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.     

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.     

Motor primitives in vertebrates and invertebrates

In recent years different lines of evidence have led to the idea that motor actions and movements in both vertebrates and invertebrates are composed of elementary building blocks. The entire motor repertoire can be spanned by applying a well-defined set of operations and transformations to these primitives and by combining them in many different ways according to well-defined syntactic rules. Motor and movement primitives and modules might exist at the neural, dynamic and kinematic levels with complicated mapping among the elementary building blocks subserving these different levels of representation. Hence, while considerable progress has been made in recent years in unravelling the nature of these primitives, new experimental, computational and conceptual approaches are needed to further advance our understanding of motor compositionality.     

Compositionality of arm movements can be realized by propagating synchrony

We present a biologically plausible spiking neuronal network model of free monkey scribbling that reproduces experimental findings on cortical activity and the properties of the scribbling trajectory. The model is based on the idea that synfire chains can encode movement primitives. Here, we map the propagation of activity in a chain to a linearly evolving preferred velocity, which results in parabolic segments that fulfill the two-thirds power law. Connections between chains that match the final velocity of one encoded primitive to the initial velocity of the next allow the composition of random sequences of primitives with smooth transitions. The model provides an explanation for the segmentation of the trajectory and the experimentally observed deviations of the trajectory from the parabolic shape at primitive transition sites. Furthermore, the model predicts low frequency oscillations (<10 Hz) of the motor cortex local field potential during ongoing movements and increasing firing rates of non-specific motor cortex neurons before movement onset.     

Kinematics of the Coordination of Pointing during Locomotion

In natural motor behaviour arm movements, such as pointing or reaching, often need to be coordinated with locomotion. The underlying coordination patterns are largely unexplored, and require the integration of both rhythmic and discrete movement primitives. For the systematic and controlled study of such coordination patterns we have developed a paradigm that combines locomotion on a treadmill with time-controlled pointing to targets in the three-dimensional space, exploiting a virtual reality setup. Participants had to walk at a constant velocity on a treadmill. Synchronized with specific foot events, visual target stimuli were presented that appeared at different spatial locations in front of them. Participants were asked to reach these stimuli within a short time interval after a “go” signal. We analysed the variability patterns of the most relevant joint angles, as well as the time coupling between the time of pointing and different critical timing events in the foot movements. In addition, we applied a new technique for the extraction of movement primitives from kinematic data based on anechoic demixing. We found a modification of the walking pattern as consequence of the arm movement, as well as a modulation of the duration of the reaching movement in dependence of specific foot events. The extraction of kinematic movement primitives from the joint angle trajectories exploiting the new algorithm revealed the existence of two distinct main components accounting, respectively, for the rhythmic and discrete components of the coordinated movement pattern. Summarizing, our study shows a reciprocal pattern of influences between the coordination patterns of reaching and walking. This pattern might be explained by the dynamic interactions between central pattern generators that initiate rhythmic and discrete movements of the lower and upper limbs, and biomechanical factors such as the dynamic gait stability.     

Jerk-cost modulations during the practice of rapid arm movements

We examined how hand-trajectory smoothness changed during the practice of a motor task where smoothness was quantified by jerk-cost. Four human subjects each moved his nondominant arm between an upper target and a lower target, while circumnavigating a barrier that extended outward from the vertical plane of the targets. The two targets and the barrier placed boundary constraints on hand trajectories, but the motion was not restrained in any other way. Arm movements were recorded on high-speed ciné film, and linear and angular kinematical data were obtained for all arm segments. In each of 100 practice trials, subjects attempted to minimize movement time. After the practice trials, subjects repeated the same motor task but at movement times corresponding to the slowest, mid-range and fastest motion that had occurred during practice. Thus, jerk-cost could be compared for movements of different speeds during practice and after practice. Because the movement task contained several changes in hand-path direction, the changes in the vector characteristics of the hand accelerations were expected to be important for explaining the modulations in jerk-cost with practice. Total jerk-cost, therefore, was calculated as well as the separate magnitudinal and directional jerk-cost components. During practice, total movement time decreased, hand paths became more parabolic in shape, and significant changes occurred in hand acceleration magnitude, direction, and timing. Total jerk-cost and the magnitudinal and directional jerk-cost components were significantly less when slowest hand movements were compared after practice versus during practice. The decrease in jerk-cost indicated an increased smoothness of the practiced movements.K. Schneider was supported by the German Research Association (Deutsche Forschungsgemeinschaft)     

