Endpoint Variance Research Articles

Increased variability but intact integration during visual navigation in Autism Spectrum Disorder

Autism Spectrum Disorder (ASD) is a common neurodevelopmental disturbance afflicting a variety of functions. The recent computational focus suggesting aberrant Bayesian inference in ASD has yielded promising but conflicting results in attempting to explain a wide variety of phenotypes by canonical computations. Here, we used a naturalistic visual path integration task that combines continuous action with active sensing and allows tracking of subjects' dynamic belief states. Both groups showed a previously documented bias pattern by overshooting the radial distance and angular eccentricity of targets. For both control and ASD groups, these errors were driven by misestimated velocity signals due to a nonuniform speed prior rather than imperfect integration. We tracked participants' beliefs and found no difference in the speed prior, but there was heightened variability in the ASD group. Both end point variance and trajectory irregularities correlated with ASD symptom severity. With feedback, variance was reduced, and ASD performance approached that of controls. These findings highlight the need for both more naturalistic tasks and a broader computational perspective to understand the ASD phenotype and pathology.

An acute application of transcranial random noise stimulation does not enhance motor skill acquisition or retention in a golf putting task

Transcranial random noise stimulation (tRNS) is a brain stimulation technique that has been shown to increase motor performance in simple motor tasks. The purpose was to determine the influence of tRNS on motor skill acquisition and retention in a complex golf putting task. Thirty-four young adults were randomly assigned to a tRNS group or a SHAM stimulation group. Each subject completed a practice session followed by a retention session. In the practice session, subjects performed golf putting trials in a baseline test block, four practice blocks, and a post test block. Twenty-four hours later subjects completed the retention test block. The golf putting task involved performing putts to a small target located 3 m away. tRNS or SHAM was applied during the practice blocks concurrently with the golf putting task. tRNS was applied over the first dorsal interosseus muscle representation area of the motor cortex for 20 min at a current strength of 2 mA. Endpoint error and endpoint variance were reduced across the both the practice blocks and the test blocks, but these reductions were not different between groups. These findings suggest that an acute application of tRNS failed to enhance skill acquisition or retention in a golf putting task.

The duration of reaching movement is longer than predicted by minimum variance.

Whether the central nervous system minimizes variability or effort in planning arm movements can be tested by measuring the preferred movement duration and end-point variability. Here we conducted an experiment in which subjects performed arm reaching movements without visual feedback in fast-, medium-, slow-, and preferred-duration conditions. Results show that 1) total end-point variance was smallest in the medium-duration condition and 2) subjects preferred to carry out movements that were slower than this medium-duration condition. A parsimonious explanation for the overall pattern of end-point errors across fast, medium, preferred, and slow movement durations is that movements are planned to minimize effort as well as end-point error due to both signal-dependent and constant noise.

Modeling Inter-trial Variability of Saccade Trajectories: Effects of Lesions of the Oculomotor Part of the Fastigial Nucleus

This study investigates the inter-trial variability of saccade trajectories observed in five rhesus macaques (Macaca mulatta). For each time point during a saccade, the inter-trial variance of eye position and its covariance with eye end position were evaluated. Data were modeled by a superposition of three noise components due to 1) planning noise, 2) signal-dependent motor noise, and 3) signal-dependent premotor noise entering within an internal feedback loop. Both planning noise and signal-dependent motor noise (together called accumulating noise) predict a simple S-shaped variance increase during saccades, which was not sufficient to explain the data. Adding noise within an internal feedback loop enabled the model to mimic variance/covariance structure in each monkey, and to estimate the noise amplitudes and the feedback gain. Feedback noise had little effect on end point noise, which was dominated by accumulating noise. This analysis was further extended to saccades executed during inactivation of the caudal fastigial nucleus (cFN) on one side of the cerebellum. Saccades ipsiversive to an inactivated cFN showed more end point variance than did normal saccades. During cFN inactivation, eye position during saccades was statistically more strongly coupled to eye position at saccade end. The proposed model could fit the variance/covariance structure of ipsiversive and contraversive saccades. Inactivation effects on saccade noise are explained by a decrease of the feedback gain and an increase of planning and/or signal-dependent motor noise. The decrease of the fitted feedback gain is consistent with previous studies suggesting a role for the cerebellum in an internal feedback mechanism. Increased end point variance did not result from impaired feedback but from the increase of accumulating noise. The effects of cFN inactivation on saccade noise indicate that the effects of cFN inactivation cannot be explained entirely with the cFN’s direct connections to the saccade-related premotor centers in the brainstem.

