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.     

Analysis of kinematic invariances of multijoint reaching movement

 There is a no unique relationship between the trajectory of the hand, represented in cartesian or extrinsic space, and its trajectory in joint angle or intrinsic space in the general condition of joint redundancy. The goal of this work is to analyze the relation between planning the trajectory of a multijoint movement in these two coordinate systems. We show that the cartesian trajectory can be planned based on the task parameters (target coordinates, etc.) prior to and independently of angular trajectories. Angular time profiles are calculated from the cartesian trajectory to serve as a basis for muscle control commands. A unified differential equation that allows planning trajectories in cartesian and angular spaces simultaneously is proposed. Due to joint redundancy, each cartesian trajectory corresponds to a family of angular trajectories which can account for the substantial variability of the latter. A set of strategies for multijoint motor control following from this model is considered; one of them coincides with the frog wiping reflex model and resolves the kinematic inverse problem without inversion. The model trajectories exhibit certain properties observed in human multijoint reaching movements such as movement equifinality, straight end-point paths, bell-shaped tangential velocity profiles, speed-sensitive and speed-insensitive movement strategies, peculiarities of the response to double-step targets, and variations of angular trajectory without variations of the limb end-point trajectory in cartesian space. In humans, those properties are almost independent of limb configuration, target location, movement duration, and load. In the model, these properties are invariant to an affine transform of cartesian space. This implies that these properties are not a special goal of the motor control system but emerge from movement kinematics that reflect limb geometry, dynamics, and elementary principles of motor control used in planning. All the results are given analytically and, in order to compare the model with experimental results, by computer simulations. Received: 6 April 1994/Accepted in revised form: 25 April 1995     

Spatial coding of visually guided arm movements in primate motor cortex

Previous studies of the motor cortex in behaving animals were focused on the relations between the activity of single cells, usually pyramidal tract neurons, and parameters of isometric contraction (e.g., intensity of force) or parameters of movement along one axis (e.g., flexion-extension) of a single joint (e.g., elbow or wrist). However, the commonly meaningful behavioral parameter is the trajectory of the hand in extrapersonal space, which is realized by simultaneous motions about two or three joints (e.g., elbow, shoulder, wrist) and concurrent engagement of several muscles. The spatial parameters of a straight trajectory are its direction and extent. We hypothesized that a major function of the motor cortex, among other possible roles, is the specification and control of the direction of the movement trajectory in space. This reference of motor cortical function to the control of spatial aspects of the trajectory differentiated our approach from the other approaches outlined above. We investigated the directional selectivity cells in the arm area of the motor cortex by recording their activity while monkeys moved their hands in various directions in space towards visual targets. There were two salient findings of these studies. First, the intensity of the discharge of single cells varies in an orderly fashion with the direction of movement in space, so that the discharge rate is highest with movements in a preferred direction, and decreases progressively with movements made in directions more and more away from the preferred one. Thus single cells are broadly tuned around a preferred direction which differs among different cells.(ABSTRACT TRUNCATED AT 250 WORDS)     

A hierarchical neural-network model for control and learning of voluntary movement

