Figure 1.
Correction of state estimation following a perturbation.
A: Overhead sketch of a perturbation evoked displacement relative to the prescribed joint location (θ0). The initial change in joint angle following a perturbation is sensed after some delay (thin trace, Δθ(t−δt)). A correction of the actual change in joint angle (thick trace, Δθ(t)) involves a prediction based on the available sensory data combined with an internal model of the perturbation (red arrow). B: Schematic representation of the sensory prediction on the joint displacement plotted as a function of time (numbers are for illustration). The sensory prediction (red arrow) estimates the present state of the limb (thick trace) based on delayed sensory feedback (thin trace) and internal assumptions about the perturbation profile. C: Illustration of overestimation resulting from updating the current state estimate based on an expected movement profile (dashed trace) that follows the same initial displacement as the actual one but diverges during the time interval corresponding to the feedback delay (solid trace).
Figure 2.
Computational models and simulations.
A: Computational model based on a state estimator that ignores feedback delays and directly integrates sensory feedback with prior beliefs about the body state. The controller outputs a motor command (ut) specified by the behavioral goal and motor costs. The forward dynamic model (Motor Prediction) predicts the consequence of the descending motor command (Predicted Consequences, ). The feedback signal (yt+1) is a function of the delayed state vector expressed in f(xt−h+1), where h represents the feedback delay in number of sample times (see Methods). B: Simulation of the model corresponding to panel A on the control of a single joint actuator during unperturbed reaching movements of varying amplitudes (left) and following perturbation loads of three selected magnitudes (right). Displays are the joint angle (top) and angular velocity (bottom). Time 0 corresponds to the onset of the reaching movement, or the perturbation onset (solid line). C: Full model with the sensory predictor highlighted in red. Following a perturbation, the output of the motor prediction still indicates that the joint displacement is zero, but the sensory prediction corrects the estimate of the joint state based on maximum-likelihood principle. This is illustrated by the conditional expectation about the present state, given delayed sensory information (Predicted Current State,
) D: Same as B with the control and state estimation corresponding to panel C.
Figure 3.
A: Overhead representation of the initial joint configuration and of a typical perturbation related movement. The initial joint configuration is shown in black. Hand path from perturbation onset until the first hand-speed minimum (TH) is represented in solid trace. The remaining portion of the corrective movement is shown in dashed trace. B: Illustration of the different torque profiles: black traces illustrate the step perturbations (1 Nm and 3 Nm are displayed with thin traces), ramp-down perturbations in red and ramp-up perturbations in blue. C: Schematic representation of the effect of a ramp-down perturbation profile on the state estimation: the overestimation (gray region) result from the difference between the expected perturbation profile (step-function, dashed) and the actual perturbation (solid). The ramp-down profile was designed to produce an overestimation as illustrated in Figure 1 C. The opposite reasoning applies to ramp up-perturbation that produces an underestimation of the present state of the joint.
Figure 4.
A: Average elbow motion from one exemplar subject when step perturbations were expected (top) and when ramp-profiles were presented in blocked manner (bottom). Displays use the same color code as in Figure 3 B. The three black traces correspond to 1 Nm, 2 Nm and 3 Nm step-torques. The vertical black dashed line is the reversal time averaged across responses to all step-torque perturbations. Blue and red dashed lines represent the reversal times of ramp-up and ramp-down perturbations. The inset shows the initial elbow displacement for the different perturbation profiles and magnitude (average ± SD across subjects). The gray rectangle emphasizes that the initial joint displacement across ramp-up (down) perturbations and the 1 Nm (3 Nm) step were similar for the first ∼10 ms. B: Same as A from the model simulations. C: Timing of the corrective response measured in joint coordinates based on the elbow reversal time (TE, black) and in Cartesian coordinate based on the first hand speed minimum (TH, gray) for each perturbation profile (1, 2 and 3 Nm step torques; RD, ramp-down; RU, ramp-up; mean ± SD across subject). Orange dots represent the joint reversal time computed from model simulation. Times were measured for each subject and simulations relative to the mean across responses to the step torque perturbations. Single (double) star(s) indicate significant differences at the level P<0.05 (P<0.01). D: Effect of context on the timing of the corrective response for the ramp-down and ramp-up torque profiles with the same color code as in panel A.
