Conceived and designed the experiments: JAT RBI. Performed the experiments: JAT. Analyzed the data: JAT. Wrote the paper: JAT RBI.
The authors have declared that no competing interests exist.
Visuomotor rotation tasks have proven to be a powerful tool to study adaptation of the motor system. While adaptation in such tasks is seemingly automatic and incremental, participants may gain knowledge of the perturbation and invoke a compensatory strategy. When provided with an explicit strategy to counteract a rotation, participants are initially very accurate, even without on-line feedback. Surprisingly, with further testing, the angle of their reaching movements drifts in the direction of the strategy, producing an increase in endpoint errors. This drift is attributed to the gradual adaptation of an internal model that operates independently from the strategy, even at the cost of task accuracy. Here we identify constraints that influence this process, allowing us to explore models of the interaction between strategic and implicit changes during visuomotor adaptation. When the adaptation phase was extended, participants eventually modified their strategy to offset the rise in endpoint errors. Moreover, when we removed visual markers that provided external landmarks to support a strategy, the degree of drift was sharply attenuated. These effects are accounted for by a setpoint state-space model in which a strategy is flexibly adjusted to offset performance errors arising from the implicit adaptation of an internal model. More generally, these results suggest that strategic processes may operate in many studies of visuomotor adaptation, with participants arriving at a synergy between a strategic plan and the effects of sensorimotor adaptation.
Motor learning has been modeled as an implicit process in which an error, signaling the difference between the predicted and actual outcome is used to modify a model of the actor-environment interaction. This process is assumed to operate automatically and implicitly. However, people can employ cognitive strategies to improve performance. It has recently been shown that when implicit and explicit processes are put in opposition, the operation of motor learning mechanisms will offset the advantages conferred by a strategy and eventually, performance deteriorates. We present a computational model of the interplay of these processes. A key insight of the model is that implicit and explicit learning mechanisms operate on different error signals. Consistent with previous models of sensorimotor adaptation, implicit learning is driven by an error reflecting the difference between the predicted and actual feedback for that movement. In contrast, explicit learning is driven by an error based on the difference between the feedback and target location of the movement, a signal that directly reflects task performance. Empirically, we demonstrate constraints on these two error signals. Taken together, the modeling and empirical results suggest that the benefits of a cognitive strategy may lie hidden in many motor learning tasks.
When learning a new motor skill, verbal instruction often proves useful to hasten the learning process. For example, a new driver is instructed on the sequence of steps required to change gears when using a standard transmission. As the skill becomes consolidated, the driver no longer requires explicit reference to these instructions. Operating a vehicle with a stiffer or looser clutch does not generally require further instruction, but rather entails a subtle recalibration, or adaptation of the previously learned skill. Indeed, the use of an explicit strategy may even lead to degradation in the expert's performance. Consideration of these contradictory issues brings into question the role of instructions or explicit strategies in sensorimotor learning.
The type of motor task and nature of the instruction can have varying effects on motor execution and learning
Various studies have examined the role of explicit strategies in tasks involving sensorimotor adaptation
In many of the studies cited above, the assumption has been that the development of awareness can lead to the utilization of compensatory strategies. However, few studies have directly sought to manipulate strategic control during sensorimotor adaptation. One striking exception is a study by Mazzoni and Krakauer
Mazzoni and Krakauer
In the present paper, we start by asking how this hypothesis could be formalized in a computational model of motor learning. State space modeling techniques have successfully described adaptation and generalization during motor learning
Current models of sensorimotor adaptation have not addressed the effect of explicit strategies. Therefore, we started with the standard state-space model (Eq 1 and 2), and incrementally modified it to accommodate the use of an explicit strategy. The standard model for target error is given as:
A −45° rotation was introduced on movement 121 and remained present for the next 322 trials. A) Simulated target error for four state space models. Black: Standard state-space model (A = 1, B = 0.02); Green: Setpoint model in which the target error is used to adapt the internal model; Red: Setpoint model with direct-feedthrough in which the aiming error is used to adapt the internal model. Drift is attenuated by either reduced adaptation rate (Cyan: A = 1, B = 0.01) or reducing the availability of the aiming error signal that selectively operates on the strategy (Magneta: K = 0.5). B) Internal model estimation of rotation when a fixed strategy (blue) is combined with either high (red, K = 1) or low (magenta, K = 0.5) certainty of aiming errors. C) Effect of a variable strategy on target error. The strategy was simulated with a low (blue; E = 1 and F = 0.01) or high (orange; F = 0.05) weighting of the target error.