Stability of Phase Relationships While Coordinating Arm Reaches with Whole Body Motion.

The human movement repertoire is characterized by the smooth coordination of several body parts, including arm movements and whole body motion. The neural control of this coordination is quite complex because the various body parts have their own kinematic and dynamic properties. Behavioral inferences about the neural solution to the coordination problem could be obtained by examining the emerging phase relationship and its stability. Here, we studied the phase relationships that characterize the coordination of arm-reaching movements with passively-induced whole-body motion. Participants were laterally translated using a vestibular chair that oscillated at a fixed frequency of 0.83 Hz. They were instructed to reach between two targets that were aligned either parallel or orthogonal to the whole body motion. During the first cycles of body motion, a metronome entrained either an in-phase or an anti-phase relationship between hand and body motion, which was released at later cycles to test phase stability. Results suggest that inertial forces play an important role when coordinating reaches with cyclic whole-body motion. For parallel reaches, we found a stable in-phase and an unstable anti-phase relationship. When the latter was imposed, it readily transitioned or drifted back toward an in-phase relationship at cycles without metronomic entrainment. For orthogonal reaches, we did not find a clear difference in stability between in-phase and anti-phase relationships. Computer simulations further show that cost models that minimize energy expenditure (i.e. net torques) or endpoint variance of the reach cannot fully explain the observed coordination patterns. We discuss how predictive control and impedance control processes could be considered important mechanisms underlying the rhythmic coordination of arm reaches and body motion.

Analysis of EPA’s endocrine screening battery and recommendations for further review

EPA’s Endocrine Disruptor Screening Program Tier 1 battery consists of eleven assays intended to identify the potential of a chemical to interact with the estrogen, androgen, thyroid, or steroidogenesis systems. We have collected control data from a subset of test order recipients from the first round of screening. The analysis undertaken herein demonstrates that the EPA should review all testing methods prior to issuing further test orders. Given the frequency with which certain performance criteria were violated, a primary focus of that review should consider adjustments to these standards to better reflect biological variability. A second focus should be to provide detailed, assay-specific direction on when results should be discarded; no clear guidance exists on the degree to which assays need to be re-run for failing to meet performance criteria. A third focus should be to identify permissible differences in study design and execution that have a large influence on endpoint variance. Experimental guidelines could then be re-defined such that endpoint variances are reduced and performance criteria are violated less frequently. It must be emphasized that because we were restricted to a subset (approximately half) of the control data, our analyses serve only as examples to underscore the importance of a detailed, rigorous, and comprehensive evaluation of the performance of the battery.

Kinematic feedback control laws for generating natural arm movements

We propose a stochastic optimal feedback control law for generating natural robot arm motions. Our approach, inspired by the minimum variance principle of Harris and Wolpert (1998 Nature 394 780–4) and the optimal feedback control principles put forth by Todorov and Jordan (2002 Nature Neurosci. 5 1226–35) for explaining human movements, differs in two crucial respects: (i) the endpoint variance is minimized in joint space rather than Cartesian hand space, and (ii) we ignore the dynamics and instead consider only the second-order differential kinematics. The feedback control law generating the motions can be straightforwardly obtained by backward integration of a set of ordinary differential equations; these equations are obtained exactly, without any linear–quadratic approximations. The only parameters to be determined a priori are the variance scale factors, and for both the two-DOF planar arm and the seven-DOF spatial arm, a table of values is constructed based on the given initial and final arm configurations; these values are determined via an optimal fitting procedure, and consistent with existing findings about neuromuscular motor noise levels of human arm muscles. Experiments conducted with a two-link planar arm and a seven-DOF spatial arm verify that the trajectories generated by our feedback control law closely resemble human arm motions, in the sense of producing nearly straight-line hand trajectories, having bell-shaped velocity profiles, and satisfying Fitts Law.

