Analysis of an optimal control model of multi-joint arm movements

 In this paper, we propose a model of biological motor control for generation of goal-directed multi-joint arm movements, and study the formation of muscle control inputs and invariant kinematic features of movements. The model has a hierarchical structure that can determine the control inputs for a set of redundant muscles without any inverse computation. Calculation of motor commands is divided into two stages, each of which performs a transformation of motor commands from one coordinate system to another. At the first level, a central controller in the brain accepts instructions from higher centers, which represent the motor goal in the Cartesian space. The controller computes joint equilibrium trajectories and excitation signals according to a minimum effort criterion. At the second level, a neural network in the spinal cord translates the excitation signals and equilibrium trajectories into control commands to three pairs of antagonist muscles which are redundant for a two-joint arm. No inverse computation is required in the determination of individual muscle commands. The minimum effort controller can produce arm movements whose dynamic and kinematic features are similar to those of voluntary arm movements. For fast movements, the hand approaches a target position along a near-straight path with a smooth bell-shaped velocity. The equilibrium trajectories in X and Y show an ‘N’ shape, but the end-point equilibrium path zigzags around the hand path. Joint movements are not always smooth. Joint reversal is found in movements in some directions. The excitation signals have a triphasic (or biphasic) pulse pattern, which leads to stereotyped triphasic (or biphasic) bursts in muscle control inputs, and a dynamically modulated joint stiffness. There is a fixed sequence of muscle activation from proximal muscles to distal muscles. The order is preserved in all movements. For slow movements, it is shown that a constant joint stiffness is necessary to produce a smooth movement with a bell-shaped velocity. Scaled movements can be reproduced by varying the constraints on the maximal level of excitation signals according to the speed of movement. When the inertial parameters of the arm are altered, movement trajectories can be kept invariant by adjusting the pulse height values, showing the ability to adapt to load changes. These results agree with a wide range of experimental observations on human voluntary movements. Received: 4 December 1995 / Accepted in revised form: 17 September 1996     

Kinematic construction of the trajectory of sequential arm movements

A kinematic construction rule determining the trajectory of human sequential movements is formulated using minimum-jerk and minimum-angular-jerk trajectories. The kinematic construction rule states that the observed trajectory of sequential movements coincides with a weighted average of the minimum-jerk trajectory and the segmented minimum-angular-jerk trajectory. This rule covers not only point-to-point movements but also simple sequential movements. Five kinds of experiments that measure the trajectories in planar, multijoint sequential arm movements were conducted. The measured trajectories coincide with the predictions made on the basis of the kinematic construction rule presented here. Moreover, predictions of previous models such as the minimum-jerk, the equilibrium-trajectory, and the minimum-torque-change models are shown to be incompatible with our observations of sequential movements. Received: 31 October 1997 /Accepted in revised form: 18 November 1998     

End-point constraints in aiming movements: effects of approach angle and speed

 The present study focuses on two trajectory-formation models of point-to-point aiming movements, viz., the minimum-jerk and the minimum torque-change model. To date, few studies on minimum-jerk and minimum torque-change trajectories have incorporated self- or externally imposed end-point constraints, such as the direction and velocity with which a target area is approached. To investigate which model accounts best for the effects on movement trajectories of such – in many circumstances – realistic end-point constraints, we adjusted both the minimum-jerk and the minimum torque-change model so that they could generate trajectories of which the final part has a specific direction and speed. The adjusted models yield realistic trajectories with a high curvature near movement completion. Comparison of simulated and measured movement trajectories show that pointing movements that are constrained with respect to final movement direction and speed can be described in terms of minimization of joint-torque changes. Received: 7 July 1999 / Accepted in revised form: 8 January 2001     

Characterizing and Predicting Submovements during Human Three-Dimensional Arm Reaches

