A Bayesian inference model can predict the effects of attention on the serial dependence in heading estimation from optic flow

Abstract

It has been demonstrated that observers can accurately estimate their self-motion direction (i.e., heading) from optic flow, which can be affected by attention. However, it remains unclear how attention affects the serial dependence in the estimation. In the current study, participants conducted two experiments. The results showed that the estimation accuracy decreased when attentional resources allocated to the heading estimation task were reduced. Additionally, the estimates of currently presented headings were biased toward the headings of previously seen headings, showing serial dependence. Especially, this effect decreased (increased) when the attentional resources allocated to the previously (currently) seen headings were reduced. Furthermore, importantly, we developed a Bayesian inference model, which incorporated attention-modulated likelihoods and qualitatively predicted changes in the estimation accuracy and serial dependence. In summary, the current study shows that attention affects the serial dependence in heading estimation from optic flow and reveals the Bayesian computational mechanism behind the heading estimation.

Introduction

Accurately perceiving our self-motion direction (i.e., heading) is vital for survival. Optic flow, the dynamic motion pattern projected on observers' retinas as they move in the world (Gibson, 1950), plays a crucial role in this process. As observers move forward along a straight line, the objects in the environment appear to move radially outward from a point called the focus of expansion. Studies have demonstrated that observers can accurately estimate their heading by localizing the position of the focus of expansion (Sun, Zhan, You, & Dong, 2020; Sun, Yan, Wang, & Li, 2022; Warren, Morris, & Kalish, 1988; Warren & Saunders, 1995; Xu, Sun, Zhang, & Li, 2022). 

Although heading estimation from optic flow is accurate, it exhibits that the current heading estimate is biased toward or away from the previously seen headings, called attractive and repulsive serial dependence, respectively (Xu et al., 2022, Sun, Zhang et al., 2020). Sun, Zhang et al. (2020) found that the serial dependence effect increased as the reliability of optic flow stimuli decreased. This was well-predicted by the Bayesian inference model developed by Xu et al. (2022). Therefore, serial dependence is also consistent with the Bayesian inference theory. 

In addition to examining the systematic biases of heading estimation, researchers have also investigated its occurrence mechanisms. Royden and Hildreth (1999) compared the heading estimates between the single-task condition (only heading estimation task) and the dual-task condition (biological motion direction task plus heading estimation task). The dual-task condition resulted in reduced attentional resources allocated to the heading estimation task compared with the single-task condition. However, there were no significant difference in heading errors between the two conditions, indicating that attention did not affect heading estimation. Thus, the researchers proposed that the heading estimation from optic flow was purely information-driven or occurred at the sensory perception level. They suggested that cognitive abilities beyond sensory perception, such as attention, working memory, and decision-making, were not involved. 

In contrast (Dubin & Duffy, 2007; Dubin & Duffy, 2009) found that the neural activities in the cortical area MSTd were modulated by attention. This area has been demonstrated the involvement of processing of heading direction from optic flow (Cardin & Smith, 2010). Comparing Dubin's and Royden's study showed that Royden and Hildreth (1999) selected heading direction from a narrow range (i.e., 4°, 6°, 8°, and 10°), whereas (Dubin & Duffy, 2007; Dubin & Duffy, 2009) were 30° away from the display center. Hence, it was expected that attention mainly affected the heading directions those were far away from the display center. Sun, Wang, and Gong (2024) directly demonstrated this proposal and suggested that estimating headings from optic flow needs the involvement of cognitive abilities, such as attention. However, none of the aforementioned studies directly examined the effects of attention on the serial dependence in heading estimation. Moreover, some studies with other physical features (e.g., oriented Gabors, size) have revealed that attention affects the on serial dependence in these features (see Cicchini, Mikellidou, & Burr, 2024; Manassi, Liberman, Kosovicheva, Zhang, & Whitney, 2023; Pascucci et al., 2023 for review). Accordingly, it can be proposed that attention also affects the serial dependence in heading perception from optic flow. 

Moreover, studies proposing that heading estimation from optic flow is a Bayesian inference process (Sun, Zhang et al., 2020; Warren & Saunders, 1995; Xing & Saunders, 2016; Xu et al., 2022) only ask participants to conduct a heading estimation task (except Xu et al., 2022). This means that the task takes all attentional resources. Although Sun et al. (2024) reveal the effects of attention on the estimation, it remains unclear whether the Bayesian inference model can predict the effects. Studies have shown that one feature's discrimination or detection sensitivity increases when it is attended (Anobile, Cicchini, & Burr, 2012; Carrasco, Ling, & Read, 2004; Fernández, Li, & Carrasco, 2019; Gobell & Carrasco, 2005; Liu, Fuller, & Carrasco, 2006; Liu, Abrams, & Carrasco, 2009). Neurophysiological studies also find that when attention is directed toward other features or positions, the neurons that selectively respond to the feature become less active (Bahrami, Lavie, & Rees, 2007; Culham, Cavanagh, & Kanwisher, 2001; Nizamoğlu & Urgen, 2022). As a result, the representations of these features became unreliable. If the heading estimation from optic flow remains consistent with the Bayesian inference theory across different attentional states, the serial dependence is expected to be changed as the attention load changes. 

Previous studies have found that serial dependence with nonoptic flow features is consistent with the Bayesian inference process when all attentional resources are allocated to the experimental task (see Cicchini et al., 2024; Manassi et al., 2023 for review). These studies found that when the reliability of previously seen features was reduced, the serial dependence was reduced; and vice versa. According to the above paragraph, the reliability of features varies among different attentional states. It, therefore, can be expected that the Bayesian inference model can predict the effects of attention on serial dependence. 