Knowing Each Random Error of Our Ways, but Hardly Correcting for It: An Instance of Optimal Performance

Random errors are omnipresent in sensorimotor tasks due to perceptual and motor noise. The question is, are humans aware of their random errors on an instance-by-instance basis? The appealing answer would be ‘no’ because it seems intuitive that humans would otherwise immediately correct for the errors online, thereby increasing sensorimotor precision. However, here we show the opposite. Participants pointed to visual targets with varying degree of feedback. After movement completion participants indicated whether they believed they landed left or right of target. Surprisingly, participants' left/right-discriminability was well above chance, even without visual feedback. Only when forced to correct for the error after movement completion did participants loose knowledge about the remaining error, indicating that random errors can only be accessed offline. When correcting, participants applied the optimal correction gain, a weighting factor between perceptual and motor noise, minimizing end-point variance. Together these results show that humans optimally combine direct information about sensorimotor noise in the system (the current random error), with indirect knowledge about the variance of the perceptual and motor noise distributions. Yet, they only appear to do so offline after movement completion, not while the movement is still in progress, suggesting that during movement proprioceptive information is less precise.

Movement trajectory smoothness is not associated with the endpoint accuracy of rapid multi-joint arm movements in young and older adults

The minimum variance theory proposes that motor commands are corrupted by signal-dependent noise and smooth trajectories with low noise levels are selected to minimize endpoint error and endpoint variability. The purpose of the study was to determine the contribution of trajectory smoothness to the endpoint accuracy and endpoint variability of rapid multi-joint arm movements. Young and older adults performed arm movements (4 blocks of 25 trials) as fast and as accurately as possible to a target with the right (dominant) arm. Endpoint accuracy and endpoint variability along with trajectory smoothness and error were quantified for each block of trials. Endpoint error and endpoint variance were greater in older adults compared with young adults, but decreased at a similar rate with practice for the two age groups. The greater endpoint error and endpoint variance exhibited by older adults were primarily due to impairments in movement extent control and not movement direction control. The normalized jerk was similar for the two age groups, but was not strongly associated with endpoint error or endpoint variance for either group. However, endpoint variance was strongly associated with endpoint error for both the young and older adults. Finally, trajectory error was similar for both groups and was weakly associated with endpoint error for the older adults. The findings are not consistent with the predictions of the minimum variance theory, but support and extend previous observations that movement trajectories and endpoints are planned independently.

A computational model for optimal muscle activity considering muscle viscoelasticity in wrist movements

To understand the mechanism of neural motor control, it is important to clarify how the central nervous system organizes the coordination of redundant muscles. Previous studies suggested that muscle activity for step-tracking wrist movements are optimized so as to reduce total effort or end-point variance under neural noise. However, since the muscle dynamics were assumed as a simple linear system, some characteristic patterns of experimental EMG were not seen in the simulated muscle activity of the previous studies. The biological muscle is known to have dynamic properties in which its elasticity and viscosity depend on activation level. The motor control system is supposed to consider the viscoelasticity of the muscles when generating motor command signals. In this study, we present a computational motor control model that can control a musculoskeletal system with nonlinear dynamics. We applied the model to step-tracking wrist movements actuated by five muscles with dynamic viscoelastic properties. To solve the motor redundancy, we designed the control model to generate motor commands that maximize end-point accuracy under signal-dependent noise, while minimizing the squared sum of them. Here, we demonstrate that the muscle activity simulated by our model exhibits spatiotemporal features of experimentally observed muscle activity of human and nonhuman primates. In addition, we show that the movement trajectories resulting from the simulated muscle activity resemble experimentally observed trajectories. These results suggest that, by utilizing inherent viscoelastic properties of the muscles, the neural system may optimize muscle activity to improve motor performance.