Classification of Domain Movements in Proteins Using Dynamic Contact Graphs

A new method for the classification of domain movements in proteins is described and applied to 1822 pairs of structures from the Protein Data Bank that represent a domain movement in two-domain proteins. The method is based on changes in contacts between residues from the two domains in moving from one conformation to the other. We argue that there are five types of elemental contact changes and that these relate to five model domain movements called: “free”, “open-closed”, “anchored”, “sliding-twist”, and “see-saw.” A directed graph is introduced called the “Dynamic Contact Graph” which represents the contact changes in a domain movement. In many cases a graph, or part of a graph, provides a clear visual metaphor for the movement it represents and is a motif that can be easily recognised. The Dynamic Contact Graphs are often comprised of disconnected subgraphs indicating independent regions which may play different roles in the domain movement. The Dynamic Contact Graph for each domain movement is decomposed into elemental Dynamic Contact Graphs, those that represent elemental contact changes, allowing us to count the number of instances of each type of elemental contact change in the domain movement. This naturally leads to sixteen classes into which the 1822 domain movements are classified.     

Assessing the Perception of Trunk Movements in Military Personnel with Chronic Non-Specific Low Back Pain Using a Virtual Mirror

Chronic pain, including chronic non-specific low back pain (CNSLBP), is often associated with body perception disturbances, but these have generally been assessed under static conditions. The objective of this study was to use a “virtual mirror” that scaled visual movement feedback to assess body perception during active movement in military personnel with CNSLBP (n = 15) as compared to military healthy control subjects (n = 15). Subjects performed a trunk flexion task while sitting and standing in front of a large screen displaying a full-body virtual mirror-image (avatar) in real-time. Avatar movements were scaled to appear greater, identical, or smaller than the subjects’ actual movements. A total of 126 trials with 11 different scaling factors were pseudo-randomized across 6 blocks. After each trial, subjects had to decide whether the avatar’s movements were “greater” or “smaller” than their own movements. Based on this two-alternative forced choice paradigm, a psychophysical curve was fitted to the data for each subject, and several metrics were derived from this curve. In addition, task adherence (kinematics) and virtual reality immersion were assessed. Groups displayed a similar ability to discriminate between different levels of movement scaling. Still, subjects with CNSLBP showed an abnormal performance and tended to overestimate their own movements (a right-shifted psychophysical curve). Subjects showed adequate task adherence, and on average virtual reality immersion was reported to be very good. In conclusion, these results extend previous work in patients with CNSLBP, and denote an important relationship between body perception, movement and pain. As such, the assessment of body perception during active movement can offer new avenues for understanding and managing body perception disturbances and abnormal movement patterns in patients with pain.     

Functions of mammalian spinal interneurons during movement

The major recent advances in understanding the role of spinal neurons in generating movement include new information about the modulation of classic reflex pathways during fictive locomotion and in response to pharmacological probes. The possibility of understanding movements in terms of spinal representations of a basic set of movement primitives has been extended by the analysis of normal reflexes. Recordings of the activity of cervical interneurons in behaving monkeys has elucidated their contribution to generating voluntary movement and revealed their involvement in movement preparation.     

Motor organization of the lateral cerebellar nucleus of the rat

The motor organization of the nucleus lateralis (NL) of the cerebellum of the rat was investigated by studying the motor effects following the electrical microstimulation. The movements evoked by the NL stimulation concerned prevalently the forelimb and the head segments. The movements of the hindlimb segments were evoked in only few cases. The NL is organized as a mosaic of zones without, or at least very little overlap. The various body segments are differently represented in the NL. Some of them are once represented (single representations). In other cases, the same movements were evoked by different NL regions (multiple representations). Finally, in a last lot of cases, various representations concerned the same body regions but from each representation a different type of movement was evoked (specific representations, i.e. displacement of an individual digit and flexion of all digits together). The topographical distribution of the representations in the NL cytological regions (magnicellularis, NLm; dorsolateral hump, DLH; subnucleus lateralis parvocellularis, slp) suggests the idea that each of them may be concerned in a specific motor activity: the NLm would control the position of the body, or of part of it, in the space; the DLH would be concerned in the oral (prevalently) and in the forelimb motor activity; the slp would be concerned in the exploration of the environment as well as in skilled movements of the distalmost forelimb segments.     

The participation of two rhythms in the leaf movements of xanthium plants given various light-dark cycles

We have shown Xanthium strumarium exhibit two distinct leaf movement rhythms with one occurring in continuous light and presumably related to an endogenous rhythm initiated by the “light-on” signal and the other occurring in continuous dark and presumably related to an endogenous rhythm initiated by the “light-off” signal. Characteristic of the light-on rhythm is a sudden and rapid downward movement of the leaf occurring about 16 hours after the light-on signal. Characteristic of the light-off rhythm is an immediate and sudden upward movement following the light-off signal. Under certain photoperiodic treatments, the two movements seem to be in conflict.     

