Overt responses refute the fixed looming detection threshold assumption

After exclusion of a small minority of trials for early (0.6%) and missing (0.2%) detection responses, and a larger number of trials for electrooculographic indications of eye blinks (15.9%; see Materials and methods for details), the final data set included 22 participants, with an average of 182 trials per participant (an average of 46 trials per looming condition). Fig 1C shows that our data replicated the kinematics-dependency reported by Lamble et al. [52], with detection occurring at lower average optical expansion rates for the larger initial distance (F(1, 3698) = 1255.48;p <.0001). Also in an absolute sense, our values at detection were similar to those observed in the test track experiment (black crosses in Fig 1C), but slightly lower, potentially due to the reduced noise in the laboratory environment and the use of a finger key press instead of a foot pedal to report response.

As initially suggested in [35], detection at lower values for larger initial distances is predicted by a looming accumulation account, because looming grows more slowly from larger distances, and because accumulation (i.e., integration) of a small over a long time is equivalent to accumulation of a large over a short time; see the shaded areas in Fig 1A. Similarly, for lower deceleration magnitudes, where looming develops even more slowly, the looming accumulation account also predicts detection at further decreased values. This was the motivation for the inclusion of the 0.35 m/s² deceleration condition, and the observed values at detection were indeed further reduced for this lower magnitude of deceleration (Fig 1C; F(1, 3698) = 810.26;p <.0001).

These behavioral findings strongly reinforce the idea that looming detection occurs at magnitudes of optical expansion rate that are dependent on the kinematics of the collision course, in contrast with the conventional LDT assumption of a situation-independent threshold for detection. The triangle symbols in Fig 1B indicate the response times that would be predicted by a situation-independent looming threshold fixed at the average at detection observed across this experiment. These LDT predictions are too early in fast looming conditions, and too late in slow looming conditions, which is precisely the qualitative pattern of errors that one would expect to see if participants’ responses were instead determined by evidence accumulation of optical expansion rate.

From a methodological point of view it is worth noting that our behavioral analyses also identified statistically significant effects of experimental block (slightly increased looming sensitivity in later blocks) and the 1.5–3.5 s pre-looming wait time (slightly increased looming sensitivity with increased pre-looming wait time). These effects were substantially smaller than the effects of looming condition and between-participant differences (see Table B in S1 Appendix), and were therefore not separated out in the subsequent model fitting described below.

A visual looming accumulator model accounts for full detection distributions

As illustrated in Fig 1B, the looming accumulation hypothesis can be computationally formalized as a single-boundary accumulator (or drift diffusion) model, with its rate of evidence accumulation (sometimes referred to as “drift rate”) at each point in time determined by the momentary optical expansion rate, multiplied by some gain, and where overt detection response occurs once an evidence threshold is reached. Noise in the evidence accumulation process (e.g., due to noisy sensory input, interference from other brain activity, or both) gives rise to variability, i.e., probability distributions of response time. It may be noted that our model, like previous evidence accumulation models of detection of intermittent, subtle changes in abstract stimuli [14, 16], effectively implements Page’s cumulative sum (’CUSUM’) technique for change detection [61].

The conventional LDT assumption is completely deterministic, and as such does not make predictions about probability distributions. However, one might consider a stochastic threshold model, positing that looming detection occurs once a noisy optical expansion rate signal first exceeds a fixed threshold. In fact, such a model would also predict the qualitative findings reported above, since in conditions where looming develops slowly, there would be more time for large noise values to occur by chance, thus eventually exceeding the threshold even if the sensory signal itself is still sub-threshold. We therefore tested also this type of model, and compared it to the accumulator model.

Due to the time-varying drift rate in our accumulator model, there is no closed-form expression for its response time distribution [62]; we estimated these distributions numerically instead. All considered models were fitted per participant, both using maximum likelihood estimation (MLE) and approximate Bayesian computation (ABC), to the per-participant response time distributions in each looming condition. The emphasis in this paper is on the MLE results, whereas the ABC results, mostly reported in the Supporting information, provide additional confirmation of the key conclusions, a more complete view of model parameter estimates, and allowed us to follow up on auxiliary questions which would have been computationally prohibitive under the MLE approach.

