In Expt 1 (n = 16), we employed a version of the ‘go-before-you-know’ task in which a combination of different FF environments was used to investigate movement planning under uncertainty. While gripping a robotic manipulandum that could apply forces to the hand (Figure 1a), participants initiated 20 cm cued-onset reaching movements towards either a single prespecified target (1-target trials; Figure 1b, left) or two potential targets (2-target trials; Figure 1b, right). On 1-target trials, the target, located at a left (+30°), right (−30°), or center (0°) eccentricity from the midline, was displayed for 1000 ms before an auditory go cue was delivered. On 2-target trials, the same pair of potential targets always appeared in the left and right target locations for 1000 ms before the auditory go cue, but one (randomly selected) was extinguished 3 cm after movement onset, leaving only the final reach target on-screen. Comparing the data from 1- and 2-target trials thus allowed us to infer how uncertainty about the final target location influences motor planning on 2-target trials.

Figure 1

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We designed a novel physical environment composed of multiple FFs that would result in very different predictions from MA vs. PO for motor planning on 2-target trials. Specifically, we trained participants to adapt to a viscous curl FF (see Materials and methods) that perturbed the left/center/right movements during 1-target trials with a FF_LATERAL/FF_CENTER/FF_LATERAL composite environment (Figure 1c), where FF_CENTER = -FF_LATERAL, and the sign of the FFs was balanced across participants. For 2-target trials, MA would predict that participants average the force patterns (FF_LATERAL in both cases) learned for the left and right (lateral) targets, which correspond to the potential target locations on 2-target trials (Figure 1d, left). In contrast, PO would predict that participants produce the force pattern (FF_CENTER) appropriate for optimizing the planned intermediate movement since this movement maximizes the probability of successful target acquisition (Hudson et al., 2007; Haith et al., 2015a; Figure 1d, right). Importantly, and in contrast to previous studies (Gallivan et al., 2017; Nashed et al., 2017; Stewart et al., 2013; Stewart et al., 2014), we were able to reliably elicit intermediate movements during uncertainty in all our experiments by controlling the reward rate with sliding scales for movement time criteria (see Materials and methods).

Participants first performed baseline 1- and 2-target trials to gain familiarity with them, and then completed an FF training block, where we imposed the multi-FF environment on 1-target trials as outlined above. Note that, because we differentially perturbed movements to the center target compared to the lateral targets, adaptation to the FFs associated with the lateral targets, FF_LATERAL in both bases, would interfere with adaptation to the FF associated with the center target, FF_CENTER. To elicit similar levels of FF adaptation for all three target locations, we included a greater number of training trials for the center target (using a ratio of 1:2:1 for left, center, and right target directions; see Materials and methods). The adaptive compensation to the FFs administered during training was measured with pseudorandomly interspersed error clamp trials (Scheidt et al., 2000). After training, participants experienced a test epoch, which included 2-target partial error clamp trials, on which the initial force pattern could be measured (see Materials and methods). The test epoch also included intermittent 1-target FF and error clamp trials, used to maintain the adaptive change in motor output developed during the training epoch and measure its ongoing state.

Training within the multi-FF environment resulted in substantial levels of motor adaptation in all three target directions throughout both the training and test epochs as demonstrated in Figure 2a. We measured this adaptation with an adaptation coefficient, a regression-based metric for comparing each movement’s force profile relative to the imposed FF perturbation (see Materials and methods) (Smith et al., 2006; Sing et al., 2009). We found that the final adaptation coefficient levels achieved at the end of the training period (defined as the last 10% of EC trials) were within ~9% for all three target directions (0.73 ± 0.15 [95% C.I.], 0.80 ± 0.15, and 0.73 ± 0.12 for training in the left, center, and right target directions, respectively). Importantly, these adaptation levels were largely maintained during the test epoch (0.71 ± 0.13, 0.68 ± 0.13, and 0.74 ± 0.09 for training in left, center, and right target directions, respectively). Therefore, we used the population-averaged force profiles measured on 1-target error clamp trials (i.e., the adaptive response) during this test epoch to construct our predictions for MA vs. PO, as shown in Figure 2b. Note that all force data are aligned to target cue onset (T_ON), the time at which one of the potential targets was extinguished for 2-target trials, which corresponds to the 3 cm excursion point shown in Figure 1b. Also note that all force profile data are shown for the time that feedforward motor output was observed on 2-target trials since forces due to feedback corrections do not reflect movement planning. We defined the time of feedback response onset (T_RESP) as the point in time where we could detect statistically significant feedback responses during 2-target trials (150 ms after target cue onset; see Materials and methods). In line with the adaptation coefficient measure we used to characterize overall learning, the more detailed analysis that we present of how force profiles evolve in time also normalizes raw force measurements by the level of the ideal compensation.

