During object manipulation, such as swinging a jump rope, we experience external forces. To manipulate the object, we need to estimate the mapping between the motion and the resulting forces, and then send appropriate commands to the muscles that compensate for the arm and object dynamics (Leib et al., 2024). While this seems trivial, it is still unknown how we represent the dynamic mapping of such tasks, as there are at least three different reference frames that can solve this problem (Figure 1A). First, we could use a Cartesian coordinate frame in which the forces are represented as vectors in a coordinate system located externally to the body. Second, we could represent the forces using the shoulder, elbow, and wrist joint torques (Shadmehr and Mussa-Ivaldi, 1994). Third, we could represent the forces as belonging to the hand or a hand-held object using the orientation vector connecting the shoulder and the object or hand in space (Berniker et al., 2014).

Figure 1

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One way to distinguish between these representations is by examining how dynamics are generalized in space. In dynamic generalization, participants learn to move under force perturbations Shadmehr and Mussa-Ivaldi, 1994 followed by testing how the learned dynamics are extrapolated to new locations in space (Figure 1B). Since the extrapolated force pattern can differ when using different reference frames, mainly due to the dependency on arm posture in space or lack of it, we can understand which reference frame was used by the brain. Although many studies have used dynamic generalization to address the topic of representation, the results do not provide a consistent view of the coordinate frame used by the motor system. Some studies support a Cartesian-based representation (Criscimagna-Hemminger et al., 2003), while others support a joint-based (Shadmehr and Mussa-Ivaldi, 1994) or object-based representation (Ahmed et al., 2008; Yeo et al., 2015; Franklin et al., 2016). In addition, some studies suggested that the system uses a mixture of the aforementioned coordinate frames (Parmar et al., 2011; Berniker et al., 2014). The lack of consistency between these results raises the question of whether the motor system alternate between different representations and if so, what triggers preferring one representation over the other.

We hypothesize that this inconsistency in results can be explained using a framework in which the motor system assigns different weights to each solution and calculates a weighted sum of these solutions. Usually, to support such a framework, previous studies found the weights by fitting the weighed sum solution to behavioral data (Berniker et al., 2014). While we treat the problem similarly, we propose the Reliable Dynamics Representation (Re-Dyn) mechanism that determines the weights instead of fitting them. According to our framework, the weights are calculated by considering the reliability of each representation during dynamic generalization. That is, the motor system prefers certain representations if the execution of forces based on this representation is more robust to distortion arising from neural noise. In this process, the motor system estimates the difference between the desired generalized forces and generated generalized forces while taking into consideration noise added to the state variables that equivalently define the forces. For example, for the Cartesian-based representation, the generalized forces depend on the end-point initial and target points for the movement. When the estimations for these points are noisy, the calculated generalized force is different from the desired forces (Figure 1C, upper panel). The difference between these noisy generalized forces and the desired forces sets how much the motor system can rely on this coordinate system. In a similar way, such assessment is made for the joint-based representation in which noise is added to the joints angle values and for the object-based representation in which noise is added to the hand orientation (Figure 1C, middle and bottom panels). When repeating this process of randomizing the added noise values, while using similar noise distributions across the different coordinate systems, we can observe different distributions for this error estimation (Figure 1D). We propose that based on these distributions, specifically the variances of each distribution, the motor system can set the relative contribution of each coordinate system. One way of doing so is by assigning different weights to each coordinate system, for example, using an inverse variance estimation, which was shown to be an optimal way of combining different estimations to minimize the variance of the combined estimation (Shadmehr and Mussa-Ivaldi, 2012). The overall generalized forces are formed by multiplying the generalized forces of each representation with the respective weight.