Fig 2A shows detection response time distributions, visualized as cross-participant averages (using the “Vincentizing”method [63]) of per-participant data and MLE model fits. The threshold model (model T) does indeed capture the qualitative effect of looming condition, but is unable to accurately reproduce the location and shape of the response time distributions. Similarly to the patterns shown in Fig 1A for the deterministic threshold model, model T has a tendency to predict responses that are substantially too late in slow looming conditions. The accumulator model (model A) does not have this problem, and achieves a noticeably better fit despite having the same number of free parameters as model T.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Model comparisons.

(A) Averaged (“Vincentized”) per-participant cumulative density functions (CDFs) of looming detection response time, for human participants and the maximum-likelihood-fitted threshold model (T), accumulator model (A), and variable-gain accumulator model (AV), in the four looming conditions. (B) Per-participant differences in Akaike Information Criterion (AIC) for the T → A and A → AV model comparisons. Negative ΔAIC values indicate preference for the latter model in the comparison.

https://doi.org/10.1371/journal.pcbi.1009096.g002

When modelling perceptual decision-making in paradigms with stationary stimulus saliency, it has often been found that assuming between-trial variability of the stationary rate of evidence accumulation is needed to closely reproduce human response time distributions [10]. Analogously, we investigated an extended version of our basic accumulator model, where the input gain applied to the optical expansion rate was not constant per participant, but instead drawn at random per trial from a normal distribution, of which the standard deviation thus becomes an additional free model parameter. This extended model (model AV) produced distributions that more closely matched those of the human detection responses. Fig 2B shows the relative goodness of fit of the three models, in terms of differences in Akaike Information Criterion (AIC). These results indicate a very strong preference for the accumulator model over the threshold model for all participants but one, with an average ΔAIC of -43.2 (a difference of more than 14 suggests “very strong support” for the preferred model [64]). This analysis also indicated that for most participants, the additional model complexity introduced by the input gain variability (model AV) was warranted given the improvements in model fit (average ΔAIC = -29.1). Per-participant fits for models T, A, and AV are shown in Fig A in S1 Appendix. Model AV had four free parameters: the non-decision time T_ND, the accumulator noise intensity σ, and mean and standard deviation K and σ_K of the looming input gain; estimated values for these parameters across participants are shown in Fig B in S1 Appendix.

The ABC analysis also favored the accumulator model over the threshold model. The geometric mean of the per-participant Bayes factor (an estimate of the expected Bayes factor for hypothetical additional participants [65, 66]) in favor of the accumulator model was 3.0–7.3 (“substantial evidence” [67]), depending on the choice of the ABC distance threshold hyperparameter ϵ_RT, with the highest Bayes factors for the more stringent ϵ_RT (see Fig E in S1 Appendix). The ABC comparison of accumulator models with and without variable gain was inconclusive (geometrical mean Bayes factor 1.4–1.5 in favor of model A), possibly because of the relatively broad priors we used, which may have excessively penalized model complexity [68].

Using both MLE and ABC methods we also investigated other model variants, incorporating gating of the looming input (requiring it to exceed a minimum threshold before contributing to the accumulation) or evidence leakage (as a form of short-term memory decay), as well as more complex models incorporating different combinations of these various model assumptions. As further described in S1 Appendix, none of these alternative models were found preferable over the variable-gain accumulator model (model AV).

As an additional test of this best-performing model, we also examined its predictions in response to variations in pre-looming wait time. As mentioned above, the model-fitting was blind to this experimental manipulation. However, as shown in Fig C in S1 Appendix, model AV nonetheless predicted the observed pattern of increased looming sensitivity with increased pre-looming wait times, with approximately correct magnitudes.

