execution and perception

The nervous system is sensitive to statistical regularities of the external world and forms internal models of these regularities to predict environmental dynamics. Given the inherently social nature of human behavior, being capable of building reliable predictive models of others' actions may be essential for successful interaction. While social prediction might seem to be a daunting task, the study of human motor control has accumulated ample evidence that our movements follow a series of kinematic invariants, which can be used by observers to reduce their uncertainty during social exchanges. Here, we provide an overview of the most salient regularities that shape biological motion, examine the role of these invariants in recognizing others' actions, and speculate that anchoring socially-relevant perceptual decisions to such kinematic invariants provides a key computational advantage for inferring conspecifics' goals and intentions.

An Uncontrolled Manifold Analysis of Arm Joint Variability in Virtual Planar Position and Orientation Telemanipulation

OBJECTIVE

In teleoperated robot-assisted tasks, the user interacts with manipulators to finely control remote tools. Manipulation of robotic devices, characterized by specific kinematic and dynamic proprieties, is a complex task for the human sensorimotor system due to the inherent biomechanical and neuronal redundancies that characterize the human arm and its control. We investigate how master devices with different kinematics structures and how different task constraints influence users capabilities in exploiting arm redundancy.

METHODS

A virtual teleoperation workbench was designed and the arm kinematics of seven users was acquired during the execution of two planar virtual tasks, involving either the control of position only or position-orientation of a tool. Using the uncontrolled manifold analysis of arm joint variability, we estimated the logarithmic ratio between the task irrelevant and the task relevant manifolds ( R_v).

RESULTS

The R_v values obtained in the position-orientation task were higher than in the position only task, while no differences were found between the master devices. A modulation of R_v was found through the execution of the position task and a positive correlation was found between task performance and redundancy exploitation.

CONCLUSION

Users exploited additional portions of arm redundancy when dealing with the tool orientation. The R_v modulation seems influenced by the task constraints and by the users possibility of reconfiguring the arm position.

SIGNIFICANCE

This paper advances the general understanding of the exploitation of arm redundancy in complex tasks, and can improve the development of future robotic devices.

Disruption of network for visual perception of natural motion in primary dystonia

In healthy subjects, brain activation in motor regions is greater during the visual perception of "natural" target motion, which complies with the two-thirds power law, than of "unnatural" motion, which does not. It is unknown whether motion perception is normally mediated by a specific network that can be altered in the setting of disease. We used block-design functional magnetic resonance imaging and covariance analysis to identify normal network topographies activated in response to "natural" versus "unnatural" motion. A visual motion perception-related pattern (VPRP) was identified in 12 healthy subjects, characterized by covarying activation responses in the inferior parietal lobule, frontal operculum, lateral occipitotemporal cortex, amygdala, and cerebellum (Crus I). Selective VPRP activation during "natural" motion was confirmed in 12 testing scans from healthy subjects. Consistent network activation was not seen, however, in 29 patients with dystonia, a neurodevelopmental disorder in which motion perception pathways may be involved. Using diffusion tractography, we evaluated the integrity of anatomical connections between the major VPRP nodes. Indeed, fiber counts in these pathways were substantially reduced in the dystonia subjects. In aggregate, the findings associate normal motion perception with a discrete brain network which can be disrupted under pathological conditions.

On the Value of Estimating Human Arm Stiffness during Virtual Teleoperation with Robotic Manipulators

Teleoperated robotic systems are widely spreading in multiple different fields, from hazardous environments exploration to surgery. In teleoperation, users directly manipulate a master device to achieve task execution at the slave robot side; this interaction is fundamental to guarantee both system stability and task execution performance. In this work, we propose a non-disruptive method to study the arm endpoint stiffness. We evaluate how users exploit the kinetic redundancy of the arm to achieve stability and precision during the execution of different tasks with different master devices. Four users were asked to perform two planar trajectories following virtual tasks using both a serial and a parallel link master device. Users' arm kinematics and muscular activation were acquired and combined with a user-specific musculoskeletal model to estimate the joint stiffness. Using the arm kinematic Jacobian, the arm end-point stiffness was derived. The proposed non-disruptive method is capable of estimating the arm endpoint stiffness during the execution of virtual teleoperated tasks. The obtained results are in accordance with the existing literature in human motor control and show, throughout the tested trajectory, a modulation of the arm endpoint stiffness that is affected by task characteristics and hand speed and acceleration.

