Prediction of time to contact under perceptual and contextual uncertainties

Abstract

Accurately estimating time to contact (TTC) is crucial for successful interactions with moving objects, yet it is challenging under conditions of sensory and contextual uncertainty, such as occlusion. In this study, participants engaged in a prediction motion task, monitoring a target that moved rightward and an occluder. The participants’ task was to press a key when they predicted the target would be aligned with the occluder's right edge. We manipulated sensory uncertainty by varying the visible and occluded periods of the target, thereby modulating the time available to integrate sensory information and the duration over which motion must be extrapolated. Additionally, contextual uncertainty was manipulated by having a predictable and unpredictable condition, meaning the occluder either reliably indicated where the moving target would disappear or provided no such indication. Results showed differences in accuracy between the predictable and unpredictable occluder conditions, with different eye movement patterns in each case. Importantly, the ratio of the time the target was visible, which allows for the integration of sensory information, to the occlusion time, which determines perceptual uncertainty, was a key factor in determining performance. This ratio is central to our proposed model, which provides a robust framework for understanding and predicting human performance in dynamic environments with varying degrees of uncertainty.

Introduction

Predictions form an integral part of our daily lives, guiding our decisions and actions by anticipating future events and modulating our responses accordingly. An example of the significance of predictions becomes apparent when we track an object that becomes obscured by other elements in the environment. Consider a situation where we are driving and notice a cyclist approaching an intersection. As the cyclist begins to cross, a large vehicle pulls up beside us and obstructs our view of the cyclist. Sometimes, the moment when the cyclist becomes occluded can be uncertain and unexpected, for example, if the large vehicle changes its speed or advances unexpectedly. However, we can extrapolate the position of the cyclist (i.e., motion extrapolation; Finke, Freyd, & Shyi, 1986; Linares, López-Moliner, & Johnston, 2007; Nijhawan, 1994) based on its velocity and position prior to the occlusion and make a prediction about when the cyclist will reappear ahead of the vehicle (Figure 1A). Therefore, in situations with high perceptual uncertainty (Glaze, Kable, & Gold, 2015), like when moving targets are occluded and the accumulation of sensory evidence is impaired (de la Malla & López-Moliner, 2015), predictions become essential. 

Figure 1.

 

Prediction motion task. (A) Example of the prediction motion paradigm, where a driver sees a cyclist (yellow area) crossing the street. When a large vehicle passes by, it momentarily obstructs the driver's view of the cyclist (gray area). Despite this, the driver can use information prior to the moment of occlusion to predict when the cyclist will reappear in view. (B) Schematic representation of the two occluder conditions. In the predictable condition (top panel), the occluder was located above the target and reliably indicated where the target would disappear. Occlusion is represented by the dotted circle. In the unpredictable condition (bottom panel), the occluder appeared below the target and did not reliably indicate where the target would disappear. The end position of the occluder remained the same in both conditions. The red arrows denote the relevant location for timing the responses.

The necessity of predictions in situations where objects are occluded has largely motivated the adoption of prediction motion (PM) tasks. In PM tasks, a moving target disappears, typically behind an occluder, and participants have to predict when the target would reach the end of the occluder, usually by means of a keypress (Battaglini, Campana, & Casco, 2013; Baurès, Oberfeld, & Hecht, 2010; Bennett, Baures, Hecht, & Benguigui, 2010; Makin, Poliakoff, Chen, & Stewart, 2008; Makin, Stewart, & Poliakoff, 2009; Peterken, Brown, & Bowman, 1991; Rosenbaum, 1975; Sokolov & Pavlova, 2003). In PM tasks, the target never reappears, entailing the responses to be executed solely on predictions. Successful predictions require that participants combine the velocity and position information available during the visible period to create an estimate of the time interval spanning from the moment of target occlusion to when they expect the target would reach the end of the occluder. This interval is often referred to as time to contact (TTC; Hecht & Savelsbergh, 2004; Tresilian, 1995). Performance in PM tasks is measured in terms of the timing error, the difference between the target's actual and predicted arrival time (i.e., timing accuracy). Several variables influence the timing error, such as the target's size and speed. Lower speeds result in lower timing errors for larger targets compared to smaller targets, but the effects of the target size are reversed with higher speeds (Battaglini et al., 2013; Sokolov & Pavlova, 2003). The target's contrast also alters the timing error according to the target speed. In this case, lower speeds lead to lower timing errors with low contrast compared to high contrast, and opposite effects with higher speeds (Battaglini et al., 2013). Another influencing variable is the type of motion, where second-order motion may result in higher timing errors than first-order motion (Aguilar-Lleyda, Tubau, & López-Moliner, 2018), and prior adaptation to real motion, which results in different timing errors depending on the duration of exposure (Battaglini, Campana, Camilleri, & Casco, 2015; Battaglini, Casco, Isaacs, Bridges, & Ganis, 2017; Gilden, Blake, & Hurst, 1995). Lastly, the visible duration of the target (Peterken et al., 1991) and the occlusion time (Battaglini & Mioni, 2019; Bennett et al., 2010) are key influencing factors that are explained in further detail below. Since the visible period enables the integration of sensory information regarding the target's movement, we will refer to this period as the integration time. 

Notably, the effect of the integration and occlusion time on temporal accuracy have often been treated independently. Nevertheless, they both directly affect the accumulation of perceptual/sensory evidence that feeds the predictive mechanism for TTC estimation. In general, longer integration times lead to more accurate responses (Peterken et al., 1991), potentially due to the increased time to consolidate target motion information (Battaglini & Ghiani, 2021) and obtain better estimates of the target velocity and its predicted future position (Aguilar-Lleyda et al., 2018). Conversely, longer occlusion times are associated with impaired performance (Battaglini et al., 2013; Bennett et al., 2010; Peterken et al., 1991), possibly due to noise accumulation that increases the uncertainty of the prediction or to a velocity/visual memory decay. Thus, while integration time directly affects evidence accumulation, occlusion time impacts perceptual uncertainty. Perceptual efficiency likely hinges on an optimal balance between these two variables, where specific combinations enable the visual system to perform accurate motion extrapolations with minimal uncertainty. Such efficiency is likely achieved through the successful utilization of available sensory information before occlusion to compensate for the absence of visual input during occlusion. In this study, we focus on this balance between sensory accumulation and perceptual uncertainty by assessing the effect of the integration to occlusion time ratio on the quality of predictions of TTC (accuracy and precision). This ratio revealed consistent timing accuracy patterns, for which we provide an explanatory model. The model parameters obtained from the timing errors (accuracy) allowed us to predict performance precision (SD of temporal errors), which is a variable that has received less attention in previous motion extrapolation work, except for in interceptive studies (e.g., Brenner & Smeets, 2015) and in parabolic motion studies (de la Malla & López-Moliner, 2015). Moreover, we sought to address how uncertainty regarding the moment of occlusion affected performance and did this by manipulating the reliability of the occluders. We also aimed to assess whether our model would capture differences caused by this uncertainty. 