We have demonstrated that 3D target-oriented human arm reaches can be represented as linear combinations of discrete submovements, where the submovements are a set of minimum-jerk basis functions for the reaches. We have also demonstrated the ability of deterministic feed-forward Artificial Neural Networks (ANNs) to predict the parameters of the submovements. ANNs were trained using kinematic data obtained experimentally from five human participants making target-directed movements that were decomposed offline into minimum-jerk submovements using an optimization algorithm. Under cross-validation, the ANNs were able to accurately predict the parameters (initiation-time, amplitude, and duration) of the individual submovements. We also demonstrated that the ANNs can together form a closed-loop model of human reaching capable of predicting 3D trajectories with VAF >95.9% and RMSE ≤4.32 cm relative to the actual recorded trajectories. This closed-loop model is a step towards a practical arm trajectory generator based on submovements, and should be useful for the development of future arm prosthetic devices that are controlled by brain computer interfaces or other user interfaces.     

Using voluntary motor commands to inhibit involuntary arm movements

A hallmark of voluntary motor control is the ability to stop an ongoing movement. Is voluntary motor inhibition a general neural mechanism that can be focused on any movement, including involuntary movements, or is it mere termination of a positive voluntary motor command? The involuntary arm lift, or ‘floating arm trick’, is a distinctive long-lasting reflex of the deltoid muscle. We investigated how a voluntary motor network inhibits this form of involuntary motor control. Transcranial magnetic stimulation of the motor cortex during the floating arm trick produced a silent period in the reflexively contracting deltoid muscle, followed by a rebound of muscle activity. This pattern suggests a persistent generator of involuntary motor commands. Instructions to bring the arm down voluntarily reduced activity of deltoid muscle. When this voluntary effort was withdrawn, the involuntary arm lift resumed. Further, voluntary motor inhibition produced a strange illusion of physical resistance to bringing the arm down, as if ongoing involuntarily generated commands were located in a ‘sensory blind-spot’, inaccessible to conscious perception. Our results suggest that voluntary motor inhibition may be a specific neural function, distinct from absence of positive voluntary motor commands.     

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     

EEG pattern decoding of rhythmic individual finger imaginary movements of one hand

The results of four-class classification of the motor imagery EEG patterns corresponding to the right hand finger movements (little finger, thumb, index and middle fingers) of eight healthy subjects are presented in this study. The motor imagery of individual right-hand finger movements was executed by the subjects in a prescribed rhythm and the trials contained no external stimuli. Classification was performed by means of a specially developed two-level committee of classifiers on the basis of support vector machine and artificial neural networks at the first level and by generalizing an artificial neural network at the second level. The area under the EEG signal curve and the curve length calculated in a sliding time window for sites F ₃, C ₃, and Cz of the International 10?20 System were selected as the key features of signals from the sensorimotor and adjoining frontal cortical areas contralateral to the movements. The average accuracy of four-class singletrial classification for all subjects was 50 ± 7 [SD] (maximum, 58%) for the pair of sites F ₃–C ₃ and 46 ± 11% [SD] (maximum 62%) for the pair of sites C ₃–Cz with a theoretical guessing level 25%.     

A model of handwriting

The research reported here is concerned with hand trajectory planning for the class of movements involved in handwriting. Previous studies show that the kinematics of human two-joint arm movements in the horizontal plane can be described by a model which is based on dynamic minimization of the square of the third derivative of hand position (jerk), integrated over the entire movement. We extend this approach to both the analysis and the synthesis of the trajectories occurring in the generation of handwritten characters. Several basic strokes are identified and possible stroke concatenation rules are suggested. Given a concise symbolic representation of a stroke shape, a simple algorithm computes the complete kinematic specification of the corresponding trajectory. A handwriting generation model based on a kinematics from shape principle and on dynamic optimization is formulated and tested. Good qualitative and quantitative agreement was found between subject recordings and trajectories generated by the model. The simple symbolic representation of hand motion suggested here may permit the central nervous system to learn, store and modify motor action plans for writing in an efficient manner.     