When informed of an appropriate strategy that will compensate for the rotation, participants immediately counteract the rotation and show on-target accuracy. The standard model as formulated above does not provide a mechanism to implement an explicit strategy. To allow immediate implementation of the strategy, we postulate that there is direct feedthrough of the strategy (s) to the target error equation (equation 1):
Direct feedthrough allows the strategy to contribute to the target error equation without directly influencing the updating of the internal model. If the strategy operated through the internal model, then the impact of the strategy would take time to evolve, assuming there is substantial memory of the internal model's estimation of the rotation (i.e., A has a high value in Eq. 2). With direct feedthrough, the implementation of an appropriate strategy can immediately compensate for the rotation. In the current arrangement, the appropriate strategy is fixed at 45° in the CW direction from the cued target.
Once the strategy is implemented, performance should remain stable since the error term is small. Indeed, a model based on Eq. 3 immediately compensates for the rotation. The target error, the difference between the feedback location and target location, is essentially zero on the first trial with the strategy, and remains so throughout the rotation block (
The experiment workspace consisted of 8 empty blue circles separated by 45° (three locations are shown here). The target was defined when a green circle appeared at one of the locations. The hand was occluded by the apparatus and on feedback trials, a red cursor appeared as soon as the participant crossed a virtual ring, 10-cm from the start location. A) In the baseline block, participants moved towards the cued green target. B) In the strategy-only block, participants moved to the blue circle located 45° in the clockwise direction. Feedback was presented at the veridical hand position. C) For the two rotation probes, participants were instructed to move to the green target, but feedback of hand position was rotated 45° in the counter-clockwise direction. D) In the rotation plus strategy block, participants were instructed to move to the blue circle located 45° clockwise direction from the target. The feedback of hand position was rotated 45° counter-clockwise. E) Two sources of movement error: a target error between the feedback location and target location and an aiming error between the feedback location and aiming location.
This model shows immediate compensation for the visuomotor rotation, and more importantly, produces a gradual deterioration in performance over the course of continued training with the reaching error drifting in the direction of the strategy (
It is important to emphasize that the error signal for sensorimotor recalibration in Eq. 4 is not based on the difference between the feedback location and target location (target error). Rather, the error signal is defined by the difference between the feedback location and aiming location, or what we will refer to as aiming error. When a fixed strategy is adopted throughout training (
A second prediction can be derived by considering that the error signal in Eq. 4 relies on an accurate estimate of the strategic aiming location. We assume that a visual landmark in the display can be used as a reference point for strategy implementation (e.g., the blue circle adjacent to the target). This landmark can serve as a proxy for the aiming location. The salience of this landmark provides an accurate estimate of the aiming location and, from Eq. 4, drift should be pronounced. However, if these landmarks are not available, then the estimate of the aiming location will be less certain. Previous studies have shown that adaptation is attenuated when sensory feedback is noisy
To evaluate the predictions of this setpoint model, participants were tested in an extended visuomotor rotation task in which we varied the visual displays used to define the target and strategic landmarks (see
Participants were assigned to one of three experimental groups (n = 10 per group), with the groups defined by our manipulation of the blue landmarks in the visual displays. For the aiming-target group (AT), the blue circles were always visible, similar to the method used by Mazzoni and Krakauer. For the disappearing aiming-target group (AT), the blue circles were visible at the start of the trial and disappeared when the movement was initiated. For the no aiming-target group (NoAT), the blue landmarks were not included in the display.