Previous studies assessing eye movements have shown that participants are able to track occluded targets (Bennett & Barnes, 2006) and that the eye position is a good indicator of motion extrapolation estimates (Jonikaitis, Deubel, & de'Sperati, 2009). Thus, we recorded eye movements to determine how participants sample the visual information to shape their responses and whether eye movements are influenced by the uncertainty of occlusion onset. We hypothesize that any differences in behaviors between occluder conditions will likely be reflected in eye movement patterns. A possible explanation for the key role of eye movements is that the brain creates dynamic representations of the target's changing position (Bosco et al., 2015) based on a velocity memory, or stored representation of the target's velocity. The eyes then attempt to keep up with the predicted position, occasionally lagging and then catching up (Bennett & Barnes, 2006; Orban de Xivry, Bennett, Lefèvre, & Barnes, 2006). It is therefore reasonable to suggest that eye movements play an essential role in the quality of TTC predictions. This is supported by several studies that have manipulated eye movements by instructing participants to either fixate or freely pursue the target. They have found that TTC estimates were more precise when participants were able to pursue the target, as opposed to when eye movements were restricted (Bennett et al., 2010; Makin & Poliakoff, 2011). This may be partly caused by more precise judgments of a target's velocity when one can pursue the target, as opposed to fixating elsewhere (de la Malla, Smeets, & Brenner, 2022). Therefore, there is likely an advantage when utilizing retinal and extra-retinal information (Bennett et al., 2010) such that eye movements allow participants to better integrate available visual motion information and make more precise predictions of TTC. In this study, we allowed participants to freely choose their gaze movements instead of providing explicit instructions of where to look. This enabled us to observe the impact of the occluder's predictability on gaze behavior. 

Methods

Participants

Twelve participants (six females, mean age 26 years) took part in the study. All participants had normal or corrected-to-normal vision and no evident motor abnormalities. All of them were naive to the study's purpose. This study was part of an ongoing research project approved by the Ethics Committee of the University of Barcelona (IRB: 00003099). All participants signed an informed consent prior to commencing. 

Apparatus and calibration

The experiment was conducted in a room with constant artificial light and no additional light source. Participants sat in a chair in front of a 24.5-in., 1,920-pixel × 1,080-pixel monitor with a refresh rate of 240 Hz in which stimuli were presented. We used a biteboard with a dental imprint to keep participants’ head at a fixed distance of ∼65 cm from the screen. Eye movements were recorded using a video-based eye tracker (Eyelink 1000 Plus; SR Research, Ottawa, Ontario, Canada) at a sampling rate of 1000 Hz. 

Before each block, the Eyelink 1000 Plus standard 9-point calibration and validation procedures were carried out where participants were instructed to fixate on the center of a white circle that appeared on screen and moved to different locations. 

Stimulus and procedure

The experiment was programmed in Python (version 3.7.2) using Pyglet, one of the OpenGL implementations in Python. At the beginning of each trial, a 1-cm diameter white circle and a black rectangle that always had the same end position but varied in length appeared on the screen on a gray background (Figure 1B). The target remained stationary for a random period between 1.0 and 1.5 seconds and then moved rightward at a constant velocity of 5, 10, or 15 cm/s. At some point during the target's rightward trajectory, the target disappeared. Participants were instructed to press the space bar with their dominant hand when they predicted the target would be aligned with the right edge of the occluder. In each trial, the target was visible (integration time) for 150, 300, 600, or 1,200 ms before disappearing. The length of the occlusion time was 100, 200, 400, 800, or 1,600 ms. This resulted in eight possible combinations of integration to occlusion time ratios, with higher ratios indicating longer integration times and decreased occlusion times. The starting horizontal -x- position of the target depended on the combination of the integration and occlusion time, and the target's vertical -y- position was always the same (in the middle of the screen). The length of the occluder depended on the occlusion time and the target's velocity, and it had a height of 1.2 cm. The occluder could be located either 0.8 cm above or below the targe's path. To indicate which condition was predictable, we explicitly told participants that when the occluder was above the target's path, it reliably indicated that the target would disappear at the onset of the occluder (predictable condition, top panel in Figure 1B). However, if the occluder was below the target's path (unpredictable condition, bottom panel in Figure 1B), there was no reliable indication of where the target was going to disappear (i.e., the position where the target disappeared could coincide or not with the position of the occluder). Both conditions were randomly interleaved. We used occluders in both conditions to ensure that timing information about the relative distances between the target and the end position, which is likely necessary to make accurate prediction of TTC (Chang & Jazayeri, 2018), was always available. At the end of each trial, participants received feedback on the screen indicating whether their response had been “too early,” “too late,” or “success.” The window of success was established as ±50 ms, with anything less than −50 ms considered “too early” and any response greater than 50 ms considered “too late.” We included feedback to avoid the large biases that result from long occlusion durations (López-Moliner, Supèr, & Keil, 2013) and to better capture the effect of accumulation and decay of evidence. Participants completed six blocks of 120 trials each (from the combination of 4 integration times × 5 occlusion times × 3 velocities × 2 occluder conditions). Trials were presented in a random order in each block and for each participant. Each block lasted approximately 10 minutes, and participants could rest for as long as they wanted between blocks. 