Figure 2

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To specifically determine how movements are planned in uncertain conditions, we examined the force patterns predicted by MA and PO for 2-target trials and compared them to the force patterns we measured on 2-target trials (Figure 2c). The MA and PO predictions are essentially opposite, consistent with the FF_LATERAL/FF_CENTER/FF_LATERAL environment we imposed during training. The MA prediction corresponds to the average of the force profiles (adaptive responses) associated with the left and right 1-target EC trials plotted in Figure 2b, and we note that because the left and right 1-target trial force profiles were remarkably similar, differential weighting for these motor plans would have little effect on the MA prediction. On the other hand, the PO prediction corresponds to the force profile associated with the center target plotted in Figure 2b as this is appropriate for optimizing a planned intermediate movement. Experimental data on 2-target trials show that participants systematically produce positive forces that are at odds with MA-based predictions, but in line with PO-based predictions for motor planning during uncertainty. We quantified the similarity between the 2-target trial data and the predictions using a prediction index that produces a value of +1 if the mean force from the experimental data were perfectly similar to the PO prediction, –1 if was perfectly similar to the MA prediction, and 0 if the data were halfway between both predictions (see Materials and methods). We measured this prediction index over two intervals. One extended out to the greatest duration that we could reasonably use to examine feedforward motor output (see Materials and methods), spanning from movement onset to feedback response onset (T_RESP), and the other, more conservative, spanned from movement onset to target cue onset (T_ON). Using the prediction index values measured during both time intervals, we found that the observed 2-target trial force pattern is markedly more consistent with the PO prediction compared to the MA prediction (T_ON: prediction index = +0.77 ± 0.21 [95% C.I.], p=1.41 × 10⁻⁶, t(15) = 7.25; T_RESP: prediction index = +0.54 ± 0.16, p=4.78 × 10⁻⁶, t(15) = 6.52; 1-sample t-test; Figure 2g, left).

Observation of participant hand trajectories, exemplified in Figure 2d, reveals that there is appreciable variability in movement directions on both 1-target FF trials and 2-target trials. This is important to consider because directional deviation from the intended target direction would lead to small but definite biases in the amount of adaptation for all three trained target locations (left/center/right), consequently biasing both the MA and PO predictions. Biases would occur, for example, because movements directed exactly in the 0° direction, towards the center target, would be associated with FF_CENTER after training; however, off-target movements on either side of the 0° direction would be associated with an FF in-between FF_LATERAL and FF_CENTER. Thus, a distribution of movement directions centered around the 0° direction would, on average, be associated with an FF in-between FF_LATERAL and FF_CENTER, rather than an FF comprised solely FF_CENTER. This effect would be greater for movements farther off-target, resulting in a small but definite variability-dependent bias in the FF intended to be associated with a given target. We thus performed an additional experiment (Expt 1-GEN, n = 10) where we measured the adaptation associated with a range of ‘off-target’ movements to determine the size of this variability-induced bias (Figure 2e). Specifically, we employed a task design identical to Expt 1, with multi-FF training in the left/center/right target directions, but we replaced 2-target trials with 1-target error clamp trials and positioned them in directions that enabled a dense sampling for generalization of the trained adaptation (every 7.5° in-between the trained target directions). The resulting adaptation levels, shown in Figure 2e, show that the multi-FF environment generalizes nonlinearly across movement directions, with noticeable changes in adaptation around the trained target directions (we found an ~34% change 7.5° from the 0° trained target and 38–42% changes 7.5° from the ±30° trained targets).

The pattern of generalization observed in Figure 2e is well approximated (R² = 0.98) by a model based on the additive combination of Gaussians centered around the three trained target locations, in line with previous work examining the generalization of motor adaptation (Hwang et al., 2003; Pearson et al., 2010). We therefore used this model to estimate the average adaptation level over the distribution of movement directions associated with left/right 1-target FF trials for the MA prediction and 2-target trials for the PO prediction from Expt 1 (see Materials and methods). Each participant’s expected level of adaptation was then used to scale the raw force profiles that comprise the predictions. We found that this refinement reduced the magnitude of the predicted peak forces by 20-25% for both the MA and PO models (see Figure 2c vs. f). Although the data were clearly more in line with the raw PO prediction than the raw MA prediction, there was still a noticeable mismatch between the PO prediction and the data (Figure 2c, g). However, taking generalization into account results in a refined PO prediction that is in even greater alignment with the data, corresponding to prediction indices that are closer to +1 at both T_ON and T_RESP (T_ON: prediction index = +1.0 ± 0.4 [95% C.I.], p=1.58 × 10⁻⁵, t(15) = 5.85; Figure 2g, left; T_RESP: prediction index = +0.76 ± 0.3, p=1.72 × 10⁻⁵, t(15) = 5.80; 1-sample t-test; Figure 2g, right). Together these findings indicate that movements performed under uncertainty arise from an action plan that optimizes task performance.

In Expt 1, the MA prediction that we tested against PO was based on the idea that the motor system performs a motor average at the level of motor output or force output, which could be viewed as a low-level version of MA. However, a higher-level version of MA is also possible, where the motor system may instead average the kinematics of the movements associated with the potential target locations, and then generate the motor output required for this averaged motion. In Expt 2, we test this higher-level version of MA against PO. To do so, we used an experimental approach based on the perturbation of movement kinematics rather than the perturbation of environmental force dynamics while assessing whether MA or PO can explain the intermediate movements that arise from motor planning in uncertain conditions. Specifically, we imposed kinematic perturbations by placing obstacles to block and deflect direct movements to targets or potential targets. We designed this experiment based on a subtle variation of an influential study that supported MA (Stewart et al., 2014). However, here we show that the original instantiation of this experiment, which we replicate (Expt 2a), fails to make readily dissociable predictions for MA vs. PO, but that a single modification of it (Expt 2b) leads to highly dissociable, and in fact essentially opposite, predictions for these two models of motor planning under uncertainty.

