The Author has declared that no competing interests exist.
Conceived and designed the experiments: RvB. Performed the experiments: RvB. Analyzed the data: RvB. Contributed reagents/materials/analysis tools: RvB. Wrote the paper: RvB.
Motor learning is driven by movement errors. The speed of learning can be quantified by the learning rate, which is the proportion of an error that is corrected for in the planning of the next movement. Previous studies have shown that the learning rate depends on the reliability of the error signal and on the uncertainty of the motor system’s own state. These dependences are in agreement with the predictions of the Kalman filter, which is a state estimator that can be used to determine the optimal learning rate for each movement such that the expected movement error is minimized. Here we test whether not only the average behaviour is optimal, as the previous studies showed, but if the learning rate is chosen optimally in every individual movement. Subjects made repeated movements to visual targets with their unseen hand. They received visual feedback about their endpoint error immediately after each movement. The reliability of these error-signals was varied across three conditions. The results are inconsistent with the predictions of the Kalman filter because correction for large errors in the beginning of a series of movements to a fixed target was not as fast as predicted and the learning rates for the extent and the direction of the movements did not differ in the way predicted by the Kalman filter. Instead, a simpler model that uses the same learning rate for all movements with the same error-signal reliability can explain the data. We conclude that our brain does not apply state estimation to determine the optimal planning correction for every individual movement, but it employs a simpler strategy of using a fixed learning rate for all movements with the same level of error-signal reliability.
Over the last few decades, many studies have examined whether optimality principles can explain human motor behaviour. Although different frameworks have been used, such as optimal (feedback) control (for reviews, see:
We will address this question for the example of determining the learning rate in motor learning. When we produce a movement error, this error can be used to improve planning of future movements. The learning rate is the proportion of the error by which planning is corrected. The learning rate does not need to be constant but could depend on factors such as the reliability of the error signal and the uncertainty of the motor system’s own state. The problem of determining the learning rate has similarities with the engineering problem of estimating the state of a system through noisy observations. Every new observation can be used to improve the state estimate, but the extent by which the estimate should be adjusted depends on the reliability of the new observation and on the uncertainty of the previous state estimate. The more reliable the observation and the more uncertain the previous state estimate, the larger the adjustment should be. Under certain conditions, such as that the system dynamics are linear and known and the noise is white and Gaussian, the Kalman filter
If the motor system uses a Kalman filter to determine its learning rate, the learning rate would not only be optimal on average, but it would be optimal in every individual movement. The aim of this study is to determine whether this is the case. The standard way to estimate learning rates is to use perturbations that disturb the motor performance, and to analyze how motor planning changes in response to induced errors. However, subjects in this paradigm face a dual task as they should both estimate the source of each error and determine an appropriate correction
It is not possible to obtain reliable estimates of the learning rate for individual movements because effects of motor noise cannot be distinguished from planning corrections in individual movements. It is nevertheless possible to test whether subjects used a Kalman filter for every individual movement in a series to the same target, as the Kalman filter makes specific predictions for the serial correlations of movement endpoints, and for how the mean squared movement error will evolve during a series.
We will first present the results of the experiment. We will then show how the Kalman filter can be used to make optimal planning corrections in this paradigm. Next, we will demonstrate that the observed behaviour is not consistent with the predictions of the Kalman filter. Finally, we will show that the data can be explained by a simpler model in which the learning rate is fixed during a session, but varies with the error-signal reliability.
Subjects made 30 successive movements to the same target in each series. One session consisted of 24 series of the same condition, each with a different target. All targets were located at 10 cm distance from a fixed start location, in equally spaced directions. Subjects could not see their hand during the movement, but they received visual information about the movement endpoint immediately after each movement (see
Subjects were seated at a table, and had no direct vision of the table and their arm because that was blocked by a black cloth (not shown) and a mirror that was placed midway between the tabletop and a projection screen. An LCD projector (not shown) projected images onto this screen. When the subject looked in the mirror, he saw the images at the location of the tabletop. In the shown situation, the subject just started the movement from the start position (pink disc) to the target (yellow disc).
The start position (pink disc), the targets (yellow discs), all the endpoints (small dots) and their 95% confidence ellipses of a representative subject (SG). Red and blue colours are used for the endpoints for different targets in alternating order. Asterisks denote the endpoints of the first movement to a target.