Data analysis

Data analysis was performed with R Statistical Software (R Core Team, 2023). For each trial, we calculated the timing error by looking at the temporal difference between when the target reached the end of the occluder and when participants pressed the space bar. Thus, positive errors meant that participants responded too late because the target had already passed the end of the occluder, and negative errors meant that participants were too early because the target had not yet reached the end of the occluder. We removed trials in which the response time was less than 0.01 ms, as these were trials in which participants pressed the space bar when the target had not yet started to move (14 trials excluded). To locate any outliers, we grouped trials by occluder condition, target velocity, occlusion time, and integration time and identified trials in which the error was larger than 3 standard deviations from the mean for each participant (0 trials excluded). 

To study the main effects of the occluder condition, target velocity, and integration to occlusion time ratio on the mean timing error, as well as interactions between these variables, we conducted an initial analysis with an analysis of variance (ANOVA) in which participants were treated as random effects. A similar approach was used to analyze the effects of the occluder condition on the parameters of the model explaining timing accuracy (further details below). 

To assess the quality of the TTC predictions as a function of the integration to occlusion time ratio, we computed the variable error by grouping trials by the occluder condition and the ratio and then calculated the standard deviation of the timing error for each participant. A high variable error suggests a lower quality prediction due to heightened perceptual uncertainty about the target's moment of occlusion or TTC. We then computed a second ANOVA to test the effects of the occluder condition, velocity, and integration to occlusion time ratio on the variable error. 

For the eye movement analyses, we averaged the estimates of where each eye was looking on the screen to get an estimated single gaze position at each moment of time. We evaluated differences in eye movement patterns between occluder conditions by analyzing where gaze was directed across time in each condition. Saccades were detected using an algorithm based on a model distribution (null model) of eye acceleration during fixation and smooth pursuit (null acceleration with a mean near zero and a measured SD). Saccades were detected when the likelihood of the measured acceleration fell below a given threshold according to the null model. We determined the number of saccades per trial and where these saccades were directed (toward the target, the beginning of the occluder, or the end of the occluder) and excluded trials with more than 10 saccades, as this is above what one would expect with trials of such short duration (21 trials excluded). We measured the pursuit gain in each trial by dividing the median gaze velocity (excluding saccades and blinks) by the target velocity and created heatmaps of participants’ gaze to provide an overview of the potential differences in eye movement patterns. The mean lateral gaze positions that were not within the range of the initial target position and the end of the occluder were discarded. 

Results

Timing accuracy and precision

Figures 2A and 2B show the timing accuracy (systematic timing error) and the timing variable error (SD, or reciprocal of the precision), respectively. Accuracy data (Figure 2A) are shown as a function of the log of the integration to occlusion time ratio for the three different target velocities and occluder conditions. The accuracy pattern is similar for the two occluder conditions and velocities. We conducted an initial ANOVA on the accuracy data (mean timing error) with occluder condition and target velocity as factors and the ratio as a numerical covariate. Both the occluder condition (F(1, 553) = 7.22, p = 0.01) and the velocity (F(1, 553) = 4.03, p = 0.02) had a significant effect on timing accuracy. Participants made more accurate predictions when they had reliable visual information about where the target was going to disappear, as shown by a smaller mean timing error when the occluder was predictable (1.73 ± 16.18 ms, mean ± standard error) compared to when it was unpredictable (20.08 ± 14.42 ms; mean ± standard error). In addition to the occluder condition, the target's velocity was an important modulator for performance, with varying effects on the error. When the target moved at its slowest velocity (5 cm/s), participants tended to underestimate TTC (t(−2.47), p = 0.03), resulting in a negative error (−21.97 ± 16.89 ms, mean error ± standard error), whereas when the target's velocity increased to 10 or 15 cm/s, there was a tendency to overestimate TTC (26.86 ± 14.50 ms, 27.68 ± 18.37 ms, respectively; mean error ± standard error). This was the case for both occluder conditions, but the overestimation was more pronounced when the occluder was unpredictable (Figure 2A). This pattern of errors contingent on the target's velocity is consistent with previous studies (Aguilar-Lleyda et al., 2018), where feedback on performance was also provided. 

Figure 2.

 

Mean timing error and variable error. (A) Mean timing error (in ms) across participants as a function of the log of the integration to occlusion time ratio. Data points are separated by target velocity (color-coded). The text shows the RMSE corresponding to the best fit of the model prediction represented by the lines (see model section). Error bars are 95% confidence intervals. (B) Mean standard deviation of the timing error as a function of the log of the integration to occlusion time ratio. The inset shows the same decay of the SD in linear coordinates.

The integration to occlusion time ratio had a considerable effect on the timing error (F(1, 553) = 20.36, p < 0.0001). When the ratio was low, meaning the target's visible trajectory was short and the occlusion time was long, participants made the greatest negative errors (i.e., underestimation of the target's TTC). However, as the ratio increased, errors decreased quickly and estimates became more accurate, remaining within the predefined “success” range. Interestingly, the peak positive value was outside the window of success and occurred when the ratio was close to a value of 1. After this peak, accuracy leveled off to an asymptotic value near zero error. This trend was similar irrespective of the occluder condition (Figure 2A) and had a distinctive shape: an initial steep phase (in which negative errors were largely reduced with small increments of the ratio), followed by a deceleration and a peak near a ratio value of 1, and then a final asymptotic error level. Only the interaction between the occluder condition and the ratio was significant (F(1, 553) = 4.79, p = 0.03). Due to the apparently nonlinear relation between the ratio and accuracy, we considered this initial ANOVA as a first step to characterize the results. Further analysis of these phases will be presented with the proposed model below. 

Although participants were more accurate with the predictable occluder, they were not more precise (16.18 vs. 14.42 ms standard error of the mean for the predictable and unpredictable conditions, respectively). Figure 2B shows the decrease of the variable error (SD) as a function of the integration to occlusion time ratio for the two occluder conditions. An ANOVA conducted on the variable error revealed a significant effect of the ratio on precision (F(1, 553) = 344.5754, p < 0.0001). No other variables (occluder condition and velocity) significantly affected precision, with F values less than 1 in all other cases. The evident relationship between both accuracy and precision with the integration to occlusion time ratio strengthens the argument that this ratio acts as a critical factor in guiding participants’ timing responses. 