Elastic deformations of the rotary double motor of single FoF1-ATP synthases detected in real time by Förster resonance energy transfer

Elastic conformational changes of the protein backbone are essential for catalytic activities of enzymes. To follow relative movements within the protein, Förster-type resonance energy transfer (FRET) between two specifically attached fluorophores can be applied. FRET provides a precise ruler between 3 and 8 nm with subnanometer resolution. Corresponding submillisecond time resolution is sufficient to identify conformational changes in FRET time trajectories. Analyzing single enzymes circumvents the need for synchronization of various conformations. F_OF₁-ATP synthase is a rotary double motor which catalyzes the synthesis of adenosine triphosphate (ATP). A proton-driven 10-stepped rotary F_O motor in the Escherichia coli enzyme is connected to a 3-stepped F₁ motor, where ATP is synthesized. To operate the double motor with a mismatch of step sizes smoothly, elastic deformations within the rotor parts have been proposed by W. Junge and coworkers. Here we extend a single-molecule FRET approach to observe both rotary motors simultaneously in individual F_OF₁-ATP synthases at work. We labeled this enzyme with two fluorophores specifically, that is, on the ε- and c-subunits of the two rotors. Alternating laser excitation was used to select the FRET-labeled enzymes. FRET changes indicated associated transient twisting within the rotors of single enzyme molecules during ATP hydrolysis and ATP synthesis. Supported by Monte Carlo simulations of the FRET experiments, these studies reveal that the rotor twisting is greater than 36° and is largely suppressed in the presence of the rotation inhibitor DCCD. This article is part of a Special Issue entitled: 17th European Bioenergetics Conference (EBEC 2012).     

Independent control of joint stiffness in the framework of the equilibrium-point hypothesis

In the framework of the equilibrium-point hypothesis, virtual trajectories and joint stiffness patterns have been reconstructed during two motor tasks practiced against a constant bias torque. One task required a voluntary increase in joint stiffness while preserving the original joint position. The other task involved fast elbow flexions over 36°. Joint stiffness gradually subsided after the termination of fast movements. In both tasks, the external torque could slowly and unexpectedly change. The subjects were required not to change their motor commands if the torque changed, i.e. “to do the same no matter what the motor did”. In both tasks, changes in joint stiffness were accompanied by unchanged virtual trajectories that were also independent of the absolute value of the bias torque. By contrast, the intercept of the joint compliant characteristic with the angle axis,r(t)-function, has demonstrated a clear dependence upon both the level of coactivation and external load. We assume that a template virtual trajectory is generated at a certain level of the motor hierarchy and is later scaled taking into account some commonly changing dynamic factors of the movement execution, for example, external load. The scaling leads to the generation of commands to the segmental structures that can be expressed, according to the equilibrium-point hypothesis, as changes in the thresholds of the tonic stretch reflex for corresponding muscles.     

Internal models and intermittency: A theoretical account of human tracking behavior

This paper concerns the use of tracking studies to test a theoretical account of the information processing performed by the human CNS during control of movement. The theory provides a bridge between studies of reaction time and continuous tracking. It is proposed that the human CNS includes neuronal circuitry to compute inverse internal models of the multiple input, multiple output, dynamic, nonlinear relationships between outgoing motor commands and their resulting perceptual consequences. The inverse internal models are employed during movement execution to transform preplanned trajectories of desired perceptual consequences into appropriate outgoing motor commands to achieve them. A finite interval of time is required by the CNS to preplan the desired perceptual consequences of a movement and it does not commence planning a new movement until planning of the old one has been completed. This behavior introduces intermittency into the planning of movements. In this paper we show that the gain and phase frequency response characteristics of the human operator in a visual pursuit tracking task can be derived theoretically from these assumptions. By incorporating the effects of internal model inaccuracy and of speed-accuracy trade-off in performance, it is shown that various aspects of experimentally measured tracking behavior can be accounted for.     