In Expts 2a (n = 8) and 2b (n = 26), participants made 20 cm cued-onset reaching arm movements (Figure 3a) towards either a single prespecified target or two potential targets, as in Expt 1. After a baseline period in which participants practiced these trial types without obstacles present (obstacle-free trials; Figure 3b), we began placing obstacles on some trials that would obstruct direct movements to a target or a potential target (Figure 3c, d). Please note that for simplicity the subsequent explanations in this section reference a left-side obstacle condition; however, the experimental design was balanced within participants to include an equal number of interspersed right-side obstacle trials. In Expt 2a, we used an obstacle with the same size and positioning as that in Stewart et al., 2014. This obstacle protruded 2 cm to the right and effectively infinitely far to the left of the direct path between the start position and the left target (i.e., the obstacle-obstructed target), and thus required rightward deflections for left 1-target trials, as shown in the upper panels of Figure 3c, d. By contrast, in Expt 2b, we used a pared-down version of this obstacle that protruded 2 cm to the right but 0 cm to the left of the direct path between the start position and left target, as shown in the lower panels of Figure 3c, d. This smaller obstacle permitted both leftward and rightward travel paths around it, but critically, promoted the less circuitous leftward deflections (see Figure 3c). Accordingly, all 8 participants in Expt 2a consistently veered rightward as required around the obstacle, and all 26 participants in Expt 2b consistently veered leftward around the obstacle, in line with the intent of the experiment design. Hand paths from example participants are shown in Figure 4a, and data indicating deflections observed for all participants are shown in Figure 5b, f.

Figure 3

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The contrasting movement deflections produced by the different obstacles on 1-target trials in Expt 2a vs. Expt 2b result in contrasting predictions for the MA hypothesis in these two experiments. On 2-target trials, where uncertainty in motor planning is present, MA predicts that participants would average the motor plans associated with obstacle-obstructed left 1-target trials and the unobstructed right 1-target trials (Equation 1 – baseline MA model prediction):

where ${\displaystyle {\hat{\mu }}_2}$ represents the predicted deflection, or safety margin, around the obstacle on 2-target trials, ${\mu }_{1A}$ and ${\mu }_{1B}$ represent the observed deflections, or safety margins, on obstacle-obstructed and unobstructed 1-target trials, respectively, and the $\frac{1}{2}$ coefficient for each term indicates that ${\mu }_{1A}$ and ${\mu }_{1B}$ are equally weighted in the motor average. Note that the baseline MA model is parameter-free in that both ${\mu }_{1A}$ and ${\mu }_{1B}$, which serve as inputs to the model, are experimentally measured on obstacle-obstructed and unobstructed 1-target trials. In Expt 2a, where movements on obstacle-obstructed left 1-target trials were deflected rightward, MA consequently predicts a rightward movement deflection on obstacle-present 2-target trials. And in Expt 2b, where movements on obstacle-obstructed left 1-target trials were deflected leftward, MA consequently predicts a leftward movement deflection on obstacle-present 2-target trials (see Figure 3d).

Importantly, the obstacles in Expts 2a and 2b were designed not only to induce opposite deflections on obstacle-obstructed 1-target trials, but also to display identical protrusions in the direction of the intermediate movements that occur during uncertainty. Thus a motor plan optimizing task performance for intermediate movements during uncertainty would not be differentially affected by these obstacles. Specifically, a model for PO would have two objectives on obstacle-present 2-target trials: (1) to reach the final target within the required timing criteria and (2) to avoid obstacle collision as these two objectives form the basis of task success (see Materials and methods). We therefore modeled the PO predictions as an average of the movement deflections that would arise if PO were to independently prioritize each objective, expressing the PO prediction as an equal balance of the two motor costs associated with the determinants of task performance. Prioritization of movement timing, or rapid target acquisition, would predict a movement direction midway between the two potential targets (i.e., a net 0° deflection) as this would maximize the probability of successful target acquisition during uncertainty (Haith et al., 2015a). However, prioritization of obstacle avoidance would predict deflections that incorporate a safety margin around the obstacle that is proportional to an internal estimate of variability (Hadjiosif and Smith, 2015). We therefore estimated the sensorimotor system’s safety margin for the obstacle avoidance priority on 2-target trials by scaling the margin observed on 1-target trials, ${\mu }_{1A,}$ based on the relative variability observed on 2-target vs. 1-target trials, $\frac{{\sigma }_2}{{\sigma }_{1A}}$:

(Equation 2a – baseline PO model prediction):

(Equation 2b – baseline PO model prediction):

where ${\displaystyle {\hat{\mu }}_2}$ again represents the predicted deflection, or safety margin, on obstacle-present 2-target trials, the $\frac{1}{2}$ coefficient on each priority term confers equal weighting of the two priorities, the 0° term corresponds to full prioritization of movement timing, and the ${\mu }_{1A}\cdot \frac{{\sigma }_2}{{\sigma }_{1A}}$ term is the safety margin corresponding to full prioritization of obstacle avoidance – within that term, ${\sigma }_2$ and ${\sigma }_{1A}$ represent the observed motor variability on obstacle-present 2-target trials and obstacle-obstructed 1-target trials, respectively, and ${\mu }_{1A}$ represents the observed deflection, or safety margin, on obstacle-obstructed 1-target trials. Like the baseline MA model, the baseline PO model is parameter-free in that ${\mu }_{1A}$, ${\sigma }_2$, and ${\sigma }_{1A}$, which serve as inputs to the model, are experimentally measured on obstacle-obstructed 1-target trials and obstacle-present 2-target trials. Note that the population-averaged variability ratio $\frac{{\sigma }_2}{{\sigma }_{1A}}$ was calculated as the ratio of the mean of the individual participant values for ${\sigma }_2$ and ${\sigma }_{1A}$. Interestingly, we found population-averaged values of 1.02 and 1.00 for this ratio in Expts 2a and 2b, respectively, indicating that at the population level motor variability on obstacle-present 2-target trials and obstacle-obstructed 1-target trials were quite similar. Thus, inclusion of the factor $\frac{{\sigma }_2}{{\sigma }_{1A}}$ in the PO model does not meaningfully influence the prediction for the population-averaged 2-target trial movement direction, ${\displaystyle {\hat{\mu }}_2}$. However, although ${\sigma }_2$ and ${\sigma }_{1A}$ are similar on average, we found that they can vary considerably across individuals and even more so across movement directions within individuals. Oddly, individuals often displayed idiosyncratic but consistent differences in motor variability for different movement directions. Thus, differences in ${\sigma }_2$ and ${\sigma }_{1A}$ are not entirely surprising given the large differences in movement direction that can be observed on obstacle-obstructed 1-target trials and obstacle-present 2-target trials. Importantly, taking this idiosyratic direction-specific motor variability into account substantially enhances the ability to predict the safety margin observed in one movement direction from motor variability observed at another across individuals. Thus the factor $\frac{{\sigma }_2}{{\sigma }_{1A}}$ is important for predicting inter-individual differences in 2-target trial movement directions (see Equations 5 and 6).

