Movement curvature planning through force field internal models

Human motion studies have focused primarily on modeling straight point-to-point reaching movements. However, many goal-directed reaching movements, such as movements directed towards oneself, are not straight but rather follow highly curved trajectories. These movements are particularly interesting to study since they are essential in our everyday life, appear early in development and are routinely used to assess movement deficits following brain lesions. We argue that curved and straight-line reaching movements are generated by a unique neural controller and that the observed curvature of the movement is the result of an active control strategy that follows the geometry of one’s body, for instance to avoid trajectories that would hit the body or yield postures close to the joint limits. We present a mathematical model that accounts for such an active control strategy and show that the model reproduces with high accuracy the kinematic features of human data during unconstrained reaching movements directed toward the head. The model consists of a nonlinear dynamical system with a single stable attractor at the target. Embodiment-related task constraints are expressed as a force field that acts on the dynamical system. Finally, we discuss the biological plausibility and neural correlates of the model’s parameters and suggest that embodiment should be considered as a main cause for movement trajectory curvature.     

Optimized torsion-angle normal modes reproduce conformational changes more accurately than cartesian modes

We present what to our knowledge is a new method of optimized torsion-angle normal-mode analysis, in which the normal modes move along curved paths in Cartesian space. We show that optimized torsion-angle normal modes reproduce protein conformational changes more accurately than Cartesian normal modes. We also show that orthogonalizing the displacement vectors from torsion-angle normal-mode analysis and projecting them as straight lines in Cartesian space does not lead to better performance than Cartesian normal modes. Clearly, protein motion is more naturally described by curved paths in Cartesian space.     

Interpretable machine learning for high-dimensional trajectories of aging health

We have built a computational model for individual aging trajectories of health and survival, which contains physical, functional, and biological variables, and is conditioned on demographic, lifestyle, and medical background information. We combine techniques of modern machine learning with an interpretable interaction network, where health variables are coupled by explicit pair-wise interactions within a stochastic dynamical system. Our dynamic joint interpretable network (DJIN) model is scalable to large longitudinal data sets, is predictive of individual high-dimensional health trajectories and survival from baseline health states, and infers an interpretable network of directed interactions between the health variables. The network identifies plausible physiological connections between health variables as well as clusters of strongly connected health variables. We use English Longitudinal Study of Aging (ELSA) data to train our model and show that it performs better than multiple dedicated linear models for health outcomes and survival. We compare our model with flexible lower-dimensional latent-space models to explore the dimensionality required to accurately model aging health outcomes. Our DJIN model can be used to generate synthetic individuals that age realistically, to impute missing data, and to simulate future aging outcomes given arbitrary initial health states.     

The human arm as a redundant manipulator: The control of path and joint angles

The movements studied involved moving the tip of a pointer attached to the hand from a given starting point to a given end point in a horizontal plane. Three joints--the shoulder, elbow and wrist--were free to move. Thus the system represented a redundant manipulator. The coordination of the movements of the three joints was recorded and analyzed. The study concerned how the joints are controlled during a movement. The results are used to evaluate several current hypotheses for motor control. Basically, the incremental changes are calculated so as to move the tip of the manipulator along a straight line in the workspace. The values of the individual joints seem to be determined as follows. Starting from the initial values the incremental changes in the three joint angles represent a compromise between two criteria: 1) the amount of the angular change should be about the same in the three joints, and 2) the angular changes should minimize the total cost of the arm position as determined by cost functions defined for each joint as a function of angle. By itself, this mechanism would produce strongly curved trajectories in joint space which could include additional acceleration and deceleration in a joint. These are reduced by the influence of a third criterion which fits with the mass-spring hypothesis. Thus the path is calculated as a compromise between a straight line in workspace and a straight line in joint space. The latter can produce curved paths in the workspace such as were actually found in the experiments. A model calculation shows that these hypotheses can qualitatively describe the experimental findings.     

Differences in Context and Feedback Result in Different Trajectories and Adaptation Strategies in Reaching

Computational models of motor control have often explained the straightness of horizontal planar reaching movements as a consequence of optimal control. Departure from rectilinearity is thus regarded as sub-optimal. Here we examine if subjects may instead select to make curved trajectories following adaptation to force fields and visuomotor rotations. Separate subjects adapted to force fields with or without visual feedback of their hand trajectory and were retested after 24 hours. Following adaptation, comparable accuracies were achieved in two ways: with visual feedback, adapted trajectories in force fields were straight whereas without it, they remained curved. The results suggest that trajectory shape is not always straight, but is also influenced by the calibration of available feedback signals for the state estimation required by the task. In a follow-up experiment, where additional subjects learned a visuomotor rotation immediately after force field, the trajectories learned in force fields (straight or curved) were transferred when directions of the perturbations were similar but not when directions were opposing. This demonstrates a strong bias by prior experience to keep using a recently acquired control policy that continues to produce successful performance inspite of differences in tasks and feedback conditions. On relearning of force fields on the second day, facilitation by intervening visuomotor rotations occurred only when required motor adjustments and calibration of feedback signals were similar in both tasks. These results suggest that both the available feedback signals and prior history of learning influence the choice and maintenance of control policy during adaptations.     