Affine differential geometry and smoothness maximization as tools for identifying geometric movement primitives

Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (e.g., direction, shape) or temporal (e.g., speed) parameters of trajectories rather than trajectory's representation as a whole. This work is about identifying geometric building blocks of movements by unifying different empirically supported mathematical descriptions that characterize relationship between geometric and temporal aspects of biological motion. Movement primitives supposedly facilitate the efficiency of movements' representation in the brain and comply with such criteria for biological movements as kinematic smoothness and geometric constraint. The minimum-jerk model formalizes criterion for trajectories' maximal smoothness of order 3. I derive a class of differential equations obeyed by movement paths whose nth-order maximally smooth trajectories accumulate path measurement with constant rate. Constant rate of accumulating equi-affine arc complies with the 2/3 power-law model. Candidate primitive shapes identified as equations' solutions for arcs in different geometries in plane and in space are presented. Connection between geometric invariance, motion smoothness, compositionality and performance of the compromised motor control system is proposed within single invariance-smoothness framework. The derived class of differential equations is a novel tool for discovering candidates for geometric movement primitives.

Micrographia, much beyond the writer's hand

INTRODUCTION

This review on micrographia aims to draw the clinician's attention to non-Parkinsonian etiologies, provide clues to differential diagnosis, and summarize current knowledge on the phenomenology, etiology, and mechanisms underlying micrographia.

METHODS

A systematic review of the existing literature was performed.

RESULTS

Micrographia, namely small sized handwriting has long been attributed to Parkinson's disease. However, it has often been observed as part of the clinical picture of additional neurodegenerative disorders, sometimes antedating the motor signs, or following focal basal ganglia lesions without any accompanying parkinsonism, suggesting that bradykinesia and rigidity are not sine-qua-non for the development of this phenomenon. Therefore, micrographia in a patient with no signs of parkinsonism may prompt the clinician to perform imaging in order to exclude a focal basal ganglia lesion. Dopaminergic etiology in this and other cases is doubtful, since levodopa ameliorates letter stroke size only partially, and only in some patients. Parkinsonian handwriting is often characterized by lack of fluency, slowness, and less frequently by micrographia. Deviations from kinematic laws of motion that govern normal movement, including the lack of movement smoothness and inability to scale movement amplitude to the desired size, may reflect impairments in motion planning, possible loss of automaticity and reduced movement vigor.

CONCLUSIONS

The etiology, neuroanatomy, mechanisms and models of micrographia are discussed. Dysfunction of the basal ganglia circuitry induced by neurodegeneration or disruption by focal damage give rise to micrographia.

Human movements and abstract motion displays activate different processes in the observer's motor system

Brain imaging studies have shown that observation of both bodily movements and abstract motion displays complying with human kinematics activate the observer's motor cortex. However, it is unknown whether the same processes are active in the two conditions. Here, we addressed this issue using transcranial magnetic stimulation (TMS) to directly compare cortico-spinal excitability during observation of actions and motion stimuli that complied with or violated normal human kinematics. We found that kinematics significantly modulated the motor-evoked potentials (MEPs) produced by TMS during observation of both human and abstract motion stimuli. However, only the temporal unfolding of cortico-spinal excitability during observation of human movements significantly correlated with instantaneous stimulus velocity. This correlation was present for normal movements and also for a subset of the movements having unnatural kinematics. Furthermore, bodily movements for which we found no correlation between MEPs and stimulus velocity produced significantly higher MEPs. Our novel results suggest a dissociation in how human movements and abstract motion displays engage the observer's motor system. Specifically, while both stimulus types significantly activate the observer's motor cortex, only bodily movements produce patterns of cortico-spinal excitability that closely follow the velocity profile of the observed movement. This internal "re-enactment" of observed bodily movements seems to be only partially attuned to normal human kinematics.

Cathodal transcranial direct current stimulation can stabilize perception of movement: Evidence from the two-thirds power law illusion