Looming accumulation explains onsets of pre-response scalp potentials

Fig 3 illustrates the main EEG findings. The response-locked scalp maps in Fig 3A show a positivity at the overt response, in line with the CPP observed by many others [13–19]. Fig 3B and 3C show stimulus-locked and response-locked ERPs, per condition, averaged over five electrodes centered on Pz. The original paper on the CPP centered its analysis on the CPz electrode location [13], but many subsequent reports have shown more parietally located CPPs, consistent with what we observe here [14–16, 18, 69]. Again in line with previous observations, Fig 3C shows that this positive wave builds up before the overt response and peaks at the response itself, including a characteristic separation at this peak, with higher CPP amplitudes for the more salient looming conditions [13, 14, 16, 18]; see the dots along the bottom of Fig 3C.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. EEG results.

(A) Grand average scalp potentials shortly before, at, and after overt detection response. (B) Event-related potentials (ERPs) relative to the looming stimulus onset, averaged over five electrodes centered at Pz (marked in A), in the four different looming conditions. (C) The same four ERPs as in (B), but instead response-locked, i.e., relative to the time of overt detection response. The dots along the bottom show where an ANOVA indicated a statistically significant main effect of looming condition on these ERPs. (D) As panel C, but without high-pass filtering and ocular artefact removal. (E) Response-locked average accumulated evidence E in each looming condition, for the best-fitting variable-gain accumulator model AV. Note that the model traces converge at the decision threshold E = 1; the exact location of this point in the plot depends on how much of the non-decision time in the model is assumed to be due to sensory and motor delays, respectively. (F) Estimated onsets of the pre-response centroparietal positivity (CPP) relative to the overt detection response, in the four looming conditions. This includes the (17 out of 22) participants for which the CPP onsets could be reliably estimated. The black markers indicate condition means. The intervals plotted in black and gray at the bottom of the panel show, respectively, the maximum observed difference between condition means, and the upper edge of a 95% confidence interval for this difference. (G) Averaged (“Vincentized”) per-participant cumulative density functions (CDFs) of CPP onset time relative to the looming stimulus onset, for the participants and the maximum-likelihood-fitted variable-gain accumulator model. (H) The top panel shows per-participant AIC differences for the T → A and A → AV model comparisons, when fitted to the CPP onset data. The bottom panel shows the total cross-participant sums of AIC differences, with 95% confidence intervals.

https://doi.org/10.1371/journal.pcbi.1009096.g003

However, the CPP we observe here differs in at least two respects from previous observations. First, the build-up of the CPP has previously been reported to be of similar duration as the overt response times, typically only 100–200 ms shorter [13–18]. This is consistent with the idea that the CPP reflects evidence accumulation that starts shortly after stimulus presentation. In contrast, average response times per looming condition in our experiment ranged from 1.1 to 2.4 s, yet it is clear from Fig 3C that the average CPP build-up duration was shorter than 0.5 s for all conditions. Second, since in previous studies CPP increase over ERP baseline has been obvious soon after stimulus onset, conditions with slower responses (typically due to less salient stimuli) have produced CPP profiles with build-up commencing earlier in time before the overt response [13–15, 17]. In contrast, in our response-locked ERP data, there is no obvious separation between conditions in when the CPP build-up commences.

Fig 3D shows that the CPP signal is subtly affected by EEG pre-processing choices (most notably, the 0.1 Hz low-pass filter we used causes a slight suppression to negative voltage before the CPP build-up; see further Fig F in S1 Appendix), but the general aspect of a late, condition-independent CPP onset remains. Meanwhile, as can be seen in Fig 3E, the onset of evidence build-up in our behavioral model was both early and condition-dependent, as expected since the model directly accumulates the looming input.

Previous studies have indicated that purely behavioral fits of evidence accumulation models can yield model evidence profiles that align qualitatively with the corresponding CPP [14, 15, 17–19], but as described above this was not the case here. One possible reason for this could be that our models were flexible enough to achieve good behavioral fits for a range of parameterizations, with a range of widely different evidence build-up profiles, and that our purely behavioral fits were therefore not enough to observe an alignment between model evidence and recorded CPP signatures. Therefore, using ABC, we investigated whether fitting the models simultaneously to the behavioral and neural data could identify a behaviorally well-fitting model which did not exhibit early separation of accumulated model evidence, thus aligning better with our CPP observations, but no such model was identified; see Fig J in S1 Appendix.

