Rapid reaching to a target is generally accurate but also contains random and systematic error. Random errors result from noise in visual measurement, motor planning, and reach execution. Systematic error results from systematic changes in the mapping between the visual estimate of target location and the motor command necessary to reach the target (e.g., new spectacles, muscular fatigue). Humans maintain accurate reaching by recalibrating the visuomotor system, but no widely accepted computational model of the process exists. Given certain boundary conditions, a statistically optimal solution is a Kalman filter. We compared human to Kalman filter behavior to determine how humans take into account the statistical properties of errors and the reliability with which those errors can be measured. For most conditions, human and Kalman filter behavior was similar: Increasing measurement uncertainty caused similar decreases in recalibration rate; directionally asymmetric uncertainty caused different rates in different directions; more variation in systematic error increased recalibration rate. However, behavior differed in one respect: Inserting random error by perturbing feedback position causes slower adaptation in Kalman filters but had no effect in humans. This difference may be due to how biological systems remain responsive to changes in environmental statistics. We discuss the implications of this work.

*L*. Reaching toward the target requires a visual estimate of its location and a movement that will bring the hand to that location. The accuracy of the reach can be assessed by comparing the visual estimates of the target location,

_{ V}, and the reach endpoint feedback,

_{ V}. If the subject observes an error,

_{ V}−

_{ V}, and adjust the visual location estimate and/or the motor command to make the error zero on the next trial:

_{ t}is the observed reach error for the current reach,

_{ t}is the visuomotor mapping estimate on the current reach, and

_{ t+1}is the visuomotor mapping estimate that will be used for the next reach.

*n*is the number of observations in the average. Of course, averaging over many observations would make the system slow to respond to abrupt changes in systematic error. In sum, the optimal strategies for dealing with systematic and random errors are quite different.

*X*

_{ t}, changing in time, the Kalman filter continuously updates the estimate of the mapping by a weighted combination of the most recent observed error and the current mapping estimate based on previous estimates:

*K*is the proportion of the observed error by which the mapping estimate will be adjusted (the Kalman gain). By using multiple observations, weighted by recency, to estimate the mapping error, the impact of random error is reduced. In this way, the system retains the ability to respond to changes in systematic error while not over-reacting to random error.

*K*:

*λ*is the exponential time constant: Faster adaptation is associated with higher gain. There is no analytical solution for

*K*(see 1 for the exact equation at steady state), but to close approximation,

*σ*

_{ $ z ^$}is the uncertainty associated with measuring the location of the feedback (measurement uncertainty) and

*σ*

_{ x}is the uncertainty of the visuomotor mapping that would produce unbiased performance (mapping uncertainty) (Korenberg & Ghahramani, 2002).

*σ*

_{ $ z ^$}and

*σ*

_{ x}should be consistent with the distributions of measurements made by the observer before the step change was introduced. If an observed error is likely to have been caused by an erroneous measurement,

*σ*

_{ $ z ^$}should be large, and the visuomotor system should adapt slowly ( Equations 4 and 5). If the visuomotor motor mapping is changing quickly,

*σ*

_{ x}should be large, and the system should adapt quickly to minimize what is likely to be systematic error. Figure 1 shows the effects of those parameters on the adaptation rate of a Kalman filter. Higher values of

*σ*

_{ $ z ^$}yield slower adaptation and higher values of

*σ*

_{ x}yield faster adaptation. The Kalman filter is provided the statistics, but humans must learn the values of

*σ*

_{ $ z ^$}and

*σ*

_{ x}from the distributions of measurements (Baddeley et al., 2003). We presented many trials before the step change so that human subjects could learn the statistics and set their internal parameters appropriately before having to respond to the step change.

*σ*

_{ $ z ^$}) changes adaptation rate in humans, as it does in a Kalman filter: specifically, whether increasing

*σ*

_{ $ z ^$}causes a slowing of adaptation. To do this, we used the set-up in Figure 2 to examine adaptation rate when the localizability of the visual feedback stimulus was varied. The feedback stimuli were blurry isotropic Gaussian blobs with different standard deviations ( Figure 3A). Using blur to reduce measurement uncertainty has two advantages for analyzing the system experimentally. First, reaching adaptation often occurs quite quickly in humans, sometimes in only a few trials (von Helmholtz, 1867). Fast rates are difficult to quantify from noisy data, so it is advantageous to work with slower adaptation rates. Second, unlike other forms of measurement uncertainty (see Experiment 3), we can use psychophysical techniques to determine exactly how uncertain a given amount of blur makes the measurement of feedback position.

