Article Figures & Data
Figures
- Figure 1.
Predictions of OFC theory on mean trajectories of a point mass moving in force fields. The objective is to arrive at the target by 0.45 s, while minimizing motor costs. a, When there are no forces acting on the mass, the optimal policy is a straight-line path (dashed line). When a velocity-dependent field pushes the mass to the right, the optimal policy is not a return to the straight path but an apparent overcompensation to the left. The amount of overcompensation depends on the accuracy of one's model of the force field. If the field is described by a viscous matrix, D, this accuracy refers to α, where D̂ = αD. The labels “1”, “0.8”, etc. refer to the value of α. The arrows are a schematic representation of the force field during an idealized reach. b, The subplots show the forces along the x- and y-direction that the controller produces (fx and fy) under an optimal policy (1 in a) compared with forces required to move the mass in a minimum-jerk path. The controller overcompensates early into the movement and undercompensates at peak velocity. This results in a total force ∫fTfdt that is ∼16% less than the minimum-jerk (straight line) path. c, The optimal policy when the controller's forward model is uncertain about the strength of the field. Left, The shaded area represents the SD of the force vectors that the mass would encounter along its path to the target. Middle, The optimum policy produces less overcompensation as the SD of the field increases from zero (σ0), to a small (σS = 0.2), to a larger (σL = 0.3) value. Therefore, overcompensation is a good policy only if one is sure that the field will be present. In these simulations, α = 0.8. Right, The speed profiles of the optimal policy (normalized to the maximum speed in the null condition) for different field variances. As the field becomes more variable, the optimal plan is a faster start, resulting in a reduced speed as the mass nears the target.
- Figure 2.
Reaching in a deterministic curl field. Subjects learned to reach in a single direction in a CW or CCW velocity-dependent force field that pushed the hand perpendicular to its direction of motion. The objective was to arrive at the target by 0.45 s. a, The arrows are a schematic representation of the force field during an idealized reach. The plots show hand paths of two representative subjects in the CW and CCW groups. The average trajectory in the null set and the first, third, and the average of the final 50 trials on days 1–3 are labeled. The dashed line is the trajectory in the null field, measured on the first day. b, Success rates (probability of arriving at target in time). Null, The last 50 trials in the null field; 1st and 3rd, the first and third trials of training on day 1; Day 1, Day 2, and Day 3, the last 50 trials of training on each day. c, A measure of overcompensation (maximum difference along the x-axis from the path of null trials). For the CCW group, this measure is positive to the left of the null trajectory and zero otherwise. For the CW group, this measure is positive to the right of the null trajectory and zero otherwise. Error bars are SEM.
- Figure 3.
Reaching in a stochastic curl field f = Dẋ. On each trial, D was drawn from a normal distribution such that f = D̄ẋ + Dδẋ, where D̄ = [0, 13;−13, 0] Nm/s (same value as in the deterministic field) and δ are a normally distributed scalar random variable with zero mean and SD σ = 0.3. The data for the deterministic and stochastic groups are labeled as σ0 and σL. a, Mean hand path (averaged for the last 50 trials of training and across subjects) for the two groups. The dashed line represents the across-group average hand path in the null condition. In the stochastic field, the overcompensation disappeared (trajectories are mean ± SE). b, Overcompensation of the hand path as measured with respect to the null trajectory. The measure refers to the maximum perpendicular displacement to the left of the null trajectory. Test, An extra set of 50 trials that is included in day 3, in which field variance was reset to zero. c, A measure of the entire shape of the hand paths in the two groups. A1–A2 areas are shown on the right. The dotted line is the trajectory in the null field. d, Success rates (arrival at target in time) for the two groups. Null, The last 50 trials in the null field; First, the first 50 trials in the force field; Day 1, Day 2, and Day 3, the last 50 trials of training on each day. e, f, Hand speed corresponding to the hand paths shown in a, normalized to the peak of hand speed in the null condition. In a zero variance field, speeds returned to near baseline, whereas in the high variance field, speeds remained elevated. All error bars are SEM.
- Figure 4.
Same as Figure 3, except for a CCW stochastic curl field. The data for the deterministic and stochastic groups are labeled as σ0 and σL. a, Mean hand path (averaged for the last 50 trials of training and across subjects) for the two groups. The dashed line represents the across-group average hand path in the null condition (trajectories are mean ± SE). b, Overcompensation of the hand path as measured with respect to the null trajectory. The measure refers to the maximum perpendicular displacement to the right of the null trajectory. Test, An extra set of 50 trials that is included in day 3, in which field variance was reset to zero. c, A measure of the entire shape of the hand paths in the two groups. Areas A1–A2 are show on the right. The dotted line is the trajectory in the null field. d, Success rates (arrival at target in time) for the two groups. Null, The last 50 trials in the null field; First, the first 50 trials in the force field; Day 1, Day 2, and Day 3, the last 50 trials of training on each day. e, f, Hand speed corresponding to the hand paths shown in a, normalized to the peak of hand speed in the null condition. All error bars are SEM.
- Figure 5.
Reaching in a stochastic environment in which mean force is zero. Here, the environmental forces were f = Dẋ, where D = [0, 13;−13, 0] Nm/s and δ are a normally distributed random variable with zero mean and variance σ2. The task is to reach to the target in 0.6 s. a, Left, Schematic representation of a low-variance zero-mean curl field. Middle, Average hand paths in the deterministic (σ0, dashed line) and small variance (σS, solid line) conditions. Right, The speed profiles. Data are for the last 50 trials in each condition. b, Same as in a, except for a high-variance condition. c, Peak speed for the two stochastic conditions during the experiment.
- Figure 6.
Reaching through a via-point in a stochastic zero-mean field. The task was to have the hand at the via-point target at 400 ms and at the final target at 1.0 s. a, The trajectory produced by the optimal controller when the environment had zero (σ0), small (σS), or large (σL) variance. The hand path was always straight. However, the speed profiles showed a segmentation of the movement with increased variance, slowing the hand as it approached the via-point. b, Mean trajectories (last 50 trials of training) of subjects in the σ0 and σS conditions. c, Trajectories in the σ0 and σL conditions. Gray bars are SEM.
Additional Files
Supplemental Data
Files in this Data Supplement:
- supplemental material - Supplemental Material