The inclusion of the safety margin in motor planning for PO predicts deflections that skew away from the obstacle in both Expts 2a and 2b. In Expt 2a, MA, like PO, predicts rightward deflections that skew away from the obstacle on 2-target trials. However, in Expt 2b, MA predicts leftward deflections that skew towards the obstacle, whereas PO continues to predict rightward deflections that skew away from the obstacle. The 15° offset term that is subtracted from the ${\mu }_{1A}\cdot \frac{{\sigma }_2}{{\sigma }_{1A}}$ safety margin term is present because the obstacle protruded 15° away from the 0°, straight-ahead movement. Because of this 15° obstacle offset, the predicted movement deflection driven by full prioritization of obstacle avoidance would be 15° less than the ${\mu }_{1A}\cdot \frac{{\sigma }_2}{{\sigma }_{1A}}$ safety margin for that priority. Relatedly, if the ${\mu }_{1A}\cdot \frac{{\sigma }_2}{{\sigma }_{1A}}$ safety margin was 15° or smaller, corresponding to $\left({\mu }_{1A}\cdot \frac{{\sigma }_2}{{\sigma }_{1A}}-15°\right)\le 0°$ in Equation 2b, then the 0° straight-ahead movement that fully satisfies the movement timing objective would already provide a safety margin at or above the level required for the obstacle avoidance objective, removing the need to compromise between the two objectives because a 0° straight-ahead movement would simulatensously optimize both. For simplicity, however, we drop this conditional secondary equation from the PO model variants presented below (Equations 4–6) as the condition required for using it is never met when applying these equations to the data from Expts 2a or 2b.

Like in Expt 1, we focused our analysis on the initial portion of motor output, measured as the initial movement direction, as this reflects feedforward motor planning. We calculated the movement direction as the direction of the hand at the midpoint of the movement (i.e., along the axis of the obstacle protrusion) relative to the direction of the hand at movement onset (but note that we obtained qualitatively similar results for all analyses when the movement direction was calculated 4 cm into the movement). In estimating each participant’s mean 2-target trial movement direction, we combined the left-side and right-side obstacle data. To facilitate a balanced comparison between the obstacle-free and obstacle-present 2-target trial conditions, we randomly assigned each obstacle-free 2-target trial a left-side or right-side obstacle condition label and combined the data accordingly. Correspondingly, on obstacle-present 2-target trials, we found that participants displayed movement directions that were consistently biased away from the obstacle compared to obstacle-free movement directions in both Expt 2a (6.84 ± 4.51° vs. 0.32 ± 0.50° [mean ± 95% CI], p=9.40 × 10⁻³, t(7) = –3.04) and Expt 2b (2.7 ± 0.80° vs. 0.056 ± 0.29°, p=8.01 × 10⁻⁷, t(25) = –6.51; paired t-test). This is consistent with the predictions of PO, but at odds with the predictions of MA. Sample trajectory data are shown in Figure 4a, and population-averaged movement direction data are shown in Figure 4b.

Figure 4 with 1 supplement see all

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As in Expt 1, we quantified the relative accuracy of the model predictions for MA and PO using a prediction index that attained a value of +1 if the experimental data matched the PO prediction, –1 if they matched the MA prediction, and 0 if they were midway between these two predictions (see Materials and methods). Correspondingly, we tested whether the data were significantly closer to the prediction of one model or the other by determining whether this prediction index was significantly above or below zero. We found that the experimentally observed movement directions on 2-target trials, which characterize motor planning in uncertain conditions, were far closer to the PO prediction than the MA prediction in Expt 2b, where gross differences in the PO and MA predictions were expected, with mean squared prediction errors of 4.24 ± 1.47 deg² (mean ± SEM) for the PO model and 163.50 ± 11.49 deg² for MA (see the 'baseline' MA and PO predictions in Figure 4b). Correspondingly, the mean prediction index was close to +1 (+0.96 ± 0.12; p=4.31 × 10⁻¹⁴, t(25) = 15.13; t-test). In Expt 2a, however, where gross differences between the MA and PO predictions were not expected, the mean obstacle-present 2-target trial data were fourfold closer to the PO prediction than the MA prediction, but the individual participant data was more variable than in Expt 2b. Furthermore, the span between the MA and PO predictions was smaller, providing less leverage to dissociate between them. These effects resulted in no significant difference between the models (prediction index = +0.59 ± 1.19; p=0.37, t(7) = 0.96; t-test; squared errors of 39.49 ± 32.18 deg² for PO and 73.04 ± 14.54 deg² for MA). Collectively, these findings suggest that during uncertainty, participants form an action plan that optimizes task performance instead of automatically averaging potential motor plans.