In order to control voluntary movements, the central nervous system (CNS) must solve the following three computational problems at different levels: the determination of a desired trajectory in the visual coordinates, the transformation of its coordinates to the body coordinates and the generation of motor command. Based on physiological knowledge and previous models, we propose a hierarchical neural network model which accounts for the generation of motor command. In our model the association cortex provides the motor cortex with the desired trajectory in the body coordinates, where the motor command is then calculated by means of long-loop sensory feedback. Within the spinocerebellum — magnocellular red nucleus system, an internal neural model of the dynamics of the musculoskeletal system is acquired with practice, because of the heterosynaptic plasticity, while monitoring the motor command and the results of movement. Internal feedback control with this dynamical model updates the motor command by predicting a possible error of movement. Within the cerebrocerebellum — parvocellular red nucleus system, an internal neural model of the inverse-dynamics of the musculo-skeletal system is acquired while monitoring the desired trajectory and the motor command. The inverse-dynamics model substitutes for other brain regions in the complex computation of the motor command. The dynamics and the inverse-dynamics models are realized by a parallel distributed neural network, which comprises many sub-systems computing various nonlinear transformations of input signals and a neuron with heterosynaptic plasticity (that is, changes of synaptic weights are assumed proportional to a product of two kinds of synaptic inputs). Control and learning performance of the model was investigated by computer simulation, in which a robotic manipulator was used as a controlled system, with the following results: (1) Both the dynamics and the inverse-dynamics models were acquired during control of movements. (2) As motor learning proceeded, the inverse-dynamics model gradually took the place of external feedback as the main controller. Concomitantly, overall control performance became much better. (3) Once the neural network model learned to control some movement, it could control quite different and faster movements. (4) The neural netowrk model worked well even when only very limited information about the fundamental dynamical structure of the controlled system was available. Consequently, the model not only accounts for the learning and control capability of the CNS, but also provides a promising parallel-distributed control scheme for a large-scale complex object whose dynamics are only partially known.     

The point of no return in planar hand movements: an indication of the existence of high level motion primitives

Previous psychophysical studies have sought to determine whether the processes of movement engagement and termination are dissociable, whether stopping an action is a generic process, and whether there is a point in time in which the generation of a planned action is inevitable (“point of no return”). It is not clear yet, however, whether the action of stopping is merely a manifestation of low level, dynamic constraints, or whether it is also subject to a high level, kinematic plan. In the present study, stopping performance was studied while nine subjects, who generated free scribbling movements looking for the location of an invisible circular target, were requested unexpectedly to impede movement. Temporal analysis of the data shows that in 87% of the movements subsequent to the ‘stop’ cue, the tangential motion velocity profile was not a decelerating function of the time but rather exhibited a complex pattern comprised of one or more velocity peaks, implying an unstoppable motion element. Furthermore, geometrical analysis shows that the figural properties of the path generated after the ‘stop’ cue were part of a repetitive geometrical pattern and that the probability of completing a pattern after the ‘stop’ cue was correlated with the relative advance in the geometrical plan rather than the amount of time that had elapsed from the pattern initiation. Altogether, these findings suggest that the “point of no return” phenomenon in humans may also reflect a high level kinematic plan and could serve as a new operative definition of motion primitives.     

Stimulus correlation and adaptive motion detection using spiking neurons

Stimulus correlation and adaptive movement detection, among other tasks can be performed with VLSI general-purpose neurons that have controllable steady and transient responses. This paper presents experimental results of simple neural primitives based on the CMOS neuron approach described in [11]. Stimulus correlation experiments illustrate the well defined behavior of the CMOS approach. This basic primitive is used to implement motion detectors with adaptive capabilities that enable it to work efficiently in a wide velocity range.     

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.     

Schema generation in recurrent neural nets for intercepting a moving target

The grasping of a moving object requires the development of a motor strategy to anticipate the trajectory of the target and to compute an optimal course of interception. During the performance of perception-action cycles, a preprogrammed prototypical movement trajectory, a motor schema, may highly reduce the control load. Subjects were asked to hit a target that was moving along a circular path by means of a cursor. Randomized initial target positions and velocities were detected in the periphery of the eyes, resulting in a saccade toward the target. Even when the target disappeared, the eyes followed the target’s anticipated course. The Gestalt of the trajectories was dependent on target velocity. The prediction capability of the motor schema was investigated by varying the visibility range of cursor and target. Motor schemata were determined to be of limited precision, and therefore visual feedback was continuously required to intercept the moving target. To intercept a target, the motor schema caused the hand to aim ahead and to adapt to the target trajectory. The control of cursor velocity determined the point of interception. From a modeling point of view, a neural network was developed that allowed the implementation of a motor schema interacting with feedback control in an iterative manner. The neural net of the Wilson type consists of an excitation-diffusion layer allowing the generation of a moving bubble. This activation bubble runs down an eye-centered motor schema and causes a planar arm model to move toward the target. A bubble provides local integration and straightening of the trajectory during repetitive moves. The schema adapts to task demands by learning and serves as forward controller. On the basis of these model considerations the principal problem of embedding motor schemata in generalized control strategies is discussed.     