Human movements conform to specific kinematic laws of motion. One of such laws, the "two-thirds power law", describes the systematic co-variation between curvature and velocity of body movements. Noticeably, the same law also influences the perception of moving stimuli: the velocity of a dot moving along a curvilinear trajectory is perceived as uniform when the dot kinematics complies with the two-thirds power law. Instead, if the dot moves at constant speed, its velocity is perceived as highly non-uniform. This dynamic visual illusion points to a strong coupling between action and perception; however, how this coupling is implemented in the brain remains elusive. In this study, we tested whether the premotor cortex (PM) and the primary visual cortex (V1) play a role in the illusion by means of transcranial Direct Current Stimulation (tDCS). All participants underwent three tDCS sessions during which they received active or sham cathodal tDCS (1.5mA) over PM or V1 of the left hemisphere. During tDCS, participants were required to adjust the velocity of a dot moving along an elliptical trajectory until it looked uniform across the whole trajectory. Results show that occipital tDCS decreases the illusion variability both within and across participants, as compared to sham tDCS. This means that V1 stimulation increases individual sensitivity to the illusory motion and also increases coherence across different observers. Conversely, the illusion seems resistant to tDCS in terms of its magnitude, with cathodal stimulation of V1 or PM not affecting the amount of the illusory effect. Our results provide evidence for strong visuo-motor coupling in visual perception: the velocity of a dot moving along an elliptical trajectory is perceived as uniform only when its kinematics closely complies to the same law of motion that constrains human movement production. Occipital stimulation by cathodal tDCS can stabilize such illusory percept.

Alpha and beta band event-related desynchronization reflects kinematic regularities

The short-lasting attenuation of brain oscillations is termed event-related desynchronization (ERD). It is frequently found in the alpha and beta bands in humans during generation, observation, and imagery of movement and is considered to reflect cortical motor activity and action-perception coupling. The shared information driving ERD in all these motor-related behaviors is unknown. We investigated whether particular laws governing production and perception of curved movement may account for the attenuation of alpha and beta rhythms. Human movement appears to be governed by relatively few kinematic laws of motion. One dominant law in biological motion kinematics is the 2/3 power law (PL), which imposes a strong dependency of movement speed on curvature and is prominent in action-perception coupling. Here we directly examined whether the 2/3 PL elicits ERD during motion observation by characterizing the spatiotemporal signature of ERD. ERDs were measured while human subjects observed a cloud of dots moving along elliptical trajectories either complying with or violating the 2/3 PL. We found that ERD within both frequency bands was consistently stronger, arose faster, and was more widespread while observing motion obeying the 2/3 PL. An activity pattern showing clear 2/3 PL preference and lying within the alpha band was observed exclusively above central motor areas, whereas 2/3 PL preference in the beta band was observed in additional prefrontal-central cortical sites. Our findings reveal that compliance with the 2/3 PL is sufficient to elicit a selective ERD response in the human brain.

The Default Mode Network Differentiates Biological From Non-Biological Motion

The default mode network (DMN) has been implicated in an array of social-cognitive functions, including self-referential processing, theory of mind, and mentalizing. Yet, the properties of the external stimuli that elicit DMN activity in relation to these domains remain unknown. Previous studies suggested that motion kinematics is utilized by the brain for social-cognitive processing. Here, we used functional MRI to examine whether the DMN is sensitive to parametric manipulations of observed motion kinematics. Preferential responses within core DMN structures differentiating non-biological from biological kinematics were observed for the motion of a realistically looking, human-like avatar, but not for an abstract object devoid of human form. Differences in connectivity patterns during the observation of biological versus non-biological kinematics were additionally observed. Finally, the results additionally suggest that the DMN is coupled more strongly with key nodes in the action observation network, namely the STS and the SMA, when the observed motion depicts human rather than abstract form. These findings are the first to implicate the DMN in the perception of biological motion. They may reflect the type of information used by the DMN in social-cognitive processing.

Implied dynamics biases the visual perception of velocity

We expand the anecdotic report by Johansson that back-and-forth linear harmonic motions appear uniform. Six experiments explore the role of shape and spatial orientation of the trajectory of a point-light target in the perceptual judgment of uniform motion. In Experiment 1, the target oscillated back-and-forth along a circular arc around an invisible pivot. The imaginary segment from the pivot to the midpoint of the trajectory could be oriented vertically downward (consistent with an upright pendulum), horizontally leftward, or vertically upward (upside-down). In Experiments 2 to 5, the target moved uni-directionally. The effect of suppressing the alternation of movement directions was tested with curvilinear (Experiment 2 and 3) or rectilinear (Experiment 4 and 5) paths. Experiment 6 replicated the upright condition of Experiment 1, but participants were asked to hold the gaze on a fixation point. When some features of the trajectory evoked the motion of either a simple pendulum or a mass-spring system, observers identified as uniform the kinematic profiles close to harmonic motion. The bias towards harmonic motion was most consistent in the upright orientation of Experiment 1 and 6. The bias disappeared when the stimuli were incompatible with both pendulum and mass-spring models (Experiments 3 to 5). The results are compatible with the hypothesis that the perception of dynamic stimuli is biased by the laws of motion obeyed by natural events, so that only natural motions appear uniform.