*σ*

_{blur}) and the ability to localize it visually. The just-noticeable difference (JND) increased monotonically with

*σ*

_{blur}( Figure 3B). Thus, blurring had the desired effect of increasing the uncertainty of the measurement of feedback position. Henceforth, our estimate of feedback uncertainty

*σ*

_{ $ z ^$}will equal the measured JND/

*X*

_{ t}, between the reach endpoint and the visual feedback ( Figure 2B) and gained experience with the environmental conditions so their internal parameters could be properly set. In the step phase, we introduced a step change in the mapping: Visual feedback was shifted 8.2° up and to the right. The step persisted until the post-step phase when the original mapping was restored. Data from a full run of 160 trials are shown in Figure 2D. Adaptation was assessed by examining reaching errors during the step phase (see Methods section for why we did not focus on post-step data). For our experiments, reaching error

*E*

_{ t}is the difference between the locations of the visual target and visual feedback on trial

*t*. To quantify adaptation rate, we fit an exponential function separately for the

*x*and

*y*directions to each subject's reaching errors across trials:

*λ*is a time constant,

*b*is a bias term,

*C*is a constant equal to the shift, and

*t*is the trial number. We defined adaptation rate as the time constant (

*λ*) of the best-fitting exponential.

*p*< 1.7*10

^{−27}; sign test). This justifies our use of exponentials to analyze individual subject data and indirectly supports the Kalman filter model.

*b,*presumably manifests a bias to either mislocalize the feedback or reach toward the center of the display screen and was not incorporated in our Kalman filter model. Because we are interested primarily in learning rate and since removing constant bias from the data has been shown not to affect estimates of rate (Cheng & Sabes, 2006), we focus on the time constants required to fit the data and not on the bias.

*σ*

_{ $ z ^$}). Like a Kalman filter, subjects adapted more slowly when the position of the feedback was less certain.

*K*is fully determined by the visuomotor-mapping uncertainty,

*σ*

_{ x}, and measurement uncertainty,

*σ*

_{ $ z ^$}( 1). We set

*σ*

_{ $ z ^$}to the value of the visual JND/

*σ*

_{ x}and determined how the filter's adaptation rate changes with

*σ*

_{ $ z ^$}. For each assumed value of

*σ*

_{ x}, there is a line relating the visual JND to adaptation rate. All the lines have approximately the same slope. The line in Figure 3E shows the behavior of the Kalman filter that fits our data best (

*R*

^{2}= 0.61). The change in adaptation rate observed in humans is similar to the pattern of rate changes in the optimal adaptor. The value of

_{ x}that provides the closest agreement is 0.08° (95% CI = 0.04°–0.11°). Clearly, humans adapted at rates consistent with the Kalman filter as the feedback was made less certain.

*σ*

_{ x}, produces faster adaptation in the Kalman filter ( Equations 4 and 5). As we said earlier, this makes sense. If the mapping changes frequently, the system should adjust its mapping estimate quickly in response to an error because the previous mapping estimate is less likely to be correct. If the mapping has been stable, adjustments should be slow because it is likely that the mapping from previous trials is correct and that the error was due to measurement error.

*N*(0,

*σ*

_{walk}) is a Gaussian random variable with mean 0 and standard deviation

*σ*

_{walk}. In the pre- and post-step phases, (

*x*′,

*y*′) = (

*x*+

*W*+

_{x}, y*W*), where (

_{y}*x*′,

*y*′) were the coordinates of the reach endpoint on the graphics tablet, (

*x*+

*W*+

_{x}, y*W*) were the screen coordinates of the feedback, and

_{y}*W*

_{ x}and

*W*

_{ y}were the horizontal and vertical components of

*W*

_{ t}. After the pre-step trials, step-phase trials were presented with a constant shift between the reach endpoint and the feedback: (

*x*′,

*y*′) = (

*x*+

*W*+ 5.8°,

_{x}*y*+

*W*+ 5.8°). We manipulated mapping variability by changing

_{y}*σ*

_{walk}. Figure 5A shows examples of how the mapping between reach endpoint and feedback changed for two random walks with different values of

*σ*

_{walk}. Figure 5B shows trial-by-trial mapping estimates from two representative subjects in response to two random walks with the same variability. Clearly, subjects adjusted their mapping estimates to follow the walk.

