1. Computational and Systems Biology
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Direction-dependent arm kinematics reveal optimal integration of gravity cues

  1. Jeremie Gaveau  Is a corresponding author
  2. Bastien Berret
  3. Dora E Angelaki
  4. Charalambos Papaxanthis
  1. Université Bourgogne Franche-Comté, INSERM CAPS UMR 1093, France
  2. CIAMS, Université Paris-Sud, Université Paris Saclay, France
  3. CIAMS, Université d'Orléans, France
  4. Baylor College of Medicine, United States
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Cite this article as: eLife 2016;5:e16394 doi: 10.7554/eLife.16394

Abstract

The brain has evolved an internal model of gravity to cope with life in the Earth's gravitational environment. How this internal model benefits the implementation of skilled movement has remained unsolved. One prevailing theory has assumed that this internal model is used to compensate for gravity's mechanical effects on the body, such as to maintain invariant motor trajectories. Alternatively, gravity force could be used purposely and efficiently for the planning and execution of voluntary movements, thereby resulting in direction-depending kinematics. Here we experimentally interrogate these two hypotheses by measuring arm kinematics while varying movement direction in normal and zero-G gravity conditions. By comparing experimental results with model predictions, we show that the brain uses the internal model to implement control policies that take advantage of gravity to minimize movement effort.

https://doi.org/10.7554/eLife.16394.001

eLife digest

Many of the activities of humans and other animals require the limbs to be moved in a coordinated manner. For a movement to be successful, the brain must generate muscle contractions that take into account factors in the environment that might affect the movement. One such prominent environmental feature is gravity, and it is broadly believed that the brain develops and uses an internal representation of gravity to anticipate its effects on the limbs.

How an internal representation of gravity helps limb movements to be made successfully is not known. Theorists have proposed that the brain could use the internal model of gravity to predict how to compensate for its mechanical effects – or, on the contrary, take advantage of them.

Flying a plane in a “parabolic” arc creates a microgravity environment inside it that produces a feeling of weightlessness. Gaveau et al. asked volunteers to perform arm movements in normal earth gravity and in microgravity conditions. Under normal gravity, the volunteers made arm movements with speed profiles that differed according to movement direction. When they first performed these movements in microgravity, the speeds still differed according to direction. However, as the participants gained more experience of making the movements in microgravity, the speed at which upward and downward arm movements were made became more similar. Eventually movements were performed at the same speed in either direction.

Comparing these results to numerical simulations revealed a sophisticated behavior where movements are organized to take advantage of the effects of gravity to minimize the effort that the muscles need to make. Further research into the neural mechanisms behind this optimization process could benefit the development of various rehabilitative and assistive technologies, such as brain-machine interfaces and robotic devices to guide and support limbs.

https://doi.org/10.7554/eLife.16394.002

Introduction

It is always fascinating to witness the ability of acrobats and dancers to accomplish complex and elegant movements, graciously interacting with gravito-inertial forces. Computational theory postulates that this captivating performance is due to the ability of the brain to learn and store internal representations of environmental dynamics (Wolpert and Ghahramani, 2000). On earth, gravity is the most ubiquitous and constant environmental feature. As such, a neural representation of gravity is created and stored through an internal model (Papaxanthis et al., 1998a; Angelaki et al., 1999; Merfeld et al., 1999; McIntyre et al., 2001; Angelaki et al., 2004; Indovina et al., 2005; Miller et al., 2008; Crevecoeur et al., 2009; Gaveau and Papaxanthis, 2011; Laurens et al., 2013a, 2013b).

The need for an internal model of gravity arises because Einstein’s equivalence principle prevents any single sensory receptor from encoding gravity without simultaneously also encoding inertial accelerations (Einstein, 1908). The neural representation of gravity is thought to solve this ambiguity by multisensory statistical inference (Angelaki et al., 1999; Merfeld et al., 1999; Angelaki et al., 2004; Laurens et al., 2013b). An internal model of gravity has been shown to benefit the anticipation of a free falling object motion (Zago and Lacquaniti, 2005; Zago et al., 2008; Lacquaniti et al., 2013), as well as the visual perception of allocentric vertical (Van Pelt et al., 2005; De Vrijer et al., 2008; Elmore et al., 2014). However, whether and how an internal model of gravity benefits the planning and execution of skilled movement remains unknown.

One influential viewpoint assumes a need for compensation; i.e., the internal model of gravity is used to predict and compensate for its mechanical effects on the body. This hypothesis has been motivated by experimental findings on arm movements in the presence of externally applied Coriolis, viscous force fields and interaction torques (Shadmehr and Mussa-Ivaldi, 1994; Gribble and Ostry, 1999; Pigeon et al., 2003). The main benefit of a compensation strategy would be the simplification of motor planning by allowing for invariant trajectories (Hollerbach and Flash, 1982; Atkeson and Hollerbach, 1985). Current research in fields as diverse as neurorehabilitation, movement perception, or motor control modularity, assumes such a compensation principle (Prange et al., 2009, 2012; Cook et al., 2013; Russo et al., 2014).

An alternative perspective is based on a need for effort optimization, i.e., the internal model of gravity could be used to predict and take advantage of its mechanical effects on the body. It has been proposed that adaptation to the Earth’s gravity field has allowed control policies to evolve that take advantage of environmental dynamics – use gravity as an assistive force to accelerate downward movements and as a resistive force to decelerate upward movements. This strategy, which has been formalized into a Minimum Smooth-Effort model, would result in movement kinematics that varies with direction (Berret et al., 2008a; Gaveau et al., 2014). Indeed, upward movements were shown to have shorter time to peak velocity and larger curvature than downward movements, but such comparisons were until now largely qualitative (Papaxanthis et al., 1998b; Gentili et al., 2007). Furthermore, upward/downward direction-dependent kinematics could also arise from the complex dynamics of the peripheral neuromuscular system. For example, the firing properties of extensor motoneurons (pulling the arm downwards when upright) are known to obey different rules from those of flexor motoneurons (pulling upwards when uptight; Cotel et al., 2009; Wilson et al., 2015). In addition, muscle force production generated by eccentric contraction (elongation, e.g. downward movements for flexors) is also known to obey differing rules from concentric contraction (e.g. upward movement for flexors; for a review see Enoka, 1996). Because of these asymmetrical neuromuscular peripheral properties, upward/downward kinematic asymmetries cannot necessarily be attributed to gravity effort optimality (i.e., minimization of muscular force to elevate and lower the arm). Thus, a solid test of the effort optimization hypothesis has been lacking.

Here, we explicitly distinguish between the compensation and effort optimization hypotheses with two critical experiments. First, we contrast predictions of optimal control models that either compensate or take advantage of gravity force effects. Then, we quantitatively compare these predictions with actual kinematic features of arm movements in multiple directions. Second, in order to discard a possible influence of peripheral neuromuscular mechanisms, we measure how differences in upward versus downward arm movement kinematics are influenced by the lack of gravity during the zero-G phase of parabolic flight. If directional asymmetries originate from asymmetric firing of flexor/extensor motoneurons, they should persist in the absence of gravity because motoneurons properties are not expected to change during the short zero-G phase of a parabolic flight (Ishihara et al., 1996, 2002; for a review see Nagatomo et al., 2014). On the other hand, if they originate from eccentric versus concentric force production differences, directional asymmetries should cease to exist instantly in 0g because movement dynamics no longer differ for upward and downward movements (Enoka, 1996). Alternatively, however, if directional asymmetries originate from neural planning processes that take advantage of the internal model of gravity, following a gradual recalibration of the gravity internal model, directional asymmetries should progressively decrease towards new optimal zero-G values (McIntyre et al., 2001; Izawa et al., 2008; Snaterse et al., 2011). In support of the effort optimization hypothesis, we show that directional asymmetries are gradually eliminated during repeated exposure to the zero-G phase of parabolic flight.

Results

We asked fifteen humans to perform arm movements around the shoulder joint in different directions (Figure 1A), whereby the work of gravity torque is systematically varied (Figure 1B), while other dynamic variables (interaction torques, Coriolis and centripetal forces) are constant (see Figure 1—figure supplement 1 for a geometrical illustration of the mechanical system and details on gravity torque computations). In single-degree of freedom movements, although the spatial shape of the endpoint trajectory is circular and constant across directions, the temporal shape of the endpoint trajectory (the form of the velocity profile) could change. As a measure of the shape of the velocity profile, we define a symmetry ratio (SR: acceleration time divided by the total movement time). SR corresponds to the relative timing of peak velocity (as in Figure 1C) and thus allows quantifying whether arm kinematics remains invariant (compensation hypothesis prediction) or changes (effort optimization hypothesis prediction) with movement direction. Here we use the optimal control framework (Bryson and Ho, 1975; Todorov, 2004) to simulate arm movement planning in different directions. Specifically, we change the cost function being minimized to understand how the neural model of gravity serves motor planning; i.e., we compare simulations of optimal control models that use the internal model of gravity to either compensate or take advantage of its mechanical effects on the body.

Figure 1 with 3 supplements see all
Task design and theoretical prediction.

(A) Participants’ initial position and projection of the 17 targets onto the frontal plane. The angle γ, representing the target inclination with respect to horizontal, was used to calculate the gravity torque projecting onto the plane of motion (see also Figure 1—figure supplement 1). (B) The Work of Gravity Torque (WGT), projected onto the plane of motion and integrated over the whole movement, is plotted as a function of movement direction (same color code as in A). This non-linear (cosine-tuned) dependence is well fitted by a sigmoidal function (black curve, average RMSE = 0.12; [min: 0.09, max: 0.15], see Materials and methods). Positive/negative values indicate that WGT has the same/opposite direction as the arm movement. (C) Mean velocity profiles (for the average subject), normalized in both amplitude and duration, illustrate the predictions of the compensation hypothesis (Jerk) and the effort optimization hypothesis (Smooth-Effort) in the vertical plane (upwards, −90; downwards, 90°; see arrows and color-coded direction definition in panel A). SR up (red) and SR down (blue) illustrate the calculation of a symmetry ratio (acceleration time / movement time) allowing quantification of kinematic differences/similarities. (D) Simulated symmetry ratio predicted by the compensation hypothesis: Minimum Jerk (triangles, Flash and Hogan, 1985); and the effort optimization hypothesis: Minimum Smooth-Effort model (dots, Gaveau et al., 2014); as a function of movement direction (−90°, upwards and 90°, downwards). Similarly to WGT (see panels B), simulated symmetry ratio obtained from the Smooth-Effort model is well fitted by a sigmoidal function (black curve). (E) Simulated symmetry ratio as a function of WGT (grey triangles, Jerk and black dots, Smooth-Effort). Each data point represents the prediction for one subject moving in one direction (n = 255 in each plot). It is noticeable that according to the effort optimization hypothesis, arm kinematics (symmetry ratio) should not be invariant but instead linearly correlate with WGT. (F) Simulated symmetry ratio predicted by two other well-known models minimizing dynamic cost functions: the Minimum Variance (Harris and Wolpert, 1998) and the Minimum Torque Change (Uno et al., 1989). It can be observed that the modulation of kinematics with movement direction is a specific feature of the effort-related optimization only.

