**We examined how human subjects acquire and represent models of visuo-motor error and how they transfer information about visuo-motor error from one task to a closely related one. The experiment consisted of three phases. In the training phase, subjects threw beanbags underhand towards targets displayed on a wall-mounted touch screen. The distribution of their endpoints was a vertically elongated bivariate Gaussian. In the subsequent choice phase, subjects repeatedly chose which of two targets varying in shape and size they would prefer to attempt to hit. Their choices allowed us to investigate their internal models of visuo-motor error distribution, including the coordinate system in which they represented visuo-motor error. In the transfer phase, subjects repeated the choice phase from a different vantage point, the same distance from the screen but with the throwing direction shifted 45°. From the new vantage point, visuo-motor error was effectively expanded horizontally by $ 2 MathType@MTEF@5@4@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIYaaaleqaaaaa@36CA@ $. We found that subjects incorrectly assumed an isotropic distribution in the choice phase but that the anisotropy they assumed in the transfer phase agreed with an objectively correct transfer. We also found that the coordinate system used in coding two-dimensional visuo-motor error in the choice phase was effectively one-dimensional.**

**Figure 1**

**Figure 1**

*isotropy bias*. Similar failure to compensate for the anisotropy in speeded reaching movements was present in the results of Hudson, Tassinari, and Landy (2010), where the anisotropy was artificially introduced into the subjects' visuo-motor error by jittering the display screen. But why should subjects' internal models have such systematic deviations from true? And is the isotropy bias specific to reaching movements, or is it present in other visuo-motor tasks?

*ϕ*(

*x*,

*y*). However, there is little reason to believe that the visuo-motor system uses a Cartesian representation, and—as we will see—our results indicate that a one-dimensional coordinate system similar to polar coordinates (

*r*,

*θ*) but with only the single parameter

*r*better accounts for our data.

*incorrect*, isotropic estimates to the new position. They were wrong, but they were consistent.

**Figure 2**

**Figure 2**

*choice position*), facing the screen, beanbag in her right hand. The task was to choose, given a pair of targets (rectangle and circle), the target that was easier to hit. The time course of the task is shown in Figure 1C. The order of the circle and the rectangle was randomized across trials. Subjects were prompted to answer the question “Which is easier to hit? First or second?” The experimenter recorded subjects' verbal responses by key press.

*new position*, to their right, that had the same distance from the center of the screen but a 45° viewpoint to the screen (Figure 1D, right). As in the choice phase, they were asked to choose which of two targets would be easier to hit if they threw from the new position. The choice trials in the transfer phase (the new position) were the same as in the choice phase (the choice position).

*x*,

*y*). We modeled (

*x*,

*y*) in the throwing task as a bivariate Gaussian random variable centered at (0, 0) with standard deviations that decreased as exponential functions of the trial number

*t*, separately for the horizontal and vertical directions.

*x*,

*y*) and only knew that the endpoint was outside the screen region

*S*. The likelihood that the endpoint

*E*was observed on a specific trial was therefore

*t*: where

*θ*,

_{x}*κ*,

_{x}*υ*,

_{x}*θ*,

_{y}*κ*, and

_{y}*υ*are free parameters, and the trial number of the

_{y}*i*th bin is defined as the central number of the bin

*t*= 8 + 15(

*i*– 1). With the fitted Equation 4, we could estimate the standard deviations for any trial.

*σ*and

_{x}*σ*are free parameters, denoted

_{y}*T*

_{2}as a logistic function of

*p*

_{1}−

*p*

_{2}following the normalized expected utility model (Erev, Roth, Slonim, & Barron, 2002): where

*τ*> 0 is a temperature parameter determining the randomness of the choice—the lower the temperature, the closer the subject to an ideal observer who always chooses the target that is more likely to be hit. We estimated

*τ*,

*T*

_{2}determined by where

*A*

_{1}and

*A*

_{2}are the areas of the targets

*T*

_{1}and

*T*

_{2}and

*τ*> 0 is a temperature parameter determining the randomness of the choice. We estimated

*τ*using maximum-likelihood estimates. We compared the Gaussian model (Equation 5) with the area-matching model in goodness of fit using nested hypothesis tests (Mood, Graybill, & Boes, 1974, p. 440). If a subject's Gaussian model did not fit better than her area-matching model at the 0.05 significance level, we excluded the subject. Two subjects were excluded at the choice position and one additional subject was excluded at the new position. These subjects might really have used an area-matching strategy so that their choices did not rely on their internal models of visuo-motor error distribution, or they might simply have assumed

**Figure 3**

**Figure 3**

**Figure 4**

**Figure 4**

*t*are denoted

*σ*, and

^{t}*β*.

^{t}*β*> 1). At the end of the training (Figure 3C), the ratio

^{T}*β*of the vertical to the horizontal standard deviation had a mean of 1.84 across the 15 subjects. The ratio varied with subject and changed across training, but except for the first few trials, the ratio for any subject across trials was well above 1 (Figure 3D).

