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.

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.

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.

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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; F_16,224=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°; F_16,224=1.13, p=0.32 and F_16,224=1.02, p=0.44, respectively). As illustrated in Figure 2A, SR shows a sigmoidal modulation as a function of movement direction (F_16,224=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.

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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 (F_5,50=1.274, p=0.29) and movement direction (F_1,10=0.549, p=0.48).

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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, F_5,50=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).

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, 2002; Cotel 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.

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).

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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.

   "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.

   "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.

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