Invariant errors reveal limitations in motor correction rather than constraints on error sensitivity

Implicit sensorimotor adaptation is traditionally described as a process of error reduction, whereby a fraction of the error is corrected for with each movement. Here, in our study of healthy human participants, we characterize two constraints on this learning process: the size of adaptive corrections is only related to error size when errors are smaller than 6°, and learning functions converge to a similar level of asymptotic learning over a wide range of error sizes. These findings are problematic for current models of sensorimotor adaptation, and point to a new theoretical perspective in which learning is constrained by the size of the error correction, rather than sensitivity to error.

(best fit A and learning rates, B e , for each clamp size). The simulated functions predict indistinguishable adaptation rates for the 3.5° and 15° clamp conditions over the initial portion of the perturbation block, in contrast to the markedly different rates observed in the actual behavior. (e) Simulation using estimates of B derived from the behavioral data. The functions approximate the early separation between learning functions, but predict divergent asymptotes. (Note: y-axis scaling was changed to fit data). (f) Based on a model from Tanaka et al. (2012), a saturating error function can capture the nonlinear effects of error size on adaptation. However, similar to the fits in d, this model also predicts similar early adaptation rates for the three error sizes, as well as a lower asymptote for the 1.75° clamp. 95% CIs for parameter estimates in brackets. Lines and shading denote mean and SEM, respectively.

Aftereffect data
In Experiment 1, there was an effect of error size on the aftereffect data obtained after 40 cycles of exposure to the clamped feedback (F 6,77 =3.25, p=.007, ƞ 2 =.20, Fig. 1g). However, this effect was driven by the 1° group, and likely due to the trivial reason that the design did not include a sufficient number of training cycles for this group to reach asymptote. A Tukey-Kramer posthoc test revealed the only significant differences were between the 1° group and larger clamp sizes. Furthermore, excluding the 1° group, there were no reliable differences among all of the other pairwise comparisons (all t 22 <2.07, all p>.05). Indeed, the magnitude of the aftereffect was not correlated with error size (r 82 =0.04, p=0.71).
These results suggested that asymptotic adaptation may be independent of error size, a hypothesis which was more rigorously tested in Experiment 2.

Comparisons between Experiment 1 and Experiment 2
Overall, the magnitudes of final aftereffects in Experiment 2 were considerably larger than the aftereffect of matched groups in Experiment 1. One reason for this is, of course, the increase in the number of clamp cycles in Experiment 2, which had 160 compared to the 40 cycles used in Experiment 1. As can be seen in Supplementary Figure 3, the magnitude of adaptation is still rising after 40 cycles of clamped feedback.
As an exploratory analysis, we performed a between-experiment 2-way ANOVA, with one factor being experiment (1 or 2) and the other error size, limiting this to the three conditions in Experiment 1 that were included in Experiment 2 (1.75°, 3.5° and 15°). We compared the aftereffect data after 40 clamp cycles (i.e., final aftereffects from Experiment 1 compared to first no feedback probe in Experiment 2). There was a significant effect of experiment (F 1,60 =5.27, p=.03, ƞ 2 =.07), a marginal effect of clamp size (F 2,60 = 2.8, p=.07, ƞ 2 =.08), and no interaction (F 2,60 =.57, p=.57, ƞ 2 =.02). These results suggest that some performance differences may have arisen from the methodological changes introduced in Experiment 2, namely, eliminating frozen endpoint feedback and utilizing a quicker method for finding the start position before each trial (see Methods). This resulted in a shorter total trial duration in Experiment 2, likely reducing the decay of learning associated with delay 16 . This point highlights that asymptotic magnitude is context specific, as adaptation is highly sensitive to variables such as temporal delay between hand and cursor movement 17,18 , viewing angle 19 , and the nature of visual feedback (e.g., online versus endpoint feedback) 20,21 . As such, our claim that the asymptote is independent of error size holds for a given context; the value will shift in other contexts, although the shift will be uniform for all error sizes.

Supplementary Note 3.
Computational models of sensorimotor adaptation predict divergence of asymptotic adaptation when early adaptation rates are different.
A single rate state-space model can generally provide a reasonable account of the performance changes observed in standard adaptation studies in which the feedback is contingent on the movement 4-6 . This model takes the following form: ŷn = -zn (2) z n represents the state estimate of the perturbation on trial n. The learning rate, B, corresponds to the proportion of the error, e n , that is corrected for, and A is a retention factor which represents the proportion of the state retained from one trial to the next. The reach direction relative to the target on trial n, ŷ n,, is opposite in sign to the state estimate. Note that unlike standard adaptation experiments in which the error changes over trials, the error remains constant with the clamp method.
We evaluated different state-space models, focusing on the data from Experiment 2 (reproduced in Supplementary Fig. 2a). For our model fitting and simulation procedures we applied standard bootstrapping techniques, constructing group-averaged hand angle data 1000 times by randomly resampling with replacement from the participant pool. Using Matlab's fmincon function, we estimated the retention and learning parameters which minimized the least squared error between the bootstrapped data and model output (ŷn). All values in brackets represent the 95% bootstrapped C.I. of the resampled means.
We first verified that a single rate state-space model, which in its basic form assumes single, constant A An alternative class of models posit that performance changes reflect the activity of multiple learning processes, each operating in a similar manner but over different time scales 7,8 . For example, in a dualrate state-space model of adaptation 8 , the output (5) is the sum of two single rate models, one in which both learning and forgetting occur over a faster time scale (3) than the other (4). Despite this added flexibility, this model is similarly constrained as the single rate model. As previously explained, the asymptotes for each state will still be equal to (Be)/ (1-A), meaning different size clamps eventually reach different asymptotes. Indeed, our simulations show that a dual rate state-space model will generate functions ( Supplementary Fig. 2c) which are qualitatively similar to the single rate model ( Supplementary   Fig. 2b).
where A f < A s , B f > B s z = z f + z s (5) As noted in the main text, various studies have suggested that the learning rate, B, may vary with error size [9][10][11][12][13][14] . To examine this class of models, we used a single rate state-space model (eqns. 1 and 2) with a common retention factor and three learning rates, one for each error (clamp) size. We again obtained in Supplementary Fig. 2d In summary, even models that account for variation in sensitivity to error fail to capture the combined effects of divergent early adaptation rates and invariant asymptotic performance (see Moazzezi 2018 for a comprehensive analysis of this issue) 22 .