Eye movements

There were clear differences in eye movement patterns between the two occluder conditions (Figures 3–5). These differences give us some insights into how eye movements could be related to predictions of TTC. Figure 3 shows the eye movement data from a representative participant (Figures 3A, 3B), as well as from all participants (Figures 3C, 3D), when the target velocity was 15 cm/s, the integration time was 1,200 ms, and the occlusion time was 1,600 ms for the predictable (Figures 3A, 3C) and unpredictable (Figures 3B, 3D) occluder. The heatmaps show normalized time as a function of normalized lateral gaze position. Normalized time represents the time interval from when the target began to move (time = 0.0) until the participant pressed the space bar (time = 1.0). Normalized lateral gaze position represents the horizontal range from the initial position of the target to the end of the occluder. The horizontal red line indicates the position where the target became occluded. Note that this corresponds to the start of the occluder in the predictable condition but not necessarily in the unpredictable condition. The colored density scale represents frequency and shows where the gaze is concentrated, with red colors indicating higher density areas and blue colors indicating lower density areas. Therefore, red clusters indicate fixation spots, lower density gaps indicate saccades, and a uniform density with no gaps indicates smooth pursuit. 

Figure 3.

 

Differences in eye movement patterns between occluder conditions. (A) Eye movement pattern of a representative participant in the predictable occluder condition. (B) Eye movement pattern of the same participant as in A for the unpredictable occluder condition. (C) Eye movement pattern of all participants in the predictable and (D) unpredictable condition. Normalized time of 0 represents the target movement onset, and the value of 1 represents the moment of the response. Normalized lateral gaze position of 0 represents the initial position of the target, and 1 represents the occluder's right edge. All four panels include data only from trials where the target moved at 15 cm/s, the integration time was 1,200 ms, and the occlusion time was 1,600 ms. The color scale represents the relative density of the data points, with blue indicating areas that participants hardly directed their gaze (low density) and red indicating areas where participants spent much more time looking (high density).

Figure 4.

 

Differences in eye movement patterns across participants with the predictable occluder as a function of trial duration. Each panel represents different trial durations (in seconds), made up by the different combinations of integration and occlusion times. The red lines indicate the position where the occlusion starts. In some cases, there is more than one red line because of different combinations of integration and occlusion times for the same trial duration.

Figure 5.

 

Differences in eye movement patterns across participants with the unpredictable occluder as a function of trial duration. Each panel represents different trial durations (in seconds), made up by the different combinations of integration and occlusion times. The red lines indicate the position where the occlusion starts. Note that in this case, the position of the start of occlusion does not necessarily coincide with where the occluder's left edge is. In some cases, there is more than one red line because of different combinations of integration and occlusion times for the same trial duration.

Figure 3 highlights the differences in eye movement patterns between occluder conditions with specific velocity, integration, and occlusion time values. To further describe the differences in eye movement patterns, we looked at data across all participants, velocities, and trial durations. When the occluder was predictable, there was pursuit of the target as it began to move, with some catch-up saccades toward the target (1.72 ± 0.03; mean count ± standard error) and saccades toward the starting position (i.e., the left edge) of the predictable occluder (1.64 ± 0.02; mean count ± standard error). On the other hand, the unpredictable occluder condition led to a higher smooth pursuit gain (0.74 ± 0.05) compared to the predictable condition (0.70 ± 0.07; mean pursuit gain ± standard error) and less saccades directed toward both the target (1.55 ± 0.02) and the onset of the unpredictable occluder (1.49 ± 0.02; mean count ± standard error). The predictable condition then showed a strong fixation at the start of the occluder, illustrated by the high-density cluster on the red line in Figures 3A and 3C, compared to a weaker fixation past the start of the unpredictable occluder (weak cluster past the red line in Figures 3B and 3D). During the occlusion period, the predictable condition had a series of saccades toward the end of the occluder, while the unpredictable condition showed less evident tracking through occlusion. This was reflected by the saccade count during occlusion toward the end of the predictable (1.59 ± 0.02) versus unpredictable (1.42 ± 0.02; mean count ± standard error) occluder. 

The eye movement differences between occluder conditions were most evident as trial duration increased. Figure 4 shows heatmaps with data from all participants, velocities, integration times, and occlusion times with the predictable occluder, and Figure 5 shows data with the unpredictable occluder. The different panels in each figure represent different trial durations in seconds from the possible combinations of occlusion and integration times. In the longest duration trials (2.8 seconds), where the differences were most apparent, the predictable condition had a higher number of saccades directed toward the target during the visible trajectory (2.68 ± 0.08) compared to the unpredictable condition (2.32 ± 0.07; mean count ± standard error) and then a series of saccades toward the end of the occluder during occlusion (2.11 ± 0.07 compared to 1.77 ± 0.07 in the unpredictable condition; mean count ± standard error). On the other hand, the unpredictable occluder had a higher smooth pursuit gain during the visible trajectory (1.00 ± 0.06), compared to the predictable condition (0.80 ± 0.06; mean pursuit gain ± standard error), and no clear series of saccades during occlusion. There was also a direct fixation at the right edge of the occluder in the unpredictable condition rather than a series of saccades throughout the occluded period. 

Proposed model

To understand the consistent accuracy pattern observed across participants (Figure 2A), we looked for an explanation in terms of a mechanistic model that could comprehensively account for these findings. Figure 6 shows individual accuracy data for six of the participants (data for the remaining participants can be found in the Supplementary Materials) separated by occluder conditions (top panel: predictable; bottom panel: unpredictable) and target velocity (color-coded). The lines denote the model fits for each participant. 

Figure 6.

 

Individual timing error. Mean timing error as a function of the log of the integration to occlusion time ratio for six participants (in different columns), for each occluder condition (in different rows), and velocities (color-coded). The text indicates the RMSE of the fits denoted by the lines, and error bars denote standard errors.