The overall picture in condition M (
In all conditions, the endpoint of the first movement to a target often differed considerably from later ones. This suggests that planning of the first movement to a target was often inaccurate. The fact that only the first endpoint differed implies that the error in this movement was used to adjust planning of the next movement. In conditions H and M, the error signal was visual, and it was reliable enough to reduce the error. In condition L, the error signal arose by comparing the felt finger location to the seen target position. Idiosyncratic biases in the proprioceptive sense of finger location relative to visual targets
Learning curves were constructed to quantify how quickly subjects shifted their endpoints in the beginning of a series towards the steady-state position. We calculated the Mahalanobis distance (“the squared number of standard deviations that an endpoint differs from the mean endpoint of the series”, see
Colored lines represent the average (across all subjects) per condition, whereas the black line denotes the mean of all conditions. The dashed line at 2 shows the expected value if the endpoint in the first trial does not, on average, differ more from the mean endpoint than the endpoints in later trials.
In conditions M and L, subjects also changed their endpoints in a couple of movements (
The observed serial correlations are plotted in
In condition L, the autocorrelations at several lags greater than 1 are also positive (
In summary, we found that both the time constant of the learning curve and the lag 1 autocorrelation of the endpoints increased with increasing error-signal uncertainty. Since the time constant and the autocorrelation increase when smaller error corrections are made, these results confirm the earlier finding
We used the Kalman filter to determine the optimal planning correction for individual movements. The task of motor planning is to generate motor commands that will bring the finger to the target. The substantial errors in the first movement to a target (
Let
where
where
We next assume that the planned motor command of the movement just executed will serve as a basis for the planning of the next movement, while a (yet to be determined) planning correction
The actual endpoint is unknown to the subject. It can be eliminated from the above equations to yield:
The first equation can be viewed as a state equation with state
We will now use the Kalman filter to determine the planning corrections
Hats denote estimates and the minus symbol indicates that these are a priori values. The measurement update equations give the a posteriori values that are obtained after the endpoint is observed:
Here,
The planning correction for the next movement should correct for the difference between the estimated planned aim point and the target location
When we substitute this expression in the first time update equation (Eq. 5a), we obtain:
This equation shows that for this planning correction, every movement is planned such that it is expected to be accurate. The second measurement update equation (Eq. 6b) then becomes:
This equation shows that after the endpoint has been observed, the estimated planned aim point of the movement just executed is corrected by an amount that is proportional to the sensed error (the difference between the sensed endpoint and the target location). When we substitute this into Eq. 7, we find that the planning correction is equal to:
The planning correction is thus proportional to the sensed error, and the Kalman gain
To complete the specification of the Kalman filter, we have to choose the initial values of the state estimate and its error covariance. The planned aim point of the first movement in a series depends on the initial state estimate. The fact that the first movement could be quite inaccurate in each series (
The model assumes that the dynamics of the system are linear, that all noise is Gaussian, and that planning corrections are proportional to the sensed error. Previous studies
We tested whether the Kalman filter can explain the data by evaluating whether it can reproduce the observed learning curves and autocorrelations. The predictions depend on the various covariance matrices defined above. Since it is not possible to obtain accurate estimates of all of these matrices, we followed a different approach in which we essentially determined whether any set of values of the covariance matrices could reproduce both the observed learning curves and the autocorrelations. To reduce the number of free parameters, we assumed that all covariance matrices were diagonal (this is justified by the observation that all endpoint ellipses had their major axis roughly aligned with the movement direction), we used literature values or estimates obtained in a control experiment for the error-uncertainty covariance matrix Σ
For each subject and each condition, we determined the values of the free parameters that minimized the difference between the predicted and observed values of the initial value of the learning curve, the learning-curve time constant, and the lag 1 and lag 2 autocorrelations of the Extent and Direction component (see
Condition |
|
|
|
H | 0.35±0.09 | – | 2.75±0.33 |
M | 0.45±0.10 | 17.6±0.7 | 2.57±0.49 |
L | 0.32±0.08 | 8.3±3.5 | 7.84±2.15 |
The serial correlations predicted for condition H (
In summary, the Kalman filter predicts faster correction for initial errors than observed and it predicts different autocorrelations for the Extent and Direction component whereas the observed ones do not differ. We conducted a sensitivity analysis to examine whether these failures of the model can be the result of incorrect assumptions in the parameterization of the covariance matrices. In this analysis we repeated the analysis above several times, where each time the value of one or two parameters was doubled or halved. The parameters that were varied were: Σ
|
|
Time cst. | ACF(1)Ext | ACF(1)Dir | |
Observed | 0.81 | 0.002 | −0.050 | ||
Baseline | 0.35 | 2.75 | 0.56 | −0.042 | −0.043 |
Σ |
0.35 | 2.75 | 0.57 | −0.042 | −0.043 |
Σ |
0.35 | 2.75 | 0.56 | −0.042 | −0.044 |
AR(Σ |
0.35 | 2.75 | 0.57 | −0.042 | −0.040 |
AR(Σ |
0.35 | 2.75 | 0.56 | −0.041 | −0.042 |
AR(Σ |
0.39 | 2.63 | 0.54 | −0.042 | −0.043 |
AR(Σ |
0.30 | 2.94 | 0.59 | −0.041 | −0.041 |
Σ |
0.40 | 2.29 | 0.59 | −0.042 | −0.042 |
Σ |
0.08 | 2.96 | 0.77 | −0.010 | 0.044 |
AR: aspect ratio.