Spatial constant equi-affine speed and motion perception

The two-thirds power law, postulating an inverse local relation between the velocity and cubed root of curvature of planar trajectories, is a long-established simplifying principle of human hand movements. In perception, the motion of a dot along a planar elliptical path appears most uniform for speed profiles closer to those predicted by the power law than to constant Euclidean speed, a kinetic-visual illusion. Mathematically, complying with this law is equivalent to moving at constant planar equi-affine speed, while unconstrained three-dimensional drawing movements generally follow constant spatial equi-affine speed. Here we test the generalization of this illusion to visual perception of spatial motion for a dot moving along five differently shaped paths, using stereoscopic projection. The movements appeared most uniform for speed profiles closer to constant spatial equi-affine speed than to constant Euclidean speed, with path torsion (i.e., local deviation from planarity) directly affecting the speed profiles perceived as most uniform, as predicted for constant spatial equi-affine speed. This demonstrates the dominance of equi-affine geometry in spatial motion perception. However, constant equi-affine speed did not fully account for the variability among the speed profiles selected as most uniform for different shapes. Moreover, in a followup experiment, we found that viewing distance affected the speed profile reported as most uniform for the extensively studied planar elliptical motion paths. These findings provide evidence for the critical role of equi-affine geometry in spatial motion perception and contribute to the mounting evidence for the role of non-Euclidean geometries in motion perception and production.

Compositionality in neural control: an interdisciplinary study of scribbling movements in primates

This article discusses the compositional structure of hand movements by analyzing and modeling neural and behavioral data obtained from experiments where a monkey (Macaca fascicularis) performed scribbling movements induced by a search task. Using geometrically based approaches to movement segmentation, it is shown that the hand trajectories are composed of elementary segments that are primarily parabolic in shape. The segments could be categorized into a small number of classes on the basis of decreasing intra-class variance over the course of training. A separate classification of the neural data employing a hidden Markov model showed a coincidence of the neural states with the behavioral categories. An additional analysis of both types of data by a data mining method provided evidence that the neural activity patterns underlying the behavioral primitives were formed by sets of specific and precise spike patterns. A geometric description of the movement trajectories, together with precise neural timing data indicates a compositional variant of a realistic synfire chain model. This model reproduces the typical shapes and temporal properties of the trajectories; hence the structure and composition of the primitives may reflect meaningful behavior.

Segmenting sign language into motor primitives with Bayesian binning

The endpoint trajectories of human movements fulfill characteristic power laws linking velocity and curvature. The parameters of these power laws typically vary between different segments of longer action sequences. These parameters might thus be exploited for the unsupervised segmentation of actions into movement primitives. For the example of sign language we investigate whether such segments can be identified by Bayesian binning (BB), using a Gaussian observation model whose mean has a polynomial time dependence. We show that this method yields good segmentation and correctly models ground truth kinematics composed of consecutive segments derived from wrist trajectories recorded from users of Israeli Sign Language (ISL). Importantly, polynomial orders between 3 and 5 yield an optimal trade-off between complexity and accuracy of the trajectory approximation, in accordance with the minimum acceleration and minimum jerk models. Comparing the orders of the polynomials best approximating natural kinematics against those needed to fit the power law ground truth data suggests that kinematic properties not compatible with power laws are also not adequately represented by low order polynomials and require higher order polynomials for a good approximation.

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.

"Biological geometry perception": visual discrimination of eccentricity is related to individual motor preferences

Background

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.

Methodology/Principal Findings

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.

Conclusions/Significance

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.

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.

No existing theory successfully accounts for several amazing properties of biological movements: dependence of movement speed on path curvature, isochrony (movement duration is nearly independent of its size) and the construction of more complex movements from simpler building blocks. Here we present a new theory of movement generation, based on movement invariance with respect to geometrical transformations. Several types of transformations are considered. Euclidian transformations preserve lengths and angles; affine transformations, which are less restricted, preserve parallelisms between lines, while equi-affine transformations preserve both parallelism and area. Each geometry is associated with a different measure of distance along curves. Movement timing is continuously prescribed by the brain by combining different “geometrical times” each assumed to be proportional to the measure of distance of the corresponding geometry. Movements are constructed by using a series of instantaneous (Cartan) coordinate frames. The predictions of the theory compared well with experimental observations of human drawing and walking. Equi-affine geometry was found to play a dominant role in both tasks and is complemented by affine geometry during drawing and by Euclidian geometry during locomotion. The proposed theory has far reaching implications with respect to brain representations of motion for both action production and perception.