Despite the apparent complexity of the relationship between the integration to occlusion time ratio and the accuracy, the observed patterns can be described as an initial acceleration phase followed by a Gaussian modulation of the timing error. The following expression captures these two aspects of the predicted TTC:  

\begin{eqnarray} TTC\left( x \right){\rm{\ }} &=& {\rm{\ }}a \cdot {\rm{\sigma }} \cdot \exp \left( { - \frac{{{{{\left( {x - \mu } \right)}}^2}}}{{2{{{\rm{\sigma }}}^2}}}} \right)\nonumber\\ && - {{{\rm{\sigma }}}^{ - b}} \cdot {{x}^{ - b}} + c\quad \end{eqnarray}

(1)

 

Equation 1 is a composite function with both Gaussian and power-law characteristics. The first part describes the Gaussian component, where µ denotes the position of the peak, and σ indicates the standard deviation, which determines the width of the bell-shaped curve. In our model, the parameter µ (peak value), which was consistently observed to be around 1 or 1.5 across participants, plays a crucial role in defining the critical values of the integration to occlusion time ratio. The consistency in µ allows us to simplify the model by reducing the number of free parameters without losing descriptive accuracy. The fact that µ converges toward these values suggests that it represents a pivotal point in the perceptual process. When µ is 1, it represents an important threshold within the interplay between the target's visible and occluded period. It highlights a significant transition point, or optimal ratio value, where the perceptual system reaches a maximum accuracy with minimal available information to predict the motion trajectory during occlusion. The observed homogeneousness in µ suggests that the perceptual system may use a common integration mechanism across individuals, revealing a ratio of integration to occlusion time that is most effective for motion extrapolation. This implies that there is an inherent preference or optimal condition within the perceptual system for processing motion information, captured by the specific balance point dictated by µ. In the context of our task, σ could be interpreted as how reliably the evidence is accumulated per step of the integration to occlusion time ratio. In this sense, a smaller σ will result in a greater reliability. Therefore, σ or aσ determines how the TTC predictions, and hence timing accuracy, evolve as a function of the ratio. The height, or amplitude, of the Gaussian over the baseline (0) is represented by the factor (a · σ). Note that the height is modulated by σ and this property captures the fact that larger uncertainties, reflected by a larger σ, will result in a larger proportion of overestimation (late response) and larger increments of the ratio to level off the timing error. The parameter a allows us to exert control on the amplitude without affecting the statistical integrity and characteristics of the Gaussian distribution represented by σ. The subtractive term σ^−b · x^−b represents a power-law function, which modulates the rate at which the initial error is reduced with increments of the ratio, that is, this function describes the whole rate of accumulation of sensory evidence provided by the ratio. The power function has been proposed (Gold & Shadlen, 2000, Gold & Shadlen, 2003) as a drift function in drift-diffusion models (Ratcliff, Smith, Brown, & McKoon, 2016) to describe the rate at which sensory evidence is accumulated at each time step. We set b = 1.6 in all cases from preliminary fits of the model to the data. The presence of σ in this term implies that this rate is modulated by the same factor that affects the spread of the Gaussian component, linking the two processes. Mathematically, σ influences the rate at which the error is initially reduced in this term: A smaller σ will result in steeper slopes due to the negative power. Finally, the constant, c, serves as a baseline or offset for the entire function, shifting it vertically on the coordinate plane. It represents a general bias to respond early or late, independently of the evidence provided by the ratio x. Figure 7A illustrates the model dynamics for two different values of σ while keeping a and c constant. The key parameter in fitting the model to the data is then σ. The inset in Figure 7A shows the contribution of the subtractive term (power-law rate). 

Figure 7.

 

Model dynamics. (A) The course of the model prediction is plotted against the log of the ratio for two different values of σ. The inset shows the power-law component of the model, which accounts for the initial decrease of the timing error with small increments of the ratio. (B) The predicted SD from Equation 2 of the two models in panel A. The inset shows the evolution of the sensitivity measure.

We assume that large changes in accuracy in response to relatively small changes in the stimulus (in this case, the ratio x) can be interpreted as an index of sensitivity, responsiveness, or precision. This conceptualization is inspired by how the measures of discriminability or sensitivity are computed in psychophysical theory (Kingdom & Prins, 2016). The derivative of Equation 1 with respect to the ratio (x) characterizes the change in accuracy as a function of the ratio (x) and is given by  

\begin{eqnarray*} dTTC\left( x \right)\ &=& \ - \left( {\frac{{x - {\rm{\mu }}}}{{{{{\rm{\sigma }}}^2}}}} \right) \cdot a \cdot {\rm{\sigma }} \cdot \exp\\ && \times \left( { - \frac{{{{{\left( {x - {\rm{\mu }}} \right)}}^2}}}{{2{{{\rm{\sigma }}}^2}}}} \right) + b \cdot {{{\rm{\sigma }}}^{ - b}} \cdot {{x}^{ - b - 1}} \end{eqnarray*}

 

An index of the precision or sensitivity of ratio x is  

\begin{eqnarray*} {\rm{\tau }} = \log \mathop \int \limits_0^{\rm{x}} \left| {{\rm{dTTC}}\left( {\rm{x}} \right)} \right|dx \end{eqnarray*}

 

This expression captures the sum of the absolute changes of accuracy, suggesting a running total of how much variation or fluctuation in accuracy has occurred up to each point x. Conceptually, this could be seen as an accumulation of sensitivity or responsiveness (i.e., change of accuracy) up to some value of the ratio x. The SD, or variable error, would be a function of the reciprocal of τ:  

\begin{eqnarray} SD\ = \ D \cdot {\rm{\ I}}\left( {{{\tau }^{ - 1}}} \right) + s{{d}_0}\quad \end{eqnarray}

(2)

 

The I function denotes a normalization step so that τ⁻¹ is between 0 and 1. D and sd₀ account for the amplitude and lower asymptote of the SD. Figure 7B shows the predicted SDs of the corresponding accuracy models in Figure 7A. The inset in Figure 7B shows the evolution of the precision or sensitivity τ with the log ratio. As can be seen, a smaller σ results in an initial faster decay of the SD with respect to the ratio. 