We conducted four dynamic generalization experiments and showed that the generalized force patterns can be predicted by combining force patterns generated using the Cartesian, joint, and object coordinate systems using weights calculated from the level of noise in each coordinate system. In each experiment, participants learned to perform reaching movements while experiencing dynamic perturbations generated by a haptic robot (Figure 2A). While performing reaching movements between two points (Figure 2B), participants experienced velocity-dependent forces that were perpendicular to the movement direction and scaled according to it. That is, the forces could have different amplitudes and could have clockwise or counter-clockwise directions as a function of the motion direction (Figure 2C). A convenient way to representing this force field is to assign positive and negative signs to the clockwise and counter-clockwise directions, respectively. By doing so, the force pattern as a function of movement direction can be illustrated as a cosine wave (Figure 2D). After participants adapted to this force field at a training workspace, we tested the generalized forces pattern at two other spatial locations in the reachable space using a force channel technique (Scheidt et al., 2001). At these test workspaces, as well as in some trials at the training workspace, participants moved inside a channel, generated by virtual walls, which constrained their movement to a straight line (Figure 2E). While moving inside the channel, participants generated forces on the virtual walls that expressed their predictions of the force pattern that they could encounter while moving in this new workspace. Using these forces, we calculated the level of force compensation participants exhibited as a ratio between the produced and unscaled ideal forces (see Methods for more details).

Figure 2

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In experiment 1, we used three arm configurations that set the training workspace, in which the participants experienced the scaled force field, and two test workspaces, in which we tested the force generalization pattern (Figure 3A). Participants adapted to the force field as indicated by both the reduction in maximum perpendicular error, measuring how much the force field perturbed the participants from moving in a straight line towards the target (Figure 3B), and the increase in predictive force compensation profile in the training workspace (Figure 3C, gray curve) that resembles the cosine tuning of the scaled force field (Figure 2D). The generalized force compensation in test workspaces 1 and 2 also exhibited a cosine pattern; however, it was shifted with respect to the curve of the training workspace or the curve representing the force field. For the curve calculated based on movements in the training workspace (Figure 3C, gray curve) we did not find a shift with respect to the curve of the force field (–0.01±0.03°, mean±STD). For the predictive force compensation curve in test workspace 1 (Figure 3C, blue curve), we found a leftward shift of –23.0±4.8° with respect to the force field curve. For the predictive force compensation curve in test workspace 2 (Figure 3C, red curve) we again found a leftward shift of the curve, however, this shift (–6.3±4.8°) was smaller compared with the shift we found for test workspace 1. In addition to the phase shift of the generalized force compensation pattern, we observed a reduction in the amplitude of the generalized forces. Based on previous studies such a decrease in generalized force magnitude might be due to the distance between the training and test workspaces (Gandolfo et al., 1996; Berniker et al., 2014).

Figure 3

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These force compensation profiles can be predicted by a mixture model of the three coordinate systems while using the Re-Dyn-based weights. For each coordinate system, i.e., Cartesian, joint, and object, we calculated the predicted force compensation profile based on the arm configuration in the training and test workspaces. For test workspace 1, both the object-based and joint-based representations predict a leftward shift of the force compensation curve while the Cartesian-based representation predicts the same curve as the original curve describing the force field (Figure 3D). Importantly, none of the individual coordinate systems can accurately capture the experimental force compensation trend of the participants. However, when combining the three coordinate systems according to Re-Dyn-based weights (Figure 3E, upper panel), we could accurately predict this force compensation trend (Figure 3E, bottom panel). To scale the magnitude of the predicted force compensation pattern, we multiplied the weights by the decay factor that was equal to the peak-to-peak amplitude of the experimental force compensation pattern. By doing so, we kept the relative contribution of each coordinate system as predicted by the simulations. Despite the fact that the Re-Dyn prediction uses no information from the experimental results with the exception of the amplitude scaling, the prediction almost perfectly fits the complex pattern of forces produced by the experimental participants. Similar to test workspace 1, the Re-Dyn model could predict the combination between force profiles of the three coordinate systems that will produce the force compensation trend for test workspace 2. The individual predictions of each coordinate system for generalization in this workspace are depicted in Figure 3F. Similar to workspace 1, these individual predictions could not capture the experimental force compensation trend in workspace 2, while the combination of these reference frames according to the Re-Dyn weights predicts this trend almost perfectly (Figure 3G).