A neurodynamic account of spontaneous behaviour

The current article suggests that deterministic chaos self-organized in cortical dynamics could be responsible for the generation of spontaneous action sequences. Recently, various psychological observations have suggested that humans and primates can learn to extract statistical structures hidden in perceptual sequences experienced during active environmental interactions. Although it has been suggested that such statistical structures involve chunking or compositional primitives, their neuronal implementations in brains have not yet been clarified. Therefore, to reconstruct the phenomena, synthetic neuro-robotics experiments were conducted by using a neural network model, which is characterized by a generative model with intentional states and its multiple timescales dynamics. The experimental results showed that the robot successfully learned to imitate tutored behavioral sequence patterns by extracting the underlying transition probability among primitive actions. An analysis revealed that a set of primitive action patterns was embedded in the fast dynamics part, and the chaotic dynamics of spontaneously sequencing these action primitive patterns was structured in the slow dynamics part, provided that the timescale was adequately set for each part. It was also shown that self-organization of this type of functional hierarchy ensured robust action generation by the robot in its interactions with a noisy environment. This article discusses the correspondence of the synthetic experiments with the known hierarchy of the prefrontal cortex, the supplementary motor area, and the primary motor cortex for action generation. We speculate that deterministic dynamical structures organized in the prefrontal cortex could be essential because they can account for the generation of both intentional behaviors of fixed action sequences and spontaneous behaviors of pseudo-stochastic action sequences by the same mechanism.     

Aberrant Neuromagnetic Activation in the Motor Cortex in Children with Acute Migraine: A Magnetoencephalography Study

Migraine attacks have been shown to interfere with normal function in the brain such as motor or sensory function. However, to date, there has been no clinical neurophysiology study focusing on the motor function in children with migraine during headache attacks. To investigate the motor function in children with migraine, twenty-six children with acute migraine, meeting International Classification of Headache Disorders criteria and age- and gender-matched healthy children were studied using a 275-channel magnetoencephalography system. A finger-tapping paradigm was designed to elicit neuromagnetic activation in the motor cortex. Children with migraine showed significantly prolonged latency of movement-evoked magnetic fields (MEF) during finger movement compared with the controls. The correlation coefficient of MEF latency and age in children with migraine was significantly different from that in healthy controls. The spectral power of high gamma (65–150 Hz) oscillations during finger movement in the primary motor cortex is also significantly higher in children with migraine than in controls. The alteration of responding latency and aberrant high gamma oscillations suggest that the developmental trajectory of motor function in children with migraine is impaired during migraine attacks and/or developmentally delayed. This finding indicates that childhood migraine may affect the development of brain function and result in long-term problems.     

Modeling motor cortical operations by an attractor network of stochastic neurons

Understanding the neural computations performed by the motor cortex requires biologically plausible models that account for cell discharge patterns revealed by neurophysiological recordings. In the present study the motor cortical activity underlying movement generation is modeled as the dynamic evolution of a large fully recurrent network of stochastic spiking neurons with noise superimposed on the synaptic transmission. We show that neural representations of the learned movement trajectories can be stored in the connectivity matrix in such a way that, when activated, a particular trajectory evolves in time as a dynamic attractor of the system while individual neurons fire irregularly with large variability in their interspike intervals. Moreover, the encoding of trajectories as attractors ensures high stability of the ensemble dynamics in the presence of synaptic noise. In agreement with neurophysiological findings, the suggested model can provide a wide repertoire of specific motor behaviors, whereas the number of specialized cells and specific connections may be negligibly small if compared with the whole population engaged in trajectory retrieving. To examine the applicability of the model we study quantitatively the relationship between local geometrical and kinematic characteristics of the trajectories generated by the network. The relationship obtained as a result of simulations is close to the 2/3 power law established by psychophysical and neurophysiological studies.     