To allow the MA and PO models to make predictions without free parameters, we assumed equal weighting for both potential targets in the MA model, in line with previous work (Chapman et al., 2010; Ghez et al., 1997; Gallivan et al., 2016; Van der Stigchel et al., 2006), and analogously, for both objectives in the PO model (obstacle avoidance and rapid target acquisition), as specified in Equations 1 and 2. While this is reasonable, it is possible that this constraint might provide an explanation for why the baseline MA model could not accurately predict the experimental data. We therefore evaluated a refined version of the baseline MA model that removed this constraint by including a weighting parameter, $\alpha$, that permitted differential weighting of the obstacle obstructed and unobstructed targets, as shown in Equation 3:

Equation 3 – refined MA model:

Thus $\alpha =1$ would indicate that all weight is assigned to the motor plan associated with the obstacle-obstructed target ${\mu }_{1A}$, $\alpha =0$ would indicate that all weight is assigned to the motor plan associated with the unobstructed target ${\mu }_{1B}$, and $\alpha =\frac{1}{2}$ would indicate equal weighting, as in Equation 1. Note that the value for $\alpha$ and that for $\beta$ (defined below) was constrained to be between 0 and 1 to avoid any negative weighting. We found weighting coefficients of 0.26 and 0.74 for the obstacle-obstructed and unobstructed targets, respectively (${\displaystyle \alpha =0.26}$ when fitting to the Expt 2a data, and 0 and 1, ${\displaystyle \alpha =0}$ when fitting to the Expt 2b data). However, to avoid the effects of overfitting, we evaluated these parameter estimates not based on their ability to explain the data on which they were fit, but instead, based on the ability of the parameter estimate obtained from one experiment’s data to predict the results of the other experiment. The findings from this cross-validation are discussed below.

Analogous to the refinement of the MA model, we assessed whether including differential weighting for the obstacle avoidance and rapid target acquisition objectives might improve the already-accurate predictions from the baseline PO model. We therefore also evaluated a refined version of the PO model that incorporated a weighting parameter, $\beta$, that allowed for differential weighting of these objectives:

Equation 4 – refined PO model:

Thus $\beta =1$ would indicate full prioritization of movement timing, $\beta =0$ would indicate full prioritization of obstacle avoidance, and $\beta =\frac{1}{2}$ would indicate equal weighting of these priorities as in Equation 2. We found weighting coefficients for movement timing and obstacle avoidance of 0.66 and 0.34, respectively $\left(\beta =0.66\right)$, when fitting to the Expt 2a data, and 0.46 and 0.54 $\left(\beta =0.46\right)$ when fitting to the Expt 2b data. These weighting parameters suggest similar priority levels for movement timing and obstacle avoidance in Expts 2a and 2b. However, like for the refined MA model, we evaluated these parameter estimates not based on their ability to explain the data on which they were fit, but instead, based on the ability to predict the results of the other experiment (see Figure 4b).

Unsurprisingly, we found that when the parameters $\alpha$ and $\beta$ were fit onto the Expt 2a data, the resulting refined versions of the MA and PO models both exactly matched the mean data from Expt 2a, corresponding to mean squared errors of 37.03 ± 14.31 deg² across participants for both refined models. However, when the parameter estimates obtained from the Expt 2b data were cross-validated on the Expt 2a data, the prediction from the PO model increased the mean squared error by only 11%, whereas that from the MA model increased it by 115%. Accordingly, the experimental data from Expt 2a were noticeably closer to the cross-validated PO prediction than the cross-validated MA prediction, as shown in Figure 4b. The difference, however, was not significant (with a prediction index of +1.11 ± 1.98; p=0.11, t(7) = 1.85; t-test; and mean squared errors of 79.52 ± 53.71 deg² vs. 40.93 ± 33.89 deg² for MA vs. PO). This finding was not surprising as Expt 2a was not expected to clearly dissociate the models.

In contrast, the data from Expt 2b, which was specifically designed to dissociate MA from PO, yielded clear results. First, due to the grossly opposite predictions for MA and PO in Expt 2b, the MA model was unable to exactly match the mean 2-target trial data when fit to the Expt 2b data, unlike the PO model (see Figure 4b). More importantly, when parameter estimates obtained from the same Expt 2a data were cross-validated on the Expt 2b data, the PO model prediction increased the mean squared error by 29%, whereas the MA model prediction increased it by 750% (squared error of 5.42 ± 1.16 deg² vs. 4.19 ± 1.59 deg² for PO compared to 58.10 ± 7.20 deg² vs. 6.78 ± 2.67 deg² for MA). Correspondingly, we found that the Expt 2b data were far better explained by the prediction from the PO model than by the prediction from the MA model (prediction index = +0.74 ± 0.19; p=5.79 × 10⁻⁸, t(25) = 7.61), with mean squared errors that were 10-fold smaller for the PO model’s prediction (squared errors of 58.10 ± 7.20 deg² vs. 5.42 ± 1.16 deg², for the MA and PO models, respectively).

These results show that the assumption of equal weighting was not the reason that the baseline MA model failed to outperform the baseline PO model, and they provide further support for the hypothesis that PO rather than MA explains motor planning under uncertain conditions. We note that to better understand each model’s sensitivity to its weighting parameter we also examined predictions that were based on 1:2 and 2:1 weightings for the obstacle-obstructed and -unobstructed targets in the MA model, and for the movement timing and obstacle avoidance objectives in the PO model. However, we found that these predictions led to a similar pattern of results (see Figure 4—figure supplement 1).