In summary, the current study conducted two experiments to investigate the effects of attention on heading estimation from optic flow. A Bayesian inference model was developed to elucidate the underlying computational mechanisms. Each experiment consisted of four blocks of trials, with each corresponding to one of the following conditions: no-load baseline, load baseline, no-load serial dependence, and load serial dependence. Because the Bayesian inference process is bidirectional, a reduction in the reliability of the current heading increases the bias toward the previously seen heading, whereas a decrease in the reliability of the previously seen heading decreases the bias toward it. In experiments A and B, we presented the attentional load task before or on the current headings in the load conditions to vary the reliability of the currently and previously seen headings, respectively. Our results show that the serial dependence is affected by the attentional load task, which is well-predicted by a Bayesian inference model. These findings suggest that, even when other tasks or features confound the heading estimation from optic flow, the estimation process is still a Bayesian inference process. The current study improves our understanding of the cognitive mechanisms underlying heading estimation from optic flow. 

Methods

Participants

Forty participants were enrolled from our university. All participants had normal or corrected-to-normal vision and were naive to the purpose of the experiment. The experiment was approved by the Scientific and Ethical Review Committee of the Department of Psychology at our university. We obtained all participants’ written informed consent form before starting the experiment. The participants were divided into two groups (group A: 8 males, 12 females; 18–26 years; group B: 8 males, 12 females; 18–27 years) and conducted experiments A and B, respectively. The sample size was decided based on previous studies (e.g., Sun, Zhang et al., 2020; Sun, Zhang et al., 2024; Xu et al., 2022). 

Stimuli and apparatus

Optic flow displays (80° H × 80° V; luminance: 0.24 cd/cm²) (Figure 1) simulated observers moving forward in a three-dimensional dot-cloud at a speed of 1 m/s. The three-dimensional dot-cloud consisted of 90 dots (diameter, 0.24°; luminance, 22.5 cd/cm²). On each trial of the (no-load and load) baseline conditions (Figures 1A and 1B), only one optic flow display (i.e., the second heading display, relative to the serial dependence conditions in Figures 1C and 1D) was presented, and its simulated motion direction (i.e., heading) was selected randomly from 0°, ±10°, and ±20°. Positive (negative) values indicate that headings are right (left) to the display center (0°). On each trial of the (no-load and load) serial dependence conditions (Figures 1C and 1D), another optic flow display (referred as the first heading) was presented before the above flow display. The simulated heading of this flow display was left or right to the current heading by 0°, 5°, or 10°. The difference between the first and second headings was named the heading offset. 

Figure 1.

 

Illustrations of trial procedures in the (A) no-load baseline, (B) baseline, (C) no-load serial dependence, and (D) load serial dependence conditions in experiments A (left) and B (right). The flow display simulated observers translating in a three-dimensional dot cloud. White dots indicate the dots’ position on the display in the first frame; white lines indicate the motion trajectories in the following frame, which are invisible in the experiments. On each trial of the no-load and load baseline conditions (A and B), only one flow display was presented; however, on each trial of the no-load and load serial dependence conditions (C and D), two flow displays were presented. Given that the flow display in the baseline conditions and the second flow display in the serial dependence conditions were presented at the same time in each trial, we named these displays as the second flow display. Accordingly, the first presented flow display in the serial dependence conditions was named as the first flow display. Additionally, the difference in the heading directions between the first and second display was named as heading offset, the value of which is 0°, ±5°, or ±10°. Negative (positive) heading offset means that the heading of the first flow is to the left (right) of the heading of the second display. On the trials of load conditions (B and D), three integers were positioned vertically on the display center. Participants were asked to sum the first two integers up and compare the sum with the third integer.

On the trial of the load conditions, three integers (RGB: [255 255 0]; 1.76° V × 1.76° H) were presented vertically on the display center. The gap between the two numbers was 0.44°. The first two integers were randomly selected from the range [11, 40]. The last integers were randomly selected from the range [40, 92]. 

The experiment was programmed using MATLAB with the Psychophysics Toolbox 3. Stimuli were displayed on a 27-inch ASUS monitor (resolution: 2,560 H × 1,440 V pixels; refresh rate: 60 Hz) with an NVIDIA GeForce GTX 1660Ti graphics card. 

Procedure

Participants were seated in a dark room in front of a computer monitor with a viewing distance of 20 cm. They viewed the display monocularly with their right eyes to decrease the conflict between the motion parallax and binocular disparity depth cues. Participants' heads were fixed using a chinrest, and they were instructed to fixate on the center of the display during the experiment. To decrease the effects of nonoptic flow information on the heading estimation, participants were instructed to keep their eyes, head, and body still, consistent with previous studies (Sun, Zhang et al., 2020; Sun et al., 2022; Warren et al., 1988; Warren & Saunders, 1995; Xu et al., 2022). 

Each experiment consisted of four blocks of trials, with each corresponding with one of following conditions: no-load baseline condition, load baseline condition, no-load serial dependence condition, and load serial dependence condition. Figure 1 shows the trial procedures for these conditions in experiments A and B. 