Trajectory formation based on physiological characteristics of skeletal muscles

Human arm trajectories in natural unrestricted reaching movements were studied. They have particular properties such that a hand path is a rather simple straight or curved line, and a tangential velocity profile of hand is bell-shaped. Also these properties are invariant, independent of movement duration and hand-held load. In this study, trajectory formation is investigated on the basis of physiological characteristics of skeletal muscles, and a criterion prescribed by a derivative of isometric muscle torque is proposed. Subsequently, optimal trajectories are formulated under various conditions of movement to account for a planning strategy of human arm trajectories. In addition to such a theoretical approach, human arm trajectories are experimentally observed by a measuring system which provides a visual sensor and a target tracking device, enabling totally unrestricted movements. Then, optimal trajectories are quantitatively evaluated in comparison with experimental data in which essential properties of human arm trajectories are demonstrated. These results support the idea that human arm trajectories are planned in order to minimize the proposed criterion which is determined from physiological aspects. Finally, the physiological advantages of human arm trajectories are discussed with regard to the analysis of observed and optimal trajectories. Received: 2 December 1997 / Accepted in revised form: 20 March 1998     

Interpolation of segment Euler angles can provide a robust estimation of segment angular trajectories during asymmetric lifting tasks

Video-based field methods that estimate L5/S1 net joint moments from kinematics based on interpolation in the sagittal plane of joint angles alone can introduce a significant error on the interpolated joint angular trajectory when applied to asymmetric dynamic lifts. Our goal was to evaluate interpolation of segment Euler angles for a wide range of dynamic asymmetric lifting tasks using cubic spline methods by comparing the interpolated values with the continuous measured ones. For most body segments, the estimated trajectories of segment Euler angles have less than 5° RMSE (in each dimension) with 5-point cubic spline interpolation when there is no measurement error of interpolation points. Sensitivity analysis indicates that when the measurement error exists, the root mean square error (RMSE) of estimated trajectories increases. Comparison among different lifting conditions showed that lifting a load from a high initial position yielded a smaller RMSE than lifting from a low initial position. In conclusion, interpolation of segment Euler angles can provide a robust estimation of segment angular trajectories during asymmetric lifting when measurement error of interpolation points can be controlled at a low level.     

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     

Are the dynamics of gastric evacuation of natural prey in a piscivorous flatfish different from what is going on in a gadoid?

Despite the fact that the stomach of turbot Psetta maxima is a curved tube that forms a half circle, it was demonstrated that gastric evacuation in this predatory flatfish fed natural prey closely followed the surface‐dependent cylinder model developed from studies on gadoids with a straight stomach. Evacuation experiments were performed on two size groups of P. maxima fed sandeel Ammodytes tobianus as well as on P. maxima fed brown shrimp Crangon crangon at three different temperatures. This enabled the provision of a gastric evacuation model for studies on P. maxima, which takes into account the effects of the explanatory variables predator size, temperature, prey energy density and resistance of prey to the digestive processes in the stomach. Basically, the cylinder model predicts that a square‐root function accurately describes gastric evacuation, which is inconsistent with the conclusion of a previous study on P. maxima that evacuation of A. tobianus is essentially linear with time. Use of the cylinder model to the values of the explanatory variables presented in the latter study, however, gave accurate predictions of the actually acquired evacuation data, which points to the generic value of the model.     

Effect of compliance on haptic perception of curvature

The haptic sense of geometric properties such as the curvature of a contour is derived from somatosensory cues about the motions and forces experienced during exploratory actions. This study addressed the question of whether compliance, the relationship between force and displacement, influences haptic perception of curvature. Subjects traced a curved 30?cm long compliant contour by grasping the handle of a manipulandum and reported whether the contour curved towards or away from them. The contour at which there was equal probability of responding either way was taken to represent one that was sensed as being straight. The compliance of the contour was varied, being constant, greatest in the middle or greatest at the ends. Subjects exhibited a bias in what they sensed to be a straight edge. However, the actual handpath that was judged to be straight did not vary across the three compliance profiles. Our results rule out a hypothetical strategy in which an intended motion is planned and the actual trajectory is then inferred by sensing force feedback. Another strategy in which the force against the contour is controlled and the handpath is inferred from proprioceptive feedback is more consistent with the observations.     