After finding that the PO model accurately predicts the group-averaged data from Expt 2, we proceeded to examine whether it might also be able to explain inter-individual differences in motor planning during uncertain conditions in Expts 2a and 2b. To study the ability of the PO model to explain inter-individual differences in motor planning, we created a version of it that combined the effects of using the group-averaged data and each individual participant’s data, as shown below:

Here, the relative weighting between the group-averaged and the individual participant data is given by a single parameter $k$, which we refer to as the individuation index. $k$ = 1 corresponds to full individuation, whereas $k$ = 0 corresponds to no individuation, that is, full weighting of the group-average data. The other variables in Equation 5 are the same as those in Equation 2, but note that here the subscript $i$ refers to individualized data from participant $i$, and that a horizontal bar over a variable indicates the group-average value. Unlike Equations 2 and 4, this version of the PO model encodes the individual participant measurements of motor variability in ${\sigma }_{2_i}$ and ${\displaystyle {\sigma }_{{1A}_i}}$, which allows the model to account for individual differences in variability between 1- and 2-target trials that result from idiosyncratic spatial differences in movement direction. Note that the left side of Equation 5 represents predicted inter-individual differences from the group-averaged obstacle-present 2-target trial movement direction because ${\displaystyle {\overline{\mu }}_2}$ is subtracted from ${\displaystyle {\hat{\mu }}_{2_i}}$, and analogously, that the right side represents inter-individual differences from the group-averaged model prediction because $K_0$, the group-averaged model prediction, is subtracted. Also note that the inclusion of this offset parameter, $K_0$, is necessary so that the model’s ability to explain variance in the data is well-posed. The partial weighting of individual and group-averaged data that $k$ allows is often used for optimal estimation with noisy data, corresponding to the fact that the experimental estimates made for variables, like the mean or variability of the initial movement directions in our data, are affected by measurement noise to a far greater extent for individual participant measurements than for group averages.

Remarkably, when we fit this one-parameter-plus-offset model to the experimental data, we found that it was able to explain 87% of the variance for inter-individual differences in movement direction on 2-target trials in the Expt 2a dataset (F(1,6) = 39.4, p=7.60 × 10⁻⁴, first bar in Figure 5a) and 57% of this variance in the Expt 2b dataset (F(1,24) = 31.9, p=8.21 × 10⁻⁶, first bar in Figure 5e). This indicates that a majority of the inter-individual variability in the initial direction of intermediate movements planned under uncertain conditions in our experiments can be explained by a PO model that takes individual differences in 1-target trial safety margins, 1-target trial motor variability, and 2-target trial motor variability into account.

Figure 5

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We next examined the extent to which the model’s ability to predict inter-individual differences accrued from each of its input variables: 1-target trial safety margins (${\mu }_{1A}$), 1-target trial motor variability (${\displaystyle {\sigma }_{1A}}$), and 2-target trial motor variability (${\sigma }_2$). To accomplish this, we compared a more complex three-parameter-plus-offset version of this model that included different individuation indices ($k_{\mu }$, $k_{{\sigma }_1}$, $k_{{\sigma }_2}$) for each input variable to two-parameter versions where the remaining individuation index was held at zero. This removed individuation for one of the three input variables and allowed us to calculate each input variable’s partial R² value, which characterizes the contribution of that variable to the explained variance in a nested model. The three-parameter-plus-offset model is shown in Equation 6:

Analysis of the three-parameter-plus-offset version of the PO model revealed two key observations. First, we found that the three-parameter-plus-offset model resulted in only marginal improvement in its ability to explain inter-individual differences in the data compared to the one-parameter-plus-offset model, suggesting that any true differences in individuation between the three input variables were relatively small (compare the first and second bars in Figure 5a, e). Second, we found that two of the three input variables – 1-target trial safety margins and 2-target trial motor variability – contributed significantly to explaining individual differences in the data, whereas 1-target trial motor variability did not. This was separately true for both Expt 2a (we found partial R² values of 0.90, 0.81, and 0.00 for $k_{\mu }$, $k_{{\sigma }_2}$, and $k_{{\sigma }_1}$, respectively; p=4.02 × 10⁻³, 1.50 × 10⁻², and 1.000; F(1,4) = 35.3, 16.7, 0.0) and Expt 2b (we found partial R² values of 0.32, 0.65, and 0.00 for $k_{\mu }$, $k_{{\sigma }_2}$, and $k_{{\sigma }_1}$, respectively; p=3.78 × 10⁻³, 3.18 × 10⁻⁶, and 0.763; F(1,22) = 10.5, 38.2, 0.09). Scatter plots of the relationships between each of the three input variables and the inter-individual differences that remain when the other two input variables are accounted for are shown in Figure 5b–d for Expt 2a and in Figure 5f–h for Expt 2b.

We were initially surprised to find that individuation using $k_{{\sigma }_1}$ did not meaningfully contribute to explaining the individual differences in 2-target trial movement directions. However, examination of the scatter plots in Figure 5 reveals that the individual differences in 1-target trial motor variability were considerably smaller than those for the other two input variables. Thus, individual differences in 1-target trial motor variability, ${\sigma }_{1A},$ had considerably less leverage to explain individual differences in the 2-target trial movement direction, ${\mu }_2$, than the other input variables.

Importantly, these individual participant results provide further evidence at odds with the MA hypothesis. First, the clear positive relationship between the 2-target trial movement direction and the 1-target trial safety margin in Expt 2b (see Figure 5e) is opposite of the negative relationship that MA would predict (in line with the opposite predictions for MA and PO for the Expt 2b data illustrated in Figure 4a, b). Second, the clear positive relationships between the mean movement direction and the amount of motor variability on 2-target trials found in both in the Expt 2a data (Figure 5d) and the Expt 2b data (Figure 5h) are consistent with PO, but are at odds with MA, which predicts that there should be no relationship between these variables.