Adaptation to sensory-motor reflex perturbations is blind to the source of errors

In the study of visual-motor control, perhaps the most familiar findings involve adaptation to externally imposed movement errors. Theories of visual-motor adaptation based on optimal information processing suppose that the nervous system identifies the sources of errors to effect the most efficient adaptive response. We report two experiments using a novel perturbation based on stimulating a visually induced reflex in the reaching arm. Unlike adaptation to an external force, our method induces a perturbing reflex within the motor system itself, i.e., perturbing forces are self-generated. This novel method allows a test of the theory that error source information is used to generate an optimal adaptive response. If the self-generated source of the visually induced reflex perturbation is identified, the optimal response will be via reflex gain control. If the source is not identified, a compensatory force should be generated to counteract the reflex. Gain control is the optimal response to reflex perturbation, both because energy cost and movement errors are minimized. Energy is conserved because neither reflex-induced nor compensatory forces are generated. Precision is maximized because endpoint variance is proportional to force production. We find evidence against source-identified adaptation in both experiments, suggesting that sensory-motor information processing is not always optimal.

Timing variability and not force variability predicts the endpoint accuracy of fast and slow isometric contractions

The purpose of the study was to determine the contributions of endpoint variance and trajectory variability to the endpoint accuracy of goal-directed isometric contractions when the target force and contraction speed were varied. Thirteen young adults (25 +/- 6 years) performed blocks of 15 trials at each of 2 contraction speeds and 4 target forces. Subjects were instructed to match the peak of a parabolic force trajectory to a target force by controlling the abduction force exerted by the index finger. The time to peak force was either 150 ms (fast) or 1 s (slow). The target forces were 20, 40, 60, and 80% of the maximal force that could be achieved in 150 ms during an MVC. The same absolute forces were required for both contraction speeds. Endpoint accuracy and variability in force and time along with intramuscular EMG activity of the agonist (first dorsal interosseus) and antagonist (second palmar interosseus) muscles were quantified for each block of trials. The principal dependent variables were endpoint error (shortest distance between the coordinates of the target and the peak force), endpoint variance (sum of the variance in peak force and time to peak force), trial-to-trial variability (SD of peak force and time to peak force), SD of the force trajectory (SD of the detrended force from force onset to peak force), normalized peak EMG amplitude, and the SD of normalized peak EMG amplitude. Stepwise multiple linear regression models were used to determine the EMG activity parameters that could explain the differences observed in endpoint error and endpoint variance. Endpoint error increased with target force for the fast contractions, but not for the slow contractions. In contrast, endpoint variance was greatest at the lowest force and was not associated with endpoint error at either contraction speed. Furthermore, force trajectory SD was not associated with endpoint error or endpoint variance for either contraction speed. Only the trial-to-trial variability of the timing predicted endpoint accuracy for fast and slow contractions. These findings indicate that endpoint error in tasks that require force and timing accuracy is minimized by controlling timing variability but not force variability, and that endpoint error is not related to the amplitude of the activation signal.

Evaluation of Trajectory Planning Models for Arm-Reaching Movements Based on Energy Cost

Computational studies have suggested that many characteristics of reaching trajectories in a horizontal plane can be effectively predicted by certain models, including, the minimum end point variance model and minimum torque change model. It has also been reported that these characteristics appear to differ from those obtained by the minimum energy cost model that has been reported to explain the characteristics of locomotor patterns. Do these results imply that the human nervous system uses different strategies to resolve the redundancy problem for different tasks? In order to reexamine the optimality of reaching trajectories from a viewpoint of energy cost, we considered the corrective submovements to compensate for positional error due to signal-dependent noise in motor commands and computed the expected value of the total energy costs required to reach a target by repetition of submovements planned by each of the following models: the minimum energy cost model, minimum end point variance model, and minimum torque change model. The results revealed that when the noise is large, the total energy cost required by the minimum end point variance model and the minimum torque change model can be lower than that required by the minimum energy cost model which assumes minimizing energy cost under noise-free condition. This result indicates that the minimization of the expected value of the energy cost would be an important factor in determining the reaching trajectories.