Each trial of the load-serial dependence condition in experiment A (Figure 1D, left) started with a 200-ms fixation, followed by a 500-ms flow display. Three integers were positioned at the center of the flow display. Participants were asked to add the first two integers up and compare the sum to the third integer as quickly and accurately as possible. If they did not respond within 500 ms, a number reminder display was presented to remind them to complete the number addition task. Following their response, a heading response display appeared, consisting of a horizontal line and a mouse-controlled vertical probe were presented. Participants were asked to adjust the position of the probe to indicate their perceived heading. Note that, participants were asked to finished these tasks within 2200 ms. If they did not take the time, a blank display was presented until the whole duration was 2,200 ms. Next, the second flow display was presented for 500 ms, followed by the heading response display. The next trial started after their response. According to our experience, the time took to report the heading direction was less than 1.5 s. Therefore, the interval time between two trials was less than 1.5 s. To differentiate between the two flow displays, we referred to them as the first and second flow displays. 

The trial procedure of the load-serial dependence condition in experiment B (Figure 1D, right) was similar to that in experiment A, except that 1) three integers were presented on the second flow display instead of on the first. Participants were asked to judge numbers first and then estimate the heading of the second flow display within 2,000 ms. 2) Participants were asked to estimate the heading of the first flow display within 1,500 ms. 

The trial procedures of the no-load serial dependence condition in the two experiments (Figure 1C) were similar to those of the load-serial dependence condition, except that no integer was presented on the flow display, and there was no number reminder display. Similarly, the trial procedure of the load baseline condition (Figure 1B) was created by replacing the first flow display with a 500-ms blank display in the load-serial dependence condition; and the procedure of the no-load condition (Figure 1A) was created by removing the integers in the load baseline condition. 

The no-load and load baseline conditions included 5 second headings. Each second heading was repeated 50 times. Thus, there were 250 trials (5 second headings × 50 trials) in each condition. In no-load and load serial dependence conditions, each second heading was accompanied by five heading offsets—the difference between the first and second headings (i.e., 0°, ±5°, or ±10°). Each heading offset was repeated 10 times. Thus, there were a total of 250 trials (5 current headings × 5 heading offsets × 10 trials). Note that ideally, each block contained 250 trials. If participants failed to respond to the number addition task or the heading estimation task within the fixed time, the trial would be added back to the trial list. As a result, some participants completed more than 250 trials in 1 block. 

Participants were given approximately 15 practice trials before each condition block to familiarize themselves with the condition. The corresponding block then started and lasted for approximately 20 minutes. The conducting sequences of the four conditions were counterbalanced across participants. 

Data analysis

In the load (baseline or serial dependence) conditions of each experiment, we first calculated the accuracy of the number addition task. Participants with an accuracy below 0.75 were removed. The results showed that all participants had an accuracy above 0.75. We then excluded trials where participants did not complete the number addition or heading estimation tasks within the specified time interval. As a result, each participant had 250 trials in each condition block. 

For the heading estimation task, we collected participants’ heading estimates (HE) and calculated the difference between HE and the actual heading (AH), named estimation error (EE), given by:  

\begin{eqnarray}EE = HE - AH. \quad \end{eqnarray}

(1)

 

The larger the absolute heading error, the lower the estimation accuracy. A two-factor (Load conditions × Actual headings) repeated-measures analysis of variance (ANOVA) was conducted on the heading error of the first flow display in the load-serial dependence condition of experiment A and the second flow display in the load baseline and the load serial dependence conditions of experiment B to examine the effects of attention on the estimation accuracy. If the interaction effect of the two factors was significant and the further simple effect analysis showed that the main effect of load conditions in each heading was significant, then attention affected the estimation accuracy. 

To examine whether serial dependence was in heading estimation from optic flow, we first calculated the heading difference between the first and second flow displays in the (no-load and load) serial dependence conditions, which was defined as the heading offset. The heading offsets were 0°, ±5°, and ±10°, as described in the stimulus and apparatus section. Negative and positive values mean that the heading of the first display is to the left or right of the heading of the second display. Therefore, each heading of the second display was paired with five heading offsets. Then, we calculated the heading error of each heading of the second display in each heading offset, indicated by the EE_{SD_no-load} and EE_{SD_load}. Moreover, the heading error of each heading in the (no-load and load) baseline conditions were also calculated, indicated by the EE_{baseline_no-load} and EE_{baseline_load}. Finally, the estimation error induced by serial dependence (REE) could be given by:  

\begin{eqnarray} && RE{E_{no - load}} = E{E_{SD\_no - load}} - E{E_{baseline\_no - load}}, \qquad \end{eqnarray}

(2.1)

 

\begin{eqnarray} RE{E_{load}} = E{E_{SD\_load}} - E{E_{baseline\_load}}. \qquad \end{eqnarray}

(2.2)

 

Equations 2.1 and 2.2 indicate the REE in the no-load and load conditions. If the sign of the residual heading error is the same as the sign of the heading offset, then the heading estimate of the second flow display is biased toward the heading of the first flow display, indicating an attractive serial dependence; if the signs are opposite, then there is a repulsive serial dependence. A repeated-measures ANOVA with three factors (Load conditions × Actual headings × Heading offsets) was conducted to examine whether attention affected serial dependence. If there was a significant interaction effect between load conditions and heading offsets, as well as a significant main effect of load conditions on each heading offset revealed by the further simple effect analysis, we could confirm the effects of attention on serial dependence. 