Cognitive Loading Affects Motor Awareness and Movement Kinematics but Not Locomotor Trajectories during Goal-Directed Walking in a Virtual Reality Environment

The primary purpose of this study was to investigate the effects of cognitive loading on movement kinematics and trajectory formation during goal-directed walking in a virtual reality (VR) environment. The secondary objective was to measure how participants corrected their trajectories for perturbed feedback and how participants'' awareness of such perturbations changed under cognitive loading. We asked 14 healthy young adults to walk towards four different target locations in a VR environment while their movements were tracked and played back in real-time on a large projection screen. In 75% of all trials we introduced angular deviations of ±5° to ±30° between the veridical walking trajectory and the visual feedback. Participants performed a second experimental block under cognitive load (serial-7 subtraction, counter-balanced across participants). We measured walking kinematics (joint-angles, velocity profiles) and motor performance (end-point-compensation, trajectory-deviations). Motor awareness was determined by asking participants to rate the veracity of the feedback after every trial. In-line with previous findings in natural settings, participants displayed stereotypical walking trajectories in a VR environment. Our results extend these findings as they demonstrate that taxing cognitive resources did not affect trajectory formation and deviations although it interfered with the participants'' movement kinematics, in particular walking velocity. Additionally, we report that motor awareness was selectively impaired by the secondary task in trials with high perceptual uncertainty. Compared with data on eye and arm movements our findings lend support to the hypothesis that the central nervous system (CNS) uses common mechanisms to govern goal-directed movements, including locomotion. We discuss our results with respect to the use of VR methods in gait control and rehabilitation.     

Learning the dynamics of reaching movements results in the modification of arm impedance and long-latency perturbation responses

 Some characteristics of arm movements that humans exhibit during learning the dynamics of reaching are consistent with a theoretical framework where training results in motor commands that are gradually modified to predict and compensate for novel forces that may act on the hand. As a first approximation, the motor control system behaves as an adapting controller that learns an internal model of the dynamics of the task. It approximates inverse dynamics and predicts motor commands that are appropriate for a desired limb trajectory. However, we had previously noted that subtle motion characteristics observed during changes in task dynamics challenged this simple model and raised the possibility that adaptation also involved sensory–motor feedback pathways. These pathways reacted to sensory feedback during the course of the movement. Here we hypothesize that adaptation to dynamics might also involve a modification of how the CNS responds to sensory feedback. We tested this through experiments that quantified how the motor system's response to errors during voluntary movements changed as it adapted to dynamics of a force field. We describe a nonlinear approach that approximates the impedance of the arm, i.e., force response as a function of arm displacement trajectory. We observe that after adaptation, the impedance function changes in a way that closely matches and counters the effect of the force field. This is particularly prominent in the long-latency (>100 ms) component of response to perturbations. Therefore, it appears that practice not only modifies the internal model with which the brain generates motor commands that initiate a movement, but also the internal model with which sensory feedback is integrated with the ongoing descending commands in order to respond to error during the movement. Received: 10 January 2001 / Accepted in revised form: 30 May 2001     

The Main Sequence of Saccades Optimizes Speed-accuracy Trade-off

In primates, it is well known that there is a consistent relationship between the duration, peak velocity and amplitude of saccadic eye movements, known as the ‘main sequence’. The reason why such a stereotyped relationship evolved is unknown. We propose that a fundamental constraint on the deployment of foveal vision lies in the motor system that is perturbed by signal-dependent noise (proportional noise) on the motor command. This noise imposes a compromise between the speed and accuracy of an eye movement. We propose that saccade trajectories have evolved to optimize a trade-off between the accuracy and duration of the movement. Taking a semi-analytical approach we use Pontryagin’s minimum principle to show that there is an optimal trajectory for a given amplitude and duration; and that there is an optimal duration for a given amplitude. It follows that the peak velocity is also fixed for a given amplitude. These predictions are in good agreement with observed saccade trajectories and the main sequence. Moreover, this model predicts a small saccadic dead-zone in which it is better to stay eccentric of target than make a saccade onto target. We conclude that the main sequence has evolved as a strategy to optimize the trade-off between accuracy and speed.     