The relation of hand and arm configuration variances while tracking geometric figures in Parkinson's disease: aspects for rehabilitation

Variances of drawing arm movements between patients with Parkinson's disease and healthy controls were compared. The aim was to determine whether differences in joint synergies or individual joint rotations affect the endpoint (hand position) variance. Joint and endpoint coordinates were measured while participants performed drawing tasks. Variances of arm configurations and endpoints were computed and statistically analyzed for 12 patients and 12 controls. The variance of arm movements for patients (both for arm configuration and endpoint) was overall higher than that for the control group. Variation was smaller for drawing a circle versus a square and for drawing with the dominant versus the nondominant hand within both groups. The ratio of arm configuration variances between groups was similar to the ratio of endpoint variances. There were significant differences in the velocity, but not in the path lengths of movements comparing the two groups. Patients presented less movement stability while drawing different figures in different trials. Moreover, the similarity of the ratios suggests that the ill-coordinated hand movement was caused by the error in the movements of individual body parts rather than by the lack of intersegmental coordination. Thus, rehabilitation may focus on the improvement of the precision of individual joint rotations.

Influence of Viscous Loads on Motor Planning

Here we computationally investigate how encumbering the hand could alter predictions made by the minimum torque change (MTC) and minimum endpoint variance hypotheses (MEPV) of movement planning. After minutes of training, people have made arm trajectories in a robot-generated viscous force field that were similar to previous baseline trajectories without the force field. We simulate the human arm interacting with this viscous load. We found that the viscous forces clearly differentiated MTC and MEPV predictions from both minimum-jerk predictions and from human behavior. We conclude that learned behavior in the viscous environment could arise from minimizing kinematic costs but could not arise from a minimization of either torque change or endpoint variance.

Optimality of the minimum endpoint variance model based on energy consumption

It has been reported that arm trajectories for reaching movements are not planned so as to minimize the energy cost. However, from a biological point of view, saving energy consumption would be important for survival. If human brains do not apply the criterion of the minimum energy consumption to reaching movements, what is the reason? If they do, what has been unrecognized in the studies of the minimum energy model? In this study, we computed energy costs required for the reaching movements planned by the minimum energy cost model and the minimum endpoint variance model which is one of the leading models for reaching movements, respectively. In the computation we considered corrective sub-movements to compensate for the positional error due to signal-dependent noise for motor commands. The results show that the trajectories planned so as to minimize energy cost without considering the effect of noise are easily affected by noise. Therefore, when noise is large, sub-movements are required to reach the target and as a result the total energy cost becomes larger than that for the minimum endpoint variance model. These results suggest that the human arm reaching movement might be optimized on the energy cost and we should consider the effect of the noise in the analysis of the reaching movement from the view point of optimization on the energy cost.

Object Grasping using the Minimum Variance Model

Reaching-to-grasp has generally been classified as the coordination of two separate visuomotor processes: transporting the hand to the target object and performing the grip. An alternative view has recently been formed that grasping can be explained as pointing movements performed by the digits of the hand to target positions on the object. We have previously implemented the minimum variance model of human movement as an optimal control scheme suitable for control of a robot arm reaching to a target. Here, we extend that scheme to perform grasping movements with a hand and arm model. Since the minimum variance model requires that signal-dependent noise be present on the motor commands to the actuators of the movement, our approach is to plan the reach and the grasp separately, in line with the classical view, but using the same computational model for pointing, in line with the alternative view. We show that our model successfully captures some of the key characteristics of human grasping movements, including the observations that maximum grip size increases with object size (with a slope of approximately 0.8) and that this maximum grip occurs at 60-80% of the movement time. We then use our model to analyse contributions to the digit end-point variance from the two components of the grasp (the transport and the grip). We also briefly discuss further areas of investigation that are prompted by our model.