The participants were initially required to reach to the green target (
Participants first practiced moving to the cued target without a rotation (black) and while using the strategy without a rotation (orange). The rotation was turned on between movements 121 and 443 (dashed vertical lines). For the first two of these trials, the rotation probes, the participants had not been given the strategy (X's). For the next 320 rotation trials, participants were instructed to use the strategy. Following this, the rotation was turned off and participants were instructed to move towards the cued target, first without endpoint feedback (X's) and then with endpoint feedback (circles). A) Aiming-Target Group (blue). B) Disappearing Aiming-Target Group (magenta). C) No Aiming-Target Group (red). Shading represents the 95% confidence interval of the mean.
Practicing the 45° CW strategy did not produce interference on a subsequent baseline block in which participants were again instructed to reach to the cued, green target (
Without warning, the CCW rotation was introduced (
The participants were then instructed to use the strategy and required to produce a total of 320 reaching movements under the CCW rotation. This extended phase allowed us to a) verify that error increased over time, drifting in the direction of the strategy, and b) determine if the magnitude of the drift would approximate the magnitude of the rotation, a prediction of the simplest form of the setpoint model. Consistent with the results of Mazzoni and Krakauer
A) Average endpoint angular error relative to the target for the three groups, binned by averaging over epochs of ten movements (AT group in blue, DAT group in magenta, NoAT group in red). B) Peak drift with respect to the eight target location for the three groups. The empty circles are the target locations. To identify peak drift, 10 bins of four movements were calculated for each direction. C) Angular error after the rotation was turned off and participants were instructed to stop using the strategy. Triangles are average of the first eight post-rotation trials, performed without visual feedback. Squares are washout block with feedback. D) Relationship of drift and aftereffect based on the estimated peak drift for each participant and the first eight post-rotation trials. For B) and D), the means and 95% confidence interval of the mean were estimated through bootstrapping.
Our rotation plus strategy block lasted 320 trials, nearly four times the number of trials used by Mazzoni and Krakauer
A and B are from the AT group; C is from the DAT group. A) Drift followed by large fluctuations in error. B) Drift followed by an abrupt change in target error. C) Continuous drift across training.
The drift persisted over the 320 trials of the rotation block for participants in the DAT group (
The availability or certainty in the estimate of the aiming location was manipulated by altering the presence of the aiming target across the groups. As predicted by the setpoint model, the degree of drift was attenuated as the availability of the aiming targets decreased. In the current implementation of our model, this decrease in drift rate is captured by a decrease in the adaptation rate (B): with greater uncertainty, the weight given to the error term for updating the internal model is reduced.
However, one prediction of this model is at odds with the empirical results. Variation in the adaptation rate not only predicts a change in drift rate, but also predicts a change in the washout period. Specifically, decreasing the adaptation rate should produce a slower washout, or extended aftereffect (
Alternatively, it is possible that the adaptation rate (B) is similar for the three groups and that the variation in drift rate arises from another process. One possibility is that the manipulation of the availability of the aiming targets influences the certainty of the desired strategy term in Equation 4, and correspondingly, modifies the aiming error term:
A value of K that is less than 1 will attenuate drift (
In sum, while variation in B or K can capture the group differences in drift rate, only the latter accounts for the similar rates of washout observed across groups. When the availability of the aiming targets is reduced, either by flashing them briefly or eliminating them entirely, the participants' certainty of the aiming location is attenuated. This hypothesis is consistent with the notion that the aiming locations serve as a proxy for the predicted aiming location.
As noted above, none of the participants showed drift approaching 45°. Even those exhibiting the largest drift eventually reversed direction such that they became more accurate over time in terms of reducing endpoint error with respect to the target location. To capture this feature of the results, we considered how participants might vary their strategy over time as performance deteriorates. It is reasonable to assume that the participant may recognize that the adopted strategy should be modified to offset the rising error. One salient signal that could be used to adjust the strategy is the target error, the difference between the target location and the visual feedback.