https://doi.org/10.7554/eLife.16394.003

Minimizing a kinematic cost only, without taking joint torque into account, such as the Jerk model (Flash and Hogan, 1985), is a perfect example of the compensation strategy. This is because the brain must compensate all perturbing dynamics to produce a consistent and invariant kinematically-defined motor plan (Hollerbach and Flash, 1982; Atkeson and Hollerbach, 1985), thus predicting constant SR values for all movement directions (Jerk prediction in Figure 1D). In contrast, the Smooth-Effort model minimizes a hybrid cost that, in addition to the Jerk, takes the external dynamics into account to minimize the muscular force needed to move the arm (absolute work of muscular torque). Thus, by design, the Smooth-Effort model implements the effort optimization strategy, whereby the brain uses the gravity internal model to predict and take advantage of gravity torques in accelerating and decelerating downward and upward movements, respectively (Berret et al., 2008a, 2008b; Gaveau et al., 2011, 2014). The effort optimization hypothesis predicts a sigmoidal dependence of SR on movement direction (Smooth-Effort prediction in Figure 1D). Critically, increasing the weight of movement smoothness (Jerk) in the Smooth-Effort hybrid cost, leads to a progressive disappearance of the gravity-torque related tuning of arm kinematics (Figure 1—figure supplement 2). Accordingly, the compensation hypothesis predicts that movement kinematics remains unchanged for various gravity torque conditions (Jerk prediction in Figure 1E), whilst the effort optimization hypothesis predicts that movement kinematics strongly correlates (average R = 0.99 [min: 0.986, max: 0.994], p<1e-07) with the amount of gravity torque engaged in the motion (Smooth-Effort prediction in Figure 1E). Importantly, the SR dependence on movement direction (i.e., on gravity torque) is a unique feature of effort-related optimization as minimizing other cost functions that take joint torques into account, but are not directly related to effort, such as the Variance (Harris and Wolpert, 1998) or Torque Change (Uno et al., 1989), predict constant SR as a function of arm movement direction (Figure 1F; see also Figure 1—figure supplement 3 for additional simulations testing the effect of including muscle dynamics into the minimization of end-point Variance). In fact, minimization of an effort-related cost is a necessary and sufficient condition to predict directional asymmetries in the vertical plane (Berret et al., 2008a, 2008b).

Participants accomplished rapid arm movements with single-peaked velocity profiles. Average duration did not vary with movement direction (0.40s ± 0.01, SD; F16,224=0.67, p=0.82). Because arm movements were visually guided, systematic and variable errors were small and independent of movement direction (-2.5°< SE < 2°; VE < 2.5°; F16,224=1.13, p=0.32 and F16,224=1.02, p=0.44, respectively). As illustrated in Figure 2A, SR shows a sigmoidal modulation as a function of movement direction (F16,224=28.36, p<0.001). The robustness of this result is illustrated in Figure 2B, which shows sigmoidal fits for individual subjects (average RMSE = 9.77e-03 [min: 4.5e-03, max: 1.53e-02]). When SR was regressed against gravity torque, correlation coefficients were high, averaging R = 0.84 [min: 0.67, max: 0.92], p=2.5e-05 (Figure 2C). Furthermore, SR was independent of movement duration and movement amplitude (average correlation coefficient for duration R = 0.22 [min: 0.04, max: 0.57], p=0.41, see Figure 2—figure supplement 1A; for amplitude R = 0.29 [min: 0.02, max: 0.69], p=0.26, see Figure 2—figure supplement 1B). Thus, arm kinematics is selectively modulated according to the gravity torque requirements of the movement, as predicted by the Smooth-Effort model and quantified by high correlation coefficients between predicted and experimental SR values (average R = 0.82 [min: 0.61, max: 0.91], p=5.6e-05; compare Figure 1D and E to Figure 2A and C, respectively). These findings support the effort optimization hypothesis, whereby the brain implements control policies that exploit gravity effects to minimize muscular efforts.

Figure 2 with 1 supplement see all
Experimental findings.

(A) Experimentally recorded symmetry ratio, averaged across all subjects, is plotted as a function of movement direction. Similarly to the effort optimization hypothesis prediction (see panels D in Figure 1), experimental symmetry ratio is well fitted by a sigmoidal function (black curve). Error bars illustrate SD. (B) Fits of a sigmoid function to symmetry ratio as a function of movement direction for data from individual subjects. (C) Symmetry ratio as a function of Work of Gravity Torque (WGT). Each data point represents the mean of 12 trials for one subject moving in one direction (n = 255; 3060 trials total). It is noticeable that, similarly to the effort optimization hypothesis prediction (Figure 1E), arm kinematics (symmetry ratio) linearly correlates with gravity torque.

https://doi.org/10.7554/eLife.16394.007

Although undeniably supportive, interpretation of these results is complicated by direction-dependent properties of the peripheral neuromuscular system (Enoka, 1996; Cotel et al., 2009; Wilson et al., 2015). As a further test, exploitation of microgravity environments (e.g., during the zero-G phase of parabolic flight) offers a powerful tool to interrogate neural vs. peripheral origins of the directional asymmetries. This is because during the repeated transitions to zero-G, gravity torque is temporarily eliminated. According to the effort optimization hypothesis, kinematic asymmetries should gradually but systematically decrease to zero, because a progressive re-optimization neural process should take place gradually over multiple zero-G transitions (McIntyre et al., 2001; Izawa et al., 2008; Snaterse et al., 2011). In contrast, a peripheral origin of kinematic asymmetries leads to different predictions. Specifically, if directional asymmetries originate from flexor/extensor motoneuron properties, SR should not change during the zero-G phase of parabolic flight (Ishihara et al., 1996, 2002; Cotel et al., 2009; for a review see Nagatomo et al., 2014; Wilson et al., 2015). Else, if directional asymmetries originate from force production properties, SR asymmetries should be eliminated instantly, not gradually, in 0g (Enoka, 1996). A second experiment was designed to test these predictions.

Eleven participants performed fast and visually guided single-degree of freedom arm reaching movements in two directions (toward the head and toward the feet, Figure 3A), in zero-G conditions during 5 parabolas (P1-P5) of a flight where centrifugal manoeuvers allow cancellation of gravity effects in the plane’s frame of reference. Arm movements were planar with comparable systematic and variable errors for different gravity and direction conditions (shoulder abduction-adduction and internal/external rotation < 3.1°; -3°< SE < 3.3°; VE < 3.4°; gravity effect on SE, p=0.12 and VE, p=0.27; direction effect on SE, p=0.41 and VE, p=0.35). Velocity profiles were single-peaked in both one-G and zero-G conditions and average movement durations ranged between 0.40s and 0.54s (on average, 0.45±0.13s), without any statistical difference between gravity conditions (F5,50=1.274, p=0.29) and movement direction (F1,10=0.549, p=0.48).

Figure 3 with 1 supplement see all
Adaptation to microgravity.

(A) Participants’ initial position and positioning of the 3 targets in the sagittal plane. Eleven participants performed fast and visually guided mono-articular arm movements (shoulder rotations) in the sagittal plane under normal gravity (one-G) and micro-gravity conditions (zero-G) during a parabolic flight (parabola 1, P1 to parabola 5, P5). (B) Symmetry ratios (acceleration time / movement time) predicted by the Minimum Smooth-Effort model in one-G and in zero-G conditions. (C) Symmetry ratios experimentally recorded before (1g) and during adaptation to zero-G (P1 to P5). (D) Mean velocity profiles, normalized in amplitude and duration. Qualitative comparisons between upward and downward arm movements illustrate the progressive decrease of directional asymmetries when subjects adapted to the new microgravity environment. (E) Vertical bars represent the average symmetry ratio (black vertical axis, left) for the recorded (black filled bars) and simulated (black open bars) data in one-G and during zero-G (P1 to P5) environments. For simulated data, the g value was fitted (−2*G<g<2*G) in order to best predict the measured symmetry ratios. These fitted g values, represented by the green dots (green vertical axis, right), reveal a progressive decrease of the g internal model value during zero-G exposure. Error bars illustrate SD and color-coded arrows denote movement direction (red = up; blue = down). See also Figure 3—figure supplement 1.

https://doi.org/10.7554/eLife.16394.009

Figure 3B shows the SR values predicted from the Smooth-Effort model for upward (red) and downward (blue) arm movements, both in one-G and zero-G environments. Optimization to the zero-G environment no longer predicts direction-dependent differences in velocity profiles, as was the case in one-G. In line with the effort optimization hypothesis, which predicts gradual adaptation to the zero-G environment, SR slowly converged towards the new direction-independent optimal value in zero-G (Figure 3C; gravity condition and movement direction interaction effect, F5,50=10.54, p<0.001). Importantly, directional asymmetries persisted early in zero-G (p=4.8e-05 for P1 and p=1.09e-03 for P2). This indicates that, during initial exposure to zero-G, the brain still uses the internal model of the one-G environment to plan arm movements.

However, directional asymmetries progressively disappeared in the following parabolas, suggesting a gradual re-optimization of motor commands through sensorimotor adaptation to the zero-G environment (post-hoc, p=0.34 for P3, p=0.92 for P4, and p=0.99 for P5). The corresponding average velocity profiles qualitatively illustrate the increase in acceleration/deceleration symmetry (Figure 3D). For the sake of clarity, these experimental findings are replotted as SR Down - SR Up in Figure 3E (black bars). To quantitatively validate the hypothesis of a gradual re-optimization of motor commands, we fitted this directional difference with the Smooth-Effort model (white bars in Figure 3E) by letting the internal model of gravity, g, being a free parameter (-2*G<g<2*G). The progressive decrease in g (internal model of gravity; green dots in Figure 3E) shows that the progressive change in arm kinematic asymmetries can be well explained as a recalibration of the gravity internal model used for optimal motor planning based on the effort optimization (minimization) model (McIntyre et al., 2001; Izawa et al., 2008; Snaterse et al., 2011).

Discussion

It is broadly believed that the brain develops and uses internal models of the sensor and effector dynamics, as well as physical laws of motion, to optimally interact with the external environment (Shadmehr and Mussa-Ivaldi, 1994; Conditt et al., 1997; Gribble and Ostry, 1999; Wolpert and Ghahramani, 2000; Pigeon et al., 2003; Todorov, 2004; Ahmed et al., 2008; Scott, 2012). Having evolved in the Earth’s gravitational environment, our brains have thus acquired an internal model of gravity (Angelaki et al., 1999, 2004; Indovina et al., 2005; Miller 2008). Here, we have conducted two critical experiments and showed that the brain takes advantage of this internal model to implement control policies that minimize movement effort under gravito-inertial constraints.

First, we have shown that humans use versatile temporal trajectories that are linearly tuned to the gravity torque requirements of the task. Simulations of a Smooth-Effort model, which minimizes a hybrid cost composed of the absolute work of muscular forces (mechanical energy expenditure) and jerk (inverse smoothness of the trajectory) can predict not only differences in upward/downward arm movements, but also the linear modulation of endpoint kinematics according to the gravity torque requirements of the movement. Because the smoothness term of the Smooth-Effort model corresponds to the Jerk, the Smooth-Effort model can be considered an expansion of the minimum Jerk requirement to include how task dynamics shapes movement kinematics. Yet, inclusion of an effort-related cost is a necessary and sufficient condition to predict directional asymmetries in the vertical plane (Berret et al., 2008a, 2008b). Furthermore, minimizing other torque-related cost functions failed to predict direction-dependent kinematics (Figure 1F). The present results therefore strongly support effort minimization in humans, further extending the growing idea that perceived effort plays an important role in the tailoring of human motor as well as non-motor behaviors (Bramble and Lieberman, 2004; Walton et al., 2006; Mazzoni et al., 2007; Carrier et al., 2011; Kurzban et al., 2013; Selinger et al., 2015; Farshchiansadegh et al., 2016; Shadmehr et al., 2016).