Optimal Control of Redundant Muscles in Step-Tracking Wrist Movements

An important question in motor neuroscience is how the nervous system controls the spatiotemporal activation patterns of redundant muscles in generating accurate movements. The redundant muscles may not only underlie the flexibility of our movements but also pose the challenging problem of how to select a specific sequence of muscle activation from the huge number of possible activations. Here, we propose that noise in the motor command that has an influence on task achievement should be considered in determining the optimal motor commands over redundant muscles. We propose an optimal control model for step-tracking wrist movements with redundant muscles that minimizes the end-point variance under signal-dependent noise. Step-tracking wrist movements of human and nonhuman primates provide a detailed data set to investigate the control mechanisms in movements with redundant muscles. The experimental EMG data can be summarized by two eminent features: 1) amplitude-graded EMG pattern, where the timing of the activity of the agonist and antagonist bursts show slight variations with changes in movement directions, and only the amplitude of activity is modulated; and 2) cosine tuning for movement directions exhibited by the agonist and antagonist bursts, and the discrepancy found between a muscle's agonist preferred direction and its pulling direction. In addition, it is also an important observation that subjects often overshoot the target. We demonstrate that the proposed model captures not only the spatiotemporal activation patterns of wrist muscles but also trajectory overshooting. This suggests that when recruiting redundant muscles, the nervous system may optimize the motor commands across the muscles to reduce the negative effects of motor noise.

The minimum endpoint variance trajectory depends on the profile of the signal-dependent noise

It has been proposed that the central nervous system determines reaching movement trajectories so as to minimize the positional variance of the endpoint in the presence of signal-dependent noise. The hypothesis well reproduces the empirical movement trajectories for noise to the control signal whose variance is proportional to the second power of the amplitude of the control signal. However, empirical studies do not necessarily exhibit such a simple signal-noise relationship. The studies exhibit a wide distribution of estimates of the value of the exponent. This discrepancy raises the question of how the minimum endpoint variance trajectory depends on the value of the exponent. To address this question, we calculated minimum endpoint variance trajectories in simulations in which the value of the exponent was varied from 0 to 3. We found that the optimal trajectories differed according to the value of the exponent, and the profiles of optimal trajectories gradually diverged from the empirical ones as the value approached 0, though this change was not remarkable for larger values. Moreover, the optimal trajectories failed to replicate Fitts' law when the value was not equal to 2. These results suggest that the acceptability of the minimum endpoint variance theory depends on the value of the exponent in our motor system.

When two eyes are better than one in prehension: monocular viewing and end-point variance.

Previous research has suggested that binocular vision plays an important role in prehension. It has been shown that removing binocular vision affects (negatively) both the planning and on-line control of prehension. It has been suggested that the adverse impact of removing binocular vision is because monocular viewing results in an underestimation of target distance in visuomotor tasks. This suggestion is based on the observation that the kinematics of prehension are altered when viewing monocularly. We argue that it is not possible to draw unambiguous conclusions regarding the accuracy of distance perception from these data. In experiment 1, we found data that contradict the idea that a consistent visuomotor underestimation of target distance is an inevitable consequence of monocular viewing. Our data did show, however, that positional variance increases under monocular viewing. We provide an alternative explanation for the kinematic changes found when binocular vision is removed. Our account is based on the changes in movement kinematics that occur when end-point variance is altered following the removal of binocular vision. We suggest that the removal of binocular vision leads to greater perceptual uncertainty (e.g. less precise stimulus cues), resulting in changes in the kinematics of the movement (longer duration movements). Our alternative account reconciles some differences within the research literature. We conducted a series of experiments to explore further the issue of when binocular information is advantageous in prehension. Three subsequent experiments were employed which varied binocular/monocular viewing in selectively lit conditions. Experiment 2 explored the differences in prehension measured between monocular and binocular viewing in a full cue environment with a continuous view of the target object. Experiment 3 required participants to reach, under a monocular or binocular view, for a continuously visible self-illuminated target object in an otherwise dark room. In Experiment 3, the participant could neither see the target object nor the reaching hand following initiation of the prehension movement. Our results suggest that binocular vision contributes to prehension by providing additional information (cues) to the nervous system. These cues appear to be weighted differentially according to the particular constellation of stimulus cues available to the participants when reaching to grasp. One constant advantage of a binocular view appears to be the provision of on-line information regarding the position of the hand relative to the target. In reduced cue conditions (i.e. where a view of the target object is lost following initiation of the movement), binocular information regarding target location appears to be particularly useful in the initial programming of reach distance. Our results are a step towards establishing the specific contributions that binocular vision makes to the control of prehension.