In experiment 2, we tested if the Re-Dyn model can predict generalized dynamics for a different set of arm configurations (Figure 4A). We found a similar adaption trend to the scaled curl force field as seen in the reduction in maximum perpendicular error (Figure 4B). The force compensation curve in the training workspace resembles the cosine tuning of the scaled force field (Figure 4C, gray curve), and we found a similar pattern for the force compensation in test workspaces 1 and 2 but for this experiment, they were shifted to the right with respect to the curve of the training workspace or the curve representing the force field. Similar to experiment 1, the predictive force compensation curve in the training workspace was not shifted with respect to the curve of the force field (–1±2.45°, mean ±STD). For the predictive force compensation curve in test workspace 1 (Figure 4C, blue curve), we found a rightward shift of 34.5±1.2° with respect to the actual force field. For the predictive force compensation curve in test workspace 2 (Figure 4C, red curve) we again found a significant rightward shift, however, this shift of 15.4±4.4° was smaller than that of test workspace 1.

Figure 4

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Similar to experiment 1, we found that none of the individual coordinate reference frames were able to predict the shifts in the force compensation in the test workspaces (Figure 4D and F). However, again we found that the experimental force compensation on both test workspaces matched the Re-Dyn prediction. That is, these trends match the mixed model prediction based on weights calculated using the Re-Dyn model. Although the joint- and object-based representations predicted a rightward shift of the curve for test workspace 1 (Figure 4D) and a smaller rightward shift of the curve for test workspace 2 (Figure 4F), they did not fully account for the experimental force compensation curves. When combining these individual predictions according to the Re-Dyn weights, we found an agreement between the predicted curve and the experimental curve for both test workspace 1 (Figure 4E) and test workspace 2 (Figure 4G). Importantly, we found a different pattern of weights in experiment 2 compared with experiment 1 which represents the changes in noise sensitivity due to the different arm configurations.

In experiment 3, we aimed to test whether the weighted combination of reference frames can predict generalized dynamics when arm configuration remains the same (Figure 5A). After adaptation to the scaled force field in the training workspace (Figure 5B), participants had to move the tip of a pole between the start and target positions in the test workspaces (Figure 5A). That is, the arm posture did not change between the adaptation and generalization parts of the experiment. We found a rightward shift of the curve for both test workspaces with respect to the training workspace. For the predictive force compensation curve in the training workspace (Figure 5C, gray curve) we did not find a shift with respect to the force field (–0.05±0.14°, mean±STD). However, for the force compensation curve in test workspace 1, we found a rightward shift of 17±5.7° (Figure 5C, blue curve) despite no changes in the arm posture. Aside from the phase shift, we also observed that the force compensation profile was not centered around the zero value as in the other experiments. In test workspace 2 we again found a rightward shift of the predictive force compensation curve, however, this shift (4.5±3.6°) was smaller compared with the shift we found for test workspace 1 (Figure 5C, red curve).

Figure 5

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In this experiment, since the arm configuration did not change between the training and test workspaces, both the Cartesian and joint-based representations predict that the force compensation curve for the test workspaces will not shift compared with the force compensation curve for the training workspace. Contrary to these predictions, due to the changed orientation of the tool, the object-based representation predicts a rightward shift for both workspace 1 and 2, with a stronger shift for workspace 1 (Figure 5D and F, dotted curves). For both test workspaces, we found that the predicted weights alternate between the strong contribution of the object-based coordinate system, the strong contribution of the Cartesian and joint-based coordinates systems, and a roughly similar contribution for all coordinate systems which depended on movement direction (Figure 5E and G upper panels). For test workspace 2, we observed a general agreement between the predicted and experimental force compensation curves (Figure 5G). However, for test workspace 1, since all individual models are centered around the zero compensation value, we could not capture the general amplitude shift of the force compensation curve (Figure 5E).