To capture this idea, we modified the setpoint model, setting the strategy as a function of target error (
The availability of the aiming targets, captured by K in Eq. 6, influences the magnitude of the drift. Greater drift occurs when the aiming error, that between the feedback location and aiming location, is salient (
This setpoint model (Eqs. 8–11) was fit by bootstrapping (see
The fits (
The setpoint model with direct-feedthrough (equations 8–11) was fit to each group's data (from
AT Group | DAT Group | NoAT Group | |
0.991±0.002 | |||
0.012±0.003 | |||
0.999±0.001 | |||
0.985±0.034 | 0.409±0.122 | 0.195±0.108 | |
0.023±0.006 | 0.002±0.003 | 0.725±0.319 | |
0.682±0.075 | 0.713±0.069 | 0.650±0.081 |
A: Retention factor of the internal model; B: Adaptation rate based upon aiming errors; E: Retention factor of the strategy; K: Availability of the strategic aiming location; F: Adjustment rate of the strategy based upon target errors (high values favor strategy change). A, B, and E were constrained to be the same across groups, while parameters K and F were allowed to vary between groups. The means and 95% confidence interval of the mean were estimated through bootstrapping.
The dynamics of the recalibration process and strategy state (Eqs. 10 and 11) are plotted in
Following the rotation block, we instructed the participants that the rotation would be turned off and they should reach to the cued green target. For the first eight trials, no endpoint feedback was presented. This provided a measure of the degree of sensorimotor recalibration in the absence of learning (
In the setpoint model, the internal model will continue to adapt even in the face of strategic adjustments adopted to improve endpoint accuracy. As such, the model predicts that the size of the aftereffect should be larger than the degree of drift. To test this prediction, we compared the peak drift during the rotation block to the aftereffect. In the preceding analysis, we had estimated peak drift for each participant by averaging over 10 movements and identifying the bin with the largest error. However, a few errant movements could easily bias the estimate of drift within a 10-movement bin. As an alternative procedure, we used a bootstrapping procedure to identify the bin with the largest angular error for each group. This method should decrease the effect of noise because the estimate of peak drift is selected from an averaged sample of the participants' data. Moreover, any bias in the estimate of the magnitude of the peak should be uniform across the three groups of participants. For consistency, we estimated the aftereffect (the first 8 trials without feedback) using the same bootstrap procedure.
For the AT group, the peak drift was 14.8±2.5° in the CW direction, occurring 64±30 movements into the rotation block. For the DAT group, the peak drift was 10.0±1.8°, occurring at a later point in the rotation block (130±106). For the NoAT group, peak drift was only 3.2±2.7° and occurred after 145±131 movements. As predicted by the model, the aftereffect was significantly larger than peak drift for the AT and NoAT groups (
Visuomotor rotation tasks are well-suited to explore how explicit cognitive strategies influence sensorimotor adaptation. Following the approach introduced by Mazzoni and Krakauer
Mathematical models of sensorimotor adaptation have not explicitly addressed how a strategy influences learning and performance. By formalizing the effect of strategy usage into the standard state-space model of motor learning, we can begin to evaluate qualitative hypotheses that have been offered to account for the influence of strategies on motor learning. Mazzoni and Krakauer
This simple setpoint model was capable of completely eliminating error on the first trial and capture the deterioration of performance with increased training. Drift arises because the error signal is driven by the difference between the internal model's prediction of the aiming location and the actual, endpoint feedback. The idea that an aiming error signal is the source of drift is consistent with the conjecture of Mazzoni and Krakauer
The attenuation of adaptation with increasing uncertainty (as reflected by reduced drift) is similar to the effects on adaptation predicted by a Kalman filter when measurement noise is large. Several studies have shown that adaptation rates can change when the certainty of sensory information is manipulated
The effect of the visual landmarks on adaptation also provides insight into why other studies have not observed drift, even when participants develop some explicit awareness of the rotation, and presumably, use that knowledge