Second, we have also shown that the direction-dependent kinematics observed in normal gravity progressively vanishes during repeated exposure to a microgravity environment. Remarkably, Smooth-Effort model simulations nicely predict this adaptation to zero-G. Results of this second experiment are of major importance to disentangle peripheral and neural mechanisms for direction-dependent kinematics. This is because, if the observed direction-dependent kinematics were due to properties of the neuromuscular system, either an abrupt change or no change at all would be expected in the new gravity environment (Enoka, 1996; Ishihara et al., 1996, 2002Cotel et al., 2009; Nagatomo et al., 2014; Wilson et al., 2015). The fact that neither happens reveals a central mechanism.

Furthermore, the fact that the observed progressive changes in kinematic asymmetries lead to new optimal values clearly supports the hypothesis of a progressive re-optimization procedure originating from planning processes (Izawa et al., 2008; Snaterse et al., 2011; Selinger et al., 2015). The present findings on arm movements extend and supplement the recent results of Selinger and collaborators on human walking (Selinger et al., 2015), suggesting that i) the brain can easily and progressively adapt motor patterns to reduce energy expenditure and ii) energy-related criteria (such as effort) are not only the result of, but actually tailor, motor patterns.

The present results and conclusions stand in contrast to a broad view that the brain uses internal models of perturbing forces for their compensation such that stereotypic trajectories can be maintained (Hollerbach and Flash, 1982; Atkeson and Hollerbach, 1985; Shadmehr and Mussa-Ivaldi, 1994). Although very influential, such a compensation hypothesis has been challenged by results of studies that quantified velocity profiles and revealed that the temporal organization of arm kinematics shows a small, yet consistent, dependence on movement direction, speed and load (Papaxanthis et al., 1998b; Gaveau et al., 2011, 2014). Furthermore, findings inconsistent with the compensation hypothesis have been largely ignored. For example, Virji-Babul et al. (1994) reported regression slopes of SR over movement amplitude that significantly differed between upward and downward movements (see Figure 2C in Virji-Babul et al., 1994). Also, the consistent observation of negative periods on the phasic activation of arm muscles, resulting from the subtraction of the hypothesized gravity-compensatory activity from the full muscle activation (Flanders et al., 1996), suggest that muscular activity does not compensate gravity torque (Gaveau et al., 2013).

Luckily, the application of the optimal control theory to the study of biological movement has given better insights into old phenomena. For example, recent studies, which framed motor adaptation as a process of re-optimization (Izawa et al., 2008; Crevecoeur et al., 2009; Gaveau et al., 2011; Cluff and Scott, 2015), have reported subtly altered trajectories – by contrast to the traditional compensation view that assumes invariant trajectories. Thus, newly constructed/calibrated internal models may serve trajectory optimization rather than external force compensation. The present results provide further support for this notion and demonstrate the propensity of the motor system for multiple control policies (i.e., trajectories) whose temporal organization shows small, but systematic, differences, such as to allow minimization of motor effort in our daily living ubiquitous gravity environment.

Although we have only used simple, mono-articular arm movements in the present experiments, our conclusions are generalizable. Specifically, directional asymmetries in the vertical plane (upwards versus downwards) have also been observed in multi-articular arm reaching, reaching to grasp, grasping, hand drawing, and whole-body sit-to-stand / stand-to-sit movements (Papaxanthis et al., 1998c, 2003, 2005; Yamamoto and Kushiro, 2014). Thus, we propose that the directional tuning of movement kinematics is a general feature of motor control that may reflect an evolutionary and/or developmental advantage for effort optimization in the Earth’s gravity field (Bramble and Lieberman, 2004; Carrier et al., 2011; Selinger et al., 2015).

Finally, it is important to speculate that the Smooth-Effort model pioneered here should not be considered solely an extension of the minimum jerk optimization model, which was proposed for planar horizontal movements that are unaffected by gravity (Flash and Hogan, 1985), to now include optimization of work against gravity for vertical movements. Even for movements in the horizontal plane, because the effort component of the cost function in the Smooth-Effort model is torque-dependent, the predictions of the Smooth-Effort model will change with the torque requirements of the movement. The minimum Jerk model predictions, however, will remain constant. Therefore, if interaction torques (for multi-degree of freedom arm movements) or additional external torques (produced by a robotic manipulandum for example) are experienced in the horizontal plane, the Smooth-Effort and Jerk model predictions would be different. Future experiments should test whether the need for optimization of work is limited to vertical movements pro or against gravity or, as we propose, represent a more general principle of motor control.

Materials and methods

Twenty-six right-handed healthy adults participated in these experiments (Experiment 1: 4♀ / 11, mean age = 24±3.2 years; Experiment 2: 2♀ / 9♂, mean age = 27±4.1 years). All gave their written informed consent. Right hand preference was evaluated by the Edinburg test (individual scores > 0.86; Oldfield, 1971). The regional ethics committee of Burgundy (C.E.R) approved the protocol of Experiment 1 and the ethics committee of INSERM (Institut National de la Santé et de la Recherche Médicale) approved the protocol of Experiment 2. All procedures were carried out in agreement with local requirements and international norms (Declaration of Helsinki, 1964).

Experiment 1

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Participants comfortably sat on a chair with their trunk in the vertical position (Figure 1A). All trials started from a fixed initial position: shoulder elevation 90°, shoulder abduction 0°, elbow joint fully-extended, hand semi-pronated and aligned with the upper arm and the forearm. From that initial position, participants carried out rapid, visually guided, single degree of freedom arm movements (rotation around the shoulder, 45° amplitude) towards 17 targets (plastic markers, diameter 1 cm) placed in the right sagittal-frontal space. Note that results from previous experimental and theoretical studies have demonstrated that directional asymmetries in the vertical plane do not originate from the existence of inertial interaction torques at the elbow and wrist joints (Le Seac'h and McIntyre, 2007; Gaveau et al., 2014). Figure 1A depicts the projection of targets' position onto the frontal plane. The inter-target angles are described in Figure 1—figure supplement 1C (Plane angles). All targets were centered on the participants’ right shoulder at a distance equal to the length of their fully-extended arm. Reaching movements required a combination of shoulder abduction and shoulder flexion or extension. Participants performed 204 trials in a random order (12 trials for each movement direction, total trials in the experiment = 3060).

Experiment 2

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This experiment took place in an aircraft during parabolic flight. Participants comfortably sat on the aircraft’s floor with their legs strapped and their trunk in the vertical position (Figure 3A). The general organization of the task was exactly as described in previous studies (Gaveau et al., 2011, 2014). Briefly, 3 targets were centered on the participants’ right shoulder (parasagittal plane) at a distance equal to the length of their fully extended arm. Participants were requested to perform fast visually guided upward and downward arm reaching movements (45° shoulder rotation). Participants first performed arm movements in normal gravity during the flight (before parabolic maneuvers started, 40 trials for each movement direction) and then in microgravity during 5 parabolas (≈75 trials, ≈15 trials per parabola). During the parabolic flight and based on pilots’ instructions about the gravity force level, the experimenter verbally instructed participants when to start a block of reaching movements and when to stop. This was important to ensure that movements started and finished within zero-G conditions; i.e., participants made no movement in one-G or two-G conditions during or between parabolic manoeuvers. At the beginning of a block of movements within a parabola, participants repetitively reached between the middle, the upward, and the downward targets as follows: middle-upward (stopped for roughly 1 s), upward-middle (pause 1 s), middle-downward (pause 1 s), downward-middle (pause 1 s) and so on for approximately 15 trials. As initial position did not influence adaptation results across parabolas, movements with different initial positions were pooled together within each direction. The experiments were carried out during 4 different flights. Each flight was composed of thirty parabolas, each parabola consisting of three successive phases: (i) hypergravity ~ 1.8 g, (ii) microgravity ~ 0 g, and (iii) hypergravity again ~ 1.8 g. Each of those three phases lasted ~30 s and the parabolas were separated by a time interval of ~ 2 min. The whole flight lasted ~ 2 hr. Here we present the results of an experiment performed during 5 parabolas for each participant. After this experiment, participants carried out a different experiment from which the results are not presented here.

Two participants were tested on each flight. Therefore, to reduce motor adaptation to zero-G before the experiment took place, participants who did not perform the experiment during the first five parabolas were restrained on the aircraft floor so as to prevent any motion. Similar results were observed for participants who did the experiment at the beginning or at the end of the flight (see Figure 3—figure supplement 1).

Data recording, analysis and modeling

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All analyses were performed using custom programs in Matlab (Matworks, Natick, MA) and have been described in details in previous studies (Gaveau et al., 2011, 2014). Arm movements in both Experiments were recorded using an optoelectronic system of motion analysis (Smart, B.T.S., Italy) with 4 TV-cameras (120 Hz). Five reflective markers (diameter: 4 mm) were placed on the shoulder (acromion), elbow (lateral epicondyle), wrist (middle of the wrist), hand (first metacarpo-phalangeal joint), and the nail of the index. Kinematics was recorded in three dimensions (X, Y and Z) and low-pass filtered (10 Hz) using a digital fifth-order Butterworth filter. The start and end of each movement was defined as the time at which finger tangential velocity went above or fell below 5% of maximum velocity. An automatic inspection of all trials revealed that shoulder angular velocity profiles were single-peaked and presented no motion (<3°) from any other joint. Following this analysis, we calculated the subsequent kinematic parameters: (i) movement duration (MD), (ii) constant and variable angular final error and (iii) symmetry ratio (SR) of the finger velocity profile, defined as the ratio of acceleration time to total movement time (a ratio equal to 0.5 indicates temporally symmetric velocity profiles). In the present study, we used SR to quantify kinematic variations with movement direction. SR is a standard parameter that has been routinely used in numerous studies, therefore ensuring an easy comparison of the present result with past ones.

We calculated the gravity torque (GT) normal to the plane of motion with the following formula:

(1) GT=mglcosθsinγ

where m is the mass of the arm (estimated for each subject from anthropometric tables, Winter, 1990), g = 9.81 m.s-2, l is the lever arm length (Winter, 1990), θ is the movement amplitude (0° to 45°) and γ is the plane of motion inclination with respect to horizontal. The geometrical illustration of the mechanical system along with details on how to derive Equation 1 are displayed in Figure 1—figure supplement 1.

In Experiment 1 we report the Work of Gravity Torque calculated as follows:

(2) WGT= 045(GT)dθ

Positive values indicate that WGT has the same direction as arm motion, whilst negative values indicate that WGT direction is opposite to arm motion direction (see Figure 1—figure supplement 1).

In Experiment 1, we also performed iterative least square minimizations to fit a sigmoid function on simulated as well as experimentally recorded SR. This was performed using the “nlinfit” Matlab function (Mathworks) and Equation 3:

(3) F=p1+ p2/(1+exp(x+p3p4))

where p1 to p4 are free parameters and x is the angular scale. The goodness of fit was assessed by the Root-Mean-Square Error (RMSE), which is essentially equivalent to Standard Deviation and has the same units as the variable being fitted.

Simulations

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We predicted SR using an optimal control framework based on the Minimum Smooth-Effort model (Gaveau et al., 2014). More sophisticated versions of the model focusing on multi-degree of freedom arm movements have been described in previous publications (Berret et al., 2008a, 2008b; Gaveau et al., 2011).