In experiment 4, we aimed to test the predicted combination between reference frames when the hand orientation remains the same between training and test workspaces while the arm configuration changes (Figure 6A). Similar to experiments 1 and 2, participants had to move at the test workspaces after adapting to the scaled force field in the training workspace. We found a similar adaptation to the scaled force field as evidenced by the reduction in MPE (Figure 6B). Again, the training workspace predictive force compensation curve (Figure 6C, gray curve) did not significantly shift with respect to the original force field curve (–0.03±0.07°, mean±STD). For test workspace 1, we found a small leftward shift of –3.7±3.1° (Figure 6C, blue curve) and for test workspace 2 we found a rightward shift of 6.8±1.8° (Figure 6C, red curve).

Figure 6

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In this experiment, we set the location of the training and test workspaces in such a way that the hand orientation did not change across workspaces. In this case, both the Cartesian and object-based representations predict that the curves extracted from movements in the test workspaces will resemble the ones extracted from the training workspace. Contrary to these predictions, the joint-based representation predicts a leftward shift for the force compensation curve extracted from test workspaces 1 and a rightward shift for the curve extracted from test workspaces 2 (Figure 6D and F). We found that in general, the Cartesian and object-based representations are the more dominant reference frames as evidenced by the predicted weights (Figure 6E and F). For most directions, the weight for these reference frames is higher than the weight for the joint reference frame, although for some directions this gap in weight value is small and could even reverse. This contribution of the joint-base reference frame explains the small phase shift of the mixture model that matches the experimental results.

In addition to the noise sensitivity model we considered two alternative mechanisms to explain the weighting between reference frames. The first suggests that the weights are set by minimizing the energy needed to generate the generalized forces. For every movement direction, the motor system will give more weight to the reference frame in which the applied forces, which are proportional to the energy, are the lowest. Similar to the Re-Dyn model, we used normalized inverse energy measurement to calculate the weight of each reference frame for the four experiments (Figure 7A). The second alternative mechanism suggests that the motor system is trying to maximize the smoothness of the force profile. Similar to other studies that suggested smoothness as a fundamental, target aim for the motor system, we tested whether maximizing the smoothness of the generalized force profile may set the weighting between reference frames. For this purpose, we calculated the smoothness, measured by the inverse sum of the magnitude of the force signal third derivative (see Methods), and set a higher weight for more smooth force profiles (Figure 7A).

Figure 7

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An example for the predicted force compensation profile of these two alternatives for the mean force compensation of participants in experiment 1 is depicted in Figure 7B. Importantly, none of these three models were directly fitted to the experimental data. To test the prediction quality of the three models, that is the Re-Dyn, energy, and smoothness-based models, we also found an optimal force compensation profile by fitting the weights of the three reference frames directly to the data. The weights were independent of the movement direction and fitted to all three reference frames since this combination was superior to all other combinations (see Appendix 1 for comparison with other combinations). That is, we found a set of three weights that combine the reference frames so it minimizes the error between the fitted and experimental force combination profile. This fitted curve quantifies how well each of these models can predict the results in comparison with a fitted optimal constant mixture of the Cartesian, joint, and object reference frames. Note that the optimal model is directly fitted to the data (unlike the other model predictions), and provides an upper bound, demonstrating how close any of the predictions are to a directly optimized model (for comparison). In this example, we observed that both the force compensation magnitude and the phase shift are better predicted by the Re-Dyn model compared with the energy or smoothness-based models. Across the eight generalization test workspaces we had in the four experiments, we found that the Re-Dyn model has less prediction error, as measured by the error between the prediction and experimental force compensation curves (Figure 7C), and better predicts the phase shift of the force compensation curve (Figure 7D). As expected, the predictions of the Re-Dyn model were not as good as the optimal weights model that was fitted to the data. This is due to the inherent additional free parameters in the optimal model. To account for the additional degrees of freedom, we calculated the Bayesian Information Criterion (BIC) for each model. We observed that the Re-Dyn was superior to the other models, including the fitted optimal model, when accounting for the additional degrees of freedom (Figure 7E). In addition, we found an additional condition in which the direction-independent optimal weights model cannot predict the experimental results despite the additional parameters (see Appendix 2). In general, these results suggest that the Re-Dyn model better captures the weighting between coordinate frames compared to all alternative models.