Our model entails two types of error signals: an aiming prediction error between the feedback location and aiming location, and performance error between the feedback location and the target location (
Interestingly, while there was an initial rise in endpoint error, this function eventually reversed, returning close to zero endpoint error by the end of the strategy phase for the AT group. We assume that at some point, the size of the endpoint error exceeded the participant's self-defined tolerance for errors and caused them to modify the strategy. Unfortunately, we do not have a direct measure of strategy change. Examination of the learning profiles revealed considerable variability across individual participants (
At a minimum, multiple processes are required to capture this nonmonotonic learning function. In our initial modeling efforts, we fixed the strategy for the entire training process. Under this assumption, the system should exhibit drift that is equal in size to the rotation, an effect never observed. Thus, the final version of our model is a variant of a two-rate state space model
It is reasonable to assume that our manipulation of the availability of the aiming locations influenced the degree of certainty associated with the desired aiming location. When certainty is reduced, adaptation arising from the aiming error signal is slower, and in our two-process model, the level of adaptation achieved by the motor system is lowered. Moreover, the model does not predict that drift will reach 45°. The strategy is adjusted, reaching a point where it offsets the drift arising from adaptation of the internal model. The interplay of these two processes is complex (
Linking the strategy adjustment to the target error signal offers a process-based approach to capture flexibility in strategy use. Our setpoint model captures this through the strategy adjustment parameter (F), a weighting term on target error. The NoAT group appears to give more weight to target error than the AT and DAT group. Interestingly, the modeling results indicate that the AT group showed more utilization of the target errors than the DAT group. We assume this arises because the AT group eventually offset the relatively large drift to restore on-target accuracy. In contrast, the DAT group never corrected for drift, suggesting that the weight given to target errors for this group was nearly zero.
It is important to highlight one difference in how we conceptualize changes in the rate of strategy adjustment (F) compared to changes in the rate of adaptation (B). Adjustments in a strategy can occur on very fast timescale; for example, once instructed, participants were able to immediately offset the full rotation. Variation in F refers to the rate at which participants change where to aim. In contrast, B reflects a gradual process, reflecting the rate of change in a system designed to reach a desired location.
In many sensorimotor adaptation tasks, variable learning rates are used to model the substantial variability observed in individual learning curves. In a similar manner, our setpoint model captures individual differences in strategy utilization by varying the strategy adjustment rate (F). Nonetheless, this formulation does not adequately capture the full range of behavior observed in the current study. In particular, this approach is insufficient to account for abrupt changes in performance. For example, the learning profile shown in
An alternative approach to model strategy change could be derived from models of reinforcement learning
A reinforcement learning approach based on a discrete set of strategies is problematic with the current data set. At one extreme, one might suppose that such values could take on the locations of the aiming targets (e.g., 0° and 45°), and perhaps some intermediary points (e.g., 22.5°, the point halfway between two aiming targets). At the other extreme, the set might consist of a large set of values. Choosing a sparse set of potential actions will result in more abrupt changes in performance, while choosing a finer set of potential actions will allow for more gradual changes. Studies designed to explore reinforcement learning models generally use a limited set of choices and performance thus entails discrete shifts in behavior. In our task, reach direction spans a continuous space, and in fact, for most of our participants, the changes in performance were gradual. Future experiments that constrain the set of potential actions and manipulate reward may be better suited for employing a reinforcement learning perspective to explore strategy change.