Behavioral results

Estimation accuracy

Figures 2A and 2B plot the heading error against the actual heading in experiments A and B. Heading error is the difference between the heading estimate and the actual heading. It clearly shows in the serial dependence conditions in experiment A (Figures 1C, left, and 1D, left), the heading error of the first flow display shows an increasing trend when attentional load was presented in the flow display (Figure 2A, lef). Note that the heading of the first flow display was left or right to the heading of second display by 0°, 5°, and 10°. Given that the headings of the second flow display included 0°, ±10°, and ±20°, there are unique 13 headings for the first flow display (i.e., 0, ±5°, ±10°, ±15°, ±20°, ±25°, and ±30°). A 2 (Load conditions: no-load vs. load) × 13 (Headings) repeated measures ANOVA showed that the interaction effect between two factors was significant, Greenhouse-Geisser corrected: F (1.52, 28.94) = 8.86, p = 0.0022, η² = 0.32. Further simple effect analysis (Bonferroni corrected) showed that the estimation errors of ±30°, ±25°, +20°, and +15° were significantly larger in the load condition than in the no-load condition (p < 0.040). Similar result pattern was also observed in experiment B for the second flow display (Figure 2B, right). A 2 (Load conditions: no-load vs. load) × 2 (Serial dependence conditions: with vs. without) × 13 (Headings) repeated measures ANOVA showed that the interaction effect between load conditions and headings was significant, F (1.32, 25.07) = 5.31, p = 0.022, η² = 0.22. Simple effect analysis showed that the estimation errors of +20° were significantly larger in the load condition than in the no-load condition (p < 0.015). However, when the attentional load and the heading stimuli were separately presented (experiment A: the load baseline condition and the effect of load on the second heading) (Figure 1B, left, and 1D, left) (experiment B: the load on the first heading) (Figure 1D, right), the heading errors are overlapped (Figure 2A, right, Figure 2B, left). Taken together, the results show that attention affects heading estimation from optic flow. When the attentional load is added, the estimation accuracy decreased. 

Figure 2.

 

Results of estimation error in experiments A and B. (A and B) Heading error is against the actual heading. Heading error is the difference between heading estimates and actual headings. Dots and shaded areas indicate the mean and standard error of heading errors across 20 participants. Left and right on the x axis (y axis) indicate the actual heading (heading estimate) left or right to the display center (actual heading). The areas surrounded by dashed rectangles indicate that the mean heading errors were significantly different between the no-load and load serial dependence conditions. The left and right panels correspond with the estimated results of previous and current headings in each trial (see Figure 1). *p < 0.05; **p < 0.01; ***p < 0.001.

Serial dependence

Figures 3A and 3B plot the heading error induced by serial dependence (REE, Equations 2.1 and 2.2) against the actual heading in the no-load (left) and load (right) conditions. Negative (positive) REEs mean that the estimate of the second heading was left (right) biased. The data in each condition (or panel) were categorized into five groups. Each group correspond to one heading offset that was the difference in the actual heading between the first and second flow displays. The negative (positive) heading offsets mean that the heading of the first flow display was to the left (right) of the heading of the second flow display. It clearly shows that on average, the signs of the REEs are the same as the heading offsets in the no-load serial dependence conditions, and the REE increases with the heading offset (Figures 3A, left, and 3B, left). When the attentional load is presented on the first flow display, the REEs of all heading offsets are overlapped (Figure 3A, right), indicating that the serial dependence disappears. In contrast, when the attentional load is presented on the second flow display, the differences in the residual heading between heading offsets tend to increase (Figure 3B, right), indicating that serial dependence increases. A 2 (Load conditions: no-load vs. load) × 5 (Heading offsets) × 5 (Headings) repeated measures ANOVA showed that the main effects of heading offsets were significant in experiment A, F (1.73, 32.81) = 65.60, p < 0.001, η² = 0.78, and experiment B, F (1.43, 27.52) = 124.96, p < 0.001, η² = 0.87. The interaction between load conditions and heading offsets was also significant in experiment A, F (1.73, 32.81) = 43.24, p < 0.001, η² = 0.70, and experiment B, F (2.05, 38.99) = 22.53, p < 0.001, η² = 0.54. Results of the simple-effect analysis were shown in Figures 3C and 3D, supporting the above descriptions. 

Figure 3.

 

Results of serial dependence in experiments A and B. (A and B) Heading error induced by serial dependence (REE) is against the actual heading. The error is given by Equations 2.1 and 2.2. Left and right on the x axis mean the actual heading left or right to the display center. Left and right on the y axis mean the heading estimate is more left or right biased in the serial dependence condition than in the baseline condition. In each condition (no-load vs. load serial dependence), the data were categorized into five groups, each corresponding to one heading offset—the difference in the heading direction between the first and second flow displays. The negative and positive heading offsets mean that the heading of the first flow display is to the left or right of the second flow display. (C and D) The REE averaged across actual headings is against the no-load and load conditions. It shows the significance of the main effect between the load and no-load condition in each heading offset, as well as the significance of the main effect between difference heading offsets in the load or no-load conditions. *p < 0.05; **p < 0.01; ***p < 0.001.

To clarify the relationship between the strength of serial dependence and heading offsets, we calculated the mean REE across different actual headings. Then, the ratios between the mean REEs and heading offsets were calculated, which indicates the strength of serial dependence. Figure 4 plots the ratio against the heading offset. It clearly shows that when no attentional load was added in the heading estimation task (daker dots), the strength of serial dependence was approximately 20% of the heading offset; whereas when the attentional load was added in the second heading (solid blue dots, experiment B), the strength of serial dependence was increased to 50% of the heading offset; in contrast, when the attentional load was added in the first heading (blue circles, experiment A), the strength of serial dependence was decreased to 4% of the heading offset. Figure 4 clearly shows the effects of attentional load on the strength of the serial dependence. 

Figure 4.