Equation 4 describes the equation of motion for a single degree of freedom limb movement, with amplitude angle  (θ), plane of motion inclination with respect to horizontal (γ), moment of inertia (I), viscous friction coefficient B=0.87 (as in Nakano et al., 1999) and gravitational torque GT(θ,γ). The net muscle torque acting at the shoulder is obtained as follows:

(4) τ=Iθ¨+Bθ˙+GT(θ,γ)

The Minimum Smooth-Effort model minimizes a combination of Effort and Smoothness. The mechanical effort related to a movement, i.e., the amount of muscular force spent to move the arm, can be computed as the absolute work of the muscle torque:

(5) Ceffort=0T|τθ˙| dt

The rationale of this effort term is that the desired trajectory must take advantage of non-muscular torques – gravity torque in the present experiment – in order to minimize the amount of muscular torque required to move the arm up to the final posture.

It has been shown that considering effort expenditure alone usually fails to account for several features of human trajectories, such as motion smoothness (Berret et al., 2011a, 2011b). Consequently, we considered that a complementary objective of motor planning was to maximize motion smoothness. This was achieved by penalizing large angle jerks. Thus an additional term that enters into the minimization is:

(6) Csmooth=0T(dθ¨/dt)2dt

The Smooth-Effort model then relies on the following composite cost function:

(7) C=Ceffort+αCsmooth

where α is a weighting factor normalizing the relative magnitude of the jerk term in the total cost function. For all simulations, but Figure 1—figure supplement 2, we set α=7e5. Figure 1—figure supplement 2 presents simulations where α was systematically varied to test the relative roles of both the effort and smoothness parts of the cost function in predicting direction dependent kinematics (i.e., the linear correlation of SR with the Work of Gravity Torque; WGT, see Figure 1E). Note that minimizing some function f(x)=b*g(x)+c*h(x) will provide the same solution x as minimizing f(x)/d=b/d*g(x)+c/d*h(x) for any d>0. Hence, we normalized our cost function by setting b=d, thereby assuming a unit coefficient in front of one component of the cost.

For a given α, the optimal control problem consists of finding a vector (here the time derivative of the muscle torque) driving the system from an initial (θ0) to a final static posture (θf), in time T (adjusted for each subject based on experimental data), and yielding a minimum cost value C. We solved the minimization numerically using a Gauss pseudospectral method and the software GPOPS (Benson et al., 2006; Garg et al., 2009; Rao et al., 2010). We verified that the control variable was smooth, the boundary values were not reached and the Pontryagin’s maximum principle necessary conditions (such as the constancy of the Hamiltonian) met. Predicted SR values were determined based on the simulated velocity profiles, as described above for experimental data.

We also derived the solution of the Minimum Smooth-Effort model for a constant gravitational torque and obtained very similar results to those presented in the main text. This means that asymmetries are not totally due to GT variations along the movement amplitude but to the presence of non-zero GT. The linearized case for similar models has been analyzed in depth in Berret et al. (2008a); where the solution of the Minimum Smooth-Effort model was derived explicitly and the origin of asymmetries in such a model was mathematically demonstrated – minimizing an effort-related cost is a necessary and sufficient condition for the production of a brief transient muscle inactivation near the peak of velocity which in turn induces the observed kinematic asymmetries with respect to movement direction.

In Figure 1 we also present the results of simulations performed with three influential alternative models: the Minimum Jerk, the Minimum Torque Change and the Minimum Variance (Flash and Hogan, 1985; Uno et al., 1989; Harris and Wolpert, 1998). Predicted SR values corresponding to the minimization of each subjective cost were obtained using the following equations and the same optimal control framework as described for the Minimum Smooth-Effort model.

The cost to minimize for the Jerk model was:

(8) Cj=0T(dθ¨/dt)2dt

As we deal with a 1-dof arm, this model accounts for both the angular and Cartesian versions of the Minimum Jerk (which indeed provide equivalent predictions in the current case). Note that the smoothness term of the Smooth-Effort model corresponds to the minimum Jerk. The Smooth-Effort model thus represents an important and non-trivial extension of the minimum Jerk model, which also accounts for how task dynamics shapes movement kinematics. Because the effort part of our model is torque dependent, the prediction of the Smooth-Effort model will change with the torque requirement of the movement whilst the minimum Jerk prediction will not. Therefore, if interaction torques (for multi-degree of freedom arm movements) or additional external torques (produced by a robotic manipulandum for example) are experienced in the horizontal plane, the minimum Smooth-Effort and the minimum Jerk would predict very different solutions.

The cost to minimize for the Torque Change model (Uno et al., 1989) was:

(9) Cτ= 0T (dτdt)2dt  

Simulations of the Minimum Torque change model for horizontal single degree of freedom arm movements performed under various dynamics by Tanaka et al. (2004) reported that the Minimum Torque Change model predicts velocity profiles that are always symmetrical, in line with Figure 1F. The Minimum Torque Change model in the gravity field has also been investigated in depth in (Berret, 2009).

We also derived the solutions of the Minimum Variance model in the gravity field (Harris and Wolpert, 1998). We first linearized the arm’s dynamics (Equation 4) in order to compute the optimal solution according to the method presented in Harris and Wolpert (1998) and Tanaka et al. (2004). Here:

(10) mglcosθmglcosθ0=k

Equation 10 formalizes that a constant gravitational torque is pushing the moving segment downwards. Note that such linearized case for similar models has been extensively used by previous studies (Harris and Wolpert, 1998; Tanaka et al., 2004; Berret et al., 2008a).

We considered the discrete time version of the Minimum Variance optimal control problem. Denoting by x=(θ,θ˙,τ)T the state vector and by u=τ˙ the control variable, the linear state-space dynamics for the Minimum Variance model was expressed as follows:

(11) xt+1=Axt+B(ut+wt)+C

where wt~N(0,σut2) is the signal dependent (multiplicative) noise at time t (σ=0.2 in all simulations) and C=(0,k/I,0)TΔt with Δt the time step size after discretization (Δt=10ms in our simulations).

We obtained the solution to the optimal control problem by iteratively computing the distribution of the state vector at time t:

(12) E[xt]=Atx0+i=0t1At1i(Bui+C)
(13) cov[ xt ]= σi=0t1(At1iB)(At1iB)Tui2

The positional variance of the endpoint at time t (denoted by Vt) is thus given by the element (1,1) of the matrix cov[ xt ] (Equation 13). We can then define a quadratic programming problem by defining a cost related to the endpoint positional variance as follows:

(14) Cvar(u0,u1,,uT+R)=t=T+1T+RVt

where R>1 is an integer defining the post-movement stabilization time, T is the a priori chosen movement time (counting only the transient phase). In our simulations, we considered R=T. Because noise accumulates through time, Cvar is indeed a function of the control variable during the transient period of the motion. This minimization problem was solved using Matlab's fmincon function (sqp algorithm).

For the sake of completeness, we also derived the minimum variance solution when modeling some basic muscle dynamics, as performed by previous studies (Harris and Wolpert, 1998; Tanaka et al., 2004). In agreement with previous results from Tanaka et al. (2004); we obtained similar results to those presented in the main text (Figure 1F), as illustrated in Figure 1—figure supplement 3. In that case, the state vector becomes x=(θ,θ˙,aag,aant)T, the control variable becomes u= (uag,uant)T and the muscles are modeled as first order low-pass filters. The equations for the associated dynamics are as follows:

(15) τagτant=Iθ¨+Bθ˙+GT(θ,γ)
(16) τagτant=ρ(aagaant)
(17) σa˙ag=uagaag
(18) σa˙ant=uantaant

Equation 15 describes the equation of motion for a single degree of freedom limb movement, with amplitude angle (θ), plane of motion inclination with respect to horizontal (γ), moment of inertia (I), viscous friction coefficient B=0.87 (Nakano et al., 1999) and gravitational torque GT(θ,γ). In Equation 16, the constant ρ is a gain factor relating agonist and antagonist muscle activations to joint torques. Equations 17 and 18 describe the muscle dynamics as a first order low-pass filter. The control variable is the motoneurons inputs uag and uant with the constraint: (uag,aant) ϵ [ 0,1 ]2. This implies the positivity of muscle activations and, therefore, muscle torque. The net torque is simply obtained by subtracting the agonist and antagonist torques.

Statistical analyses

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For each participant, we calculated mean values for all recorded variables and checked for normal distribution (Shapiro-Wilk tests) and sphericity (Mauchly tests). In Experiment 1, statistical effects were accessed by within-subjects one-way repeated-measures ANOVA (factor: 17 angles). In Experiment 2, statistical comparisons were carried out by within-subjects two-way repeated-measures ANOVA. The factors were direction (2 levels) and gravity-conditions (six levels: 1g and 5 parabolas in 0g). For all statistical analyses, post-hoc differences were assessed with Scheffé's tests and significance was accepted at p<0.05.

References

  1. 1
  2. 2
    Computation of inertial motion: neural strategies to resolve ambiguous otolith information
    1. DE Angelaki
    2. MQ McHenry
    3. JD Dickman
    4. SD Newlands
    5. BJ Hess
    (1999)
    Journal of Neuroscience  19:316–327.
  3. 3
  4. 4
    Kinematic features of unrestrained vertical arm movements
    1. CG Atkeson
    2. JM Hollerbach
    (1985)
    Journal of Neuroscience  5:2318–2330.
  5. 5
  6. 6
    Intégration de la force gravitaire dans la planification motrice et le contrôle des mouvements du bras et du corps
    1. B Berret
    (2009)
    Université de Bourgogne.
  7. 7
  8. 8
  9. 9
  10. 10
    How humans control arm movements
    1. B Berret
    2. J-P Gauthier
    3. C Papaxanthis
    (2008b)
    Proceedings of the Steklov Institute of Mathematics 261:44–58.
    https://doi.org/10.1134/S0081543808020053
  11. 11
  12. 12
    Applied Optimal Control
    1. AE Bryson
    2. YC Ho
    (1975)
    New York: John Wiley & Sons, Inc.
  13. 13
  14. 14
  15. 15
    The motor system does not learn the dynamics of the arm by rote memorization of past experience
    1. MA Conditt
    2. F Gandolfo
    3. FA Mussa-Ivaldi
    (1997)
    Journal of Neurophysiology 78:554–560.
  16. 16
  17. 17
  18. 18
  19. 19
  20. 20
    Uber das Relativita¨tsprinzip und die aus demselben gezogenen Folgerungen
    1. A Einstein
    (1908)
    Jber Radiokt 4:411–462.
  21. 21
    Perception of planar surface orientation relative to earth vertical
    1. LC Elmore
    2. RM Cassidy
    3. A Rosenberg
    4. GC Deangelis
    5. DE Angelaki
    (2014)
    Program No. 822.02. 2014 Neuroscience Meeting Planner.
  22. 22
    Eccentric contractions require unique activation strategies by the nervous system
    1. RM Enoka
    (1996)
    Journal of Applied Physiology 81:2339–2346.
  23. 23
  24. 24
  25. 25
    The coordination of arm movements: an experimentally confirmed mathematical model
    1. T Flash
    2. N Hogan
    (1985)
    Journal of Neuroscience   198:1688–1703.
  26. 26
  27. 27
  28. 28
  29. 29
    Motor planning in the gravitational field: how do monkeys reach?
    1. J Gaveau 
    2. C Papaxanthis
    3. DW Mordan
    4. DE Angelaki
    (2013)
    Program No. 650.03. 2013 Neuroscience Meeting Planner.
  30. 30
  31. 31
  32. 32
    Compensation for interaction torques during single- and multijoint limb movement
    1. PL Gribble
    2. DJ Ostry
    (1999)
    Journal of Neurophysiology 82:2310–2326.
  33. 33
  34. 34
  35. 35
  36. 36
  37. 37
    Succinate dehydrogenase activity in rat dorsolateral ventral horn motoneurons at L6 after spaceflight and recovery
    1. A Ishihara
    2. Y Ohira
    3. RR Roy
    4. S Nagaoka
    5. C Sekiguchi
    6. WE Hinds
    7. VR Edgerton
    (2002)
    Journal of Gravitational Physiology  9:39–48.
  38. 38
  39. 39
  40. 40
  41. 41
  42. 42
  43. 43
  44. 44
  45. 45
  46. 46
  47. 47
  48. 48
    Effects of exposure to microgravity on neuromuscular systems: a review
    1. F Nagatomo
    2. M Terada
    3. N Ishioka
    4. A Ishihara
    (2014)
    Internal Journal of Microgravity Science Application 31:66–71.
  49. 49
    Quantitative examinations of internal representations for arm trajectory planning: minimum commanded torque change model
    1. E Nakano
    2. H Imamizu
    3. R Osu
    4. Y Uno
    5. H Gomi
    6. T Yoshioka
    7. M Kawato
    (1999)
    Journal of Neurophysiology 81:2140–2155.
  50. 50
  51. 51
  52. 52
  53. 53
  54. 54
  55. 55
  56. 56
  57. 57
  58. 58
  59. 59
    Gpops: A matlab software for solving multiple-phase optimal control problems using the gauss pseudospectral method
    1. A Rao
    2. D Benson
    3. C Darby
    4. M Patterson
    5. C Francolin
    6. I Sanders
    7. G Huntington
    (2010)
    ACM Transactions on Mathematical Software 37:9.
  60. 60
  61. 61
  62. 62
  63. 63
  64. 64
    Adaptive representation of dynamics during learning of a motor task
    1. R Shadmehr
    2. FA Mussa-Ivaldi
    (1994)
    Journal of Neuroscience  14:3208–3224.
  65. 65
  66. 66
  67. 67
  68. 68
    Formation and control of optimal trajectory in human multijoint arm movement. Minimum torque-change model
    1. Y Uno
    2. M Kawato
    3. R Suzuki
    (1989)
    Biological Cybernetics 61:89–101.
  69. 69
  70. 70
  71. 71
  72. 72
  73. 73
    Biomechanics and Motor Control of Human Movement
    1. D Winter
    (1990)
    New York: John Wiley and Sons.
  74. 74
  75. 75
  76. 76
  77. 77