Computational model of motor learning and perceptual change

Motor learning in the context of arm reaching movements has been frequently investigated using the paradigm of force-field learning. It has been recently shown that changes to somatosensory perception are likewise associated with motor learning. Changes in perceptual function may be the reason that when the perturbation is removed following motor learning, the hand trajectory does not return to a straight line path even after several dozen trials. To explain the computational mechanisms that produce these characteristics, we propose a motor control and learning scheme using a simplified two-link system in the horizontal plane: We represent learning as the adjustment of desired joint-angular trajectories so as to achieve the reference trajectory of the hand. The convergence of the actual hand movement to the reference trajectory is proved by using a Lyapunov-like lemma, and the result is confirmed using computer simulations. The model assumes that changes in the desired hand trajectory influence the perception of hand position and this in turn affects movement control. Our computer simulations support the idea that perceptual change may come as a result of adjustments to movement planning with motor learning.     

The fractional-order dynamics of brainstem vestibulo-oculomotor neurons

The vestibulo-ocular reflex (VOR) and other oculomotor subsystems such as pursuit and saccades are ultimately mediated in the brainstem by premotor neurons in the vestibular and prepositus nuclei that relay eye movement commands to extraocular motoneurons. The premotor neurons receive vestibular signals from canal afferents. Canal afferent frequency responses have a component that can be characterized as a fractional-order differentiation (d ^k x/dt _k where k is a nonnegative real number). This article extends the use of fractional calculus to describe the dynamics of motor and premotor neurons. It suggests that the oculomotor integrator, which converts eye velocity into eye position commands, may be of fractional order. This order is less than one, and the velocity commands have order one or greater, so the resulting net output of motor and premotor neurons can be described as fractional differentiation relative to eye position. The fractional derivative dynamics of motor and premotor neurons may serve to compensate fractional integral dynamics of the eye. Fractional differentiation can be used to account for the constant phase shift across frequencies, and the apparent decrease in time constant as VOR and pursuit frequency increases, that are observed for motor and premotor neurons. Fractional integration can reproduce the time course of motor and premotor neuron saccade-related activity, and the complex dynamics of the eye. Insight into the nature of fractional dynamics can be gained through simulations in which fractional-order differentiators and integrators are approximated by sums of integer-order high-pass and low-pass filters, respectively. Fractional dynamics may be applicable not only to the oculomotor system, but to motor control systems in general.     

Alien hand syndrome: neural correlates of movements without conscious will

Background

The alien hand syndrome is a striking phenomenon characterized by purposeful and autonomous movements that are not voluntarily initiated. This study aimed to examine neural correlates of this rare neurological disorder in a patient with corticobasal degeneration and alien hand syndrome of the left hand.

Methodology/Principal Findings

We employed functional magnetic resonance imaging to investigate brain responses associated with unwanted movements in a case study. Results revealed that alien hand movements involved a network of brain activations including the primary motor cortex, premotor cortex, precuneus, and right inferior frontal gyrus. Conscious and voluntary movements of the alien hand elicited a similar network of brain responses but lacked an activation of the inferior frontal gyrus. The results demonstrate that alien and unwanted movements may engage similar brain networks than voluntary movements, but also imply different functional contributions of prefrontal areas. Since the inferior frontal gyrus was uniquely activated during alien movements, the results provide further support for a specific role of this brain region in inhibitory control over involuntary motor responses.

Conclusions/Significance

We discuss the outcome of this study as providing evidence for a distributed neural network associated with unwanted movements in alien hand syndrome, including brain regions known to be related to movement execution and planning as well as areas that have been linked to inhibition control (inferior frontal gyrus) and experience of agency (precuneus).     

Submovement Composition of Head Movement

Limb movement is smooth and corrections of movement trajectory and amplitude are barely noticeable midflight. This suggests that skeletomuscular motor commands are smooth in transition, such that the rate of change of acceleration (or jerk) is minimized. Here we applied the methodology of minimum-jerk submovement decomposition to a member of the skeletomuscular family, the head movement. We examined the submovement composition of three types of horizontal head movements generated by nonhuman primates: head-alone tracking, head-gaze pursuit, and eye-head combined gaze shifts. The first two types of head movements tracked a moving target, whereas the last type oriented the head with rapid gaze shifts toward a target fixed in space. During head tracking, the head movement was composed of a series of episodes, each consisting of a distinct, bell-shaped velocity profile (submovement) that rarely overlapped with each other. There was no specific magnitude order in the peak velocities of these submovements. In contrast, during eye-head combined gaze shifts, the head movement was often comprised of overlapping submovements, in which the peak velocity of the primary submovement was always higher than that of the subsequent submovement, consistent with the two-component strategy observed in goal-directed limb movements. These results extend the previous submovement composition studies from limb to head movements, suggesting that submovement composition provides a biologically plausible approach to characterizing the head motor recruitment that can vary depending on task demand.     