 

Ratio between the mean REE and the heading offset is against the heading offset. This figure shows the change of the strength of serial dependence along with the heading offset. Error bar is the standard error across all participants.

Therefore, an attractive serial dependence is in the heading estimation, which can be affected by attention. Specifically, when the attentional load is added on the previous heading, participants would reduce their reliance on previous headings to estimate current headings. Conversely, when the attentional load is added on the current heading, participants would increase their reliance on previous headings to estimate current headings. 

Bayesian modeling and results

According to the Bayesian inference theory (Knill & Pouget, 2004; Knill & Richards, 1996), observers interpret feature values by optimally combining different information. If one source of information becomes unreliable, they will rely more on other sources to make an accurate interpretation. This results in a final estimate that is more biased toward the more reliable information. The predicted trend aligns with our finding that the heading estimate of the second flow display is more biased toward the heading of the first flow display as attentional load is added to the second flow display. In contrast, this bias is reduced as attentional load is added to the first flow display. To investigate whether attention's effects on serial dependence follow a Bayesian inference process, we developed a Bayesian inference model. Specifically, the model consisted of two layers: layer 1 predicted heading estimates in the (load and no-load) baseline conditions, which provides the likelihoods of different headings without serial dependence; layer 2 optimally combined the likelihoods between the headings of the first and second flow displays to predict the heading estimate of the second flow display in the serial dependence condition. The heading estimate was then subtracted from that without serial dependence in layer 1, resulting in the predicted estimation error induced by serial dependence. 

Layer 1 consisted of two steps. First, model 1 was developed to predict the estimation errors in the no-load condition, which provided the prior distribution (p(θ)) and the likelihoods (p(m|θ)) in the no-load condition. Second, model 2 was developed to predict the estimation errors in the load condition. Note that we assumed that the prior distribution in the load condition was the same as to that in the no-load condition because the prior distribution of headings is learned in everyday life and has been encoded in our sensory cortical areas (e.g., MST) (Gu, Fetsch, Adeyemo, DeAngelis, & Angelaki, 2010). Therefore, model 2 provided the likelihoods (p(m|θ)) in the load condition. 

Previous studies have shown that heading estimation from optic flow is a Bayesian inference process in the no-load condition (Sun, Zhang et al., 2020; Warren & Saunders, 1995; Xing & Saunders, 2016; Xu et al., 2022), so we first built model 1 to predict the heading estimates (m) of the first flow display in the no-load serial dependence condition of experiment A and the second flow display in the no-load baseline condition of experiment B. In experiment A, the actual heading (θ) randomly selected from the range [−30°, 30°] with a step of 5°. In experiment B, θ is randomly selected from the range [−20°, 20°] with a step of 10°. The Bayesian inference model in the no-load conditions can be given by:  

\begin{eqnarray}p(\theta |m) \propto p\left( \theta \right)p(m|\theta ), \,\,\,\,\,\,\,\, \qquad \end{eqnarray}

(Model 1)

 

\begin{eqnarray}p\left( \theta \right) = \frac{1}{{\sqrt {2\pi } {\sigma _p}}}{e^{\left( {\frac{{ - {{\left( {\theta - 0} \right)}^2}}}{{2{\sigma _p}^2}}} \right)}}, \qquad \end{eqnarray}

(3.1)

 

\begin{eqnarray}p(m|\theta ) = \frac{1}{{\sqrt {2\pi } {\sigma _{l\_noload}}}}{e^{\left( {\frac{{ - {{\left( {m - \theta } \right)}^2}}}{{2{\sigma _{l\_noload}}^2}}} \right)}}. \qquad \end{eqnarray}

(3.2)

 

In Equation 3.1, p(θ) is the prior distribution that is assumed to be a Gaussian distribution with a 0° center (i.e., display center) and a standard deviation σ_p (σ_p > 0). In Equation 3.2, p(m|θ) is the likelihood distribution that is also assumed to be a Gaussian distribution with a center at θ and a standard deviation \({\sigma _{l\_noload}}\) (\({\sigma _{l\_noload}}\)>0). σ_p and \({\sigma _{l\_noload}}\) are free parameters. Note that \({\sigma _{l\_noload}}\) was varied among different headings. p(θ|m) is the posterior distribution. Hence, the predicted heading estimates can be given by the means (\(\hat{m}\)) of the posterior, and the predicted estimation error (\(\widehat {EE}\)) can be given by \(\widehat {EE} = \hat{m} - \theta \). 

To decide the values of σ_p and \({\sigma _{l\_noload}}\), we adopted Markov chain Monte Carlo methods with 5,000,000 iterations. In each iteration, we select a positive value for σ_p and \({\sigma _{l\_noload}}{\rm{\ randomly }}\), and calculate the sum loglikelihood of all participants’ estimates in the posterior distributions. If the sum loglikelihood of the current iteration was larger than that of the previous iteration, then the new values were kept, and vice versa. The final parameter values were given by the mean of the 450,000 first iterations to the end with a step of 500. Note that 5,000,000 iterations are sufficient to obtain stable parameter values. The step number has little effect on the final result, except using a very large value leaving few effective iterations. The same Markov chain Monte Carlo method was adopted for the parameters in other models. After determining the final values of parameters, we calculated the predicted estimates (\(\hat{m}\)) based on the posteriors and the predicted estimation error (\(\widehat {EE} = \hat{m} - \theta \)). 

Figures 5A and 5B show that the predicted estimation errors (black circles) match participants’ estimation errors (black solid dots) well, indicating that heading estimation from optic flow can be predicted by a Bayesian inference model (also see Xu et al., 2022). 

