Fig 1.
A: Illustration of a choice trial. The participant was asked to make a center-out reaching movement (dashed white arrow) toward one of three potential targets (blue circles with central dot). During reaching, the cursor position was hidden (dashed white circle shows hidden location) and a white expanding circle indicated the distance (but not direction) of the cursor from the central location (cross). B: At the end of the movement, visual feedback of the static cursor position was displayed. The cursor position was displayed offset from the actual position (dashed circle, not shown to participants) by an angular perturbation. C: Perturbation range for targets with different absolute angular distances from the least-noisy direction (ϕ). The perturbation was composed of a rotation offset α (constant in all directions, here shown for α = 20°) and a target-direction-dependent noise. The noise was drawn from a uniform distribution with a range that linearly increased with the angular distance of the selected target from direction ϕ. D: Possible feedback range (pink) given the selected target (blue dots) and the reach direction (dashed red lines). For each participant, a direction ϕ (solid black line) was randomly selected and was fixed throughout the experiment. The farther the selected target is from this direction, the greater the variability of the cursor perturbation.
Fig 2.
The interplay of learning the bias angle and learning the least-noisy direction.
A: Mean absolute feedback error (red-yellow gradient) as a function of target direction (abscissa) and the reach angle relative to the target (ordinate) for a visuomotor bias angle α and a least-noisy direction ϕ. The dashed black line shows the solution manifold for which the expected signed error is 0, that is the reach angle (−α) compensates for the bias angle. Absolute error increases away from the least-noisy direction (ϕ) and away from the solution manifold, with the black circle marking the optimal solution given the optional target directions in the trial (θ1, θ2 and θ3). Green and purple curves show example manifolds (reach angle as a function of target direction) for two notional participants who have selected different distributions of target choices (each tick in the top two rows denoting a selected target direction). B: Corresponding mean absolute feedback error as a function of selected target direction for the solution manifold and for the two notional participants shown in Panel A.
Fig 3.
Examples of participants’ task performance.
Target choices on individual trials were shown by the blue dots and color of the yellow-to-red lines associated with each dot represents the feedback error amplitude. Participant A performed at an average level in learning the least-noisy direction (direction ϕ, thick black line) among participants who experienced 0° bias. Participants B and C performed the best and worst in learning direction ϕ among participants who experienced the 40° bias.
Fig 4.
Participants’ reaching and target selection behavior across bias angles.
A: Reach angle error as a function of trial number (in consecutive 10-trial bins). Solid curves show empirical results (mean ± SEM across participants) and dashed curves show model fits. B: Reach angle error as a function of the deviation from the participants’ preferred direction for probe trials. Solid curves show empirical results (45° bins, mean ± SEM across participants) and dashed curves show model fits. C: Change in target selection error as a function of trial number (in consecutive 10-trial bins). Solid curves show empirical results (mean ± SEM; for clarity SEM only shown at the end of each block) and dashed curves show model fits. D: Mean change in target selection error across the experiment. Colored dots show individual participant’s data. Black solid curve shows mean ± SEM across participants. Black dashed curve shows mean of model fits.
Fig 5.
Model of reaching and target selection.
Top row: model of reaching. A: On each trial, the model maintains an estimate of the reach perturbation (blue curve) for all target directions. The reach angle counteracts the expected perturbation in the selected target direction (blue dot). B: After movement, the observed perturbation (red dot) is used to calculate the model prediction error (difference between the observed and predicted perturbation, purple). C: The model updates the reach perturbations using the prediction error. The dashed curve shows the previous estimate and the solid curve shows the updated estimate. D: Generalization of the learning is controlled by a kernel function (orange) centered on the selected target direction. Bottom row: model of target selection. E: The model maintains the expected loss (absolute feedback error) as a function of target direction. The probability of selecting each potential target (green numbers) is calculated by applying a softmax function to the expected loss values of the three targets. F: After movement, the observed loss is used to calculate the model prediction error (difference between the observed and predicted loss, purple). G: The model updates the loss function using the prediction error. H: Amount of the loss function update in different directions is controlled by a kernel function (orange) centered on the selected target direction.
Fig 6.
Predictive performance of the target selection model.
A: Target choices of two typical participants. Black dots show the selected target direction on each trial. The target choices of participant D gradually became densely concentrated as trial number increased, while the target choices of participant S were relatively scattered until the end of the experiment. The blue gradient shows model-predicted probability of selecting a target if it is located in that specific direction and on that specific trial. The predicted probabilities matched nicely to the actual distribution of target choices for both participants D and S. B: Probabilistic fraction correct of the model predictions on the last 50 target choices of each participant. Boundary of gray shaded area shows the estimated upper bound on the predictive performance as a function of concentration of target choices. Blue curve was fit to all data points (using Gaussian Process Regression) under different bias angles. The mean predictive performance of the model was significantly above chance. The two typical participants D and S (see Panel A) are marked in the figure.
Fig 7.
Simulated target selection results across bias angles.
A: Change in target selection error as a function of trial number (in consecutive 10-trial bins). Dashed curves show mean across all simulated runs in each bias angle condition (100 runs for each participant using that participant’s best-fit parameters). B: Sample run under a single set of model parameters for all bias angle conditions. Dashed curves show mean across all simulated runs in each bias angle condition (1000 runs under each different bias angle, all using a single set of parameters fit to data pooled across all participants in all conditions in Experiment 1). Solid curves show empirical results (mean ± SEM across participants). Both types of simulations qualitatively reproduced the major behavioral result that the learning of the least-noisy direction was inhibited by large bias angles. C: Estimated model parameters for each participant. Upper row shows parameters for the reach model and lower row shows parameters for the target selection model.
Fig 8.
Comparison of different target selection models.
A: Average probabilistic fraction correct of model predictions on the last 50 target choices in each bias angle condition and for each model. Dashed line shows value at chance level (1/3). Error bars show SEM across participants. B: Mean change in target selection error across the experiment for different bias angle conditions. Black curve shows empirical results (mean ± SEM across participants). Blue curves of different brightness show mean across all simulated runs in each bias angle condition (100 runs for each participant) for different target selection models combined with the same reach model.
Fig 9.
A: Mean reach angle error in the last 50 training trials for all participants in Experiment 2. Participants with a mean error less than 20° were categorized as good learners (purple filled bars). B: Reach angle error by target direction in the last 50 training trials. Thin curves show data for individual participants. For good learners (purple), reach angle error was reduced to a relatively low level across all directions near the end of the training trials. Curves obtained by applying a periodic kernel smoother. Thick curves show mean for the good learners and the poor learners. Error shadings show SEM. C: Reach angle error as a function of trial number (in consecutive 10-trial bins, aligned at the onset of choice trials). Solid curves show empirical results (mean ± SEM across participants) and dashed curves show model fits. Good learners (purple) gradually adapted to the bias during training, so that their reach angle errors were smaller than the untrained group (green) during the first 200 choice trials. D: Learning of the least-noisy direction in the good learners (purple) and untrained group (green). Left: Change in target selection error as a function of choice trial number (in consecutive 10-trial bins). Solid curves show empirical results (mean ± SEM; for clarity SEM only shown at the end of each block) and dashed curves show model fits. Right: Mean change in target selection error across the first 200 choice trials. Colored dots show individual participant’s data. Black solid curve shows mean ± SEM across participants. Black dashed curve shows mean of model fits.
Table 1.
Information of participants included in data analyses.
Table 2.
Percentage of trials excluded from data analyses.