Model fits

We treated a, σ, and c as free parameters to closely align the model with the accuracy data points. The parameter µ was always set to 1.5, a decision informed by a qualitative inspection of the peak across participants. This inspection revealed a consistent peak location, suggesting that a µ value of 1.5 accurately captured the central tendency of the data and corresponded to the most significant point of interest in the integration to occlusion time ratio, as commented above. The parameter b was set to 1.6, guided by preliminary fits of the model to the data. These initial analyses demonstrated that variations in b away from 1.6 led to a qualitative divergence of the model predictions from the observed data patterns. Specifically, values different from 1.6 resulted in model behaviors that significantly deviated from the empirical evidence, indicating that a b value of 1.6 provides a critical balance point for accurately capturing the rate of acceleration of accuracy in the data. In addition, for the model's optimization process, we utilized the optim function within the R software to minimize the root mean square error (RMSE) between the observed timing error and the model's predictions. We fitted the model to individual participant data as well as to data aggregated across participants, but independently for each occluder condition and velocity. In order to predict the SD of the timing responses, we used the same σ parameter obtained from the fits to the accuracy data. We therefore fitted Equation 2 to the variable error (SD) data with D, sd₀, and a as free parameters to minimize the RMSE. 

The fits of the model for six participants are shown in Figure 6. We fitted each target velocity and occluder condition independently, and each panel shows the average RMSE across velocities obtained from the best fit. In addition, the lines in Figure 2A indicate the best fits for the average timing error across participants for the different velocities. The obtained RMSEs denote relatively good fits in the two occluder conditions. The relatively worse fit in the predictable occluder condition (left panel: RMSE 17.7 vs. 14.9 ms) was primarily due to the data of the fastest speed (15 cm/s), where the peak shifted to the right of the fixed parameter value of the fitted model (µ). 

TTC accuracy and model predictions

Figure 8 plots a summary of the parameter values and their 95% confidence intervals for the fits shown in Figure 2A. The additive parameter (c) was modulated by the speed: As the target velocity increased, so did the value of c, indicating later responses for faster velocities. The values of σ show a more complex pattern affected by both target velocity and occluder condition. We conducted ANOVAs to analyze the effects of velocity and occluder condition on the different parameters and to corroborate the observations in Figure 8. The scaling parameter, a, was only marginally affected by velocity (F(2, 55) = 3.06, p = 0.055), while the occluder condition (F(1, 55) = 0.8903, p = 0.35) and the interaction between the velocity and occluder condition (F(2, 55) = 1.74, p = 0.18) were not significant. The additive parameter, c, was significantly affected by the target velocity (F(2, 55) = 41.19, p < 0.0001) but not by the occluder condition (F(1, 55) = 0.13, p = 0.71) or the interaction between the velocity and occluder condition (F(2, 55) = 1.73, p = 0.18). Since the distribution of σ was not normal (positively skewed) and it was always larger than zero, we used a general linear model based on a Gamma distribution to obtain the significance of the parameters representing the velocity, occluder condition, and their interaction. The parameter for the target velocity was not different from 0 (estimate = 0.067 ± 0.04, t = 1.45, p = 0.14), but both the occluder condition (estimate = 0.53, t = 2.4, p = 0.013) and the interaction between the occluder condition and velocity (−0.064 ± 0.019, t = −3.272, p = 0.001) significantly affected the value of σ. Provided that σ inversely correlates with the efficiency at which the perceptual evidence is accumulated, the decreasing trend in σ from 5 cm/s to 10 cm/s suggests that as velocity increased, participants accumulated evidence more efficiently up to a certain point. For the unpredictable occluder condition, σ also decreased from 5 cm/s to 10 cm/s but then increased at 15 cm/s. This suggests that participants experienced a decline in the efficiency of evidence accumulation at the fastest velocity tested. Moreover, when the target moved at 15 cm/s, the σ value for the unpredictable condition rose, suggesting that participants struggled to accumulate evidence as efficiently when the movement was fast, and the occlusion onset was uncertain. These results imply that the uncertainty of the occluder significantly impacted the ability to accumulate evidence for motion extrapolation, particularly as the speed of the moving target increased. This influence was likely mediated by the differences in gaze behavior in the two occluder conditions. We resume this point in the discussion. 

Figure 8.

 

Model parameters. The values of the fitted parameters (a, c, and σ), and their 95% confidence interval as a function of the target's velocity and the occluder conditions. The parameter values correspond to the model fitted to the average timing accuracy data from Figure 2A.

The significant interaction between the occluder condition and velocity on the parameter σ, as determined by the general linear model, stands in contrast with the results from the ANOVA on the timing error. The ANOVA results on timing accuracy did not reveal a significant interaction between the occluder condition and velocity, but this discrepancy may be explained by the different aspects of the perceptual process that each variable and model capture. The parameter σ may be more sensitive to subtle aspects of evidence accumulation dynamics, especially concerning how effectively information is processed across distinct velocities and occluder conditions. In contrast, timing accuracy, while related to evidence accumulation, may be a broader measure that is influenced by additional decision-making processes not directly modulated by velocity. 

Moreover, the significant interaction between the occluder condition and the integration to occlusion time ratio in the ANOVA on timing accuracy highlights the ratio's critical role in participants’ responses. This suggests that while velocity influences the rate of evidence accumulation, as indicated by the parameter σ, it is the proportion of visible to occluded time that participants predominantly use to guide their timing decisions. The ratio likely represents a composite measure of the temporal constraints imposed by the occluder and the demands of processing motion at varying speeds. It is also possible that the discrepancy points to the presence of nonlinearities or threshold effects in the perceptual mechanisms underlying timing accuracy, which are not as pronounced when examining the parameter σ alone. These nonlinearities could be masked in an ANOVA but captured by the model. 