Decision letter

  1. Eve Marder
    Reviewing Editor; Brandeis University, United States

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

Thank you for submitting your article "Direction-dependent arm kinematics reveal optimal integration of gravity cues" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by Eve Marder as the Senior Editor. We apologize for the delay in coming to a decision, but it took a while for the reviews to come in, and then the editor and reviewers went through a fairly protracted consultation process to arrive at a decision. Basically, the reviewers all found the topic interesting and important, and felt that your work had a lot to contribute. At the same time, they felt that you have gone too far to claim its novelty and haven't pushed the analyses as far as might be suitable and useful. All of the reviewers agree that a substantial revision is warranted. The comments during the consultation session largely reiterated the comments in the reviews, so I am taking the unusual tack of forwarding the initial reviews, lightly edited, to you, and asking you to do the most thoughtful revision that addresses the substantive issues that the reviewers raised. This will probably require some additional data analysis and rewriting to do more justice to the models and accomplishments already in the literature.

Reviewer #1:

The goal of the present experiment was to settle the long-lasting debate of how the gravitational constraint is integrated in movement control. More specifically, the authors tested whether the motor system attempts to overcome the gravitational force (compensation hypothesis) or rather exploits this passive force to control movement (optimization hypothesis). To disentangle between both hypotheses, the authors analyzed the kinematics of movements performed by human participants in different directions on Earth with predictions made by a mathematical model built on the basis of the optimal control framework. Furthermore, the authors had the unique opportunity to test these hypotheses by asking participants to perform similar movements in absence of gravitational constraint (microgravity), i.e. during parabolic flights. The results of both experiments (normogravity and microgravity) and the comparisons of real and simulated movements in both environments clearly supported the optimization hypothesis.

This study continues the series of nice papers published by the authors on the effects of gravitoinertial constraints on sensorimotor processes. It addresses an important issue related to the integration of gravity during motor actions and provides significant advance over previous studies. However, the paper could be easily improved with some changes.

The problem is that other published papers, some of them referenced in the manuscript, provided convincing evidence in favor of the compensation hypothesis. Unfortunately, the authors do not provide explanations as to why the earlier evidence for the compensation hypothesis is less convincing than theirs. They kind of "kicked the ball into touch" by simply stating that "advent of the optimal control theory has cast strong doubts on this more traditional view of motor control". This was done perhaps on a voluntary basis because critics, even those that lead to progress, are not always welcome. However, because the goal of the study was "to explicitly distinguish between the compensation and optimization hypotheses", the discussion should be more explicit about problems regarding the compensation hypothesis. Despite the very nice results, without such a discussion, the impact of the present paper will be dampened.

The writing style would benefit from smoothing. Currently, there are too many long sentences, many of them are interrupted by long lists of references that sometimes interfere with understanding.

Reviewer #2:

The authors provide interesting evidence of gravitational influence on human single-joint movements. The directional dependence on the ground, and acquired loss of dependence in repeated parabolic flight, is substantive and novel. This work has implication for how the brain can ordinarily (on the ground) and flexibly (via parabolic flight) be influenced by gravity.

The particular analytical, theoretical, and hypotheses offered, however, have several under-developed and contradictory elements that cause great concern. The chief concern, with regard to the overall hypothesis, regards the sureness that these results confirm a "Minimum Smooth Effort" model. The authors choose to cite papers that develop the model in full, rather than doing so here; that is a great disadvantage, as the reader cannot even consider the actual hypothesis suggested. What is suggested throughout is that "effort" is dominated by forces (here, gravity) (e.g. "The behavioral signature of the optimal integration of gravity hypothesis…") The equations offered, however, show that jerk, a purely kinematic measure, is included in the authors' model. The sensitivity analysis, shown only in the discussion, is very incomplete, as it explores the calculated cost only on one side of the chosen coefficient. My best conclusion is that the jerk inclusion is crucial, and even the most generous reading of the work concludes with a synthesis, that both kinematics and dynamics are important in the present behavior.

Several other elements of the analysis, modeling, and interpretation are of concern.

– The presented smooth effort model includes separate terms for torques generated by agonists and antagonists. The authors have no way to measure these; they measure only kinematics; undoubtedly on the ground, and absolutely in flight, there will be co-contraction. The central variables in their model, therefore, are unmeasurable.

– The absolute flat lines calculated from minimum torque change and minimum variance hypotheses (Figure 1F) is highly questionable. Similar work (indeed the origin of these theories) suggest that load should play an important role. Much more detail on the calculation of these cost functions is needed to justify this very counter-intuitive result. The minimum variance model, in particular, needs a level of neural activation (drive) that seems complete absent from the present consideration.

– Passengers that are strapped in the flight clearly have cues that gravity is very strong, absent, and strong again. The notion that strapping will lead to no information before movement is vexing. Similarly, the notion that hypergravity for the not-strapped-down has no learning effect is unusual.

– The authors provide scant information on how they calculated just the biomechanics of the task. Meaningful quantitative consideration of kinematics versus dynamics requires more information here.

On the whole, the authors have shown that in a vertical plane, target direction, on the ground, plays a role in human trajectory control; that in early flights, an asymmetry in velocity profiles exist; and in later flights, that asymmetry disappears. All these point to a simple conclusion: the dynamics of gravity can be used by the brain to influence trajectory. All further conclusions made by the manuscript have under-developed evidence, and therefore provide very unclear impact on the field of motor control and neural computation.

Reviewer #3:

General Assessment

This manuscript presents new data and computational analysis supporting the hypothesis that the motor system generates arm movements to satisfy a tradeoff between optimizing smooth kinematics and control effort. The issue of what is being optimized in motor control has been attracting considerable attention through the last decades and no final resolution has been reached to date. Model predictions tend to be consistent with data within specific domains and in some ways this may not be an exception. The authors are considering the impact of gravity in the formation of arm trajectories and they frame the issue in the dichotomy between compensation and exploitation of gravitational torques in the generation of movement. To investigate how the motor system may take advantage of gravity in an optimal control approach, the authors consider a cost function expressing a tradeoff of effort and smoothness terms. They test the model in normal gravity and in microgravity and find evidence for a slow adaptive response when subjects perform repeated movements in the absence of gravity. The key metric considered in this work is the symmetry ratio (SR) expressing the temporal symmetry of acceleration and deceleration phases of simple reaching movements. The authors show that the symmetry ratio (as a function of movement direction and of gravitational work) follows a sigmoidal trend both in the model and in the experiments. Most importantly, the trend is gradually abolished when movements are carried out at zero-g. This is important, in my opinion, because it is not consistent with the possibility that the observed trend in normal gravity is due to an imperfect controller, attempting to implement a kinematic plan with a poor model of the gravitational field. The manuscript is well written, the comparison of the data with the model is done carefully and the conclusion that the data are supporting the idea that the motor controller exploits gravity to shape movement kinematics is convincing.

Substantive concerns

1) Small effect in a limited domain of motor behavior. The effect considered is a relatively subtle asymmetry in the speed profile that is consistent with exploiting the gravitational torque in the upward and downward phase of movement. While the data are consistent with the postulated cost function, the argument about the evolutionary advantage does not seem to match the small size of the described effect. Perhaps the effect on higher dimensional motions would be more substantial.

2) The model does not refute the minimum jerk optimization but it combines it with the optimization of work against gravity. The smoothness model was proposed for planar horizontal movements that are unaffected by gravity. In this respect the model of Gaveau and colleagues does not seem an "alternative" to the minimum jerk model but an extension to cases in which gravity also comes into play. In a similar vein, I found not quite convincing and a bit artificial the suggested contrast between internal models and optimal control. In fact, internal models can be considered in the framework of optimal control, since they enable to predict the consequences of motor commands (forward models) and to form different optimal solutions of ill-posed inverse problems. What the authors consider as kinematics compensations (i.e. the formation of minimum jerk or minimum torque change trajectories) should not be described as alternative to optimization since they are also forms of optimization.

3)The alternative between "compensation" and "optimization" is not clear as compensation is often described (or describable) as optimization.

4)"SR dependence on movement direction (i.e., on gravity torque) is a unique feature of effort related optimization." Is there a proof for this? Could some similar dependence result from different coordinate representation (as for example min-jerk expressed in angular coordinates)?

[Editors' note: further revisions were requested prior to acceptance, as described below.]

Thank you for resubmitting your work entitled "Direction-dependent arm kinematics reveal optimal integration of gravity cues" for further consideration at eLife. Your revised article has been favorably evaluated by Eve Marder (Senior editor), a Reviewing editor, and three reviewers. We are sorry for the delay in returning this to you, but your manuscript has engendered a lot of discussion among the reviewers (meaning of course that they care about the work!).

The manuscript has been improved but there are some remaining issues that need to be addressed. I am giving you the reviews in full, so you can see the issues that are being raised by the reviewers:

Reviewer #1:

The authors have significantly improved the manuscript, and I am satisfied with the revision.

Reviewer #2:

The authors have adequately addressed several concerns. The manuscript now more completely specifies the present smooth effort model, which makes the whole argument easier to understand. The additional modeling results, including demonstration of their torque change model generating varying symmetry ratios under varying duration and load, is heartening.

At this point there are two competing considerations for the manuscript as a whole.