Qualitative changes in performance may also indicate that the participants have fundamentally changed their conceptualization of the task. For example, rather than view the task goal as one involving reaching to targets, the participant may have switched to an orientation in which the task goal involved mastering a game in which the hand is a tool
The reconceptualization hypothesis would predict that peak drift should equal or be greater than the aftereffect. This follows from the idea that adaptation of the internal model should cease at the time the task goal changes from reaching to tool mastery. Once the participant switches from learning about their arm to learning how to play the visuomotor game, then there the internal model would not continue to learn. The target error gains emphasis and the aiming error falls out. As such, the aftereffect should equal the drift value or be lower if there is some time-dependent decay of the adaptation effects
While this hypothesis is plausible, there are also some limitations. First, it is important to keep in mind that in almost all adaptation studies, the only visual signals are the target location and a feedback cursor. Under such conditions, aftereffects are prominent, indicating adaptation of an internal model and not just learning a game. One would have to assume that tool conceptualization was more pronounced in the present study because of the strategic instructions. Second, our estimate of the aftereffect is actually larger than the peak drift for two of the three groups (
While future research will be required to explore the mechanisms of strategy change, the current study advances our understanding of the interactions that arise between explicit, strategic processes and implicit, motor adaptation. Consistent with Mazzoni and Krakauer
The study was conducted according to the principles expressed in the Declaration of Helsinki and the protocol was approved by the University's IRB. Thirty right-handed participants with no known neurological conditions were recruited from the University of California research participation pool. All participants provided informed consent prior to the start of the experiment.
The participant was seated in front of a table with her right hand comfortably positioned on a table surface. A horizontal, back-projection screen was positioned 48 cm above the table and a mirror was placed halfway between this screen and the table surface. The displays were presented via an overhead projector. By having the participant view the mirror, the stimuli appeared to be presented on the table surface. The mirror occluded vision of the hand; thus, feedback, when provided, was given in the form of a small red circular “cursor”. Movements were tracked by a 3D motion tracking system (miniBIRD, Ascension Technology, Burlington, VT, USA). A sensor was placed on the tip of the index finger, and position information was sampled at 138 Hz. The miniBirds have an approximate spatial resolution of 0.05 cm.
On each trial, the participant made a horizontal reaching movement to a visually displayed target, sliding their hand along the surface of the table. The target was defined by the appearance of a green circle at one of eight possible locations and the eight locations were separated by 45° on a virtual ring with a radius of 10 cm, centered on the starting position. The targets were not at cardinal directions, but started at 22.5° and increased in 45° steps. Participants were instructed to move quickly and were not provided with online visual feedback during the movement. Once the hand crossed the virtual target ring, a stationary red feedback cursor was displayed for 1000 ms. Subsequent to the feedback interval, the participant was visually guided back to the starting location. A white circle appeared, with the diameter corresponding to the distance of the hand from the starting position. The participant was trained to move so as to reduce the diameter of this circle. When the hand was within 10 pixels (8.8 mm) of the starting position, the circle changed to a cursor that the participant then moved into the start location. When this position had been maintained for 500 ms, the next target appeared. The target, start region, and feedback cursor were all 8 pixels (7 mm) in diameter.
Testing began with a familiarization block in which participant was trained to make rapid reaching movements from the start location toward the target location (
Following the familiarization block, participants were trained to use a 45° clockwise (CW) strategy. Eight blue circles (termed “aiming target”— see below), indicating the possible target locations, were always visible during these trials; one of these turned green to indicate the target. The participants were instructed to aim to the neighboring CW blue circle (
The strategy-only block was followed by a 40-trial baseline block in which participants were instructed to reach directly towards the green target. The feedback remained veridical and thus, the participant's goal was to align the feedback cursor with the green target.
Following these 40 trials, a visuomotor rotation was introduced without warning. For these trials, the position of the feedback cursor was shifted −45° (CCW rotation) from the actual hand position (
At the end of this rotation (plus strategy) block, there was a brief pause so that the experimenter could instruct the participants the rotation would no longer be present and that they should resume moving directly to the cued target location. For the next eight trials (1/target location), no endpoint feedback was provided. The purpose of this short block was to quantify the aftereffects of the rotation training, while not inducing any learning based on visual errors
The experiment concluded with a washout block that was identical to the baseline block. The rotation remained off and participants were reminded to continue reaching towards the green target. The feedback cursor was again visible, now providing an error in terms of the distance between this cursor and the green target.
There was a short temporal delay (less than 1 minute) between the blocks so that the experimenter could load the new block.