Hormonal and developmental regulation of glycosylated alpha 2u-globulin synthesis

We have investigated the regulation of glycosylated α_2u-globulin synthesis by examining the appearance of these molecules in the medium of primary monolayer cultures of hepatocytes. Hepatocytes were isolated from male and female rats of various ages, as well as from castrated or ovariectomized animals. α_2u-Globulin was immunoprecipitated from the culture medium with rabbit antibody specific for α_2u-globulin, and the dissociated precipitates were electrophoresed on sodium dodecyl sulfate-polyacrylamide gels. We found that prepubescent male and female rats synthesized only the high molecular weight glycosylated forms of α_2u-globulin. Hepatocytes from 50-day-old intact and ovariectomized female rats, as well as from ovariectomized rats treated with 17β-estradiol, secreted only glycosylated α_2u-globulin. Hepatocytes from castrated male rats treated with dihydrotestosterone in vivo synthesized the 20,000-dalton nonglycosylated form of α_2u-globulin; the rate of glycosylated α_2u-globulin synthesis was reduced in these cells. The rate of synthesis of glycosylated α_2u-globulin by male rat hepatocytes declined concomitant with increases in the age of the rats, the level of serum testosterone, and the rate of synthesis of nonglycosylated α_2u-globulin. Under our conditions, dexamethasone administration to castrated male rats or ovariectomized female rats in vivo did not alter the species of α_2u-globulin that were synthesized subsequently by hepatocytes in vitro. Our results suggest that the synthesis of glycosylated α_2u-globulin is regulated differently than the synthesis of the 20,000-dalton nonglycosylated form of α_2u-globulin.     

Adiabatic Heating of Electrons in the Magnetospheric Current Sheet

Electron dynamics and acceleration in an electromagnetic field configuration modeling the current sheet configuration of the Earth’s magnetotail region is investigated. A focus is made on the role of the dawn?dusk magnetic field component B_y in the convection electron heating by an electric field E_y. For numerical integration of a large number of test particle trajectories over long time intervals, the equations of motion written in the guiding center approximation are used. It is shown that the presence of a B_y ≠ 0 magnetic field significantly changes the electron heating and allows electrons with small pitch angles to gain energy much more efficiently than the equatorial electrons. As a result, the convection heating in the current sheet with B_y ≠ 0 leads to the formation of an accelerated anisotropic population of particles with energies higher than a few hundred electronvolts. The obtained results and spacecraft observations in the Earth’s magnetotail are compared, and possible limitations in the proposed model approaches are discussed.     

Unscented Kalman Filter for Brain-Machine Interfaces

Brain machine interfaces (BMIs) are devices that convert neural signals into commands to directly control artificial actuators, such as limb prostheses. Previous real-time methods applied to decoding behavioral commands from the activity of populations of neurons have generally relied upon linear models of neural tuning and were limited in the way they used the abundant statistical information contained in the movement profiles of motor tasks. Here, we propose an n-th order unscented Kalman filter which implements two key features: (1) use of a non-linear (quadratic) model of neural tuning which describes neural activity significantly better than commonly-used linear tuning models, and (2) augmentation of the movement state variables with a history of n-1 recent states, which improves prediction of the desired command even before incorporating neural activity information and allows the tuning model to capture relationships between neural activity and movement at multiple time offsets simultaneously. This new filter was tested in BMI experiments in which rhesus monkeys used their cortical activity, recorded through chronically implanted multielectrode arrays, to directly control computer cursors. The 10th order unscented Kalman filter outperformed the standard Kalman filter and the Wiener filter in both off-line reconstruction of movement trajectories and real-time, closed-loop BMI operation.