1) The connection between theory and experiment in Figures 1E and 2C do suggest that "effort" likely plays a role in movement asymmetry. The gradual reduction of this asymmetry in parabolic flight does suggest that different environmental conditions lead to alteration of neural control. On the whole, then, this does forward the field and provide interesting insight.

2) Several concerns remain. Even with the additional exploration of their own implementation of the torque change model, the absolute flatness of model results in Figure 1F seems unrealistic. The authors could make a much more compelling case: they could replot the y-axis data (experimental symmetric ratio) versus calculated torque change. If this experimental result is just as flat at 1F, that would be convincing. If not, then the authors need to reconsider the uniqueness of their model at explaining the result.

The reconsideration of the minimum variance model is far more concerning. Equation 10 removes any angular dependence from gravity. This step seems to prevent any meaningful outcome from their exploration of how angle influences control.

One comment from another reviewer has resonated with me. There is an alternate interpretation to early movements in parabolic flight, in which asymmetries mimic those on the ground. The reviewer rightly points out that actually experienced dynamics are very different, so the replication of asymmetry in kinematics is very odd. At the very least, this is evidence that actually-experienced gravity (or lack thereof) does not meaningfully influence control, and that the whole interpretation here is questionable. (It is a very curious result, at least).

Taking the clear with the unclear, I do think that the overall results forward the field, although I do have ongoing concerns with the models and math, as presently implemented.

Reviewer #3:

The authors have addressed several of the concerns in the revised manuscript, resulting in an improvement of the final product. The main element of novelty and interest of this work remains the demonstration – through the 0 g experiment – that the asymmetry in the kinematics of vertical reaching reflect a process of adaptation to the gravitational field. The effect is small, but I agree with the authors that this relatively small and specific effect is likely to reflect a general evolutionary (but also ontogenetic) process that rewards the ability of an organism to take advantage of the characteristics of its environment.

My major concerns remain with the attempt to overgeneralize this findings and suggest that it disproves what the authors call "compensation hypothesis" in a broader sense. This sounds a bit like a political argument, where one takes the sides of a dynamical optimization against geometrical optimization. The fact is that the literature has shown evidence of both in different contexts. A case in point can be found in a recent study by Farshchansadegh and colleagues (PLoS Computational Biology 2016) who found evidence for both dynamic ad kinematic optimization under the same mechanical conditions, depending upon the agreement between visual and haptic feedback. That said, I am normally convinced that the essential value of a manuscript should be based on the results, not on how the authors want to organize the discussion. Broad speculations about the findings are a matter of choice and intellectual freedom of the authors. However, one should avoid statements that are inaccurate, such as the suggestion that "the compensation hypothesis has critically lacked quantitative support." Perhaps this statement could only be justified by the fact that most researchers proposing that there are geometrical constraints in shaping the planning of multi articular reaching movements have not referred to this ide as "the compensation hypothesis".

https://doi.org/10.7554/eLife.16394.014

Author response

[…]

Reviewer #1:

[…]

This study continues the series of nice papers published by the authors on the effects of gravitoinertial constraints on sensorimotor processes. It addresses an important issue related to the integration of gravity during motor actions and provides significant advance over previous studies. However, the paper could be easily improved with some changes.

The problem is that other published papers, some of them referenced in the manuscript, provided convincing evidence in favor of the compensation hypothesis. Unfortunately, the authors do not provide explanations as to why the earlier evidence for the compensation hypothesis is less convincing than theirs. They kind of "kicked the ball into touch" by simply stating that "advent of the optimal control theory has cast strong doubts on this more traditional view of motor control". This was done perhaps on a voluntary basis because critics, even those that lead to progress, are not always welcome. However, because the goal of the study was "to explicitly distinguish between the compensation and optimization hypotheses", the discussion should be more explicit about problems regarding the compensation hypothesis. Despite the very nice results, without such a discussion, the impact of the present paper will be dampened.

We have now modified the Discussion to address this comment. We state that “Although very influential, such compensation hypothesis has lacked quantitative support. In contrast, other studies that quantified velocity profiles revealed that the temporal organization of arm kinematics shows a small, yet consistent, dependence on movement direction, speed and load (Gaveau et al., 2014, Gaveau and Papaxanthis, 2011, Papaxanthis et al., 1998b). Furthermore, findings inconsistent with the compensation hypothesis (e.g., negative phasic EMG activity during vertical arm movements; Flanders et al., 1996) have been ignored” (Discussion paragraph five). Previous studies that attempted to quantitatively validate the compensation hypothesis reported complex results that were not evaluated quantitatively with alternative hypotheses. Thus, even results inconsistent with the compensation hypothesis have been ignored. The recent application of optimal control theory to biological movement have allowed better quantification and interpretation of complex data sets, including the notion that motor adaptation corresponds to a re-optimization process giving rise to altered trajectories.

The writing style would benefit from smoothing. Currently, there are too many long sentences, many of them are interrupted by long lists of references that sometimes interfere with understanding.

We have tried to make the writing smoother throughout the paper.

Reviewer #2:

The authors provide interesting evidence of gravitational influence on human single-joint movements. The directional dependence on the ground, and acquired loss of dependence in repeated parabolic flight, is substantive and novel. This work has implication for how the brain can ordinarily (on the ground) and flexibly (via parabolic flight) be influenced by gravity.

The particular analytical, theoretical, and hypotheses offered, however, have several under-developed and contradictory elements that cause great concern. The chief concern, with regard to the overall hypothesis, regards the sureness that these results confirm a "Minimum Smooth Effort" model. The authors choose to cite papers that develop the model in full, rather than doing so here; that is a great disadvantage, as the reader cannot even consider the actual hypothesis suggested.

We thank the reviewer for pointing this out. In the revised manuscript, we have developed the “Simulations” section more carefully such that the reader can easily follow the logic that is provided with all required materials to reproduce the simulations. The length of the “Simulations” section was increased. More specific remarks on simulations are addressed in the following comments.

What is suggested throughout is that "effort" is dominated by forces (here, gravity) (e.g. "The behavioral signature of the optimal integration of gravity hypothesis…") The equations offered, however, show that jerk, a purely kinematic measure, is included in the authors' model. The sensitivity analysis, shown only in the discussion, is very incomplete, as it explores the calculated cost only on one side of the chosen coefficient. My best conclusion is that the jerk inclusion is crucial, and even the most generous reading of the work concludes with a synthesis, that both kinematics and dynamics are important in the present behavior.

The reviewer’s intuition is not true. Figure 1—figure supplement 2 shows that the asymmetry is gradually reduced as the relative weight of the jerk over the effort is increased. This demonstration is sufficient for the following reason: When resolving an optimal control problem, the cost function is defined up to a positive scalar factor. More precisely, if one minimizes some function

f(x)=bg(x)+ch(x)

the solution x will be the same as if one minimizes

f(x)/d=b/dg(x)+c/dh(x)

for any d>0. Therefore, we can normalize our cost function by setting b=d, thereby assuming a unit coefficient in front of the absolute work component in our case. Varying our unique weighting factor is sufficient to explore the two extreme cases, namely jerk alone and absolute work alone. When the weighting factor is zero (or very small in practice), we minimize the absolute work alone. When the weighting factor is infinite (or very large in practice), we minimize the jerk alone. Hence, our sensitivity analysis is sufficient to demonstrate that the capacity of the smooth-effort model in predicting the sigmoidal effect of direction on SR emerges from the effort part of the cost only. We now clearly explain this point in the subsection “Simulations”.

Several other elements of the analysis, modeling, and interpretation are of concern.

– The presented smooth effort model includes separate terms for torques generated by agonists and antagonists. The authors have no way to measure these; they measure only kinematics; undoubtedly on the ground, and absolutely in flight, there will be co-contraction. The central variables in their model, therefore, are unmeasurable.

Because we measure kinematics only, we indeed have no measure of agonist versus antagonist muscles participation to the motion of the arm; we can only infer the net torque. However, since agonist and antagonist torques are opposed and angular velocity keeps a constant sign (e.g. non-negative for upward motion), minimizing the absolute work of the muscle torque whilst separating agonist and antagonist torque or not (working on the net torque directly) is equivalent: this does not affect the prediction of the Minimum Smooth-Effort model. We have already shown this in a previous paper (Berret et al. 2008, PLoS Comp Biol). This is a logical result since, in both cases, minimizing the absolute work of forces will result in a control policy that produces as little co-contraction as possible. Separating agonist and antagonist torques would be interesting to test the effect of introducing some muscular dynamics (as we additionally did for the Minimum Variance Model, please see below), however, to allow meaningful interpretation of our results and reduce the number of unknown/tunable parameters, we used the simplest minimum smooth effort model without any muscular contraction dynamics. In doing so, the predicted directional asymmetries can surely be attributed to the rules of physics, i.e. gravity effects, and not to the specific muscular dynamics that we would implement. Accordingly, in the new version of the manuscript, we rewrote the equations of the model with the net torque only (please see Equations 4, 5 and 6). We thank the reviewer for this constructive remark as it allows a simpler interpretation of our findings.

– The absolute flat lines calculated from minimum torque change and minimum variance hypotheses (Figure 1F) is highly questionable. Similar work (indeed the origin of these theories) suggest that load should play an important role. Much more detail on the calculation of these cost functions is needed to justify this very counter-intuitive result. The minimum variance model, in particular, needs a level of neural activation (drive) that seems complete absent from the present consideration.

The minimum Variance model is based on the principle that some signal-dependent noise – multiplicative noise that scales with the muscle contraction level (no constant noise because duration is defined a-priori here) – corrupts muscle activation patterns and thus end-point accuracy. In our initial version of the manuscript, we did mention that our simulations incorporate signal dependent noise, however, our description of the Minimum Variance model was very brief and did not even define the control variable. Here-below we provide the necessary information to prove the effectiveness of our minimum variance simulations.

We took advantage of the study from Tanaka and collaborators (2004, Neural Computation, paper communicated by Daniel Wolpert) where the authors have compared the predictions of the minimum Variance and the Minimum Torque change models for mono-articular arm movements performed under various conditions of movement duration and external force fields. First, in agreement with our results, Tanaka and colleagues found that the minimum Torque Change model predicts velocity profiles that are always symmetrical (i.e. Symmetry Ratio = 0.5). Second, the minimum Variance was found to predict symmetry ratios that increased with movement duration and that decreased when increasing the system degree of stability (when movements are performed in hybrid viscous and elastic force fields).

Author response image 1 here below presents the results of our own equivalent simulations. Proving the validity of our minimum Variance formulation, it can be observed that our simulations reproduce both previously described effects (see Tanaka et al. 2004, figure 2 and 3).

Given the effect of the hybrid viscous and elastic force fields on SR, we understand the reviewer’s surprise with regard to our result of invariant SR predicted by the minimum Variance when movement direction changes in the gravity field. However, the gravitational field is very different from the hybrid viscous and elastic force field of Tanaka et al. (2004) in the sense that the system’s degree of stability does not change with movement direction. The signal dependent noise (i.e. variance), accumulating with the muscular contractions responsible for movement acceleration and deceleration, actually reaches a constant final level for all movement directions in the gravity field (for a given duration and system’s stability). As a result, velocity profiles, quantified by SR, are similar in all directions. This is a coherent result since the minimum variance penalizes large muscular torques that would be necessary to reproduce our experimental results; i.e. a short SR when moving upward (requiring strong acceleration against gravity and therefore high noise) and a long SR when moving downward (requiring strong deceleration against gravity and therefore high noise).