Role of motor cortex in coordinating multi-joint movements: is it time for a new paradigm?

Reaching movements to spatial targets require motor patterns at the shoulder to be coordinated carefully with those at the elbow to smoothly move the hand through space. While the motor cortex is involved in this volitional task, considerable debate remains about how this cortical region participates in planning and controlling movement. This article reviews two opposing interpretations of motor cortical function during multi-joint movements. On the one hand, studies performed predominantly on single-joint movement generally support the notion that motor cortical activity is intimately involved in generating motor patterns at a given joint. In contrast, studies on reaching demonstrate correlations between motor cortical activity and features of movement related to the hand, suggesting that the motor cortex may be involved in more global features of the task. Although this latter paradigm involves a multi-joint motor task in which neural activity is correlated with features of movement related to the hand, this neural activity is also correlated to other movement variables. Therefore it is difficult to assess if and how the motor cortex contributes to the coordination of motor patterns at different joints. In particular, present paradigms cannot assess whether motor cortical activity contributes to the control of one joint or multiple joints during whole-arm tasks. The final point discussed in this article is the development of a new experimental device (KINARM) that can both monitor and manipulate the mechanics of the shoulder and elbow independently during multi-joint motor tasks. It is hoped that this new device will provide a new approach for examining how the motor cortex is involved in motor coordination.     

Chaotic and ordered processes mediating the knee-jerk reflex

In nerve chains that mediate tendon reflexes one can distinguish structured activity that represents the development of reflex excitation in response to muscle stretch and stochastic activity which is the result of fluctuations emerging in nerve structures. In analyzing neuron processes, one should consider also the factor of information whose carriers in nerve chains are structured and stochastic activities. A putative generator of fluctuations of muscle responses (possibly the main one) are cortex levels. Stochastic processes in neuron chains cause instability of the motor system at rest and in movement. Stochastic activity maintains the tone of skeletal muscles and affects the sensitivity of sensory systems via the mechanism of stochastic resonance.     

Impedance characteristics of a neuromusculoskeletal model of the human arm II. Movement control

The modulation of neuromusculoskeletal impedance during movements is analysed using a motor control model of the human arm. The motor control system combines feedback and feedforward control and both control modes are determined in one optimization process. In the model, the stiffness varies at the double movement frequency for 2-Hz oscillatory elbow movements and has high values at the movement reversals. During goal-directed two-degrees-of-freedom arm movements, the stiffness is decreased during the movement and may be increased in the initial and final phases, depending on the movement velocity. The stiffness has a considerable curl during the movement, as was also observed in experimental data. The dynamic stiffness patterns of the model can be explained basically by the α−γ coactivation scheme where feedback gains covary with motor control signals. In addition to the modulation of the gain factors, it is argued that the variation of the intrinsic stiffness has a considerable effect on movement control, especially during fast movements. Received: 14 October 1997 / Accepted in revised form: 18 May 1999     

Identification of neuromuscular synergies in natural upper-arm movements

 In this paper we study a natural upper-arm movement composed of sequential phases of reaching, grasping, and retrieval in the horizontal plane; in the present context, natural means that the spatiotemporal constraints active in the execution of the movement are low. The proposed method of quantitative analysis, which is applied to data obtained from healthy subjects, combines quantification of arm kinematics and kinesiologic electromyography assessment of neuromuscular control for each subject; to this aim, we propose using factor analysis. Despite the existence of a time-profile invariance of velocity traces, the identified factors may differ in the muscles that load on them; the interpretation is that different neuromuscular synergies are involved in the execution of the movement, which leads to cluster the subjects into two distinct groups. Compared with group-2 subjects, group-1 subjects present a more lateral hand path direction and a less smooth hand trajectory in their approach to the grasped object. To explain the identified factors and the kinematic findings, it is hypothesized that the feedforward motor commands involved in trajectory planning are different. Received: 28 June 2001 / Accepted in revised form: 6 November 2001