Predicting precision performance

As commented above, the dynamics of accuracy across the ratio are mainly determined by σ and its scaling parameter a, while b remains fixed (Equation 1). Since σ presumably influences the precision with which sensory evidence is accumulated, we propose that the SD can be primarily predicted by how the accuracy changes as a function of the ratio. Figure 9A shows the average SD across participants, separated by the occluder condition and as a function of the log of the ratio. The curve illustrates the predicted SD based on Equation 2 that incorporates the σ parameter from accuracy fits (σ  =  0.67 and σ = 0.81 for predictable and unpredictable occluder conditions, respectively). The inset bar graph displays two parameters: D (the amplitude) and sd₀ (the asymptotic SD), the minimum variability or precision that participants’ responses asymptotically approach as the ratio increases. These parameters are fitted to the actual SD data in addition to a (not shown in the inset), while σ is held constant from the accuracy model. The underlying assumption is that the rate of decay in timing variability is directly related to the rate at which accuracy increases, as determined by the accuracy fits. The RMSE value displayed in Figure 9A indicates the average deviation of the predicted SDs from the observed data points. An RMSE of 21.8 ms suggests that the predictions are reasonably close to the observed data, providing a good fit. The strong capacity of the model to capture the two apparent trends of the SD data are also shown in Figure 9A before and after a ratio value of 1. These results imply a robust link between accuracy and precision in the context of timing responses. The relationship shown in Figure 9 supports the idea that there is a common underlying process or set of processes affecting both accuracy and precision in timing tasks, influenced by the ratio of integration to occlusion time. This link is also revealed when we plot the mean SD per participant as a function of the aσ value obtained from the accuracy model fit to the timing responses. Figure 9B shows a significant relationship between the scaled σ and the mean SD, extracted from the slope of the model. 

Figure 9.

 

Variable error prediction. (A) Observed (points) and predicted (curve) SD across participants for the two occluder conditions. The inset shows the fitted values for the D and sd₀. The parameter a was treated as a free parameter, and we did not use the parameter values from the accuracy fits. (B) Mean SD per participant as a function of the aσ value obtained from the accuracy model fit to the timing responses. Each point represents one participant and one occluder condition. The statistics represent the significance of the slope from the model.

Discussion

Differences in accuracy between occluder conditions

Participants’ accuracy clearly differed between the predictable and unpredictable conditions, and their eye movements revealed the adoption of alternative patterns contingent on the reliability of the visual information and, by extension, the attributed relevance to spatial locations. When the occluder reliably indicated where the target was going to disappear, participants made more accurate predictions (timing errors close to 0, Figure 2A). In this condition, participants initially followed the target with their eyes and then made strong fixations at the start of the occluder (Figures 3A, 3C, and 4), suggesting that they utilized the occluder onset information and that this position was relevant to try to successfully complete the task. During occlusion, participants executed a sequence of saccades to the end of the occluder, presumably tracking the hidden target. This observation is in line with studies describing that participants perform a series of saccades during occlusion, reflecting the extrapolated target movement (Bennett & Barnes, 2006; Orban de Xivry et al., 2006). These eye movement patterns differed from those in the unpredictable condition. Therefore, the eye movement pattern variations may provide insights for the observed differences in accuracy between conditions. There are several gaze-related explanations that may account for these differences in accuracy; we will outline three below. 

First, the reliability of the visual information may have dictated its relevance for the task and guided the subsequent eye movements. In the unpredictable condition, once the target became occluded, participants chose to shift their gaze directly to the end of the occluder (rather than tracking the target through occlusion). In this condition, the end of the occluder was the reliable position and relevant to the task. This is similar to findings described by de la Malla, Smeets, and Brenner (2017) in a study where participants had to intercept a moving target in a specified location, and participants ended up making a saccade toward the specific, relevant position where they had to hit the target. In our study, we propose that participants may have opted to rely more heavily on online information in the unpredictable condition and closely followed the visible target, as seen by the high smooth pursuit gain. This is consistent with work by de la Malla and López-Moliner (2015), who explain that when estimates are not reliable, people tend to adopt prospective control since later online input would provide the most beneficial information for making predictions. On the other hand, the different eye movements in the predictable condition suggest that when the information reliably indicated where the target would disappear, participants likely exploited it to make more accurate predictions. Since the integration of temporal cues with velocity information has been shown to optimize TTC predictions (Chang & Jazayeri, 2018), this may have contributed to differences in accuracy between conditions. 

Second, in the predictable condition, participants first smoothly tracked the target (briefly) and then made a saccade to the onset of the occluder. During fixation, the moving target generates a foveopetal (toward the fovea) retinal motion signal, and these signals are known to influence the perceived position of moving objects to be further ahead than they really are compared to foveofugal motion (Mateeff & Hohnsbein, 1988). Therefore, this retinal motion signal may have been used in combination with the estimated target velocity (based on the initial pursuit) to enhance TTC. Although we cannot directly compare the magnitude of extrapolation between the two occluder conditions, the differences in eye movement patterns suggest that the predictable condition resulted in eye movements that favored greater motion extrapolation, leading to the observed earlier responses in this condition compared to the unpredictable one. This was likely due to the combination of the estimated target velocity from the smooth pursuit with the foveopetal signal that then accentuated extrapolation during occlusion (as seen by the marked tracking during occlusion in this condition). This is in line with work that suggests that extrapolation of a target that has disappeared occurs after it has been pursued and that this extrapolation is due to a persistent image of the target on the retina in the direction of motion (Kerzel, 2000). Based on the eye movement patterns observed, extrapolation was likely not as strong in the unpredictable condition since there was a clear saccade to the end of the occluder, whereas the tracking during occlusion in the predictable condition was more indicative of a larger extrapolation. 

Finally, the reduced accuracy and the TTC overestimation in the unpredictable condition may be related to a slower perceived target velocity caused by the high smooth pursuit gain, consistent with the Aubert–Fleischl phenomenon (Aubert, 1886, Aubert, 1887; Turano & Heidenreich, 1999; von Fleischl, 1882; Wertheim & Van Gelder, 1990). In this phenomenon, visual objects appear to move slower when making smooth pursuit eye movements. 