_{ x}is equal to

*σ*

_{walk}, the initial error would be reduced by ∼97% in two trials when the blur is small. Thus, we expect to find a large effect of walk variability with large blur and little if any effect when blur is small.

*σ*

_{walk}, and with small blur, there was no discernible effect. There was also a clear effect of the localizability of the feedback: Adaptation was slower with large blur. Because of the previously mentioned difficulty of measuring changes in adaptation rate at high rates, we could not determine whether human rates slowed with increasing

*σ*

_{walk}in the small-blur conditions of this experiment. But for the conditions in which we can observe rate changes reliably, the pattern of human and filter rate changes were similar. This supports the idea that humans adjust adaptation rate with mapping variability changes in a manner consistent with a Kalman filter.

*σ*

_{blur}) made the feedback difficult to localize and resulted in slower adaptation, an effect probably caused by an increase in the system estimate of measurement uncertainty (

*σ*

_{walk}) resulted in faster adaptation; an effect probably caused by an increase in the system estimate of mapping variability (

_{ x}). There is a third source of variability in rapid reaching (

*σ*

_{perturb}): random trial-by-trial variation in the reach endpoints due, for example, to noise in reach execution (motor noise). We next asked how the system responds to such variations.

_{ t}and the mapping necessary to produce unbiased performance

*X*

_{ t}(Baddeley et al., 2003). Random perturbation should therefore affect the measurement uncertainty parameter

*σ*

_{$z^$}. The adaptation rate of an optimal adaptor, assuming stationary statistics (i.e., constant measurement and mapping uncertainties), should therefore slow down. This is similar to the effect of blurring the feedback (Experiment 1), but unlike blur, the amount of random perturbation cannot be estimated from a single observation, so

*σ*

_{perturb}may have more or less effect depending on how the system estimates measurement uncertainty.

*σ*

_{ x}. This would be non-optimal, however, because with random perturbation the mean of the mapping never changes, so the variance of the change in the mapping producing unbiased performance is unaffected by

*σ*

_{perturb}( 2). In fact, the variance of the change in the mapping yielding unbiased performance is 0. Thus, one expects that random perturbation should not cause an increase the system's estimate of mapping uncertainty.

*x*′,

*y*′) generated feedback at (

*x*+

*P*

_{ x},

*y*+

*P*

_{ y}), where

*P*

_{ t}=

*N*(0,

*σ*

_{perturb}) and

*N*is Gaussian with mean 0 and standard deviation,

*σ*

_{perturb}. As before, the pre-step phase allowed subjects to learn the properties of the mapping and feedback. During the step phase, reach position (

*x*′,

*y*′) generated feedback at (

*x*+

*P*

_{ x}+ 5.8°,

*y*+

*P*

_{ y}+ 5.8°). In Experiment 3a,

*σ*

_{blur}was fixed and small, and the values of

*σ*

_{perturb}were essentially equivalent to the visual JNDs for isotropic and anisotropic blobs in Experiment 1. Interestingly,

*σ*

_{perturb}had no measurable effect on adaptation rate ( Figure 6A). The spatial error profiles across trials were therefore roughly linear whether

*σ*

_{perturb}was anisotropic or not ( Figure 6B). Experiment 3b was similar to Experiment 2 except that the mapping was changed via random perturbation instead of a random walk. The blur of the feedback was also varied. There was again no measurable effect of

*σ*

_{perturb}. Adaptation rates were determined only by feedback localizability: They were fast with small blur and slow with large blur ( Figure 6C). The same pattern of results was observed in Experiment 1 when

*σ*

_{perturb}was 0°.