Our results strongly suggest that the motor system alternates between different reference frames according to their reliability. To further test the ability of the Re-Dyn combination of reference frames to predict dynamic generalization, we examined whether this model can also explain past experimental results. Previous studies did not provide a consistent view regarding the way the motor system represents dynamics. That is, each of these studies provided evidence for representation using the Cartesian, joint, and object-based systems or some mixture of them. Our simulations examined whether this inconsistency is a result of the experimental design that was used, which elevated the noise level in one coordinate system resulting in the motor system using other coordinate systems.

First, we examined the study by Shadmehr and Mussa-Ivaldi, 1994. Similar to the experiments reported here, the authors examined the generalized dynamics of participants in a test workspace after adaptation to a skew viscous force field in the training workspace (Figure 8A). To understand how the brain represents the learned forces, the authors examined the movement profiles in the test workspace in which participants did not experience any forces, i.e., moving in free space. In this case, the movement path will divert from the straight line due to the additional counteractive forces that participants generate to overcome anticipated external forces. Therefore, using different reference frames to represent the external dynamics will result in different movement paths during these free-space movements and the trajectories can reveal the reference frame the motor system used during movements. Based on these paths, the authors suggested that the motor system used the joint coordinate system to represent novel dynamics. Simulating this experimental protocol, we found that the force pattern of the joint and object-based coordinate system is similar (Figure 8B) and that the Re-Dyn predicted weights of these coordinate systems have a higher value compared with the weight of the Cartesian coordinate system (Figure 8C). Since the authors did not consider the object-based coordinate system, they concluded that the observed generalization is a result of the motor system using the joint-based coordinate system, although it is impossible to differentiate between the joint or object coordinate systems based on this design. We used the predicted weights, including an object-based representation, and simulated the free-space movement at the test workspace after adaptation to the force field in the training workspace (Figure 8D). We found that the simulation results, specifically the direction of deviation from the straight-line path, can capture the experimental results (Figure 8E).

Figure 8

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Other influential studies also supported the conclusion of Shadmehr and Mussa-Ivaldi, 1994. For example, Shadmehr and Moussavi, 2000 and Malfait et al., 2005 examined dynamic generalization using a similar protocol in which participants adapted to a force field in the training workspace and generalization was tested in a different workspace location. For both studies, our simulations of the generalized force pattern showed great similarity between the object and joint coordinate systems and that Re-Dyn predicted weights support that these coordinate systems have higher values compared with the Cartesian reference frame suggesting that participants will rely more on these force representations (Figure 8F and G). Similar to Shadmehr and Mussa-Ivaldi, 1994, the object reference frame was not considered by the authors of the two studies which explains why they concluded that representation is based on the joint-based coordinate system.

Next, we examined predictions of the Re-Dyn model for studies that had different conclusions to Shadmehr and Mussa-Ivaldi, 1994. In the study of Criscimagna-Hemminger et al., 2003 the authors tested generalization between hands. Participants who adapted to a force field while using their dominant hand generalized these forces to the non-dominant hand and the force profile supported representation using the Cartesian reference frame. The authors used a mirror configuration for the hands (Figure 8H), that is, the training and test workspaces were at the same spatial location. In this case, the generalized force profiles predictions for both Cartesian and object-based representations are the same. According to our simulations for the Re-Dyn predicted weights, we found that the weights for the Cartesian and object-based representations are indeed higher than the weights for the joint-based representation which explain the reason the authors concluded that participants used the Cartesian coordinate system for inter-limb dynamic generalization.