Integration to occlusion time ratio as the main accuracy modulator

The integration to occlusion time ratio effectively captured the dynamic interplay between the accumulation of sensory evidence (integration) and the fluctuations in uncertainty (occlusion). When plotting the timing error as a function of the ratio, we observed a consistent pattern across participants (Figure 6). Importantly, the ratio value close to 1 (1.5 in our experiment) repeatedly indicated the positive peak value of the timing error, and different combinations of integration and occlusion time converged to similar values of accuracy. This is the case with all participants and irrespective of the occluder condition. Therefore, we suggest that the ratio is the most pertinent variable on which perceptual responses are based. Not only were participants more accurate as the ratio increased, but they were also more precise, as seen by the marked reduction in the variable error (Figures 2B and 9). This likely occurred because higher ratios provided participants with more time to extract and integrate important sensory information (Battaglini & Ghiani, 2021) and to make more accurate velocity/visual memories (Tsuda & Saiki, 2019). Also, the shorter occlusion time may have afforded participants less time to forget previously assimilated visual information (visual memory decay; Aguado & López-Moliner, 2021; de la Malla & López-Moliner, 2015) and/or limited noise accumulation (Kuuramo, Saarinen, & Kurki, 2022; Lyon & Waag, 1995; Makin, 2018) that could have biased predictions. 

We developed a model to explain the observed accuracy pattern relative to the ratio and to predict response variability from the accuracy data. The key parameter, σ, indicated the signal quality, which resulted in enhanced evidence accumulation as the ratio increased. According to the model, small σ values can account for both the initial rapid decrease in timing errors with small increments of the ratio and the subsequent stabilization after the peak of the timing error. The good fit of the model suggests that the modulation of accuracy is primarily caused by perceptual dynamics rather than by response criteria (Kingdom & Prins, 2016). Although high occlusion durations (resulting from lower ratios) may drive conservative and early responses due to heightened uncertainty, the results across participants suggest that perceptual dynamics had a greater influence on TTC predictions than response criteria. This interpretation is supported by findings that illustrate that perceived speed varies according to duration, with shorter durations leading to faster perceived velocities (Algom & Cohen-Raz, 1984) probably due to short integration times. This could explain why smaller ratios appear to accelerate the perceived velocity, leading to a TTC underestimation. The model's predictive power for variable errors (SD) from accuracy further supports this interpretation. It is worth noting that our model is designed for contexts where cognitive factors are less influential. It is better suited for situations with brief visible periods (hundreds of milliseconds), minimal delays between target disappearance and the response initiation, and relatively highly practiced participants, thereby reducing and potentially eliminating the impact of cognitive processes (Tresilian, 1995). 

The role of feedback

The predictive success of our model was likely enhanced by the feedback provided in the experiment, which informed participants if their response was too early, late, or successful within a temporal window of 100 ms centered on the zero-error mark. Prior research, such as Landwehr, Brendel, and Hecht (2013), suggests that in prediction motion tasks without feedback, responses tend to be delayed (TTC overestimated). Therefore, feedback allows for the adjustment of timing based on past outcomes, refining responses throughout the experiment. On the other hand, the absence of feedback necessitates reliance on internal timing mechanisms, often resulting in greater variability and in delayed responses. Examples of this have been reported in studies like Bennett et al. (2010) for occlusions under 1 second and in Battaglini and Mioni (2019) for longer occlusions. Despite the feedback provided, the persistent late responses at specific integration to occlusion time ratios—evident in the pronounced positive peaks outside the accuracy window in the unpredictable condition (Figure 2A)—emphasize the influence of perceptual dynamics. Note that these positive peaks are nestled between more accurate responses. Thus, even with calibration through feedback, the mechanics of perceptual processing assert themselves. 

Accuracy and precision are linked

The observed positive correlation between accuracy and precision (Figure 9), alongside the significant interaction between the occluder condition and velocity on σ, suggests that these factors may affect timing performance in more nuanced ways than previously thought. Although the main effect on variability is credited to the integration to occlusion time ratio, the variability in σ insinuates more intricate dynamics at play. In the predictable condition, especially at higher speeds, the smaller σ values imply a tighter, more concentrated temporal distribution of responses. Notably, these smaller values correlate with less TTC overestimation compared to the unpredictable condition. This pattern suggests that a larger signal to noise ratio (presumably inversely represented by σ) facilitates finer adjustments and heightened sensitivity to feedback, thereby enabling participants to respond more consistently. Given that faster speeds are likely to introduce more signal-dependent noise (according to Weber's law), high-speed conditions may amplify the need for more precise temporal resolution. The discrepancy between the ANOVA results on SD and the nuanced σ values could stem from the different aspects of timing behavior captured by each measure. Whereas the SD indicates the overall variability in timing responses, σ measures the sensitivity of the accuracy to changes in the integration to occlusion time ratio. The absence of effects of the occluder condition and velocity in the ANOVA on the SD suggests that these factors do not affect the dispersion of timing responses independently. However, the interaction effects on σ within the accuracy model reveal that they do play a role in modulating how velocity signals are processed perceptually, which is important for estimating TTC. 

Conclusions

In this study, we addressed the estimation of TTC under conditions of uncertainty regarding the position of occlusion. Estimates were more accurate when the occlusion onset could be reliably predicted, and eye movement patterns may have modulated the differences in accuracy between conditions. Moreover, the accuracy and variability of TTC predictions were best described by the balance between the motion integration time and occlusion time, rather than by either dimension alone. The proposed mechanistic model accurately predicts the pattern of accuracy and highlights the balance between integration time (evidence accumulation) and the decay of information during occlusion as the crucial variable for understanding time-to-contact estimations under uncertainty. Future research should aim to further validate this model in more complex scenarios. 

Acknowledgments

Funded by grants PID2020-116400GA-I00 and PID2020-114713GB-I00 by MCIN/AEI/10.13039/501100011033 to CM and JLM, respectively. PV was supported by grant FPI PRE2021-097890 from the Spanish Ministry of Science and Innovation. 

Data availability statements: The data are available in the following link: https://osf.io/s6wr5 

Commercial relationships: none. 

Corresponding author: Joan López-Moliner. 

Email: [e-mail]. 

Address: Vision and Control of Action Group, Department of Cognition, Development, and Psychology of Education, Institute of Neurosciences, Universitat de Barcelona, Barcelona, Catalonia 08007, Spain.