_{x}is adjusted based on experience with previous trials. It therefore seems unlikely that previous trials are not considered for the estimation of

*σ*

_{$z^$}. (2) Random perturbation may increase the system's estimates of both

*σ*

_{$z^$}and

*σ*

_{x}such that their opposing effects cancel and no measurable change in adaptation rate occurs. This also seems unlikely because there is no obvious reason both estimates should be affected in the same way by random perturbation. (3) The speed at which changes in environmental statistics can be learned may differ for random perturbations and random walks. The rate for learning changes in the statistics should depend on how reliably the changes can be determined. Changes in uncorrelated noise (

*σ*

_{perturb}) that must be estimated from past measurements may be more difficult (i.e., slower) to learn than changes in correlated noise (

*σ*

_{walk}). If this is the case, the pre-step phase of

_{x}. Further research is needed to determine how the visuomotor system learns the statistics of the environment.

*k*estimators providing sensory estimators of environmental property

*S*:

_{c}is the combined estimate and

_{i}is the estimate from the

*i*th estimator (

_{i}=

*f*

_{i}(

*S*)). The resultant,

_{c}, is more precise (less variable) and more accurate (closer to correct) than the individual estimates alone (Ernst & Banks, 2002; Ghahramani et al., 1997; Oruç, Maloney, & Landy, 2003) if the individual estimators are calibrated. Substantial experimental evidence supports the idea that signals from different sensory modalities (Alais & Burr, 2004; Bresciani, Dammeier, & Ernst, 2006; Ernst & Banks, 2002; Gepshtein, Burge, Ernst, & Banks, 2005; Roach, Heron, & McGraw, 2006) and from within a modality (Helbig & Ernst, 2007; Hillis et al., 2002; Landy & Kojima, 2001; Saunders & Knill, 2003) are combined in this fashion.

_{1}and

_{2}produced by estimators

*f*

_{1}and

*f*

_{2}and a current mapping estimate between the estimators of

_{t}. The error with each new observation of

_{1}and

_{2}is

_{t}=

_{1t}−

_{2t}; the filter sets the mapping

_{t + 1}to minimize the error over time (Equation 3). The measurement uncertainty is

*σ*

_{x}, represents how variable over time measurements from two sensory estimators are relative to each other. A long-standing stable relationship between estimates should yield a low value for

*σ*

_{x}; a historically unstable relationship should yield a high value.

^{2}. Subjects' heads were restrained by a chin-and-forehead rest. A graphics tablet (AIPTEK™ Hyper Pen 1200 USB) was placed horizontally below the chin rest and in front of the subject's torso where subjects could not see it. The tablet's active area was 30.4 × 22.8 cm ( Figure 2A).

*σ*

_{blur}) and aspect ratios. Target and feedback stimuli were presented for 500 ms.

*E*

_{ t}was defined as the difference between visual target and feedback locations. To perform the task, subjects had to learn the visuomotor mapping,

*X*

_{ t}, from tablet coordinates to screen coordinates ( Figure 2B). None had difficulty doing so.

*x*′,

*y*′) generated feedback at screen position (

*x*+

*X*

_{ t x},

*y*+

*X*

_{ t y}). In the step phase (50 trials), the feedback was displaced up and to the right by 8.2° from the corresponding position of the reach endpoint: tablet position (

*x*′,

*y*′) produced feedback at screen position (

*x*+

*X*

_{ t x}+ 5.8°,

*y*+

*X*

_{ t y}+ 5.8°). In the post-step phase (50 trials), the step change was removed: tablet position (

*x*′,

*y*′) again generated feedback at screen position (

*x*+

*X*

_{ t x},

*y*+

*X*

_{ t y}) ( Figure 2C).

_{ x}(see Figure 3E).

*σ*

_{blur}= 8° × 8°, 12° × 12°, 16° × 16°, and 20° × 20°. (B) Average adaptation profiles for isotropic feedback stimulus. Light pink represents the data when

*σ*

_{blur}= 20° × 20°, dark pink for

*σ*

_{blur}= 16° × 16°, blue for

*σ*

_{blur}= 12° × 12°, and olive green for

*σ*

_{blur}= 8° × 8°. Because blobs were isotropic within a condition, error was calculated in the direction of the constant shift (rather than in

*x*and

*y*separately). An ANOVA showed a significant effect of blur (

*F*(3,23) = 2.8,

*p*< .047).