Another experimental result that can be explained using the Re-Dyn model suggested a combination of reference frames. In the study by Berniker et al., 2014, the authors suggested that generalized forces after adaptation to skew force field are a result of combining the Cartesian, joint, and object-based coordinates systems. Simulations of the experimental setup (Figure 8I) explain this result as we found that for one of the tested configurations, the generalized force patterns of the joint and object coordinate system are similar while for another tested workspace, the generalized force pattern of the joint coordinate system is the mean of the Cartesian and object force patterns. These links between the force patterns did not allow the authors to clearly indicate which reference frame is used by the motor system. This is supported by the Re-Dyn model since the combination between reference frames according to the predicted weights generates a force pattern that can be interpreted using multiple and different combinations of reference frames.

In this study, we showed that during dynamic generalization the motor system alternates between different reference frames according to their reliability. Since motor commands sent to the muscles are corrupted by neural noise, there is an increased risk of missing the movement goal. For example, during reaching movement in the presence of external force perturbations, producing inaccurate counteractive forces can result in missing the movement target. To overcome this error, we suggest that the motor system chooses to control these forces by representing and producing forces according to a coordinate system in which it can lower the effect of neural noise. To support this theory, we compared between the Re-Dyn model predicted generalized force patterns and experimental patterns outside the training workspace where participants adapted to a scaled curl force field. The scaling of the force field allowed to compare both the error in force magnitude and the general phase shift of force pattern. We found that the generalized forces resembled the prediction of a mixture model combining Cartesian, joint- and object-based representations. Importantly, we show that the weight for each coordinate system was best predicted by a model that asserts that the weights are inversely proportional to the force error produced when using a specific coordinate system. Importantly, these Re-Dyn model weights were directly determined from the experimental conditions and were not fit to the participant data. Thus, each experiment provides a direct test of the model predictions. Moreover, we further show that this way of calculating weights between coordinate systems can solve the inconsistent results of previous studies examining dynamic representation in the motor system.

Our idea suggests that the motor system uses a mix of force representation with a greater weight to the representation that is less variable in extrapolated force production. While the idea of trajectory planning while minimizing the effect of noise was presented by Harris and Wolpert, 1998, we show that the motor system alternates between controlled variables, that is, end-point, joints, or object-based forces, in order to reduce the effect of noise aside from optimizing the trajectory. By doing so, the motor system can ensure that even in the case of elevated noise, the effect will be kept to a minimum.

We focused on additive noise while the real noise in the motor system is more likely to be signal-dependent noise (Jones et al., 2002; Todorov and Jordan, 2002). For signal-dependent noise, the noise characteristics change as a function of the forces the muscles need to produce. In this case, since different reference frames predict different levels of force for different movement directions, elevated noise can increase the planned forces corruption of one reference frame compared to others, changing the weighting between coordinate systems. While this can result in better resemblances of the model prediction to the experimental data as the number of degrees of freedom for the model increases, we found that even when keeping the same noise characteristics across reference frames and movement directions we could explain the generalization trends in multiple cases.

We used simulations of neural noise added to different state variables that define the controlled forces to estimate the error between desired and corrupted force output during generalization. Using this error distribution, we calculated the weight for each reference frame based on the inverse variance estimation. While we show that this weight estimation can predict the force compensation patterns under different conditions, our study does not aim to support the inverse variance framework. Indeed, there is an ongoing debate regarding the relevance of the inverse variance in perception and motor control (Rahnev and Denison, 2018). In contrast, we aimed to show that the motor system takes into consideration the possible errors in force representation using different coordinate systems.

We expect that these representations and related weights might be stored in areas related to forward and inverse models. During motor adaptation, the sensorimotor system might change both the controller (inverse model) and the predictor (forward model) to modify the control so as to overcome changes in the environment (Flanagan et al., 2003; Franklin et al., 2008). The cerebellum has been proposed to host both types of internal models (Wolpert and Kawato, 1998; Smith and Shadmehr, 2005), however, other areas might also be involved during adaption to altered environment dynamics. For example, changes in premotor cortex (PMd) activity were also evident during force field adaptation (Perich et al., 2018). Generalization of the learned dynamics to different locations in space might require both the inverse and forward models to use dynamics transformation such as the ones considered in this study. Thus, we speculate that such transformation might be evident in the neural activity of these areas during adaptation to novel dynamics.