In summary, we find that PO provides an explanation not only for population-averaged motor planning for the intermediate movements that result from uncertainty about the target location, but also for a majority of the variance in inter-individual differences in this planning observed in Expts 2a and 2b. A detailed analysis reveals that individualized information about 1-target trial safety margins and 2-target trial motor variability both contribute significantly to the ability to explain these individual differences in motor planning.

Our finding that PO underlies movements executed during uncertainty challenges the long-standing idea that the intermediate movements executed during uncertain conditions reflect MA. However, many studies suggesting that either sensory or motor representations of movement plans are averaged did not dissociate these possibilities from selecting a single plan that optimizes motor performance. For example, the Stewart et al., 2014 study that we replicate in Expt 2a attempted to dissociate whether motor output might reflect an average of sensory or motor representations of movement plans (i.e., sensory averaging vs. MA) by using obstacles to perturb these motor plans, but did not dissociate either sensory averaging or MA from PO.

Another study, by Gallivan et al., 2017, applied a visuomotor rotation to movements towards one potential target to perturb motor planning without affecting the sensory representation of the target. This resulted in initial movement directions that were altered during 2-target trials in accordance with the perturbed motor plans, and thus in line with the prediction for MA rather than sensory averaging. However, PO predicts a movement direction identical to that predicted by MA: since the imposed visuomotor rotation shifts the final hand position associated with acquisition of the potential target to which it was applied, a corresponding shift in the movement direction that accounts for this visuomotor rotation would optimize performance by minimizing the cost of corrective movements following disclosure of the final target location. Therefore, this experimental manipulation, like that in Stewart et al., 2014 and a number of other studies (Chapman et al., 2010; Stewart et al., 2014; Gallivan et al., 2011; Gallivan et al., 2016; Gallivan and Chapman, 2014; Stewart et al., 2013; Chou et al., 1999; Van der Stigchel et al., 2006), fails to dissociate MA from PO.

Haith et al., 2015a examined motor planning under uncertainty using a timed-response reaching task where the target suddenly shifted on a fraction (30%) of trials (150–550 ms) before movement initiation. The authors observed intermediate movements when the target shift was modest (±45°), but direct movements towards either the original or shifted target position when the shift was large (±135°). The authors argued that because intermediate movements were not observed under conditions in which they would impair task performance, that motor planning under uncertainty generally reflects PO. This interpretation is somewhat problematic, however. In this task, like in the current study, the goal location was uncertain when initially presented. However, the final target was presented far enough before movement onset that this uncertainty was no longer present during the movement itself, as evidenced by the direct-to-target motion observed when the target location was shifted by ±135°. Therefore, the intermediate movements observed when the target location shifted by ±45° are unlikely to reflect motor planning under uncertain conditions. Instead, these intermediate movements likely arose from a motor decision to supplement the plan elicited by the initial target presentation with a corrective augmentation when the plan for this augmentation was certain. The results thus provide beautiful evidence for the ability of the motor system to flexibly modulate the correction of existing motor plans, ranging from complete inhibition to conservative augmentation, when new information becomes available, but provide little information about the mechanisms for motor planning under uncertain conditions.

The challenge of examining motor planning under uncertainty has persisted even for studies that made deliberate attempts at dissociating MA from PO. For example, in Wong and Haith, 2017, motor planning under uncertainty was examined under conditions where participants were required to reach towards one of two potential targets with a movement completion time that was small enough to preclude movements with corrective adjustments from being successful. The results showed that under these conditions participants frequently abandoned intermediate movements that subsequently require corrective adjustments on 2-target trials in favor of direct movements towards one of the potential targets. These findings indicate that MA is not obligatory, and that conditions can be created where intermediate movements themselves are abandoned. This stands in contrast to the present study, in which intermediate movements were allowed, yet were found to be inconsistent with MA, suggesting that MA is unlikely to occur at all. In addition, it is likely that the task conditions used in Wong and Haith simply induced participants to make a strategic decision to guess the final target location and aim at that guess. While this would indeed improve performance and could therefore be considered a type of performance optimization, such strategic decision making does not provide information about the implicit neural processing involved in programming the motor output for the intermediate movements that are normally planned under uncertain conditions.

In another study, performed by Nashed et al., 2017, motor planning under uncertainty was studied using a task that was based on grip force control. Specifically, participants grasped an object, capable of measuring grip force, and made reaching movements while environmental dynamics were experimentally manipulated with robotically applied load forces that affected the required grip force. This task design resembles Expt 1 from the current study in which different environmental dynamics were used to perturb movements towards the left/right vs. center targets in order to dissociate MA from PO. However, grip force control is known to be substantially more sensitive to the variability in environmental dynamics than to the mean dynamics (Hadjiosif and Smith, 2015), yet this study examined the MA and PO hypotheses for motor planning under uncertainty using predictions that were based entirely on the mean environmental dynamics. Unfortunately, this study did not provide information about motor variability or its effect on required grip forces, obscuring the driver for the observed changes in grip force. Thus, it is unclear how this study can elucidate the mechanisms that drive motor planning under uncertainty.

An important consideration for the present results is that sensorimotor control engages both implicit and explicit adaptive processes to generate motor output (Mazzoni and Krakauer, 2006). Because motor output reflects combined contributions of these processes, determining their individual contributions can be difficult. In particular, the experiments in the present study used environmental perturbations to induce adaptive changes in motor output, but these changes may have been partially driven by explicit strategies, and thus the extent to which the motor output measured on 2-target trials reflects implicit vs. explicit feedforward motor planning requires further investigation. One method for examining implicit motor planning during goal uncertainty might take inspiration from recent work showing that in visuomotor rotation tasks, restricting the amount of time available to prepare a movement appears to limit explicit strategization from contributing to the motor response (Haith et al., 2015b; Huberdeau et al., 2019; Leow et al., 2017; Fernandez-Ruiz et al., 2011). Future work could dissociate the effects of MA and PO on intermediate movements in uncertain conditions at movement preparation times short enough to isolate implicit motor planning.