A new child-friendly 3D bimanual protocol to assess upper limb movement in children with unilateral cerebral palsy: Development and validation

Unilateral cerebral palsy (uCP) causes upper limb movement disorders that impact on daily activities, especially in bimanual condition. However, a few studies have proposed bimanual tasks for 3D motion analysis. The aim of this study was to validate the new version of a child-friendly, 3D, bimanual protocol for the measurement of joint angles and movement quality variables. Twenty children with uCP and 20 typically developing children (TDC) performed the five-task protocol integrated into a game scenario. Each task specifically targeted one or two upper limb degrees of freedom. Joint angles, smoothness and trajectory straightness were calculated. Elbow extension, supination, wrist extension and adduction amplitudes were reduced; hand trajectories were less smooth and straight in children with uCP compared to TDC. Correlations between the performance-based score and kinematic variables were strong. High within and between-session reliability was found for most joint angle variables and lower reliability was found for smoothness and straightness in most tasks. The results therefore demonstrated the validity and reliability of the new protocol for the objective assessment of bimanual function in children with uCP. The evaluation of both joint angles and movement quality variables should increase understanding of pathological movement patterns and help clinicians to optimize treatment.ClinicalTrials.gov identifier: NCT03888443.     

Biomechanics of phalangeal curvature

Phalangeal curvature has been widely cited in primate functional morphology and is one of the key traits in the ongoing debate about whether the locomotion of early hominins included a significant degree of arboreality. This study examines the biomechanics of phalangeal curvature using data on hand posture, muscle recruitment, and anatomical moment arms to develop a finite element (FE) model of a siamang manual proximal phalanx during suspensory grasping. Strain patterns from experiments on intact cadaver forelimbs validated the model. The strain distribution in the curved siamang phalanx FE model was compared to that in a mathematically straight rendition in order to test the hypotheses that curvature: 1) reduces strain and 2) results in lower bending strains but relatively higher compression. In the suspensory posture, joint reaction forces load the articular ends of the phalanx in compression and dorsally, while muscle forces acting through the flexor sheath pull the mid-shaft palmarly. These forces compress the phalanx dorsally and tense it palmarly, effectively bending it 'open.' Strains in the curved model were roughly half that of the straight model despite equivalent lengths, areas, mechanical properties, and loading conditions in the two models. The curved model also experienced a higher ratio of compressive to tensile strains. Curvature reduces strains during grasping hand postures because the curved bone is more closely aligned with the joint reaction forces. Therefore, phalangeal curvature reduces the strains associated with arboreal, and especially suspensory, activity involving flexed digits. These results offer a biomechanical explanation for the observed association between phalangeal curvature and arboreality.     

Effect of compliance on haptic perception of curvature

The haptic sense of geometric properties such as the curvature of a contour is derived from somatosensory cues about the motions and forces experienced during exploratory actions. This study addressed the question of whether compliance, the relationship between force and displacement, influences haptic perception of curvature. Subjects traced a curved 30 cm long compliant contour by grasping the handle of a manipulandum and reported whether the contour curved towards or away from them. The contour at which there was equal probability of responding either way was taken to represent one that was sensed as being straight. The compliance of the contour was varied, being constant, greatest in the middle or greatest at the ends. Subjects exhibited a bias in what they sensed to be a straight edge. However, the actual handpath that was judged to be straight did not vary across the three compliance profiles. Our results rule out a hypothetical strategy in which an intended motion is planned and the actual trajectory is then inferred by sensing force feedback. Another strategy in which the force against the contour is controlled and the handpath is inferred from proprioceptive feedback is more consistent with the observations.     

New plot showing the existence of cooperativity, anticooperativity, and an arbitrary value of the neighbor-exclusion parameter in systems of ligand binding to linear biopolymers

A new way of plotting the isotherm for ligand binding to linear biopolymers is presented. In this plot the isotherm for noncooperative binding of ligands of length n (n-mers) becomes a straight line and the existence of cooperativity and anticooperativity between bound ligands is detected by appearance of opposite convexity of the curved isotherm. It is also usable for cases of fractional n values. Usefulness of the new plot in determining precise mechanisms of binding is shown using experimental data.     