The results we present here support the idea that the motor system can use multiple representations during adaptation to novel dynamics. Specifically, we suggested that we combine three types of coordinate systems, where each is independent of the other. Other combinations that include a single or two coordinate system can explain some of the results but not all of them, suggesting that force representation relies on all three with specific weights that change between generalization scenarios. However, we can also consider that force is represented in other coordinate systems. For example, when we reach to a target, we need to represent the location of the target in eyes/head coordinate system and transform this representation through body/world-based coordinate system to hand coordinate system (Flanders et al., 1992; Andersen et al., 1993; Buneo et al., 2002). While we considered body/world (Cartesian) and hand (joint) coordinate systems, the gaze-based coordinate system could also be a potential coordinate system in which we can represent forces. However, since in our experiments, participants were not required to gaze at a specific point in the environment and usually were gazing at the cursor, the sagittal axis orthogonal to the center of this coordinate system was always aimed at the workspace in which the participants were moving. This will result in no change in the generalized force profile, similar to the Cartesian-based coordinate system prediction. Alternatively, making participants gaze at a specific point will result in a generalized force profile that is similar to the object-based coordinate system. That is, since the difference in location between workspaces on the retina can be described using a rotation operation, the predictions in such case will be similar to the object coordinate system which is based on the same operation. Thus, while in both cases of fixed or free gaze, we cannot distinguish between the gaze and other coordinate systems, the identical predictions of these coordinate systems were still taken into consideration in the Re-Dyn model.

A different approach to try and examine the possible representations that are present in the motor system can be to try and limit the use of some of them during adaptation to novel dynamics. Since the force field is ambiguous (i.e. the force field appeared in a small workspace with no constraints on which coordinate frame to use for representation of the dynamics) it can be represented in any of the suggested coordinate frames including a mixture of them. In this case, the motor system extrapolates the pattern of forces to other locations in space and based on our suggestion it does that using the most reliable reference frame. However, this can change in the case of unambiguous force fields that extend over a larger area of the workspace according to one coordinate frame (Franklin et al., 2016). Adaptation to unambiguous force fields might result in using a single coordinate frame for dynamics representation since the prediction of forces is based on interpolation rather than extrapolation.

Force generalization has been an important way to test the coordinate system the motor system uses to represent information (Franklin and Wolpert, 2011). The initial idea that the motor system uses intrinsic, joint-based representation (Shadmehr and Mussa-Ivaldi, 1994) is more questionable nowadays as more evidence accumulated showing that the motor system can use other coordinates systems (Saha et al., 2015) as well as using a mixture of models (Brayanov et al., 2012; Berniker et al., 2014). We propose that these discrepancies suggest different weighting between reference frames based on their reliability rather than a direction-independent reference frame that is used for all cases of dynamics representation. As we show, the Re-Dyn framework can explain generalization between different locations and types of force fields as well as other types of generalization tests, such as inter-limb transfer (Criscimagna-Hemminger et al., 2003), and assist in characterizing possible failure of information representation, such as brain-related disorders (Haswell et al., 2009).

The data generated and/or analysed during the current study are available in Open Science Framework.

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Author details

1. Raz Leib

   Neuromuscular Diagnostics, TUM School of Medicine and Health, Department Health and Sport Sciences, Technical University of Munich, Munich, Germany

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing – original draft, Project administration, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-3940-2651

2. David Franklin

   1. Neuromuscular Diagnostics, TUM School of Medicine and Health, Department Health and Sport Sciences, Technical University of Munich, Munich, Germany
   2. Munich Institute for Robotics and Machine Intelligence (MIRMI), Technical University of Munich, München, Germany
   3. Munich Data Science Institute (MDSI), Technical University of Munich, Munich, Germany

   Contribution

   Conceptualization, Supervision, Investigation, Methodology, Writing – original draft, Project administration, Writing – review and editing

   For correspondence

   david.franklin@tum.de

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-9530-0820

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