We note that Gallivan et al., 2017 attempted to control for the effects of explicit strategies by (1) applying the perturbation gradually, so that it might escape conscious awareness, and (2) enforcing a 325 ms preparation time. Intermediate movements persisted under these conditions, suggesting that intermediate movements during goal uncertainty may indeed be driven by implicit processes. However, it is difficult to be certain whether explicit strategy use was, in fact, effectively suppressed as the study did not assess whether participants were indeed unaware of the perturbation. On one hand, the single study we are aware of that assessed the extent to which reduced movement preparation times can suppress strategization during motor learning, showed that participants can successfully execute a re-aiming strategy when specifically instructed to do so, even down to movement preparation times as small as 150 ms. On the other hand, when a specific strategy was not instructed, the same study found that a 250 ms preparation time yielded performance that was indistinguishable from implicit learning measured using an aim-report paradigm with a long (1000 ms) preparation time, suggesting that active strategization was effectively suppressed. It should be noted, however, that the difference between 250 and 325ms is considerable and its effect unclear, and also that any claim of full suppression in the 250ms data would depend on a negative statistical comparison (note further that the 250ms and 325ms nominal preparation times quoted in these papers are larger than the true preparation times as the quoted times include both display latencies and data-acquisition latencies; neither of these were reported in Gallivan et al., and only the former was reported in Leow et al. (27.6 ± 1.8 ms)).

Studies suggesting motor or sensory averaging have been motivated by reports of simultaneous deliberation of competing potential goals in sensorimotor brain areas (i.e., parallel planning) (Chapman et al., 2010; Stewart et al., 2014; Gallivan et al., 2011; Stewart et al., 2013; Gallivan et al., 2015). These studies argue that the motor plans prepared in parallel are averaged, resulting in the intermediate movements observed when uncertainty is present during motor planning. The neural evidence for parallel planning is based on studies of delay period activity associated with motor planning when multiple potential targets were present (Cisek and Kalaska, 2005; Cisek, 2007; Coallier et al., 2015; Pastor-Bernier and Cisek, 2011). However, because this activity was measured using single-electrode recordings, only a small number of cells could be recorded simultaneously, and thus data from same-type trials were aggregated to make population-based estimates of the planned motion. The results suggested that this aggregate delay period activity was tuned to both potential targets in sensorimotor areas, in particular dorsal premotor cortex (PMd) and parietal reach region, in monkeys. However, the observed tuning could arise from simultaneous parallel planning for both potential targets, as the authors argued, or from planning-related activity associated with one target on some trials and with the alternate target on other trials.

Recently, Dekleva et al., 2018 studied parallel planning using an electrode array to record simultaneously from 100 to 160 neurons in PMd, allowing planning-related neural activity to be examined for individual movements when two potential targets were present. Critically, they found that neural activity during the delay period encoded motor plans directed to either one target or the other on individual trials, rather than simultaneous parallel plans for both potential targets. While these results cannot rule out parallel planning in other brain areas, they call into question the evidence for parallel planning from previous studies that relied on trial-aggregated data. Taken together with the results from the current study, it is plausible that during uncertainty the brain forms a single motor plan that optimizes task performance rather than multiple parallel motor plans for MA.

In summary, the current findings indicate that motor planning during uncertain conditions does not proceed from averaging parallel motor plans, but instead, incurs the creation of a motor plan that optimizes task performance given knowledge of the current environment. These findings are compatible with the current neurophysiological data and offer a mechanistic framework for understanding motor planning in the nervous system.

26 participants (25 right-handed; 15 females; age range 18–42) performed the multi-force-field (multi-FF) adaptation experiments, with 16 participants in Expt 1 and 10 participants in Expt 1-GEN. 34 participants (32 right-handed; 18 females; age range 18–33) performed the obstacle avoidance experiments, with 8 participants in Expt 2a and 26 participants in Expt 2b. Participants were assigned to experiments based on when they responded to advertisements and their availability for scheduling. When different experiments were running concurrently, participants were randomly assigned for participation. The sample sizes used for each experiment were determined based on pilot data and existing literature (Stewart et al., 2014; Smith et al., 2006; Sing et al., 2009). All participants used their right hands to perform the experiments, were naïve to the purpose of the experiments, and were without known neurological impairment. The study protocol was approved by the Harvard University Institutional Review Board (protocol number: IRB16-2128), and all participants provided written informed consent.

t-tests were used for most statistical comparisons as indicated in the Results section. A one-sided t-test was used to compare obstacle-present versus baseline obstacle-free 2-target trial movement direction data in Expt 2 (Figure 4b), and the remaining t-tests were two-sided. When we assessed nested forms of the full PO model in Expt 2 (see Equations 5 and 6), we used F-tests. In all statistical tests, the significance level (alpha) was set to 0.01, and normality assumptions for the tests were verified with Kolmogorov–Smirnov tests. Experiments were programmed in C++ or MATLAB, and all data were analyzed in MATLAB (RRID:SCR_001622).

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

We thank Ashleigh Victoria Conroy-Zugel for help with experiments. We also thank J Ryan Morehead for creating the experimental setup illustrations (Figures 1a and 3a). This work was supported by a grant from the National Institute on Aging (R01 AG041878) to MAS.

Human subjects: The study protocol was approved by the Harvard University Institutional Review Board (protocol number: IRB16-2128), and all participants provided written informed consent.

© 2021, Alhussein and Smith

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