Figure 1.
A) The experimental setup. Subjects steer a cursor by shifts in center of pressure (COP) along the anterior-posterior axis. Noisy feedback of the cursor position (small, medium or large variance) is given while subjects are incentivized to steer the cursor to be close to the midline of the screen (target). B) Subject's movements affect the center of pressure, which is measured by a force plate. The resulting sensor readings then steer the on-screen cursor. Subjects receive noisy visual feedback about the cursor position and react to reduce errors. C) COP, cursor velocity and cursor position are shown as a function of time during one trial for a typical subject (red). The observed feedback (noisy dots) is shown in blue. D) The phase portrait of cursor position and velocity is shown for 10 successive trials. Data from (C) are highlighted in red.
Figure 2.
Task errors across time and across feedback conditions.
A) Average errors across subjects over the course of the experiment binned in blocks of 10 trials. B) The influence of feedback type on task errors. All comparisons between feedback uncertainty levels were significant (one-sided t-test). In both plots errorbars denote SEM across 10 subjects. * denotes p<0.05. *** denotes p<0.001.
Figure 3.
Cross-correlations between process noise and COP.
A) Cross-correlation between the fluctuations in cursor acceleration (process noise, ) and the center of pressure with time lag for each feedback uncertainty level. The inset shows the cross-correlation between fluctuations in the cursor acceleration and the Kalman update in a simulation. The results have been smoothed to mimic postural responses (Gaussian smoothing,
= 250 ms) B) Peak amplitude for each feedback uncertainty level. C) Peak time for each feedback uncertainty level. Confidence intervals denote SEM (N = 10). * denotes p<0.05; *** denotes p<0.001 (one-sided paired t-test).
Figure 4.
A) Distributions of the cursor position (left), cursor velocity (center) and center of pressure normalized by the standard deviation (right), averaged across subjects for the low-gain (top row) and high-gain (bottom row) conditions. In the low-gain condition, note the bimodal distribution of the center of pressure, despite the unimodal distribution of errors. This may indicate a bang-bang-like strategy. B) Policy-maps of the center of pressure averaged across subjects as a function of the true cursor position and velocity for two different levels of feedback uncertainty and across all conditions. Note that in the low-gain condition subject's responses saturate at large cursor velocities and positions. In the high-gain condition responses are much more linear.
Figure 5.
Policies as a function of position.
A) Center of pressure as a function of cursor position for typical subjects in the low and high-gain conditions. Black lines denote median responses for a given range of cursor positions. Red and blue points denote samples along the COP trajectory. B) Average responses across subjects with thin lines denoting the responses of individual subjects. C) The predicted responses from the LQR, Bang-bang, and Non-linear LQR models. Error bars denote SEM across subjects (in B and C) and sample points (in A).
Figure 6.
A) Observed center of pressure for a typical subject and trial along with the center of pressure predicted by each of the three ideal observer models. Note that the linear-quadratic-regulator and the bang-bang controllers produce qualitatively very different estimates. Note also that the non-linear LQR model has some ability to interpolate between the two. B) Cross-validated fraction of variance explained for each model for both the low and high-gain experiments (two-fold cross validation). In the low-gain condition the ideal optimal observer models explain a significantly larger fraction of variance than the PID controller (p<0.05, one-sided paired t-test), and the non-linear LQR explains a significantly larger fraction of variance than all others (p<0.001, one-sided paired t-test). Error bars denote SEM across subjects.