A different possible explanation for our late, rapid, and seemingly condition-independent CPP could be that (i) the looming accumulation process indicated by the behavioral modeling results is not directly reflected in the CPP, and (ii) the observed CPP instead reflects a second stage of the response decision process, which begins only once the looming accumulation process reaches threshold. To further investigate this hypothesis, we estimated full distributions of CPP onset times from the EEG data, by averaging ERPs over trials with similar response times, to increase signal-to-noise ratio, and identifying the last time where the averaged ERP exceeded 30% of its value at the overt response. We found that this allowed us to reliably estimate CPP onsets for 17 out of the 22 participants. We then investigated our hypothesis using this dataset in two ways: Firstly, as illustrated in Fig 3F, we analyzed the distributions of CPP onset time relative to overt response. In line with our hypothesis that the process reflected in our CPP has a duration that is independent of looming condition, we found that the largest difference in the average CPP onset between two conditions was 22 ms (interval marked in black at bottom of Fig 3F) and that the upper edge of a 95% confidence interval for this difference was 79 ms (interval marked in gray). In other words, any effect of looming condition on CPP onset relative to detection response was small in magnitude (and not statistically significant in our dataset; F(3, 377) = 1.94;p = .133). Secondly, we refitted, using MLE, the threshold and accumulator models to the estimated CPP onsets instead of to the overt responses. We found that the accumulator models could account well for the distributions of CPP onset, and reliably better than the threshold model; see Fig 3G and 3H (and see Fig I in S1 Appendix for more detailed results for models T and A). Because the CPP onset dataset being fitted to was reduced compared to the behavioral dataset, a reduced statistical power of the model comparisons should be expected, and the obtained per-participant ΔAIC values (Fig 3H, top) were indeed smaller than for the corresponding behavioral model comparisons. However, for the overall test of whether to prefer A over T across our entire experiment, the total difference in AIC was still very large (-81.4 and -31.0 for the T → A and A → AV comparisons; see Fig 3H, bottom, also showing 95% confidence intervals for these differences). The results illustrated in Fig 3F–3H were robust to variations in EEG pre-processing and our CPP onset estimation method; see further Materials and methods, and Figs F, H, and I in S1 Appendix.

Ethics statement and open software/data

All procedures were approved by the School of Psychology Research Ethics Committee, University of Leeds, reference number PSC-484. The primary research data for this study, as well as the software code implementing the experimental paradigm, data analyses, and computational models, are available here: https://doi.org/10.17605/OSF.IO/KU3H4.

Experimental design

The objective of the experiment was to observe participants’ detection of a visually looming object, to what extent this detection was influenced by the kinematical details of the object’s approach, and whether traces of the response process could be observed in participant scalp potentials.

The basic task was a computer-simulated replication and extension of the foveal looming conditions of the test track experiment in [52]. The paradigm was implemented in MathWorks MATLAB using PsychToolbox v3.0.14 [85, 86]. The stimulus was a photographic image of the back of a 1.85 m wide and 1.43 m high passenger car (used with permission from Volvo Car Corporation), as shown in Fig 1A. This image was displayed over a dark gray color on a 24 inch (0.53 m × 0.30 m) 60 Hz TFT screen at 1920 × 1080 pixels resolution. The original image was at higher resolution than shown on screen, and was scaled to appropriate size and displayed with antialiasing using the OpenGL trilinear filtering provided by PsychToolbox.