Trunk Muscle Activation at the Initiation and Braking of Bilateral Shoulder Flexion Movements of Different Amplitudes

The aim of this study was to investigate if trunk muscle activation patterns during rapid bilateral shoulder flexions are affected by movement amplitude. Eleven healthy males performed shoulder flexion movements starting from a position with arms along sides (0°) to either 45°, 90° or 180°. EMG was measured bilaterally from transversus abdominis (TrA), obliquus internus (OI) with intra-muscular electrodes, and from rectus abdominis (RA), erector spinae (ES) and deltoideus with surface electrodes. 3D kinematics was recorded and inverse dynamics was used to calculate the reactive linear forces and torque about the shoulders and the linear and angular impulses. The sequencing of trunk muscle onsets at the initiation of arm movements was the same across movement amplitudes with ES as the first muscle activated, followed by TrA, RA and OI. All arm movements induced a flexion angular impulse about the shoulders during acceleration that was reversed during deceleration. Increased movement amplitude led to shortened onset latencies of the abdominal muscles and increased level of activation in TrA and ES. The activation magnitude of TrA was similar in acceleration and deceleration where the other muscles were specific to acceleration or deceleration. The findings show that arm movements need to be standardized when used as a method to evaluate trunk muscle activation patterns and that inclusion of the deceleration of the arms in the analysis allow the study of the relationship between trunk muscle activation and direction of perturbing torque during one and the same arm movement.     

How does microgravity affect the muscular and kinematic synergies in a complex movement?

The planning and the execution of voluntary movement relies on sensorimotor transformations in which representations of the external environment are integrated into motor programs. We studied executions of Whole Body Pointing movements, in normal and in transient microgravity (parabolic flights) conditions. Three processes could lead to adaptation to the new environmental condition: a radical change of terrestrial synergies, their partial modification or preservation. By applying a multivariate analysis on kinematic and electromyographic (EMG) data and by comparing the 1g and 0g conditions, our findings hint the hypothesis the descending information from vestibular system may be directed to change the synergies' modulation. An analogous analysis was performed on the kinematics: the invariance of intersegmental coordination among the segments' elevation angles suggests that these kinematic waveforms are used as reference signals to determine the appropriate muscle synergies in a subordinate and flexible manner in order to adapt to the novel mechanical constraints.     

Apparent Motion from Outside the Visual Field,Retinotopic Cortices May Register Extra-Retinal Positions

Observers made a saccade between two fixation markers while a probe was flashed sequentially at two locations on a side screen. The first probe was presented in the far periphery just within the observer''s visual field. This target was extinguished and the observers made a large saccade away from the probe, which would have left it far outside the visual field if it had still been present. The second probe was then presented, displaced from the first in the same direction as the eye movement and by about the same distance as the saccade step. Because both eyes and probes shifted by similar amounts, there was little or no shift between the first and second probe positions on the retina. Nevertheless, subjects reported seeing motion corresponding to the spatial displacement not the retinal displacement. When the second probe was presented, the effective location of the first probe lay outside the visual field demonstrating that apparent motion can be seen from a location outside the visual field to a second location inside the visual field. Recent physiological results suggest that target locations are “remapped” on retinotopic representations to correct for the effects of eye movements. Our results suggest that the representations on which this remapping occurs include locations that fall beyond the limits of the retina.     

Bumblebee Homing: The Fine Structure of Head Turning Movements

Changes in flight direction in flying insects are largely due to roll, yaw and pitch rotations of their body. Head orientation is stabilized for most of the time by counter rotation. Here, we use high-speed video to analyse head- and body-movements of the bumblebee Bombus terrestris while approaching and departing from a food source located between three landmarks in an indoor flight-arena. The flight paths consist of almost straight flight segments that are interspersed with rapid turns. These short and fast yaw turns (“saccades”) are usually accompanied by even faster head yaw turns that change gaze direction. Since a large part of image rotation is thereby reduced to brief instants of time, this behavioural pattern facilitates depth perception from visual motion parallax during the intersaccadic intervals. The detailed analysis of the fine structure of the bees’ head turning movements shows that the time course of single head saccades is very stereotypical. We find a consistent relationship between the duration, peak velocity and amplitude of saccadic head movements, which in its main characteristics resembles the so-called "saccadic main sequence" in humans. The fact that bumblebee head saccades are highly stereotyped as in humans, may hint at a common principle, where fast and precise motor control is used to reliably reduce the time during which the retinal images moves.     