The participants were divided into three experimental groups. The only difference between the groups was the status of the blue circles, the visual landmarks that provide an aiming target during the strategy-only and rotation blocks. In the Aiming-Target group (AT group; 4 Female/6 Male, ages 19–25), the blue circles were present throughout the experiment. In the No Aiming-Target group (NoAT group; 5 Female/5 Male, ages 18–28), the blue circles were never presented except for the strategy-only practice block. In this block, the blue circles were presented on half of the trials to assist the participants in learning where 45° was in relation to the cued, green target. During the other blocks (familiarization, baseline, rotation, and washout blocks), the blue circles were absent at all times.
The third group was the Disappearing Aiming-Target group (DAT group; 5 Female/5 Male, ages 19–23). For these participants, the blue circles were presented at the start of the trial, followed shortly by a green target at one of the locations. The blue circles remained visible until the hand had been displaced 1 cm from the starting position (approximately 30 ms into the movement) at which point they disappeared. Thus, the aiming targets were not visible when the feedback cursor appeared. As in the other conditions, the green target remained on the screen until the end of the feedback interval. We opted to use blue circles on a black background as the visual landmarks to minimize visual aftereffects for the DAT group.
Kinematic information was analyzed with Matlab (MathWorks, Natick, MA). Movement duration was defined as the interval from when the hand was 1 cm from the start position until it passed through the virtual target ring (10 cm radius). We determined the heading of the hand at the point of intersection and used this to compute the endpoint hand angle, defined as the difference between this heading and a straight line connecting the starting position and the target (green circle except for the strategy-only block). When there was no rotation, the target error was identical to the endpoint hand angle. When the rotation was present, the target error was the endpoint hand angle plus 45°. The angular endpoint error was used to infer the motor plan (plus noise) since the movements were made without on-line feedback and at a speed that minimized corrective movements. Since there was a substantial difference between groups in terms of drift, we measured the aftereffect relative to the target location.
For the analyses of movement accuracy, movements within each block were averaged over 10-trial bins. However, we did not bin the first two movements when the rotation was first introduced (pre-strategy), nor did we bin the first two movements after the strategy was introduced. Rather, these two movement pairs were averaged separately to quantify error introduced by the rotation and the initial success of the participant using the strategy, respectively.
A key dependent measure in this study is the magnitude of the drift exhibited during the rotation block. Estimating peak drift is difficult, not only because of noise in performance, but also because some participants exhibited non-monotonic drift functions. To minimize these problems, we used a boostrapping
We used a similar method to compute the aftereffect. Here we focused exclusively on the first 8 trials following the end of the strategy plus rotation phase, trials in which no visual feedback was provided. The bootstrapping method here produces only a slightly different estimate of the aftereffect compared to a simple averaging across the observed data from these 8 trials.
To quantify the deadaptation rate during the washout phase, we fit an exponential function
To statistically evaluate the results of the bootstrapping procedures, the mean statistics of each resampled iteration were calculated and then used to determine p values
Occasionally participants did not move to the cued, green target (on baseline and washout blocks), mistakenly implemented the strategy in the wrong direction (i.e., went CCW instead of CW on rotation blocks), or moved to a location far from the target. We eliminated trials in which the movement heading was more than three standard deviations from the mean for that block. This resulted in an average removal of less than 1% of the movements per participant and the number of such erroneous movements was similar across the three groups (F2,27 = 1.58, p = 0.22).
The Nelder–Mead method or simplex method
A, B, and E, the parameters characterizing the internal model memory, adaptation gain, and strategy memory, were fit for all the groups collectively. K the parameter characterizing the availability of the aiming target (strategic aiming location) and F, the influence of target errors, were estimated separately for each group through bootstrapping. The group's averaged data was computed by resampling with replacement the participant pool, repeating this 1000 times, and fitting the setpoint model (Eqns 8–11) to each resampled average.
We would like to thank Azeen Ghorayshi for help with data collection. Thanks to John Schlerf for setting up the experimental equipment, Arne Ridderikhoff for help with data analysis, and Greg Wojaczynski for helpful comments. We are grateful to John Krakauer for his many comments throughout the course of this project and for suggesting the reconceptualization hypothesis.