^{T}*β*= 1.17 versus

^{C}*β*= 1.85, paired-sample two-tailed

^{T}*t*test,

*t*(11) = −4.08,

*p*= 0.002. This underestimation could not be explained as a delayed estimation of the true distribution: As shown in Figure 3D, the median vertical-to-horizontal ratio was even larger at earlier stages of training. Subjects were effectively assuming an isotropic distribution, i.e., the value of

*β*was indistinguishable from 1 (one-sample two-tailed

^{C}*t*test),

*t*(11) = 1.12,

*p*= 0.29.

*β*against

^{C}*β*for each subject. The data point of one subject (circled) was outside the regression line of

^{T}*β*over

^{C}*β*of all data points at the 0.05 significance level. When this outlier was excluded, there was a marginally significant correlation between

^{T}*β*and

^{C}*β*, Pearson's

^{T}*r*= 0.56,

*p*= 0.072. The positive correlation between

*β*and

^{C}*β*was higher for the last three quarters of trials in training than in the first one quarter of trials (Figure 5B). That is, the vertical-to-horizontal ratio assumed in subjects' model reflected the true ratio in recent experience, though the magnitude of the true ratio was considerably underestimated.

^{t}**Figure 5**

**Figure 5**

*β*was constantly around

^{N}*β*. This observation was confirmed by the following model comparison procedure.

^{C}*ε*is a zero-mean Gaussian error term whose variance is a free parameter. We fitted the four models using maximum-likelihood estimates to predict the values of

*β*

^{N}. H_{1}was the best among the four models according to the estimated likelihood: When the outlier subject (circled in Figure 5C, same as in Figure 5A) was excluded,

*H*

_{1}was 289 times more likely than the second best model to produce the observed values of

*β*; when the outlier subject was not excluded, the likelihood ratio of

^{N}*H*

_{1}to the second best model was 1,321. We conclude that subjects assumed a different vertical-to-horizontal ratio in their internal model at the new position from that of the choice position, but instead of transforming the ratio they had learned to the new position, they transformed an isotropic model in the objectively correct way to the new position.

*r*= 0.89,

*p*< 0.001), as were

*r*= 0.92,

*p*< 0.001), while the Pearson's correlation between

*r*= 0.58,

*p*= 0.048 (computed for the valid subjects at the choice position; the correlation for the valid subjects at the new position was

*r*= 0.62,

*p*= 0.041).

*p*= 0.084 and

*p*= 0.065, respectively, for the choice and new positions). The two results taken together would be expected no more than

*p*= 0.0055 of the time if the null hypothesis of independence of horizontal and vertical directions were true.

*r*controls both vertical and horizontal standard deviations. Indeed, if the visuo-motor system uses an isotropic representation, then there would be no reason to have two separate parameters

*σ*and

_{x}*σ*for horizontal and vertical variances—they must be equal. Moreover, if the isotropy assumption were correct, then the visuo-motor system could combine estimates of vertical and horizontal deviations across trials to get a more accurate estimate of

_{y}*r*.

*σ*,

^{T}*σ*, and

^{C}*σ*) as the variance estimate for the true distribution and subjects' models. Results based on the horizontal and vertical standard deviations were similar.

^{N}*t*(Pearson's correlation

*p*> 0.52). The standard deviations assumed in subjects' internal models at the new position could not be predicted by those at the old position either: There were no significant correlations between

*p*s > 0.23).

*t*: horizontal rate

**Figure 6**

**Figure 6**

*r*controls both the horizontal and vertical standard deviations of the distributional representation. One possibility is that polar coordinates (

*r*,

*θ*) rather than Cartesian coordinates (

*x*,

*y*) are used in coding two-dimensional visuo-motor error distributions. If so, the isotropy bias is due to the visuo-motor system's bias to base judgments on

*r*alone, ignoring direction

*θ*. This possibility echoes the fact that humans typically use polar coordinates to code spatial locations (Huttenlocher & Lourenco, 2007).

*θ*remains to be determined.

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*Decision**σ*and

^{C}/σ^{T}*t*. There was a significant negative correlation between

*σ*and

^{C}/σ^{T}*p*s < 0.05). In contrast, for any trial

*t*, the correlation between

*σ*and

^{C}/σ^{T}*p*s > 0.41).

*σ*as a linear function of

^{C}/σ^{T}*σ*and

^{C}/σ^{T}*σ*decreased with

^{C}/σ^{T}*σ*negatively correlated with

^{C}/σ^{T}*σ*with

^{C}/σ^{T}*σ*and computed the correlation between

^{C}/σ^{t}*σ*and

^{C}/σ^{t}*t*(1–300). Inconsistent with the prediction of delayed estimation, the correlation was negative for trials as early as trial 5.

*σ*and

^{C}/σ^{T}*t*. There was a significant positive correlation between

*p*s < 0.05). In contrast, for any trial

*t*, the positive correlation between

*p*s > 0.21).