For all of these simulations, we assumed
The sensitivity analysis for condition M (
|
|
|
Time cst. | ACF(1)Ext | ACF(1)Dir | |
Observed | 1.04 | 0.102 | 0.103 | |||
Baseline | 0.45 | 17.6 | 2.57 | 0.65 | 0.032 | 0.158 |
Σ |
0.22 | 14.1 | 2.80 | 0.82 | 0.062 | 0.190 |
Σ |
0.66 | 17.5 | 2.39 | 0.57 | 0.013 | 0.121 |
AR(Σ |
0.56 | 17.4 | 2.45 | 0.62 | 0.058 | 0.124 |
AR(Σ |
0.30 | 19.1 | 2.78 | 0.71 | 0.008 | 0.178 |
Σ |
0.63 | 17.5 | 2.36 | 0.58 | 0.011 | 0.115 |
Σ |
0.22 | 14.1 | 2.80 | 0.80 | 0.063 | 0.190 |
AR(Σ |
0.39 | 18.9 | 2.69 | 0.67 | 0.003 | 0.170 |
AR(Σ |
0.49 | 16.5 | 2.58 | 0.65 | 0.063 | 0.130 |
AR(Σ |
0.55 | 17.4 | 3.02 | 0.60 | 0.039 | 0.217 |
AR(Σ |
0.52 | 15.6 | 2.78 | 0.65 | 0.059 | 0.118 |
Σ |
0.56 | 17.9 | 1.98 | 0.68 | 0.039 | 0.182 |
AR: aspect ratio.
|
|
|
Time cst. | ACF(1)Ext | ACF(1)Dir | |
Observed | 1.29 | 0.314 | 0.295 | |||
Baseline | 0.32 | 8.3 | 7.84 | 0.80 | 0.200 | 0.354 |
Σ |
0.12 | 5.1 | 6.65 | 1.43 | 0.205 | 0.305 |
Σ |
0.73 | 13.9 | 3.61 | 0.59 | 0.147 | 0.340 |
AR(Σ |
0.38 | 7.6 | 4.61 | 0.85 | 0.255 | 0.335 |
AR(Σ |
0.39 | 10.6 | 7.53 | 0.78 | 0.179 | 0.419 |
Σ |
0.72 | 13.9 | 3.69 | 0.60 | 0.148 | 0.339 |
Σ |
0.10 | 3.6 | 8.80 | 1.55 | 0.219 | 0.318 |
AR(Σ |
0.50 | 12.9 | 4.21 | 0.74 | 0.148 | 0.410 |
AR(Σ |
0.29 | 7.2 | 6.32 | 0.88 | 0.240 | 0.316 |
AR(Σ |
0.76 | 15.0 | 4.46 | 0.62 | 0.241 | 0.427 |
AR(Σ |
0.36 | 6.0 | 5.63 | 0.94 | 0.283 | 0.260 |
Σ |
0.71 | 15.9 | 2.91 | 0.72 | 0.236 | 0.414 |
AR: aspect ratio.
In summary, the sensitivity analysis demonstrates that the failure of the Kalman filter to explain the data cannot be the result of making incorrect assumptions about the underlying covariance matrices for conditions H and M, whereas this model can explain the results of condition L only if at least three parameters are changed to extreme values. Together, this suggests that the motor system does not use a Kalman filter to determine the learning rate in every individual movement.
