A Dynamic Bayesian Observer Model Reveals Origins of Bias in Visual Path Integration

Abstract

Path integration is a strategy by which animals track their position by integrating their self-motion velocity. To identify the computational origins of bias in visual path integration, we asked human subjects to navigate in a virtual environment using optic flow and found that they generally traveled beyond the goal location. Such a behavior could stem from leaky integration of unbiased self-motion velocity estimates or from a prior expectation favoring slower speeds that causes velocity underestimation. Testing both alternatives using a probabilistic framework that maximizes expected reward, we found that subjects' biases were better explained by a slow-speed prior than imperfect integration. When subjects integrate paths over long periods, this framework intriguingly predicts a distance-dependent bias reversal due to buildup of uncertainty, which we also confirmed experimentally. These results suggest that visual path integration in noisy environments is limited largely by biases in processing optic flow rather than by leaky integration.

Keywords: Bayesian model; leaky integration; optic flow-based navigation; path integration bias; virtual reality.

Figure 1. Task structure and behavioral response

A. Subjects use a joystick to navigate to a cued target (yellow disc) using optic flow cues generated by ground plane elements (orange triangles). The ground plane elements appeared transiently at random orientations to ensure that they cannot serve as spatial or angular landmarks (Methods). B. Left: The time course of linear (top) and angular (bottom) speeds during one example trial. Time is also encoded by line color. Right: Aerial view of the subject’s spatial trajectory during the same trial. C. Top: Aerial view of the spatial distribution of target positions across trials. Bottom: Subject’s movement trajectories during a representative subset of trials. D. Left: Target location (solid black) and subject’s steering response (colored as in B) during a representative trial. Red arrow represents the error vector. Right: Vector field denoting the direction of errors across trials. The tail of each vector is fixed at the target location and vectors were normalized to a fixed length for better visibility. The grayscale background shows the spatial profile of the error magnitude. E. Top right: Comparison of the radial distance $\stackrel{\sim }{r}$ of the subject’s response (final position) against radial distance r of the target across all trials for one subject. Bottom right: Angular eccentricity of the response $\stackrel{\sim }{\theta }$ vs. target angle θ. Black dashed lines have unity slope and the red solid lines represent slopes of the regression fits. Left: Geometric meaning of the quantities in the scatter plots. F. Radial and angular biases were quantified as the slopes of the corresponding regressions and plotted for individual subjects. Error bars denote 95% confidence intervals of the respective slopes. Dashed lines indicate unbiased radial or angular position responses. Solid diagonal line has unit slope. See also Figures S1-S2 and Movies S1-S2.

Figure 2. Dynamic Bayesian observer model

Subjects combine noisy sensory evidence from optic flow with prior expectations about self-motion speed to perform probabilistic inference over their movement velocity. The resulting noisy velocity estimates are integrated to generate beliefs about one’s position. Bias in position estimation might come about from two extreme scenarios. Slow-speed prior (green): A velocity prior that favors slower speeds coupled with perfect integration. Leaky integration (purple): A uniform prior over velocity coupled with leaky integration. For simplicity, this schematic shows the one-dimensional case. For general planar motion, both linear and angular velocity must be inferred and integrated to update position in two dimensions.

Figure 3. Model comparison and validation

A. Posterior probability distribution over position implied by the best-fit slow-speed prior (left, green) and leaky integrator (right, purple) models, swept over time during an example trial for the subject with the largest bias. The distributions at different time points were rescaled to the same height, so these plots reflect this subject’s relative beliefs about his location across the duration of the trial. Target location (yellow dot) and the actual trajectory (black line) have been overlaid. Yellow ellipses depict an isoprobability contour (68% confidence interval) of the model posteriors over position at the end of the trial. B. Vector field of errors in the mean estimate of final position across trials, for the two models. Error vectors of both models were rescaled to minimize overlap. The spatial profiles of the error magnitude (distance between target and mean estimated final position) for the two models are shown by the grayscale background. C. Subject’s internal estimates of the radial distance $(\widehat{r},\mathit{left})$ and angle $(\widehat{\theta },\mathit{right})$, not to be confused with the subject’s actual response $\stackrel{\sim }{r}$ and $\stackrel{\sim }{\theta }$ in Fig 1E, are plotted against target distances and angles for the subject in (A,B). Internal estimates implied by the slow-speed prior and leaky integrator models are shown in green and purple respectively. Model estimates for each trial are shown as vertical bars centered on the mean, and ±1 standard deviation in length. D. Bias in model estimates (termed ‘residual bias’) of radial distance and angle for the two models, obtained by cross-validation (Methods). Error bars denote ±1 standard error of the mean obtained via bootstrapping. Dashed lines indicate unbiased radial or angular position estimates. Solid diagonal line has unit slope. See also Figure S3.

Figure 4. Test of model predictions

A. Reliability of optic flow was manipulated by altering the density of ground plane elements. Decreasing the density will increase subjects’ bias only if they have a slow-speed prior. B. Scatter plots showing the effect of density on radial and angular bias of one subject. Dashed line represents unity slope (unbiased performance) and solid lines represent slopes of regression fits. Trials are colored according to density – red: high density trials, blue: low density trials. C. Effect of density manipulation on radial (left) and angular (right) biases of individual subjects. Asterisks denote significant difference between means (see text). D. Subjects’ speed limit was manipulated by altering the gain of the joystick. The leaky integrator model predicts that subjects’ biases will be reduced in the high-speed condition. E. Scatter plots showing the effect of speed on distance and angle bias of one subject. Trials are colored according to speed – red: high speed trials, blue: low speed trials. F. Speed manipulation does not affect subjects’ biases in a systematic way. n.s. stands for not significant (see text). See also Figure S4.

Figure 5. Model explains bias reversal with distance

A. The width of the subjects’ probability distribution over their position (black) is modeled as a power law with exponent λ. The overlap (grey shade) of the probability distribution with the target (orange) corresponds to the subjects’ expected reward. B. Evolution of subjects’ expected reward when steering to a nearby (left) and distant (right) target for λ = 1.5 and proportionality constant equal to one. Insets show probability distributions over position at three different locations indicated by solid circles of the corresponding color on the reward curve. The peaks of the reward curves correspond to the optimal response distance. Orange bars denote the width of the target and dashed vertical lines the target center. C. The optimal response distance as a function of target distance, for the above case. D. The effect of position uncertainty (top) and the effect of slow-speed prior (bottom) combine to determine the model prediction for path integration bias, shown for various values of the power-law exponent (right). The interaction scales the optimal response distance by the slope Γ of the relation between actual and perceived distance moved. E. Mean net distance moved by one subject in response to targets at five different distances. Grey solid line corresponds to the best-fit model. F. Grey circles denote mean responses of individual subjects. Black line corresponds to the subject-averaged response. G. Mean response of one subject under conditions of low-density (blue) and high-density (red) optic flow. Asterisks denote a significant difference between mean responses under the two conditions (2m: p=0.029, 4m: p=0.007, 32m: p=4.1×10⁻⁴, paired t-test). H. Mean responses of individual subjects under the two conditions. Asterisks denote a significant difference between mean responses (across subjects) under the two conditions (2m: p=0.035, 32m: p=0.013, paired t-test). Solid lines correspond to subject-averaged response. (C-H) Black dashed lines have unit slope; (E, G) Error bars denote standard error of mean across trials. See also Figure S5.