Fig 1.
(A) Participants wear an HTC VIVE headset along with the handheld controllers to immerse themselves in a virtual room (B, C). First person view of the virtual environment at the beginning of a trial. (C) Top down view of the virtual environment. (D, E) Trial timeline for No Feedback (D) and Feedback (E) trials. At the start of each trial they face the door of the room and turn through −α degrees with visual feedback present. Visual feedback is then removed (gray squares) and they must turn back α degrees to face the door again. At the end of the turn participants stop at heading angle θt and report their confidence ς by adjusting the size of a red rectangle. The only difference between the No Feedback and Feedback conditions is the presence of a brief flash of visual information part way through the turn in the Feedback condition (E). Overall participants completed 300 trials of the Feedback condition and 100 trials of the No Feedback condition over the course of the experiment (F).
Table 1.
Glossary of key variables in the paper.
Fig 2.
Schematic of the Path Integration model.
During path integration, participants keep track of a probability distribution over their heading, which is centered at mean mt. To respond they compare this estimated heading to their remembered target location, A, halting their turn when mt = A. The experimenter observers neither of these variables, instead we quantify the measured error as the difference between the true target angle, α, and the true heading angle, θt.
Fig 3.
Model predictions for the Path Integration, Kalman Filter, Cue Combination, and Hybrid models.
In (A-C) the red lines correspond to the mean of the response error predicted by the model. In (D) the two lines correspond to the mean response when the model assumes the feedback is true (red) and false (blue). The thickness of the red and blue lines in (D) corresponds to the probability that the model samples from a distribution with this mean, i.e. ptrue for red and 1 − ptrue = pfalse for blue.
Fig 4.
Schematic of the Kalman Filter model.
Similar to the Path Integration model, this models assumes that participants keep track of a probability distribution over their heading that, before the feedback, is centered on mean mt. When the feedback, f, is presented, they combine this visual information with their path integration estimate to compute a combined estimate of heading . They then stop turning and register their response when
, their remembered target. As with the Path Integration model, none of these internal variables are observed by the experimenter, who instead measures the error as the difference between the true target, α, and heading angle θt.
Fig 5.
Schematic of the Cue Combination model.
The Cue Combination model combines two estimates of heading, the path integration estimate, mt, and the Kalman Filter estimate, , to compute a combined estimate
. The response is made when this combined estimate matches the remembered target A.
Fig 6.
Schematic of the Hybrid model.
The Hybrid model bases its estimate of heading, , either on the path integration estimate or the Kalman Filter estimate. Here we illustrate the case where the model chooses the Kalman Filter estimate. The response is made when the hybrid estimate matches the remembered target angle.
Table 2.
Parameters, their ranges and values, in the different models.
The presence of a parameter in a model is indicated by either a check mark (when it can be fit on its own), a ratio (when it can be fit as a ratio with another parameter), or = 1 when it takes the value 1.
Fig 7.
Error vs target angle for the No Feedback condition.
Each plot corresponds to data from one participant and plots are ordered from most negative slope (top left) to most positive slope (bottom right). The red circles correspond to human data, the solid blue to the mean error from the Path Integration model fit, and shaded blue area to the mean ± standard deviation of the error from the Path Integration model fit.
Fig 8.
Parameter values for the Path Integration model fit to the No Feedback data.
Histograms show the distribution parameter values across participants. The counts are the number of participants, whose fitted parameter values fall within each bin.
Fig 9.
Examples of human behavior on the feedback trials.
Fig 10.
(A) BIC scores for each model relative to the BIC score for the Hybrid model for each participant. For each model, each circle corresponds to one participant. Positive numbers imply the fit favors the Hybrid model, negative numbers imply that the fit favors the other model. A ΔBIC value of > 10 (indicated by the dotted line) is considered “very strong” evidence implied by very high posterior odds (P(MHybrid ∣ Data) > 0.99) [48]. (B) The number of participants best fit by each model. 28 out of 30 participants were best fit by the Hybrid model, suggesting that this model best describes human behavior.
Fig 11.
Comparison between data and the Hybrid model for four participants.
Four participants’ data (open grey dots) are overlaid hybrid model’s mean responses when the model assumes the feedback is true (red) and false (blue). The size of the dots corresponds to the probability that the model samples from a distribution with this mean, i.e. ptrue for red and 1 − ptrue = pfalse for blue.
Fig 12.
Computed Kalman gain for all participants and all trials.
The Kalman gains computed for each trial for each participant are shown as gray dots. The mean Kalman gain and 95% confidence intervals are shown in red.