To obtain a better understanding of the actual learning rate, we compared the observed behaviour to the predictions of a second model. This model is almost the same as the Kalman filter, but rather than using the time-varying, optimal Kalman gain as learning rate, it uses a learning rate
The only difference between these equations and equation 11 of the Kalman filter is that the Kalman gain has been replaced by learning rate
We tested the EPAPC model in the same way as the Kalman filter: by examining whether it can reproduce the observed time constants and autocorrelations. The same parameters
Condition |
|
|
|
|
H | 0.20±0.05 | – | 0.33±0.03 | 0.40±0.05 |
M | – | 13.3±3.5 | 0.29±0.04 | 0.33±0.04 |
L | – | 7.2±2.0 | 0.15±0.03 | 0.29±0.06 |
These results suggest that subjects used a fixed learning rate for all trials in an experimental condition. The estimated learning rates for the Extent and Direction component in condition H were 0.33±0.03 and 0.40±0.05 (mean of all subject ± standard error), respectively. These values were not significantly different (two-sided paired
We conducted a sensitivity analysis to determine how the estimates of the learning rates depend on the assumptions made regarding the covariance matrices. We varied the same parameters as for the sensitivity analysis of the Kalman filter. For conditions M and L we also halved and doubled the assumed value of
|
|
|
Time cst. | ACF(1)Ext | ACF(1)Dir | |
Observed | 0.81 | 0.002 | −0.050 | |||
Baseline | 0.33 | 0.40 | 0.197 | 0.83 | 0.005 | −0.049 |
Σ |
0.33 | 0.40 | 0.195 | 0.85 | 0.004 | −0.048 |
Σ |
0.33 | 0.40 | 0.195 | 0.84 | 0.006 | −0.050 |
AR(Σ |
0.33 | 0.40 | 0.195 | 0.84 | 0.006 | −0.050 |
AR(Σ |
0.33 | 0.40 | 0.195 | 0.84 | 0.005 | −0.049 |
AR(Σ |
0.27 | 0.50 | 0.155 | 0.79 | −0.006 | −0.026 |
AR(Σ |
0.42 | 0.32 | 0.252 | 0.83 | 0.015 | −0.063 |
Σ |
0.29 | 0.53 | 0.369 | 0.82 | −0.017 | −0.042 |
For all of these simulations, we assumed
AR: aspect ratio.
|
|
|
Time cst. | ACF(1)Ext | ACF(1)Dir | |
Observed | 1.04 | 0.102 | 0.103 | |||
Baseline | 0.29 | 0.33 | 13.3 | 0.95 | 0.058 | 0.084 |
0.21 | 0.25 | 10.9 | 1.21 | 0.039 | 0.066 | |
0.41 | 0.43 | 19.1 | 0.75 | 0.089 | 0.111 | |
AR(Σ |
0.34 | 0.33 | 10.6 | 0.89 | 0.029 | 0.066 |
AR(Σ |
0.27 | 0.34 | 17.0 | 0.96 | 0.073 | 0.104 |
Σ |
0.28 | 0.31 | 10.6 | 1.00 | 0.060 | 0.069 |
Σ |
0.30 | 0.35 | 16.5 | 0.91 | 0.061 | 0.108 |
AR(Σ |
0.27 | 0.34 | 14.9 | 0.97 | 0.071 | 0.091 |
AR(Σ |
0.30 | 0.29 | 12.1 | 0.97 | 0.058 | 0.087 |
AR(Σ |
0.24 | 0.49 | 15.5 | 0.86 | 0.063 | 0.117 |
AR(Σ |
0.34 | 0.20 | 11.9 | 0.99 | 0.060 | 0.043 |
Σ |
0.24 | 0.37 | 10.2 | 1.01 | −0.016 | 0.081 |
AR: aspect ratio.
|
|
|
Time cst. | ACF(1)Ext | ACF(1)Dir | |
Observed | 1.29 | 0.314 | 0.295 | |||
Baseline | 0.15 | 0.29 | 7.2 | 1.32 | 0.249 | 0.293 |
0.17 | 0.36 | 3.1 | 1.25 | 0.196 | 0.359 | |
0.20 | 0.28 | 18.8 | 1.21 | 0.317 | 0.298 | |
AR(Σ |
0.17 | 0.27 | 5.6 | 1.30 | 0.249 | 0.261 |
AR(Σ |
0.15 | 0.29 | 10.5 | 1.36 | 0.242 | 0.319 |
Σ |
0.15 | 0.27 | 6.0 | 1.35 | 0.234 | 0.234 |
Σ |
0.15 | 0.28 | 11.6 | 1.38 | 0.255 | 0.321 |
AR(Σ |
0.15 | 0.29 | 8.4 | 1.37 | 0.237 | 0.308 |
AR(Σ |
0.17 | 0.28 | 5.8 | 1.28 | 0.247 | 0.269 |
AR(Σ |
0.14 | 0.34 | 15.8 | 1.25 | 0.209 | 0.308 |
AR(Σ |
0.21 | 0.26 | 2.9 | 1.20 | 0.298 | 0.324 |
Σ |
0.23 | 0.41 | 3.2 | 1.08 | 0.176 | 0.352 |
AR: aspect ratio.