Figure 5.

 

Predicted results of estimation errors. (A and B) Heading error plots against the actual heading. Solid dots show the mean of participants’ raw data; circles show the mean of predicted heading errors randomly selected from the posterior distributions.

Next, we built model 2, which was similar to model 1, except that the standard deviation of the prior distribution (σ_p) was determined by model 1. The standard deviation of the likelihood distribution (\({\sigma _{l\_load}}\)) was the only free parameter, which was determined by the above Markov chain Monte Carlo procedure. The blue circles in Figures 5A and 5B show that the predicted estimation errors agree well with the participants’ estimation errors (blue solid dots), suggesting that the effects of the attentional load on heading estimation from optic flow can be predicted by the Bayesian model. The similar results are also observed in experiment B (Figure 5B). 

Figures 6A and 6B show the prior distributions (p(θ)) used in experiments A and B. Although different participants were enrolled in the two experiments, the standard deviations of the two priors (σ_p) were very close. Figures 6C and 6D show the likelihood distributions. The likelihood distributions in the no-load condition (black lines) tend to be higher and narrower than those in the load condition (blue lines). Figures 6e and 6F plot the standard deviation against the actual heading. It is clear that the standard deviations in the no-load condition (black dots) are smaller than that in the load condition (blue dots), suggesting that the internal heading representations become less reliable in the load condition. 

Figure 6.

 

(A and B) Prior distributions in experiments A and B. (C and D) Likelihood distributions of previous headings in experiment A, and likelihood distributions of current headings in experiment B. (E and F) A quadratic function captures well the changing trend of the standard deviation of the likelihood distribution along with the actual heading.

Moreover, the standard deviation of the likelihood distribution increased with the heading, which is well-captured by a quadratic function (solid lines in Figures 4E and 4F), consistent with the previous studies (Crowell & Banks, 1996; Gu et al., 2010). 

Next, in layer 2, we examined whether the effects of attention on serial dependence could be predicted by the Bayesian inference model. It is known that the serial dependence is the estimation bias caused by the previously presented stimuli. It is the difference between the estimate derived from the mean of the posterior that was the optimal combination of the likelihoods of the first and second headings and the actual second heading. The posterior can be given by p(m|θ_p)p(m|θ_c), where θ_p and θ_c represent the standard deviations of the likelihoods of the first and second headings and are given by model 1 and model 2 (Figures 6C and 6D). 

The dashed lines in Figure 7 show the predicted the estimation error induced by serial dependence. First, the predicted error increased with increasing heading offset in the no-load condition (dashed lines in the left), consistent with the participants’ data (solid lines in the left) (root mean square errors > 3.85). When the attentional load is added in the first flow display, the increasing trend is evidently reduced (dashed lines in Figure 7A right), but does not disappear, which is unlike the participants’ data (solid lines in Figure 7A right) (root mean square error = 5.39). When the attentional load is added in the second flow display, the increasing trend is evidently increased (dashed lines in Figure 7B right), but the increasing amplitude is less than the participants’ data (solid lines in Figure 7B right) (root mean square error = 1.42). 

Figure 7.

 

Predicted results of serial dependence. (A and B) Heading error induced by serial dependence plots against the actual heading. Solid lines show the mean of participants’ raw data and dashed lines show the mean of predicted heading errors randomly selected from the posterior distributions.

Together, the prediction results suggest that the Bayesian model can predict the serial dependence in heading perception from optic flow in the no-load condition and partially predict the effects of attention on the serial dependence in heading estimation. 

Discussion

The current study examined whether and how attention affected the estimation accuracy and serial dependence in the heading estimation from optic flow. The results showed that when the attentional resources were diverted to the number addition task, the estimation accuracy decreased, consistent with Sun et al. (2024). Additionally, attractive serial dependence increased as the attentional resources allocated to the currently (or previously) presented flow display were increased (or decreased). These findings suggest that attention plays a crucial role in heading estimation from optic flow. Moreover, the results were qualitatively predicted by our Bayesian inference models, indicating that the estimation of heading from optic flow aligns with the Bayesian inference account, regardless of the observer's attentional state. 

Attention affects heading estimation from optic flow

We further demonstrate the effects of attention on heading estimation from optic flow (Sun et al., 2024), supporting the idea that cognitive abilities are necessary for this process, contrary to the solely information-driven claim (Royden & Hildreth, 1999; Xing & Saunders, 2016). Xu et al. (2022) were the first to find evidence of the involvement of cognitive abilities. However, their methods intertwined working memory and attention. Sun et al. (2024) asked participants to estimate heading direction from optic flow. Importantly, in the attentional load condition, they also conducted a number-addition task, which distracted participants’ attention. They found that the estimation accuracy of the headings far away from the display center was reduced in the attentional load condition. Our finding further repeated their findings and confirmed the information-driven and cognitive nature of heading estimation from optic flow, as attention was also found to be involved. 