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).

Parabolic movement primitives and cortical states: merging optimality with geometric invariance

Previous studies have suggested that several types of rules govern the generation of complex arm movements. One class of rules consists of optimizing an objective function (e.g., maximizing motion smoothness). Another class consists of geometric and kinematic constraints, for instance the coupling between speed and curvature during drawing movements as expressed by the two-thirds power law. It has also been suggested that complex movements are composed of simpler elements or primitives. However, the ability to unify the different rules has remained an open problem. We address this issue by identifying movement paths whose generation according to the two-thirds power law yields maximally smooth trajectories. Using equi-affine differential geometry we derive a mathematical condition which these paths must obey. Among all possible solutions only parabolic paths minimize hand jerk, obey the two-thirds power law and are invariant under equi-affine transformations (which preserve the fit to the two-thirds power law). Affine transformations can be used to generate any parabolic stroke from an arbitrary parabolic template, and a few parabolic strokes may be concatenated to compactly form a complex path. To test the possibility that parabolic elements are used to generate planar movements, we analyze monkeys' scribbling trajectories. Practiced scribbles are well approximated by long parabolic strokes. Of the motor cortical neurons recorded during scribbling more were related to equi-affine than to Euclidean speed. Unsupervised segmentation of simulta- neously recorded multiple neuron activity yields states related to distinct parabolic elements. We thus suggest that the cortical representation of movements is state-dependent and that parabolic elements are building blocks used by the motor system to generate complex movements.

Complex Unconstrained Three-Dimensional Hand Movement and Constant Equi-Affine Speed

One long-established simplifying principle behind the large repertoire and high versatility of human hand movements is the two-thirds power law—an empirical law stating a relationship between local geometry and kinematics of human hand trajectories during planar curved movements. It was further generalized not only to various types of human movements, but also to motion perception and prediction, although it was unsuccessful in explaining unconstrained three-dimensional (3D) movements. Recently, movement obeying the power law was proved to be equivalent to moving with constant planar equi-affine speed. Generalizing such motion to 3D space—i.e., to movement at constant spatial equi-affine speed—predicts the emergence of a new power law, whose utility for describing spatial scribbling movements we have previously demonstrated. In this empirical investigation of the new power law, subjects repetitively traced six different 3D geometrical shapes with their hand. We show that the 3D power law explains the data consistently better than both the two-thirds power law and an additional power law that was previously suggested for spatial hand movements. We also found small yet systematic modifications of the power-law's exponents across the various shapes, which further scrutiny suggested to be correlated with global geometric factors of the traced shape. Nevertheless, averaging over all subjects and shapes, the power-law exponents are generally in accordance with constant spatial equi-affine speed. Taken together, our findings provide evidence for the potential role of non-Euclidean geometry in motion planning and control. Moreover, these results seem to imply a relationship between geometry and kinematics that is more complex than the simple local one stipulated by the two-thirds power law and similar models.

Hypothesis regarding the transformation of the intended direction of movement during the production of graphic trajectories: a study of drawing movements in 8- to 12-year-old children

Children from 8 to 12 years of age drew figure-eights and ellipses at a self-chosen tempo on a digitizing tablet. Global aspects (perimeter and average speed) and local aspects (relation between instantaneous speed and curvature) of performance were analyzed across age groups and types of figures. We tested the predictions of the transformation model, which is based on the hypothesis that changing the intended direction of movement is a time-consuming process that affects the evolution in time of the movement trajectory, and compared how well it fitted the data relative to the power law. We found that the relation between speed and curvature was typically better described by the transformation model than by the power law. However, the power law provided a better description when ellipses were drawn at a fast speed. The analyses of the parameters of the transformation model indicate that processing speed increased linearly with age. In addition, the results suggest that the effects of the spring-like properties of the arm were noticeable when ellipses were drawn at a fast speed. This study indicates that both biomechanical properties and central processes have an effect on the kinematics of continuous movements and particularly on the relation between speed and curvature. However, their relative importance varies with the type of figure and average movement speed. In conclusion, the results support the hypothesis that a time-consuming process of transformation of the intended direction of movement is operating during the production of continuous movements and that this process increases in speed between 8 to 12 years of age.

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.

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.