Figure 5.
State estimation following unexpected perturbations.
A: Illustration of the actual (solid) and estimated (dashed) state variables. From top to bottom, displays are the external torque (TE), the joint velocity and the joint displacement. 2 Nm step perturbation is represented in black, ramp-down in red and ramp-up in blue. B: Estimation error for each perturbation profile, defined as the difference between the estimated state and the true state.
Figure 6.
Experiment 1: Muscle responses.
A: Top: Average shoulder (dashed) and elbow (solid) joint displacements following extension torques in three different cases, 3 Nm step torques (black), and ramp-down profiles in catch and blocked conditions (thin and thick red traces, respectively) averaged across subjects. Shoulder motion was identical across all conditions and the corresponding traces are superimposed. Middle: Perturbation-evoked response recorded from an elbow flexor (brachioradialis) with the same color code as in the top panel. Bottom: Difference between the 3 Nm step response and the ramp-down in the catch condition (black), and between ramp-down torques in the block and catch (red). Arrows indicate the onset of divergence from the 3 Nm evoked response. B: Response evoked by step-torque perturbations averaged across subjects. The shaded area represents the standard error. The vertical arrow depicts the latest divergence onset across all pair-wise comparisons. C: Changes in evoked response across the conditions where ramp-down (dark gray) or ramp-up (light gray) perturbations were presented in catch or blocked design. The grand average of Brachioradialis and Triceps Lateralis responses was considered for this analysis. Positive designates that the evoked activity decreased in the block condition.
Figure 7.
A: Average hand path from one representative subject following step torques and ramp-down torques sorted by trial n-1. The blue traces are the average across all trials preceded by a step perturbation and the red traces are the average across all trials preceded by a ramp-down perturbation. B: Trial-by-trial modulation of the corrective response depending on the preceding trial for step (open symbols) and ramp-down (filled symbols) flexion and extension perturbations (disk and diamond, respectively). Significant differences in data from single perturbation profile (*) and from all perturbation pooled together (†) are shown at the level P<0.05 (one symbol) and P<0.01 (two symbols). C: Grand average across the four muscles of interest and across subjects of trials sorted by trial n-1 with identical color code as in panel A for step torques (left) and ramp-down (right) torque profiles. The black traces are the differential signals between blue and red responses. The dashed vertical lines represent the different epochs of rapid motor responses (see Methods). D: Binned analysis of the difference in activation across responses preceded by step torques and responses preceded by ramp-down torques for the four muscles of interests following step perturbations (left) and ramp-down perturbations (right). As in panel C, positive designates an increased activity when the trial is preceded by a step perturbation. Bars represent one standard error across the 12 subjects. From dark gray to light gray, displays are Brachioradialis (Br.), Biceps (Bc.), Triceps Lateralis (Tl.), and Triceps Long (Tg.). Stars indicate significant modulation of the corresponding muscle at the level P<0.05 and the dagger (†) indicates significant modulation from all muscle samples pooled together (one symbol, P<0.05; two symbols, P<0.01).
Figure 8.
Effect of muscle pre-activation.
A: Changes in reversal times relative to the average across all step responses from the block condition (similar as in Figure 4 C). Black and gray illustrate catch and block conditions while round and square items correspond to ramp-down and step responses, respectively. B: Response evoked by ramp-down perturbation when these profiles are expected (black) or presented as catch trials (gray). Data are from Brachioradialis and the trace is the average across subjects. Stars indicate that the response averaged in the corresponding time window differed significantly across catch and block conditions (P<0.05). C: Same as B for the responses following step perturbations (2 Nm).
Figure 9.
Effect of unexpected changes in perturbation magnitude.
A: Reversal times and hand speed minima across the tested values of step magnitudes as presented in Figure 4 C. We used the same scale as in Figure 4 C to emphasize that unexpected changes in step magnitude could not account for the effect of ramp profiles on the times of reversal or hand speed minima. B: Perturbation evoked response from Brachioradialis averaged across subjects. The vertical arrows illustrate the latest onset of divergence across all pair wise comparisons based on ROC analysis (35 ms).