Uncovering the effects of attention on heading estimation from optic flow opens several avenues for future studies. This discovery may prompt researchers to reevaluate the influence of other cognitive abilities, such as decision-making, on heading estimation. Sun et al. (2022) arranged the heading following uniform and single-modal distributions. The center of the single-model distribution is right to the center of the uniform distribution. They found the effects of distributions on heading estimation (also see Sun et al., 2023). In particular, when some headings were presented less frequently, participants biased their heading estimates more toward those headings, known as the prevalence-induced concept change effect (Levari, 2022; Levari et al., 2018). The results indicated that participants learned the heading distribution during the experiment and might have employed different decision strategies based on it. In addition to the underestimation bias revealed in the current study, three previous studies also found an overestimation bias in heading estimation (Crane, 2012, 2014; Cuturi & MacNeilage, 2013; Gu et al., 2010). After comparing these studies, we found that the underestimation bias studies typically asked participants to respond on a line with a limited range (see current study), whereas the overestimation bias studies typically asked participants to respond on a circle that was unlimited. Participants may adopt a conservative strategy when seeing a bounded response range and underestimate their estimates. In contrast, participants may adopt a loose strategy when seeing a circle and overestimate their estimates. Taken together, it can be inferred that decision-making also plays a role in heading estimation from optic flow. However, this question cannot be addressed by the existing studies. 

Moreover, the information drive and cognitive occurrence mechanisms imply that heading estimation from optic flow is a feed-forward and feed-backward bidirectional process. Previous heading perception studies using neurophysiological and brain-imaging techniques have mainly focused on the feed-forward process. This involves the projection of visual information from primary visual cortical areas (V1, V2, V3B/KO) to high-level visual cortical areas (e.g., MT/MST, VIP, CSv) (see Sun, 2020 for a review). The discovery of the attentional effect suggests that the activities of neurons selectively responding to headings from optic flow can be modulated by the high-level cognitive cortical areas (e.g., prefrontal cortex) (Bichot, Xu, Ghadooshahy, Williams, & Desimone, 2019; Kane & Engle, 2002; Rossi, Pessoa, Desimone, & Ungerleider, 2009). Additionally, studies have shown that the neural activities of visual cortical areas decrease when attention is diverted to other tasks (e.g., Bahrami et al., 2007; Culham et al., 2001; Nizamoğlu & Urgen, 2022). Our Bayesian model also suggests that the standard deviation of the likelihood distribution is increased in the attentional load condition, consistent with the above findings. However, none of these studies discussed the effects of high-level cortical areas on these low-level visual cortical areas. Addressing the relationship between the high-level and low-level cortical areas can help us to build a comprehensive neural network and understand the brain globally. 

Previous studies have shown that the features stored in working memory are more likely to be attended (Awh & Jonides, 2001; Awh, Jonides, & Reuter-Lorenz, 1998; Theeuwes, Belopolsky, & Olivers, 2009). Additionally, the attended features are more likely to be remembered (Belopolsky, Kramer, & Godijn, 2008; Schmidt, Vogel, Woodman, & Luck, 2002; Souza, 2016). Many studies have also found that attention and working memory share some cortical areas (Awh & Jonides, 2001; Gazzaley & Nobre, 2011; Ikkai & Curtis, 2011). That is, attention and working memory are connected closely. Therefore, the involvement of attention and working memory in heading estimation from optic flow inspires researchers to consider how they interactively affect heading estimation, improving our understanding of the processing mechanisms underlying heading estimation from optic flow. 

Heading estimation is a Bayesian inference process in different attentional states

Another important contribution of the current study is that heading estimation from optic flow is a robust Bayesian inference process independent of the observer's attentional state. The Bayesian inference account is tested when all attentional resources are allocated to the heading perception task (Denison, Adler, Carrasco, & Ma, 2018; Rao, 2005; Whiteley & Sahani, 2012). Studies have shown that the tuning profiles of neurons become broader in the high-load condition than in the low-load condition, suggesting that neural sensitivities decrease with increasing the attentional load (Bahrami et al., 2007; Culham et al., 2001; Nizamoğlu & Urgen, 2022). Hence, the attentional load can decrease the internal representation of physical features. In accordance with this, we well-predicted participants’ heading estimates in the attentional load condition by increasing the standard deviation of the likelihood distribution. This result implies that the heading estimation from optic flow remains a Bayesian inference process even when the observer's attention is diverted to another task. 

Note that our models are classical Bayesian inference models, proposing that the final estimate is the mean of the posterior distribution, which is an optimal combination of prior and likelihood distributions (Jazayeri & Shadlen, 2010; Knill & Richards, 1996; Körding & Wolpert, 2004; Stocker & Simoncelli, 2006; van den Berg, Vogel, Josić, & Ma, 2012). The prior distribution is accumulated in our daily life and cannot be changed easily. In contrast, the certainty of the likelihood distribution can be affected by the physical features and neuronal fatigue levels. These factors generate noise, which can be categorized as external/physical and internal/neural noise. However, the classical Bayesian inference theory cannot distinguish between changes in the likelihood distribution caused by internal or external noise. Wei and Stocker, 2015, Wei and Stocker, 2017 developed a Bayesian efficient coding model by combining the efficient coding theory (Dan, Atick, & Reid, 1996; Lewicki, 2002) and the Bayesian-decoding theory (Knill & Richards, 1996; Körding & Wolpert, 2004) to address this question. The model predicted the estimation bias of many features (Wei & Stocker, 2017), including the peripheral bias in heading estimation from optic flow (Crane, 2012; Crane, 2014; Cuturi & MacNeilage, 2013; Gu et al., 2010). However, this model suggests that, although increasing external noise can lead to an increase in center bias, increasing internal noise can result in a peripheral bias. In the current study, the optic flow stimuli remained constant across all conditions, ensuring stable external noise. The increase in the standard deviation of the likelihood distribution can only be attributed to the internal noise induced by the attentional load. Our results are inconsistent with the prediction of the Bayesian efficient-coding model. This model predicts that the underestimation bias would be decreased in the attentional load condition that increases the internal noise. Therefore, further examination of the effectiveness of the Bayesian efficient-coding model is necessary to explain our findings. 