We will now look in more detail at some other aspects of the observed learning and compare these to the predictions of the EPAPC model.
A comparison of the middle and right columns of
The curves represent the estimated ACF(1) of the Extent and Direction components, as found by simulations for a range of series lengths for each experimental condition. The black dashed line indicates the series length used in the experiment (30).
Although we found that the learning rate was not optimal for each individual movement (i.e., the data are not consistent with the Kalman filter), it is possible that learning rates were optimal under the restriction that they were the same for all trials of the same condition, as in the EPAPC model. With optimal, we mean that they had the value that minimized the endpoint variance. To examine whether this was the case, we derived an expression for the endpoint variance Var(
Endpoint variance (Eq. 28) is plotted as a function of the learning rates in the Extent and Direction component. The best estimates (mean of all subjects) of the model parameter were used to generate these plots. The value of
The learning rates estimated from the data (indicated in
It is surprising that learning rates were near-optimal in two conditions, but not in the third. This could be a result of drift in the proprioceptive sense of hand location in condition L. Although care was taken to minimize such drift by giving subjects visual feedback about their finger location at the beginning of each trial
We analyzed the time-series statistics of repeated reaching movements with different levels of error-signal reliability to determine the learning rate used by the motor system for updating motor planning on the basis of observed errors. We found that the learning rate increases with increasing error-signal reliability, which agrees with the results of earlier studies
One could argue that the comparison between the Kalman filter and the EPAPC model is not fair because the EPAPC model had one free parameter more. However, we parameterized the Kalman filter such that it had the highest possible number of free parameters (three): The variance of the process noise, of the measurement noise and of the initial state estimate fully determine the time constant of the learning curve and the steady-state lag 1 autocorrelation. The fact that this model is unable to reproduce both the time constant and the autocorrelation simultaneously demonstrates that subjects cannot have used a Kalman filter to determine the learning rate in every individual trial. This is confirmed by the sensitivity analysis, as this analysis showed that the Kalman filter is also unable to reproduce the data when the underlying variances are varied by large amounts. Extending the Kalman filter from a single-state to a two-state model in which the two states have different learning and retention rates
We first emphasize that we cannot exclude that learning rates were not completely fixed per condition. They could have decreased at the beginning of a series, but to a smaller extent than predicted by the Kalman filter. Wei and Körding
A next question is why our motor system chooses the specific learning rate that is chooses for a given error-signal reliability. It may be chosen to minimize the endpoint variance as it was close to the, for this purpose, optimal value in two of the three conditions. However, it was not optimal for the condition with the lowest error-signal reliability. We showed that this difference cannot be the result of proprioceptive drift, and the sensitivity analysis suggests that it cannot be explained by incorrect assumptions about model parameters either. Why did the learning rate differ from the optimal value in this condition? One possibility is that subjects used an incorrect estimate of the error-signal reliability, and used a learning rate that was optimal for this incorrect value. Using Eq. 29, one can show that this would mean that subjects overestimated the error-signal variance of the Extent component by more than a factor 2, and underestimated that of the Direction component by more than a factor 10. It is unlikely that subjects misestimated these variances by such large amounts as studies on visual-proprioceptive integration
We found that the learning rate was the same for the Extent and Direction components for conditions H and M. This confirms that the scalar learning rate that was used in the model of van Beers
Our main conclusion is that our brain does not determine the optimal learning rate for every individual movement, as a Kalman filter would do. Instead, the learning rate is approximately the same for all movements with the same level of error-signal reliability. The average behaviour is thus near optimal, but individual movements are not optimal. This conclusion applies to the particular case of determining the learning rate in motor learning, but the issue is relevant to many other cases in the sensorimotor domain as well. Examples include feedback control of on ongoing movement, integration of sensory information and bimanual coordination. Since neural algorithms that produce behaviour that is optimal for every individual movement may be different, probably more complicated, from algorithms that produce behaviour that is only optimal on average, much insight into our motor system can be gained from addressing the issue for other sensorimotor tasks. The present results suggest that our motor system may not try to optimize individual movements but cares more about the average behaviour.