Live cell imaging of the assembly, disassembly, and actin cable-dependent movement of endosomes and actin patches in the budding yeast, Saccharomyces cerevisiae

Using FM4-64 to label endosomes and Abp1p-GFP or Sac6p-GFP to label actin patches, we find that (1) endosomes colocalize with actin patches as they assemble at the bud cortex; (2) endosomes colocalize with actin patches as they undergo linear, retrograde movement from buds toward mother cells; and (3) actin patches interact with and disassemble at FM4-64–labeled internal compartments. We also show that retrograde flow of actin cables mediates retrograde actin patch movement. An Arp2/3 complex mutation decreases the frequency of cortical, nonlinear actin patch movements, but has no effect on the velocity of linear, retrograde actin patch movement. Rather, linear actin patch movement occurs at the same velocity and direction as the movement of actin cables. Moreover, actin patches require actin cables for retrograde movements and colocalize with actin cables as they undergo retrograde movement. Our studies support a mechanism whereby actin cables serve as “conveyor belts” for retrograde movement and delivery of actin patches/endosomes to FM4-64–labeled internal compartments.     

Functional and Structural Correlates of Motor Speed in the Cerebellar Anterior Lobe

In athletics, motor performance is determined by different abilities such as technique, endurance, strength and speed. Based on animal studies, motor speed is thought to be encoded in the basal ganglia, sensorimotor cortex and the cerebellum. The question arises whether there is a unique structural feature in the human brain, which allows “power athletes” to perform a simple foot movement significantly faster than “endurance athletes”. We acquired structural and functional brain imaging data from 32 track-and-field athletes. The study comprised of 16 “power athletes” requiring high speed foot movements (sprinters, jumpers, throwers) and 16 endurance athletes (distance runners) which in contrast do not require as high speed foot movements. Functional magnetic resonance imaging (fMRI) was used to identify speed specific regions of interest in the brain during fast and slow foot movements. Anatomical MRI scans were performed to assess structural grey matter volume differences between athletes groups (voxel based morphometry). We tested maximum movement velocity of plantarflexion (PF-V_max) and acquired electromyographical activity of the lateral and medial gastrocnemius muscle. Behaviourally, a significant difference between the two groups of athletes was noted in PF-V_max and fMRI indicates that fast plantarflexions are accompanied by increased activity in the cerebellar anterior lobe. The same region indicates increased grey matter volume for the power athletes compared to the endurance counterparts. Our results suggest that speed-specific neuro-functional and -structural differences exist between power and endurance athletes in the peripheral and central nervous system.     

Endogenous Short Period Rhythms in the Movements of Unifoliate Leaves of Phaseolus angularis Wight

Rhythmic rotational movements with the midvein as the axis have been observed in the unifoliate leaves of Phaseolus angularis Wight grown under controlled environmental conditions with continuous light. The mean period of this movement for all leaves was 53.2 ± 4.3 minutes and remained constant as the leaf matured, except after removal of the apical meristem and emerging trifoliate leaf when the period increased by about 5 minutes. The amplitude of the movement also remained constant as the leaf matured. These rotational movements were pronounced when the leaf blade was in a horizontal position and were not evident during the downward or “sleep” movements of the leaf. This movement began 3 days after leaf unfolding and continued for at least 6 days. It was most pronounced at the time of inflection of the leaf length growth curve after the logarithmic phase of growth.     

Action Recognition Depends on Observer’s Level of Action Control and Social Personality Traits

Humans recognize both the movement (physical) goals and action (conceptual) goals of individuals with whom they are interacting. Here, we assessed whether spontaneous recognition of others’ goals depends on whether the observers control their own behavior at the movement or action level. We also examined the relationship between individual differences in empathy and ASD-like traits, and the processing of other individual’s movement and action goals that are known to be encoded in the “mirroring” and “mentalizing” brain networks. In order to address these questions, we used a computer-based card paradigm that made it possible to independently manipulate movement and action congruency of observed and executed actions. In separate blocks, participants were instructed to select either the right or left card (movement-control condition) or the higher or lower card (action-control condition), while we manipulated action- and movement-congruency of both actors’ goals. An action-congruency effect was present in all conditions and the size of this effect was significantly correlated with self-reported empathy and ASD-like traits. In contrast, movement-congruency effects were only present in the movement-control block and were strongly dependent on action-congruency. These results illustrate that spontaneous recognition of others’ behavior depends on the control scheme that is currently adopted by the observer. The findings suggest that deficits in action recognition are related to abnormal synthesis of perceived movements and prior conceptual knowledge that are associated with activations in the “mirroring” and “mentalizing” cortical networks.