Figure 8. Speed and distance between neural trajectories reflect animals’ internal estimates.

a) Interval estimates derived from EKF model (same format as in Figure 1g). b) Interval estimate inferred from the speed of neural trajectories $\left({\widehat{t}}_e\right)$ as a function of t_s. Left: the procedure for inferring ${\widehat{t}}_e$ for each t_s. The colored traces show ${\mathbf{t}}^{\left[t_s\right]}$ as a function of t^[800] for a random draw from the bootstrap distribution (see Online Methods). To infer ${\widehat{t}}_e$ for each t_s, we measured the slope of the regression line relating ${\mathbf{t}}^{\left[t_s\right]}$ to ${\mathbf{t}}^{[800]}\left(\mathrm{\Delta }{\mathbf{t}}^{\left[t_s\right]}/\mathrm{\Delta }{\mathbf{t}}^{[800]}\right)$ and then scaled the slope by the duration of the reference interval (800 ms). The dotted line is the regression slope for t_s=900 ms. Middle and right: ${\widehat{t}}_e$ as a function of t_s inferred from the speed of neural trajectories in the S2-S3 and S3-Go epochs. The colored dots show multiple estimates of ${\widehat{t}}_e$ derived from bootstrapping (n=100). The solid curves show interval estimates derived from EKF model fits (t_e(S2) in the middle panel and t_e(S3) in the right panel). Unity indicates perfect estimates of t_s and the horizontal line represents the mean of the prior. Insets: difference in RMSE between models that assume that the speed of neural trajectories reflects t_s versus t_e(S2) (middle) or t_e(S3) (right). Positive values indicate neural data more accurately captured by EKF. c) The value of ${\widehat{t}}_e$ inferred from the distance between neural trajectories as a function of t_s. Left: the procedure for inferring ${\widehat{t}}_e$ for each t_s. The colored traces show ${\delta }^{t_s}$ as a function of time for a random draw from the bootstrap distribution (see Online Methods). We inferred ${\widehat{t}}_e$ for each t_s using a linear transformation of the average distance $\left(\alpha \langle {\delta }^{t_s}\rangle +\xi \right)$. The dotted line shows the average distance, $\langle {\delta }^{t_s}\rangle$, for t_s=900 ms. Middle and right: ${\widehat{t}}_e$ as a function of t_s inferred from the distances between neural trajectories in the S2-S3 and S3-Go epochs (same format as panel b). Insets: differences in RMSE between models assuming the distances of neural trajectories reflect t_s versus t_e(S2) (middle) or t_e(S3) (right). d) Comparison of the degree of BIAS in speed (top) and distance (bottom) during the S2-S3 and S3-Go epochs. Dashed lines are unity. See also Supplementary Figure 13.