Straight and curved conformations of FtsZ are regulated by GTP hydrolysis

FtsZ assembles in vitro into protofilaments that can adopt two conformations-the straight conformation, which can assemble further into two-dimensional protofilament sheets, and the curved conformation, which forms minirings about 23 nm in diameter. Here, we describe the structure of FtsZ tubes, which are a variation of the curved conformation. In the tube the curved protofilament forms a shallow helix with a diameter of 23 nm and a pitch of 18 or 24 degrees. We suggest that this shallow helix is the relaxed structure of the curved protofilament in solution. We provide evidence that GTP favors the straight conformation while GDP favors the curved conformation. In particular, exclusively straight protofilaments and protofilament sheets are assembled in GMPCPP, a nonhydrolyzable GTP analog, or in GTP following chelation of Mg, which blocks GTP hydrolysis. Assembly in GDP produces exclusively tubes. The transition from straight protofilaments to the curved conformation may provide a mechanism whereby the energy of GTP hydrolysis is used to generate force for the constriction of the FtsZ ring in cell division.     

Coordinate Representations for Interference Reduction in Motor Learning

When opposing force fields are presented alternately or randomly across trials for identical reaching movements, subjects learn neither force field, a behavior termed ‘interference’. Studies have shown that a small difference in the endpoint posture of the limb reduces this interference. However, any difference in the limb’s endpoint location typically changes the hand position, joint angles and the hand orientation making it ambiguous as to which of these changes underlies the ability to learn dynamics that normally interfere. Here we examine the extent to which each of these three possible coordinate systems—Cartesian hand position, shoulder and elbow joint angles, or hand orientation—underlies the reduction in interference. Subjects performed goal-directed reaching movements in five different limb configurations designed so that different pairs of these configurations involved a change in only one coordinate system. By specifically assigning clockwise and counter-clockwise force fields to the configurations we could create three different conditions in which the direction of the force field could only be uniquely distinguished in one of the three coordinate systems. We examined the ability to learn the two fields based on each of the coordinate systems. The largest reduction of interference was observed when the field direction was linked to the hand orientation with smaller reductions in the other two conditions. This result demonstrates that the strongest reduction in interference occurred with changes in the hand orientation, suggesting that hand orientation may have a privileged role in reducing motor interference for changes in the endpoint posture of the limb.     

The straight and curved conformation of FtsZ protofilaments-evidence for rapid exchange of GTP into the curved protofilament

Bacterial cell division protein FtsZ assembles into protofilaments, which can adopt a straight or curved conformation, similar to its eukaryotic homolog, tubulin. The straight protofilaments can assemble into sheets with a lattice similar to the microtubule wall. The curved protofilaments can form rings when adsorbed to a lipid monolayer, but in solution they form helices. 4 helices assemble together to make a tube, the characteristic polymer of the curved protofilament. GTP favors the straight conformation, while GDP favors the curved. We show here that addition of EDTA and GTP to tubes causes a rapid transformation to straight protofilament sheets. Apparently when the magnesium is chelated the GDP in the curved protofilaments dissociates rapidly and is replaced with GTP, and this GTP induces the transition to straight protofilaments.     

On Gibbs sampling for state space models

We show how to use the Gibbs sampler to carry out Bayesian inferenceon a linear state space model with errors that are a mixtureof normals and coefficients that can switch over time. Our approachsimultaneously generates the whole of the state vector giventhe mixture and coefficient indicator variables and simultaneouslygenerates all the indicator variables conditional on the statevectors. The states are generated efficiently using the Kalmanfilter. We illustrate our approach by several examples and empiricallycompare its performance to another Gibbs sampler where the statesare generated one at a time. The empirical results suggest thatour approach is both practical to implement and dominates theGibbs sampler that generates the states one at a time.     

How morphological constraints affect axonal polarity in mouse neurons

Neuronal differentiation is under the tight control of both biochemical and physical information arising from neighboring cells and micro-environment. Here we wished to assay how external geometrical constraints applied to the cell body and/or the neurites of hippocampal neurons may modulate axonal polarization in vitro. Through the use of a panel of non-specific poly-L-lysine micropatterns, we manipulated the neuronal shape. By applying geometrical constraints on the cell body we provided evidence that centrosome location was not predictive of axonal polarization but rather follows axonal fate. When the geometrical constraints were applied to the neurites trajectories we demonstrated that axonal specification was inhibited by curved lines. Altogether these results indicated that intrinsic mechanical tensions occur during neuritic growth and that maximal tension was developed by the axon and expressed on straight trajectories. The strong inhibitory effect of curved lines on axon specification was further demonstrated by their ability to prevent formation of multiple axons normally induced by cytochalasin or taxol treatments. Finally we provided evidence that microtubules were involved in the tension-mediated axonal polarization, acting as curvature sensors during neuronal differentiation. Thus, biomechanics coupled to physical constraints might be the first level of regulation during neuronal development, primary to biochemical and guidance regulations.