In the revised manuscript, we have provided further details in the Methods related to the minimum Variance simulations. The reader should now appreciate that our formulation of the minimum Variance model is similar to previous studies (Subsections “Simulations”). We have also added simulations showing that including some simple muscle dynamics (making the muscle torque a low-pass filtered version of the control signal) into the optimal control problem (as performed by Harris and Wolpert, 1999) did not change the conclusions made from data presented in the main text (see subsection “Simulations” and Figure 1—figure supplement 3). Note that Tanaka et al. (2004) also reported that including muscle dynamics into their simulations did not change the conclusions of their analysis.

Author response image 1
Additional simulations testing the validity of our minimum Variance simulations by comparison to previously published results from Tanaka et al. 2004.

(A) Normalized velocity profiles obtained for various movement durations. (B) Symmetry Ratio was shown to increase with movement duration (C) Normalized velocity profiles obtained for various degrees of system’s stability. The right-most curve corresponds to the case of no external force. The other four curves, from right to left, are for critical damping system with increased stability via external viscous and elastic forces (see Tanaka et al. 2004). (D) Symmetry Ratio was shown to decrease when the force field level increased.

https://doi.org/10.7554/eLife.16394.011

– Passengers that are strapped in the flight clearly have cues that gravity is very strong, absent, and strong again. The notion that strapping will lead to no information before movement is vexing. Similarly, the notion that hypergravity for the not-strapped-down has no learning effect is unusual.

Our rationale was that preventing subjects from performing any voluntary movements (while they were strapped on the floor and between 0G phases for the not-strapped-down subject) would allow reducing sensory-motor adaptation by preventing motor interaction with the various unwanted gravito-inertial force fields. We observed that subjects similarly adapted across parabolas independently of whether they started the experiment at the beginning of the flight or waited strapped on the floor (as stipulated in the manuscript subsection “Experiment 2”). These results therefore demonstrate the effectiveness of the parabolic flight experimental paradigm to answer our question. In the revised manuscript, we replaced “avoid” by “reduce” in the same section. We also added a supplemental figure (Figure 3—figure supplement 1) allowing the reader to compare results of the 1st and 2nd subject of each flight. Furthermore, a Kruskal-Wallis ANOVA on ranks did not reveal any difference between the two groups (subjects passing 1st or 2nd).

– The authors provide scant information on how they calculated just the biomechanics of the task. Meaningful quantitative consideration of kinematics versus dynamics requires more information here.

We updated Figure 1—figure supplement 1A to provide full details on how Gravity torque (Equation 1) could be computed from arm kinematics. We refer to it in the manuscript: “The geometrical illustration of the mechanical system along with details on how to derive Equation 1 are displayed in Figure 1—figure supplement 1“.

On the whole, the authors have shown that in a vertical plane, target direction, on the ground, plays a role in human trajectory control; that in early flights, an asymmetry in velocity profiles exist; and in later flights, that asymmetry disappears. All these point to a simple conclusion: the dynamics of gravity can be used by the brain to influence trajectory. All further conclusions made by the manuscript have under-developed evidence, and therefore provide very unclear impact on the field of motor control and neural computation.

We hope the reviewer finds that the revised paper addresses his concerns and supports our conclusion that the brain optimizes the mechanical effects of gravity to minimize movement effort. We believe this is an important finding which may have strong implications in fields as diverse as neurorehabilitation, movement perception or motor control modularity, which for the most part assume the compensation principle (Prange et al. 2009; Prange et al. 2012; Cook et al., 2013; Russo et al. 2012).

Reviewer #3:

[…]

Substantive concerns

1) Small effect in a limited domain of motor behavior. The effect considered is a relatively subtle asymmetry in the speed profile that is consistent with exploiting the gravitational torque in the upward and downward phase of movement. While the data are consistent with the postulated cost function, the argument about the evolutionary advantage does not seem to match the small size of the described effect. Perhaps the effect on higher dimensional motions would be more substantial.

Because human muscles present temporal constraints regarding their force production capacity (Winters JM and Stark L, IEEE Trans Biomed Eng, 1985; Zajac FE, Crit Rev Biomed Eng, 1989), SR cannot be expected to change too drastically. Although the maximal difference between upward and downward SR is small, all subjects showed a similar trend (Figure 2B) and the statistical significance was high.

Although beyond the scope of this paper, for an example of more substantial effects in higher dimensional motion, please see preliminary results presented in Author response image 2 (explained below).

In the present study, we focused on single degree of freedom arm movements (thus on the shape of speed profile) to specifically isolate gravity effects. Previous studies, however, uncovered similar effects of movement direction (upward/downward) both on spatial and temporal aspects of multi-degree of freedom movements at various speeds and amplitudes, for arm and for whole body (sit to stand versus stand to sit) movements (Papaxanthis et al., 2005, Yamamoto and Kushiro, 2014, Papaxanthis et al., 2003, Papaxanthis et al., 1998c). Importantly, the minimization of movement effort in the gravity field can explain these differences (Berret et al. 2008; Gaveau et al. 2011, 2014).

Because results in the current study were highly significant and because previous studies have uncovered similar effects in various domains of motor behavior, we believe it is reasonable to propose that the versatility of motor planning in the gravity field results from an evolutionary and/or developmental advantage for motor economy. This argument is developed in the Discussion section. We have now replaced “argue” by “propose” and added “and/or developmental”.

2) The model does not refute the minimum jerk optimization but it combines it with the optimization of work against gravity. The smoothness model was proposed for planar horizontal movements that are unaffected by gravity. In this respect the model of Gaveau and colleagues does not seem an "alternative" to the minimum jerk model but an extension to cases in which gravity also comes into play.

We now make this point at the end of the Discussion. It is important to note that, for horizontal movements, our model makes predictions that are close but not exactly equal to those of the minimum Jerk model. For example, because the effort part of our model is torque dependent, the prediction of the Smooth-Effort model in the horizontal plane changes with the anthropometric characteristics of each subject (itself changing the inertial torque level). In general, the prediction of the Smooth-Effort model will change with the torque requirement of the movement whilst the minimum Jerk prediction will not. Therefore, if interaction torques (for multi-degree of freedom arm movements) or additional external torques (produced by a robotic manipulandum for example) are experienced in the horizontal plane, the minimum Smooth-Effort and the minimum Jerk would predict very different solutions. This is now mentioned in the Methods that develop the models (see subsection “Simulations”).

Actually, we have collected preliminary data to test this hypothesis: subjects perform 60 multi-articular reaching arm movements (all joints free of constraint; movements performed between classical sets of two targets; i.e. starting and final) in the horizontal plane while handling a robotic manipulandum applying external assistive, resistive or no force on the subject’s arm. Author response image 2 shows the stabilized performance (average of last 30 trials, +SD) for the first three subjects who performed this experiment. It is striking that the temporal organization of the reaching movement is adapted such that symmetry ratios are smaller for resistive (same as upward in the current paper) and larger for assistive (same as downward in the current paper) compared to the no force condition (similar to horizontal in the current paper).

Therefore, although we agree that the present version of the minimum Smooth-Effort model does not refute the minimum Jerk optimization, as it combines the minimum Jerk with an Effort cost, we believe that our model does not only extend to cases in which gravity comes into play but actually represents a generalization that accounts for how task dynamics shapes movement kinematics.

Author response image 2
Symmetry ratios (average of 30 trials, +SD) obtained for three subjects performing fast arm reaching movements in the horizontal plane under three conditions of external force.

Assistive: the manipulandum pushes the arm toward the final target. No force: the manipulandum does not exert any force. Resistive: the manipulandum pushes the arm toward the starting target. Movement amplitude was 30 cm and duration 300ms.

https://doi.org/10.7554/eLife.16394.012

In a similar vein, I found not quite convincing and a bit artificial the suggested contrast between internal models and optimal control. In fact, internal models can be considered in the framework of optimal control, since they enable to predict the consequences of motor commands (forward models) and to form different optimal solutions of ill-posed inverse problems. What the authors consider as kinematics compensations (i.e. the formation of minimum jerk or minimum torque change trajectories) should not be described as alternative to optimization since they are also forms of optimization.

It was not our intention to contrast internal models with optimal control. We completely agree with the reviewer about the necessary consideration of internal models in the optimal control framework. We have now modified the Introduction and introduce the two opposed hypothesis (compensation vs effort optimization), both within the optimal control framework. In both cases we explicitly state that the internal model of gravity will benefit one strategy or the other by allowing anticipation of gravity effects.

3)The alternative between "compensation" and "optimization" is not clear as compensation is often described (or describable) as optimization.

The text has been modified to avoid confusion. We use the optimal control framework to disentangle between two alternative hypotheses on how the brain uses the gravity internal model when planning arm movements – i.e. we use various minimization criteria to simulate control policies that compensate or optimize the mechanical effects of gravity. A purely kinematic optimization (such as the minimum Jerk) must take gravity effects into account in order to compensate them and therefore produce the desired trajectory (the smoothest one). On the other hand, an optimization criterion that considers movement dynamics will take advantage of gravity effects in order to produce trajectories that satisfy the desired minimization: endpoint variance, muscular effort or else. Therefore, in this study, we contrast kinematic and dynamic costs to disambiguate the compensation versus optimization hypothesis of gravity effects. Then we compare the predictions of various dynamic costs to emphasize the importance of effort in motor planning (Effort versus Variance or Torque Change).

We realize that contrasting an optimal control model (the minimum jerk) with the “optimization” hypothesis of gravity effects was confusing. For the sake of clarity in the new version of the manuscript we replaced the name “optimization hypothesis” by the “effort optimization hypothesis”. Additionally, in the Introduction and Results section we explicitly state that we use optimal control models to illustrate both gravity effect compensation and gravity effect optimization (i.e. the effort optimization hypothesis).

4)"SR dependence on movement direction (i.e., on gravity torque) is a unique feature of effort related optimization." Is there a proof for this? Could some similar dependence result from different coordinate representation (as for example min-jerk expressed in angular coordinates)?

Amongst the various models simulated to test whether they can reproduce kinematic asymmetries in the vertical plane, only models that minimized an effort-related cost succeeded (Berret et al. 2008a,b; Crevecoeur et al. 2009; Gaveau et al. 2014). Furthermore, in previous studies (Berret et al. 2008a,b) an inverse optimal control approach mathematically demonstrated that the presence of an effort-related criterion in the cost function is a necessary and sufficient condition for the production of a brief transient muscle inactivation near the peak of velocity which induces the observed kinematic asymmetries with respect to movement direction. We clearly stipulate this in the revised manuscript. In addition, direct optimal control simulations confirmed that other costs such as the integrals of squared torque change, torque, and jerk (Cartesian or angular) cannot consistently reproduce the robust empirical up/down asymmetries. Regarding the use of different coordinate representations, Cartesian and angular are identical for a single degree of freedom movement. This last point was added to the manuscript subsection “Simulations”.

[Editors' note: further revisions were requested prior to acceptance, as described below.]

[…]

Reviewer #2:

The authors have adequately addressed several concerns. The manuscript now more completely specifies the present smooth effort model, which makes the whole argument easier to understand. The additional modeling results, including demonstration of their torque change model generating varying symmetry ratios under varying duration and load, is heartening.

At this point there are two competing considerations for the manuscript as a whole.

1) The connection between theory and experiment in Figures 1E and 2C do suggest that "effort" likely plays a role in movement asymmetry. The gradual reduction of this asymmetry in parabolic flight does suggest that different environmental conditions lead to alteration of neural control. On the whole, then, this does forward the field and provide interesting insight.

We thank the reviewer for her/his positive comments.