_{1}and

_{2}of the same environmental property. Equation 8 in the main text is the maximum-likelihood estimate for a linear system with Gaussian-distributed noise and conditional independence between the estimates. Rewriting that equation:

*X*drifting in time and an estimate of that mapping

_{ t}

^{−}, and the measurement of the mapping at the current time-step is

_{ t}. Again assuming Gaussian noise and conditional independence, the best estimate of the mapping (immediately after the current measurement) is

*w*

_{ $ Z ^$ t}

_{ t}

^{−}yields

*w*

_{ $ Z ^$ t}+

*w*

_{ $ X ^$ t −}= 1 yields

*K*=

*w*

_{ $ Z ^$ t}and defining

_{ t}=

_{ t}−

_{ t}

^{−}:

*u*+

*x,*where

*u*is the average change in mapping and

*x*is the uncertainty associated with the average change in mapping. Assuming unit time steps, the best estimate of the mapping and the variance of that estimate just before the next measurement are

*σ*

_{ $ X ^$ t+1 −}

^{2}=

*σ*

_{ $ X ^$ t −}

^{2}so by Equation A10:

*K*from

*σ*

_{ x}

^{2}and

*σ*

_{ $ Z ^$ t}

^{2}by substituting Equation A11 into Equation A7 and rearranging:

*σ*

_{ $ X ^$ t +}

^{2}, we use the quadratic formula. One positive value is yielded:

*K*

_{ t}, based only on the measurement and mapping uncertainties, once the system has reached steady state.

*u*+

*x,*where

*u*is the expected change in mapping with time, and

*x*is the uncertainty associated with those mapping changes. The expected value of

*u*is 0 whether the mapping change is a random walk or a random perturbation. The difference between random walks and random perturbations appears in

*x*. Changes in mapping do not accumulate over time with perturbations—mapping is not a function of time—so the variance of

*x*is 0 and

*σ*

_{ x}= 0. With random walks, mapping changes do accumulate over time, so

*σ*

_{ x}=

*σ*

_{walk}

*t*

_{2}−

*t*

_{1}= 1; thus, the standard deviation of the mapping estimate

_{ t}should increase by

*σ*

_{walk}from one trial to the next when the mapping is changed via a random walk.

_{filter}and MSE

_{subject}are the mean squared reaching errors of a Kalman filter and human subject, respectively. The filter's parameters,

_{x}and

*σ*

_{$z^$}, were set equal to

*σ*

_{walk}and

*σ*

_{perturb}, respectively. Therefore, as

*σ*

_{perturb}was increased, filter adaptation rate decreased (Equations 4 and 5). Their results are plotted in Figure C1A. When

*σ*

_{walk}was large, efficiency was reasonably high and did not vary with

*σ*

_{perturb}. To explain the constant efficiency at large

*σ*

_{walk}, the authors reasoned that the Kalman filter and humans must have responded similarly to increases in

*σ*

_{perturb}by increasing the

*σ*

_{$z^$}estimate. For small

*σ*

_{walk}, the increasing efficiency with

*σ*

_{perturb}was attributed to motor noise, which was present in humans but not in the model. From this, they concluded that the subjects' adaptation rate slowed with increases in perturbation.

*σ*

_{ $ z ^$}was set equal to

*σ*

_{perturb}, while the human estimate of

*σ*

_{ $ z ^$}was unaffected by

*σ*

_{perturb}, and (2) the human had motor noise while the filter had none. With those assumptions, we conducted a simulation to calculate efficiencies. The results are shown in Figure C1B. When

*σ*

_{walk}was large, efficiency was constant for a wide range of

*σ*

_{perturb}. When

*σ*

_{walk}was small, efficiency increased with increasing

*σ*

_{perturb}. These results are similar to those of Baddeley and colleagues. It is therefore possible that perturbation had no effect on adaptation rates in their experiments, as we observed. We conclude that their observation of constant efficiency under some conditions does not necessarily support the conclusion that the nervous system's estimate of feedback uncertainty is affected by random perturbation.