Eight subjects (three female, five male, 18–24 years old) participated in all experimental conditions. None of them reported any sensory or motor deficits, and all had normal or corrected-to-normal vision, reported being right handed, and were unaware of the purposes of the study.
All subjects gave verbal informed consent (which was then documented) before participation. All experiments were conducted in agreement with the ethics and safety guidelines of the Science Faculty of Utrecht University, where the experiment was conducted, and was part of a program that received blanket approval of the Medical Ethical Test Committee of the University Medical Centre Utrecht. All data were encoded and analyzed anonymously.
The same set up was used as in
The task was to move the tip of the right index finger from a start position to visual targets. The start position was a pink disc (4 mm radius) at a fixed location approximately 35 cm in front of the waist. A red cursor (a 4 mm radius disc) was shown at the fingertip location when it was within 3 cm from the start position. This enabled subjects to place their finger quickly and accurately on the start position, and it also prevented drift of the perceived finger location throughout an experimental session
Condition H (high error reliability): A red disc (4 mm radius) was shown at the endpoint location. It was shown alongside the target so that the error signal was highly reliable. A score was awarded based on the distance from the target (see
Condition M (medium error reliability): A cloud of 15 red circular dots (0.8 mm radius) was shown. The dot locations were drawn independently from a circular Gaussian distribution with the actual endpoint as the mean and a standard deviation of 15 mm (see
Condition L (low error reliability): Subjects received no visual feedback about their movement endpoints (see
In all conditions, the visual feedback, if any, was shown for 1 s. After that, subjects moved their finger back to the start position to begin the next trial.
A session consisted of 24 series of 30 movements each, all in the same experimental condition. The targets were located at 10 cm distance from the start position in equally spaced directions. A blocked design was used, in which the same target was used for all movements in a series. The target of the first series was randomly chosen exactly to the left of right of the start location. Each later target direction differed 105 degrees from the previous direction in the counter clockwise direction. There were breaks of at least 10 seconds between series. At the start of a session, each subject practiced the task in the condition of that session for several minutes before starting the experiment proper (with a different target than in the first series). A session lasted approximately one hour. Each subject performed one session of each condition, each on a different day. The order of conditions was randomized between subjects.
We analyzed the two-dimensional movement endpoints. A small fraction of the movements (0.56%, 0.42% and 0.21% in conditions H, M and L, respectively) was discarded from the analysis because the recording had failed. Endpoints were transformed into an Extent component (the component parallel to the vector from the start location to the mean endpoint of the series) and a Direction component (the component orthogonal to the Extent component). To characterize error-corrective learning, we determined two measures: Mahalanobis distance and serial correlations.
The Mahalanobis distance was calculated to construct learning curves. At first sight, a plot of the mean error magnitude as a function of the movement number in the series could serve as a suitable learning curve. However, since endpoint distributions were anisotropic (
where
A learning curve was constructed for each subject by calculating the Mahalanobis distance of each endpoint, and then averaging these across series as a function of the trial number in the series. Time constants of the learning curves, and their 95% confidence intervals, were estimated for individual subjects using nonlinear least-squares regression. We fit exponentials of the form
Whereas learning curves are informative about correction for large errors in the beginning of a series, the serial correlations focus on error correction in the “steady state” when errors are small. Serial correlations were calculated from the last 25 endpoints of each series, to avoid them being influenced by the correction for the large initial errors. Serial correlations express the statistical relationship between the endpoints of movements separated by a certain lag (number of movements)
Two models for trial-by-trial motor learning are described in the Results section. We ran Monte Carlo simulations to determine the predictions of each model. Each simulation consisted of 2,000 sets of 24 simulated series of 30 movements, corresponding to 2,000 subjects performing a full experiment. Random vectors were drawn from Gaussian distributions to simulate the effects of planning, execution and sensory noise, as specified in the Results section.
For condition H, the sensory-noise covariance matrix Σ
The potentially most powerful method to estimate the parameters of a linear dynamic system from time-series data is maximum likelihood estimation using the expectation-maximization algorithm
To find the best parameter estimates, we ran simulations for a range of parameter values. For instance, to fit the EPAPC model to the data of condition H,
The best estimates of the parameter values for the Kalman filter are given in
We use
The expected endpoint is
Then we have from (19):
The covariance matrix function
Γ(0) is the covariance matrix of the deviations. Since the deviations differ a constant vector from the endpoints, it is also the covariance matrix of the endpoints. When we define the endpoint variance Var(
Note that the variance and autocorrelations are independent of sensory bias
The fact that all matrices in the above equations are diagonal (see
where: