Previous work has revealed that the heading perception from optic flow can be either attracted to the straight-ahead direction showing a center bias or repelled away from the previously seen heading (i.e., repulsive serial dependence) after ruling out the center bias accounting for perceptual errors. Recent studies have debated whether the serial dependence occurs at the perceptual or postperceptual stages (e.g., working memory). Our current study reexamined the serial dependence in heading perception and investigated whether the serial dependence occurred at perceptual or postperceptual stages. Additionally, an ideal observer model was developed to explore whether observers optimally combined the straight-ahead direction and previous and current headings to perceive headings. Our results showed that after ruling out the center bias, the perceived heading was biased toward the previous heading, suggesting an attractive serial dependence in heading perception. This attractive serial dependence occurred at both perceptual and postperceptual stages. Importantly, the perceived heading was well predicted by an ideal observer model, suggesting that observers could optimally combine their perceptual observations (current heading) with their prior information about the straight-ahead direction and previous headings to estimate their heading.

^{1}According to the Bayesian inference theory (Bernardo & Smith, 1994; Jaynes, 1986; MacKay, 2003) and the ideal observer theory (see Geisler, 2011, for review), the visual system would rely more on prior (e.g., straight-ahead direction) as the reliability of optic flow decreased. Thus, the size of center bias increased with the heading eccentricity.

^{2}However, Sun et al. (2020) fitted the perceived heading as a linear function of the actual heading, assuming that the size of the center bias was constant with the heading eccentricity. The linear model overestimated the size of the center bias for the central headings while underestimating the size of center bias for the peripheral headings (see Figures 2b in Sun et al., 2020). After ruling out the center bias predicted by a linear function, the residual heading errors could be from the estimation loss of center bias instead of serial dependence. Therefore, the finding of the repulsive serial dependence revealed by Sun et al. (2020) should be further examined.

*RHE*). We also calculated the distance between the

*n*th (

*n*= 1, 2, etc.) previous and current heading, named relative heading (

*RH*). To evaluate the size of the serial dependence, we fitted the residual heading error as a linear function of the relative heading. A positive slope of the linear function represented an attractive serial dependence in heading perception, suggesting that the perceived heading was biased toward the heading direction of the previous stimulus. In contrast, a negative slope represented a repulsive serial dependence, suggesting that the perceived heading was biased away from the heading direction of the previous stimulus. Additionally, a slope significantly different from zero indicated the existence of the serial dependence in the perceptual stage. Next, to explore whether the serial dependence occurred at the postperceptual stages, we compared the slopes of the fitted lines in the perceptual condition with the postperceptual conditions. If the slopes were significantly different between the two conditions, then the serial dependence occurred at the postperceptual stage.

^{2}) simulated observers translating at 1 m/s in a three-dimensional dot-cloud space (depth range: 0.20–5 m) consisting of 90 dots (diameter: 0.28°; luminance: 22.5 cd/cm

^{2}) (Figure 1a). The simulated self-motion direction (i.e., heading direction) was selected from ±30°, ±25°, ±20°, ±15°, ±10°, ±5°, or 0°. Positive and negative values corresponded to the headings to the right or left of the display center (i.e., 0°).

*PH*) as a linear function of the actual heading (

*AH*) for each participant (the

*AH*was the second heading stimulus in each trial of the perceptual and postperceptual conditions), given as

*s*represented the slope caused by center bias. Specifically, if

_{CB}*s*was smaller than 1, then the perceived heading was smaller than the actual heading, indicating a center bias was in the heading perception from optic flow.

_{CB}*HE*) and the relative heading (

*RH*), given as

*HE*) was the difference between the perceived and actual headings of current heading stimuli. The relative heading (

*RH*) was the relative distance between the actual heading in the

*n*th (

*n*= 1, 2, etc.) previous and current stimuli. A negative

*s*suggested a repulsive serial dependence was in heading perception. In contrast, a positive

_{SD}*s*suggested an attractive serial dependence was in heading perception.

_{SD}*RHE*). The residual heading error included the differences in the perceived heading between the perceptual (postperceptual) and baseline conditions. We fitted the residual heading error as a linear function of the relative heading (

*RH*), given as

*s*suggested an attractive serial dependence in heading perception. In contrast, a negative

_{SD}*s*suggested a repulsive serial dependence. The absolute value of

_{SD}*s*reflected the size of the serial dependence. The larger the absolute value of

_{SD}*s*, the larger the serial dependence. Additionally, if the

_{SD}*s*of the perceptual condition was significantly different from 0, then the serial dependence occurred at the perceptual stage. If the

_{SD}*s*in the postperceptual condition was significantly different from that in the perceptual condition, then the serial dependence occurred at the postperceptual stage.

_{SD}*n*th trial of the baseline condition corresponded to the second heading stimulus in the

*n*th trial of the perceptual and postperceptual conditions. The heading in the

*n*–1th trial of the baseline condition (i.e., first previous heading) corresponded to the second heading in the

*n*–trial of the perceptual and postperceptual conditions (i.e., second previous heading). Since the previous headings were in different trials from the current heading, these previous headings were named between-trial preheadings. Additionally, in the perceptual and postperceptual conditions, the first previous heading stimulus and the current heading stimulus were in the same trial. These previous headings were named within-trial preheadings.

*RH*was the difference in the actual heading between the previous

*n*th (

*n*= 1, 2, etc.) trial and the current trial. In this analysis, for the trials where participants also responded to the

*n*th previous heading stimulus, we replaced the

*RH*with the

*PRH*that was the difference between the perceived heading of the previous

*n*th (

*n*= 1, 2, etc.) heading stimulus and the actual heading of the current heading stimulus, given as

*RHE*was the mean of residual heading errors in each bin with a width of 5° within the range of [−57.5°, 57.5°]. This analysis also examined whether the serial dependence occurred at the postperceptual stages. However, what differed from relative heading in Equation 3 was that the perceived relative heading in Equation 4 might include working memory and decision-making. That is, Equation 4 revealed more complex postperceptual stages than Equation 3. If the value of

*s*′

_{SD}was significantly different from the value of

*s*in the perceptual condition and significantly different from 0, then the serial dependence occurred at the postperceptual stage.

_{SD}*t*test revealed that the slope (

*s*) of the fitted line was significantly below 1 (

_{CB}*t*s(17) < −3.47,

*p*s < 0.0029, Cohen's

*d*s > 1.15), indicating significant center bias effects.

*t*test revealed that the slope (

*s*) of the fitted line was significantly larger than 0 (

_{CB}*t*(17) = −17.39,

*p*< 0.001, Cohen's

*d*= 5.80), suggesting that participants followed the instruction to memorize the heading of the first stimulus in the postperceptual condition.

*t*test revealed that the slope (

*s*) of the fitted line was significantly larger than 0 (

_{SD}*t*s(17) > 3.89,

*p*s < 0.001, Cohen's

*d*s > 1.30), indicating a significant attractive serial dependence in heading perception.

*t*test showed that the slopes (

*s*) were not significantly different between the first and second previous headings in the baseline condition (

_{SD}*t*(17) = −0.56,

*p*s = 0.58, Cohen's

*d*s > 1.30), suggesting a robust serial dependence in the baseline condition.

*F*(1.00, 17.08) = 7.85,

*p*= 0.012, η

^{2}= 0.32) and postperceptual (Greenhous–Geisser corrected:

*F*(1.00, 17.07) = 9.11,

*p*= 0.0077, η

^{2}= 0.35) conditions. Newman–Keuls post hoc analysis showed that in the two conditions, the

*s*of the first previous heading was significantly larger than that of the second and third previous headings (

_{SD}*p*s < 0.0036), while the difference in the

*s*was not significant between the second and third previous headings (

_{SD}*p*s > 0.60). These results suggested that the size of the serial dependence decreased with the increase of the interval time between the previous and current stimuli in the perceptual and postperceptual conditions.

*n*th (

*n*= 1, 2, etc.) previous and current stimuli. In Figures 4c and 4d, the x-axis was the perceived relative heading (

*PRH*), meaning the difference between the perceived heading of the

*n*th previous stimuli and the actual heading of current stimuli. It clearly shows attractive serial dependence in the first previous heading. The linear regression analysis showed that the fitted line accounted for over 42.5% variance in the residual headings. One-sample

*t*test showed that the

*s*and

_{SD}*s*′

_{SD}were significantly larger than 0 (

*t*s(17) > 3.40,

*p*s < 0.0033, Cohen's

*d*s > 1.13). However, the size of the serial dependence significantly decreased in the second and third previous headings. One-sample

*t*test showed that the

*s*and

_{SD}*s*′

_{SD}were all not significantly different from 0 (

*t*s(17) < 1.56,

*p*s > 0.14, Cohen's

*d*s < 0.52) in the second and third previous headings. These results suggest that the serial dependence occurred within one trial and disappeared across the trials after ruling out the center bias. Additionally, the significant serial dependence in the perceptual condition suggested that the serial dependence occurred at the perceptual stage.

*t*test showed that there was no significant difference in the

*s*of the first previous heading between the perceptual and postperceptual condition (

_{SD}*t*(17) = 0.72,

*p*= 0.48, Cohen's

*d*= 0.21) and between the

*s*of the perceptual condition and the

_{s}*s*′

_{SD}of the postperceptual condition (

*t*(17) = 1.56,

*p*= 0.14, Cohen's

*d*= 0.46). These suggested that the serial dependence did not occur at the postperceptual stage. Additionally, the difference between the

*s*and

_{SD}*s*′

_{SD}of the postperceptual condition was not significant (

*t*(17) = 0.18,

*p*= 0.86, Cohen's

*d*= 0.030), suggesting that the previously perceived heading did not cause a stronger serial dependence than the previous actual heading.

*RH*) between the previous first and current heading stimuli included two levels: ±10° (i.e.,

*RH*= ±10°). We calculated the slope by using the difference in the heading error or residual heading error between the two relative headings divided by the differences between the two relative headings, given as

*t*test showed that the slopes (

*s*or

_{SD}*s*′

_{SD}) were all significantly larger than 0 (

*t*s(19) > 3.53,

*p*s < 0.0022, Cohen's

*d*s > 1.12). However, the

*s*and

_{SD}*s*′

_{SD}significantly decreased in the second and third previous headings. One-sample

*t*test showed that neither

*s*nor

_{SD}*s*′

_{SD}was significantly different from 0 (

*t*s(19) < 1.52,

*p*s > 0.14, Cohen's

*d*s < 0.48). These results suggested that the serial dependence occurred within trial and disappeared between trials after ruling out the center bias. Additionally, the significant serial dependence in the perceptual condition suggested that the serial dependence occurred at the perceptual stage.

*t*test showed that the

*s*of the perceptual condition was significantly smaller than the

_{SD}*s*and

_{SD}*s*′

_{SD}of the postperceptual condition (

*t*(19) = −2.46,

*p*= 0.024, Cohen's

*d*= 0.61;

*t*(19) = −2.90,

*p*= 0.0092, Cohen's

*d*= 0.80), suggesting that the serial dependence occurred at the postperceptual stage. The difference was not significant between the

*s*and

_{SD}*s*′

_{SD}of the postperceptual condition (

*t*(19) = −0.12,

*p*= 0.91, Cohen's

*d*= 0.0034), suggesting that the previously perceived heading did not cause a stronger serial dependence than the previous actual heading.

*PH*) and the actual heading (

*AH*), given as

*CB*was the size of the center bias. If

*CB*was significantly larger than 0, then a center bias was in heading perception. The larger the

*CB*, the stronger the center bias.

_{C}means the heading of current trials, \({\theta }_{P\_i}\) means the heading of previous

*n*th trials (

*n*= 1, 2, etc.),

*M*is the perceived heading of the current trial, and

*S*is straight-ahead direction (0°). Note that, θ

_{C}, \({\theta }_{P\_i}\), and

*S*are independent. Equation 8 means that the posterior probability of a particular currently presented heading (θ

_{C}) given a particular sensory measurement (

*M*) and straight direction (

*S*= 0) is proportional to the product of the likelihood of that measurement given a current heading (θ

_{C}), previous headings (θ

_{P}), and straight-ahead direction (

*S*) and the prior probability of that measurement given the straight-ahead direction (

*S*) and a serial of previous headings (\({\theta }_{P\_i}\)). When preheading was absent (i.e., baseline conditions),

*P*(θ

_{C}|θ

_{P},

*S*= 0) =

*P*(θ

_{C}|

*S*= 0) and

*P*(

*M*|θ

_{C}, θ

_{P},

*S*= 0) =

*P*(

*M*|θ

_{C},

*S*= 0)

*P*(

*M*|

*S*= 0), so Equation 8 could be expressed as

*S*, θ

_{P}, and θ

_{C}are constants.

*M*is the mean perceived headings across 18 participants in different conditions.

*k*,

_{C}*k*, and

_{P}*k*are free parameters that decided the width of the Von-Mises probability distributions.

_{S}*P*(

*M*|θ

_{C})) and the straight-ahead direction (

*P*(θ

_{C}|

*S*)) to perceive heading and, in the perceptual condition, whether observers optimally combined the current heading, the straight-ahead direction, and the first previous heading (\(P(M|{\theta }_{P\_1})\), \({\theta }_{P\_1}\) = −10 or −30 (10 or 30)° when the current heading was −20° (20)°) that were in the same trial with the current heading.

*k*,

_{C}*k*, and

_{P}*k*). We used a 1,000,000 iterations as a burn-in period. Before starting the iteration, we randomly selected start-point values for

_{S}*k*,

_{C}*k*, and

_{P}*k*. For example, in our codes,

_{S}*k*= [3, 3], corresponding to 45 and 135 dots in the flow field, and

_{C}*k*= [3, 3], corresponding to ±30° and ±10° preheadings,

_{P}*k*= 1. The start-point values did not affect the final results. From the second iteration, we added a random number (Δ) selected from the standard normal distribution to the parameter values of preiteration. We compared the log-likelihood posterior (

_{S}*LLP*) of the current iteration with the

_{c}*LLP*

_{c − 1}of the pre–first iteration (

*c*is the index of iternation). For each dot number condition, each participant's \(LL{P}^{\prime}\) was given by

*i*,

*j*indicates three preheading conditions under one current heading (e.g., as

*i*is 1, meaning that the current heading is −20°); and the corresponding preheading conditions include three cases (

*j*= 1, 2, 3): −10°, −30°, and no preheading conditions. We then summarized the \(LL{P}^{\prime}\) of 18 participants and two dot-number conditions, given by

*s*represents the

*s*th participants, and

*k*represents the

*k*th dot-number condition. If the

*LLP*of the current iteration was larger than the

_{c}*LLP*

_{c − 1}of the pre–first iteration, the parameter values of the current iteration were kept; if not, one random number (ε) was generated. If ε > 0.5, then the parameter values of the current iteration were kept; if ε ≤ 0.5, the parameter values of the current iteration were equal to the parameter values of the preiteration. The same procedure was repeated a million times. The final values of

*k*,

_{C}*k*, and

_{P}*k*were the mean of the 18,000 iterations selected from the 100,001

_{S}^{st}iteration to the end with step 50.

^{3}

*j*= 1, 2, …, 6, that is, the combination of three within-trial preheadings (10°, 30°, and no preheading) and two between-trial preheadings (±20°).

*CB*) against the actual heading. It clearly shows that the sizes of the center bias are larger than 0 in all conditions supported by sample

*t*tests (

*t*s(17) > 2.97,

*p*s < 0.0087, Cohen’s

*d*s > 0.99), suggesting a center bias in heading perception from optic flow. Additionally, it also shows that the sizes of the center bias are generally larger in 45-dot condition than in 135-dot condition. One 2 conditions (baseline vs. perceptual) × 2 dot numbers (45 vs. 135) × 2 actual headings (±20°) repeated-measures ANOVA showed that the main effect of dot numbers was significant (

*F*(1,17) = 9.20,

*p*< 0.001, η

^{2}= 0.35). Specifically, the size of the center bias of 45 dots (black dots,

*M*±

*SE:*0.23 ± 0.025) was larger than that of 135 dots (blue dots, 0.16 ± 0.029). The interaction effect between conditions and actual headings was also significant (

*F*(1,17) = 5.80,

*p*= 0.028, η

^{2}= 0.25). The other factors’ main effects and interaction effects were not significant (

*p*s > 0.85). The results suggested that the size of the center bias increased with the decrease of the dot number, indicating that observers relied more on the straight-ahead direction to perceive their heading as the reliability of heading stimuli decreased. This was consistent with an ideal model in which the straight-ahead direction worked as the prior.

*n*th (

*n*= 1, 2, 3, etc.) previous and current stimuli. All figures show when the previous heading is left (right) to the current heading, the perceived heading is biased toward the left (right) side of the actual heading, indicating an attractive serial dependence in heading perception. One-sample

*t*test showed that all slopes (

*s*) were significantly larger than 0 (

_{SD}*t*s(17) > 2.52,

*p*s < 0.022, Cohen's

*d*s > 0.84), suggesting an attractive serial dependence in all conditions.

*s*) against different previous headings in the baseline condition. In the baseline conditions, the dot numbers of the previous and current stimuli were the same. One 2 previous headings (first vs. second) × 2 dot numbers (45 vs. 135) repeated-measures ANOVA showed that the main effects of dot numbers were significant (

_{s}*F*(1,17) = 6.62,

*p*= 0.020, η

^{2}= 0.28), Specifically, the

*s*of 45 dots (

_{s}*M*±

*SE*: 0.14±0.018) was larger than that of 135 dots (0.098 ± 0.017), suggesting that the size of the serial dependence increased, with the decrease of the dot number indicating that observers relied more on previous headings as the reliability of current heading decreased. Neither the main effects of previous headings nor their interaction effect with dot numbers were significant (

*F*s < 1,

*p*s > 0.36, η

^{2}s < 0.048).

*F*(1, 17) = 6.25,

*p*= 0.023, η

^{2}= 0.27). Specifically, the

*s*of 45 dots (

_{s}*M*±

*SE:*0.14 ± 0.039) was larger than that of 135 dots (0.098 ± 0.013). Neither the main effects of previous headings nor their interaction effect with dot numbers were significant (

*F*s < 1,

*p*s > 0.36, η

^{2}s < 0.048). Paired sample

*t*test showed that the

*s*of 45 dots was significantly larger than that of 135 dots (

_{SD}*t*(17) = 3.11,

*p*= 0.0064, Cohen's

*d*= 0.63) in the second previous heading. These results suggested that the size of the serial dependence increased with the decrease of the dot density, indicating that observers relied more on previous headings as the reliability of current heading decreased.

*P*(θ

_{C}|

*M*)) predicted by the ideal observer model. Panels (a), (d), and (e) show the result of the ideal observer model, the priors of which do not include the between-trial preheadings; other panels show the result of the ideal observer model, the priors of which include the between-trial preheadings. Panels (a), (b), and (c) show the result of the ideal observer model, the priors of which do not include the within-trial preheading; other panels show the result of the ideal observer model, the priors of which include the within-trial preheading. The black and blue markers and lines correspond to 45- and 135-dot conditions. The figure clearly shows that the participants’ perceived headings (diamond markers) are well covered by the posterior distribution. The 95% CIs of the participants’ perceived headings and the posterior distributions were very close (see Table 1 for the descriptive statistics), suggesting that the ideal observer model can well predict the performance of the heading perception from optic flow.

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^{nd}one was left or right to the heading of the 1

^{st}one. We fitted the proportion of rightward responses as a psychometric function of the actual heading. The standard deviation (σ) of the psychometric function reflected the heading discrimination threshold. A larger standard deviation indicated the lower reliability of optic flow stimuli. The results showed that the standard deviation increased with the heading eccentricity (Figure A1), consistent with the Crowell and Banks (1993).

^{2}(1) = 11.45,

*p*< 0.00010), suggesting that the cubic function worked better to explain heading perception from optic flow than linear function.

*F*(1.99,23.83) = 1.48,

*p*= 0.96,

*η*= 0.012). The interaction between the two factor was significant (

^{2}*F*(7.54,90.17) = 2.29,

*p*= 0.96,

*η*= 0.031). Further analysis showed that none of the main effects of delay was significant under each heading (

^{2}*p*s > 0.069). The results suggested that the delay time had no effect on heading perception, indicating that working memory was not involved in heading perception.

^{nd}heading stimulus. The figures clearly shows that the perceived headings were systematically compressed towards the display center showing a center bias. The lines were the best fitting of Equation 1 and accounted for over 99.1% variance in the perceived heading. One sample

*t*-test showed that the slopes of the four conditions were all significantly below 1 (

*t*s (19) < 4.45,

*p*s < 0.001, Cohen's

*d*s > 1.41), indicating a center bias in these conditions.

^{st}heading stimuli against the actual heading in the post-perceptual load condition. It clearly shows that the perceived heading increases with the actual heading. The fitted line accounted for 99.2% of variance in the perceived heading. One sample

*t*-test showed that the slope of the fitted line was significantly larger than 0 (

*t*(19) = 19.06,

*p*< 0.001, Cohen's

*d*= 6.03), indicating that participants followed the instruction to memorize the 1

^{st}heading stimuli's heading direction in the post-perceptual load condition.

*t*-test showed that the accuracies were all significantly larger than the chance level (0.5) (

*t*(19) > 19.00,

*p*< 0.001, Cohen's

*d*s > 6.00), suggesting that participants followed the instruction to adding the integers and well finished the number comparison task.

^{st}heading in the perceptual load and post-perceptual load conditions, we calculated the slope with Equations 5 and 6 rather than fitted a linear function. For the conditions, we fitted the heading error as a linear function of the relative heading (Equation 2). The linear regression analysis showed that the fitted lines accounted for over 94.8% variance in the heading error. One sample

*t*-test revealed that the slope (

*s*) was significantly larger than 0 (

_{SD}*t*s(19) > 3.53,

*p*s < 0.0022, Cohen's

*d*s > 1.43), indicating a significant serial dependence in heading perception and the SD could lasted for several seconds.