Effects of attention on serial dependence and its Bayesian inference account

In terms of serial dependence, we found an attractive serial dependence in heading estimation, consistent with previous studies (Sun et al., 2022; Xu et al., 2022). In addition, we directly revealed that attention affected serial dependence, supporting the notion that it also occurs at the cognitive stage (see Cicchini et al., 2024; Manassi et al., 2023 for review). Previous serial dependence studies have been highly controversial regarding its mechanism of occurrence. Some studies suggest that serial dependence occurs purely at the sensory stage and does not require cognitive abilities (Cicchini et al., 2017; Fischer & Whitney, 2014; Manassi et al., 2018). In contrast, other studies argued that serial dependence occurs at the cognitive stage (Ceylan, Herzog, & Pascucci, 2021; Pascucci et al., 2019; Fornaciai & Park, 2020). The current study found that, when attentional resources allocated to the current heading stimulus were decreased, observers increased their reliance on the previous stimuli, showing a strong serial dependence. Conversely, when attentional resources allocated to the previous heading stimulus were reduced, observers could decrease their reliance on the previous stimuli, showing a weak serial dependence. Accordingly, combined with the involvement of working memory (Sun et al., 2023), we are more convinced that the nature of serial dependence is both information driven and cognitive. 

These trends are consistent with the prediction of the Bayesian inference process (see Cicchini et al., 2024; Manassi et al., 2023 for review), suggesting that observers optimally combine the previous experiences (including the long-term experience learned in everyday life (Sun et al., 2023) and short-term experience (i.e., the previously presented heading) and current headings to estimate heading directions. In previous studies that propose the Bayesian inference account for serial dependence, researchers only took one short-term experience (i.e., the previously presented feature) into their Bayesian model, ignoring the effects of long-term experience on perception (e.g., Cicchini et al., 2018; see Cicchini et al., 2024 for review). Our current model provided a comprehensive computational process for the serial dependence. 

Moreover, we have further demonstrated that serial dependence is well consistent with the Bayesian inference account (Cicchni et al., 2018; Fritsche, Spaak, & De Lange, 2020; Xu et al., 2022), regardless of the observer's attentional state. Especially, we found as long as the reliability of one stimulus (e.g., previous) was decreased, observers would increase their reliance on the other stimulus (e.g., current), this is a bidirectional Bayesian inference process. However, recently, Gallagher and Benton (2022) found that the serial dependence was only affected by the reliability of currently presented stimuli, which is a unidirectional Bayesian process These differences can be due to two factors. One is the complexity of the stimuli. Gallagher and Benton (2022) showed a serial of low-level Gabor patches to participants. However, the current study showed optic flow contains rich low- and high-level information (Gibson, 1950). Additionally, it also can be due different the attentional states. All attentional resources in Gallagher and Benton (2022) were allocated to the orientation judgment task, whereas we varied the distribution of the resources to the heading estimation task. These factors may affect the Bayesian process, which can be tested in future studies. 

Previous studies have demonstrated that one classical feature of serial dependence is its spatial and temporal tuning (see Manassi et al., 2023 for review). The methods used in the current study cannot reveal this feature. Therefore, we used the methods in previous studies (e.g., Fischer & Whitney, 2014) and fitted the estimation error in the second heading in the no-load baseline condition (Figure 1A) as a first derivative of Gaussian function (DoG) of the relative heading that was the difference in the actual heading between the previous n^th trial and the current trial (n = 1, 3, 5, etc.). The results were shown in Figure 8. It clearly shows a tuning curve in each previous nth trial, and the amplitude gradually decreases with the increase of time interval between trials. Therefore, the current study well replicated the serial dependence in heading estimation from optic flow (e.g., Xu et al., 2022). 

Figure 8.

 

Serial dependence in the no-load baseline condition of experiments A and B. The data analysis methods were similar to those in Fischer and Whitney (2014). The x axis was the relative heading that was the difference in the actual heading between the previous n^th trial and the current trial (n = 1, 3, 5, etc.). The y axis is the estimation error of current trial that was the difference between the heading estimate and the actual heading. Each dot was the mean estimation averaged across 40 participants (20 in experiment A and 20 in experiment B). The error bar was the standard error across 40 participants. We fitted a first derivative of Gaussian (DoG) function. The amplitude of the DoG curve indicates the strength of the serial dependence.

Summary

The current study reveals the effects of attention on heading estimation from optic flow. Both the estimation accuracy and serial dependence are varied with the observer's attentional state. Importantly, our Bayesian model qualitatively captures these changes, suggesting that the Bayesian inference account for heading estimation from optic flow is reasonable regardless of the attentional state. The current study demonstrates that cognitive abilities are involved in heading estimation from optic flow, prompting following researchers to understand its processing mechanisms from a more comprehensive perspective. 

Acknowledgments

Supported by National Natural Science Foundation of China, China (No. 32200842) to Qi Sun. 

Qi Sun designed the study; Qian Sun discussed, proofread, and revised the manuscript; SYW collected and initially analyzed the data; QS analyzed the data and wrote the paper; LZZ and FHY proofread the paper. 

Data and code: We have made data and code available for review purposes. https://osf.io/ps2xc/?view_only=57ff3341312146168776cea90c029a23. 

Commercial relationships: none. 

Corresponding authors: Qi Sun and Qian Sun. 

Emails: [e-mail] and [e-mail]. 

Address: Zhejiang Normal University, 688 Yingbin Road, Wucheng District, Jinhua 321000, P. R. China.