2) Several concerns remain. Even with the additional exploration of their own implementation of the torque change model, the absolute flatness of model results in Figure 1F seems unrealistic. The authors could make a much more compelling case: they could replot the y-axis data (experimental symmetric ratio) versus calculated torque change. If this experimental result is just as flat at 1F, that would be convincing. If not, then the authors need to reconsider the uniqueness of their model at explaining the result.

The reviewer is still not convinced by our simulations of the minimum Torque Change model. We would like to underline that other authors (Tanaka et al. 2004) also demonstrated that optimizing the torque change predicts constant velocity profiles (i.e., symmetry ratios) for various movement conditions (changing orientation, speed or else…). To clarify this result, we provide, at the end of the response letter, a formal demonstration that the Torque-change model does not predict varying symmetry ratios.

During our investigations, we did question ourselves about why other models do not account for changes in symmetry ratio as a function of movement direction. What the reviewer is asking for, plotting the experimental symmetric ratio versus the calculated torque change derived from the experimental velocity profiles (if we understood correctly), would not help. Correlating variables that do not result from a model prediction is inappropriate to test whether a theoretical model (here the Torque Change model) can explain our experimental data or not; i.e., to reconsider the uniqueness of our model at explaining the experimental results. This would only show that the torque change (computed from experimental data) correlates with arm kinematics. This would not mean that the torque change variations are optimal and thus correspond to a planning principle, which is the aim of our study. Our formal demonstration below, along with the simulations presented in the manuscript and previous results published by Tanaka et al. (2004) prove that the torque-change predictions are “flat”.

The reconsideration of the minimum variance model is far more concerning. Equation 10 removes any angular dependence from gravity. This step seems to prevent any meaningful outcome from their exploration of how angle influences control.

Equation 10 does not totally remove angular dependence from gravity. Indeed, there are two different angular dependences regarding gravity: one from the movement plane orientation (𝛾 in our manuscript) and one from the time-varying movement amplitude (𝜃 in our manuscript). The work of gravity torque (as presented in Figure 1B) encompasses both of these angular dependences.

Equation 10 removes angular dependence from movement amplitude only; i.e., the modulation of the projected gravity torque (PGT) that is observed between 𝜃 = 0° and 𝜃 = 45° from each plot in Figure 1—figure supplement 1 (panel B). However, the cosine function varies from 1.0 to 0.7 on that interval and it is not this relatively small variation that is expected to account for the values of the symmetry ratio. The influence of movement plane orientation is actually much more important. This is illustrated by Figure 1—figure supplement 1 (panel B), where it can be observed that the angular dependence of PGT on movement amplitude mostly concerns movements performed in vertical movement planes (𝛾 > 45° 𝑎𝑛𝑑 𝛾 < −45°). For the other movement planes (𝛾 > 45° 𝑎𝑛𝑑 𝛾 < −45°), there is almost no modulation of PGT through movement amplitude. The mean (over the 17 targets) modulation of PGT, which we neglect with Equation 10, is equal to 11.88% (1.42N.m on average) of the PGT experienced on the starting target. For comparison, reversing movement orientation from upwards (𝛾 = 90°) to downwards (𝛾 = −90°) produces a 200% PGT modulation. Importantly, Equation 10 does not neglect this plane orientation angular dependence from gravity. Author response image 3A illustrates how small effects the approximation given in Equation 10 has on the Work of Gravity Torque, the experimentally manipulated parameter in our single-joint movement paradigm. It is important to note that, in our experiment, the main effect of gravity is due to movement plane orientation (given by angle 𝛾), not to the time-varying movement amplitude 𝜃.

To further illustrate this effect on optimal control simulation, we also present the results of two sets of simulations performed with the Smooth-Effort model in the case where we neglect the angular effect of movement amplitude (as in Equation 10 for the Variance model) and in the case where we do not neglect the angular effect of movement amplitude (Author response image 3B). It can be appreciated that this approximation does not qualitatively change the results; i.e., the symmetry ratio still linearly correlates with the work of gravity torque. In conclusion, the approximation of Equation 10 does not remove the main angular dependence from gravity, i.e., according to the plane of motion, and the differences between the results obtained from the Smooth-Effort and the Variance models cannot be attributed to the approximation made in Equation 10.

Author response image 3

(A) Effect of Equation 10 approximation on the work of gravity torque. Empty black circles depict the work of gravity torque (as a function of movement direction; i.e., plane angle 𝛾) computed with the same approximation as Equation 10; i.e., when assuming that gravity torque is constant throughout movement amplitude. Colored circles depict the work of gravity torque without any approximation (same data as in Figure 1B). (B) Effect of Equation 10 approximation on Smooth-Effort simulated symmetry ratios as a function of WGT. Black dots depict simulated data with the same approximation as Equation 10; i.e., when assuming that gravity torque is constant throughout movement amplitude. Red dots depict simulated data without approximation (same data as in Figure 1E). Each data point represents one subject moving in one direction (n=255 in each plot).

https://doi.org/10.7554/eLife.16394.013

One comment from another reviewer has resonated with me. There is an alternate interpretation to early movements in parabolic flight, in which asymmetries mimic those on the ground. The reviewer rightly points out that actually experienced dynamics are very different, so the replication of asymmetry in kinematics is very odd. At the very least, this is evidence that actually-experienced gravity (or lack thereof) does not meaningfully influence control, and that the whole interpretation here is questionable. (It is a very curious result, at least).

We agree with the reviewer that the in-flight early persistence of directional asymmetries is a non-trivial result. However, to understand how the brain integrates gravity torque, one must consider the whole adaptation pattern, i.e., the initial values in 0g along with the following ones, not only the initial or final ones. Existing literature on motor adaption to microgravity or, more generally, to a new dynamical environment, allows better interpreting this result.

In the light of previous microgravity studies (Crevecoeur F. McIntyre J., ThonnardJ.-L and Lefèvre P., 2010 and 2014), such behavior (the persistence of 1g behavior in early 0g followed by progressive adaptation toward new values) could be interpreted as a stabilization of movement performance under internal model uncertainty. For example, in their 2010 paper, the above-mentioned authors observed that the grip force required to holding an object in the hand (pinched between thumb and index fingers) was kept similar to 1g values at the beginning of 0g exposition. Then, grip force was observed to slowly decrease toward new optimal values across parabolas. Theoretical simulations revealed that this adaptation dynamics reflects the stabilization of movement performance when the internal model of gravity is changing (uncertain). More generally, this stabilization concords with the first of a dual adaptation process previously proposed by Izawa J., Rane T., Donchin O. and Shadmehr R., 2008: “on the one hand, adaptation produces a more accurate estimate of the sensory consequences of the motor commands (i.e., learn an accurate forward model), and on the other hand, our brain searches for a better movement plan so to minimize an implicit motor cost and maximize rewards (i.e., find an optimum controller)”.

Reviewer #3:

[…]

My major concerns remain with the attempt to overgeneralize this findings and suggest that it disproves what the authors call "compensation hypothesis" in a broader sense. This sounds a bit like a political argument, where one takes the sides of a dynamical optimization against geometrical optimization. The fact is that the literature has shown evidence of both in different contexts. A case in point can be found in a recent study by Farshchansadegh and colleagues (PLoS Computational Biology 2016) who found evidence for both dynamic ad kinematic optimization under the same mechanical conditions, depending upon the agreement between visual and haptic feedback. That said, I am normally convinced that the essential value of a manuscript should be based on the results, not on how the authors want to organize the discussion. Broad speculations about the findings are a matter of choice and intellectual freedom of the authors. However, one should avoid statements that are inaccurate, such as the suggestion that "the compensation hypothesis has critically lacked quantitative support." Perhaps this statement could only be justified by the fact that most researchers proposing that there are geometrical constraints in shaping the planning of multi articular reaching movements have not referred to this ide as "the compensation hypothesis".

We understand the reviewer’s remark and we thank her/him for the respect of our intellectual freedom. To avoid inaccurate statements, we modified the sentence. We now say: “such a compensation hypothesis has been challenged by results of studies that quantified velocity profiles and revealed that the temporal organization of arm kinematics shows a small, yet consistent, dependence on movement direction, speed and load”.

We also thank the reviewer for pointing out the very interesting and newly published study from Farshchiansadegh and colleagues (2016). We believe this study does not support the existence of a kinematic optimization per se but shows that different sensory perception of the environment (visual vs proprioceptive) can lead to energy optimization that is not always dissociable from kinematic optimization. We referenced this study as well as another newly published one (Shadmehr et al. 2016) on the importance of effort in tailoring motor behaviors.

https://doi.org/10.7554/eLife.16394.015

Article and author information

Author details

  1. Jeremie Gaveau

    Université Bourgogne Franche-Comté, INSERM CAPS UMR 1093, Dijon, France
    Contribution
    JG, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article
    For correspondence
    jeremie.gaveau@u-bourgogne.fr
    Competing interests
    The authors declare that no competing interests exist.
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-8827-1486
  2. Bastien Berret

    1. CIAMS, Université Paris-Sud, Université Paris Saclay, Orsay, France
    2. CIAMS, Université d'Orléans, Orléans, France
    Contribution
    BB, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article
    Competing interests
    The authors declare that no competing interests exist.
  3. Dora E Angelaki

    Department of Neuroscience, Baylor College of Medicine, Houston, United States
    Contribution
    DEA, Prepared the manuscript, Drafting or revising the article
    Contributed equally with
    Charalambos Papaxanthis
    Competing interests
    The authors declare that no competing interests exist.
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-9650-8962
  4. Charalambos Papaxanthis

    Université Bourgogne Franche-Comté, INSERM CAPS UMR 1093, Dijon, France
    Contribution
    CP, Conception and design, Drafting or revising the article
    Contributed equally with
    Dora E Angelaki
    Competing interests
    The authors declare that no competing interests exist.

Funding

Institut National de la Santé et de la Recherche Médicale

  • Jeremie Gaveau
  • Charalambos Papaxanthis

Agence Nationale de la Recherche (projet MOTION, 14-CE30-007-01)

  • Charalambos Papaxanthis

National Institute of Neurological Disorders and Stroke (R21-NS-075944-02)

  • Jeremie Gaveau
  • Dora E Angelaki
  • Charalambos Papaxanthis

Centre National d’Etudes Spatiales

  • Jeremie Gaveau
  • Bastien Berret
  • Charalambos Papaxanthis

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

This work was supported by the Institut National de la Santé et de la Recherche Médicale (INSERM), by the Agence National de Recherche (ANR, project MOTION ANR-14-CE30-0007-01), by the Centre National d'Etudes Spatiales (CNES) and by National Institute of Neurological Disorders and Stroke Grant R21-NS-075944-02. J Gaveau was supported by grants from the Ministère de l’Éducation Nationale, de l’Enseignement Supérieur et de la Recherche. We would like to thank Yves Ballay, Fabien Nicol and Cyril Sirandré for their help with data acquisition and technical support.

Ethics

Human subjects: Informed consent, and consent to publish, was obtained from all participants. The regional ethics committee of Burgundy (C.E.R) and the ethics committee of INSERM (Institut National de la Santé et de la Recherche Médicale) approved experimental protocols. All procedures were carried out in agreement with local requirements and international norms (Declaration of Helsinki, 1964).

Reviewing Editor

  1. Eve Marder, Brandeis University, United States

Publication history

  1. Received: March 26, 2016
  2. Accepted: November 1, 2016
  3. Accepted Manuscript published: November 2, 2016 (version 1)
  4. Version of Record published: November 21, 2016 (version 2)

Copyright

© 2016, Gaveau et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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