A central fixation target (a red dot, diameter 6 pixels, 0.095 degrees visual angle) was displayed throughout each experimental block. In each trial, initially only this fixation target was shown for 3 s, then the stimulus image appeared centrally on the screen, accompanied by an auditory tone, displayed at a size corresponding to an initial distance of either 20 or 40 m (subtending 5.30 and 2.65 degrees horizontal visual angle, respectively). Some trials were catch trials without any looming, at which the stimulus remained at the same size for 7 s before disappearing. In non-catch trials, the stimulus remained at the same size during an initial pre-looming wait time, one of 1.5, 2, 2.5, 3, 3.5 s, whereafter the size of the stimulus was gradually increased to reproduce the looming visual input from a car decelerating at either 0.35 or 0.7 m/s²; as shown in Fig 1A. The participants were instructed to keep their eyes on the fixation target, refrain from blinking while the car was being shown and until their response, which they were instructed to give by pressing the space bar on a computer keyboard with their right hand “as soon as you see the car coming closer, in other words when it is growing on the screen”. Trials terminated once participants either (a) made a correct looming detection response, after which the stimulus continued looming for another 0.5 s before disappearing, to avoid the impact of this visual transient interfering with the EEG measurements at time of response, (b) made no response before the looming stimulus had reached a trial expiry threshold of 0.03 rad/s (about ten times the threshold typically stated in the literature), or (c) made an incorrect, early detection response before the onset of visual looming; in this last case a distinct auditory tone was played to inform the participant of their incorrect detection response.

The participants viewed the stimulus screen at a distance of 1.00 m, meaning that each screen pixel subtended a visual angle of 0.95 arcmin (0.016 degrees), lower than the 1.6 arcmin threshold reported in [87] for maximum Vernier acuity with antialiased stimuli. To further reduce the risk of pixel effects, and to mimic the conditions of the replicated test track experiment [52], the stimulus was displayed with small horizontal and vertical oscillatory perturbation throughout, generated by moving the simulated viewport as if the participant themselves were sitting in a car, with perturbation spectra based on measurements from real driving; see Table A in S1 Appendix for details.

Procedure

Participants provided written informed consent before taking part in the experiment, which was carried out in a dark room with the participant sitting in front of the stimulus display, supported by a chin rest. In a first demonstration block of four trials, the experimenter demonstrated the task, including the auditory tone given upon incorrect, early responses, as well as a feedback screen that was shown after each block. This feedback screen listed average response times and frequency of correct responses for all blocks so far, and encouraged participants to rest if their response times were increasing. The participants then decided themselves when to start the next block. The participants first completed a practice block of 12 trials, two of each of the four looming conditions (2 initial distances × 2 acceleration levels), and four catch trials. Then followed the five experimental blocks, each with a total of 48 trials, eight catch trials and ten repetitions of each of the four looming conditions (two repetitions for each of the five pre-looming wait times), making for a total of 5 × 40 = 200 looming trials per participant, 50 for each looming condition. Trial order was fully randomized per block and participant.

Participants

The target initial sample size was twenty-five participants, to provide a comfortable margin over the total number of trials collected in previous studies reporting on the CPP [13, 14]. Twenty-six right-handed participants were recruited from a local pool of participants, all with normal or corrected-to-normal vision, and with no history of psychiatric diagnosis, severe brain injury, motor diseases or any skin conditions. EEG recording was incomplete for one participant, and data from the remaining twenty-five participants, of ages between 20 and 46 years (mean 26.5), 12 male and 13 female, were retained for further analysis.

Data acquisition and preprocessing

Behavioral responses were recorded at the 60 Hz refresh rate of the display screen. Out of the 25 × 40 = 1000 catch trials, there were 61 (6.1%) with false detection responses. The catch trials were not further considered in the analyses or modeling. Out of the 25 × 200 = 5000 trials with looming stimuli, participants responded before looming onset in 32 trials (0.6%), and non-responses were observed in 8 trials (0.2%); these early and non-responses were included in the behavioral model fits, but were excluded from all EEG analyses.

EEG data were recorded at 1024 Hz, using a 64 electrode 10–20 international cap Biosemi system. Electro-oculogram (EOG) electrodes were placed above and below the left eye and at the outer canthus of each eye. EEG preprocessing was done using EEGLAB v14.1.1 [88], first resampling to 512 Hz, then using the PREP pipeline EEGLAB plugin v0.55.3 [89] for robust re-referencing to average channel and interpolation of noisy channels. PREP interpolated one of the five channels analyzed as part of the pre-response positivity analyses here (Pz and surrounding channels CPz, POz, P1, P2) for only three participants (one of which was later excluded due to ocular artefacts; see below), in each case only one of the five channels was interpolated. Then, bandpass filtering was done using EEGLAB’s sinc FIR filter with a Kaiser window, with pass/stopband ripple of 0.001, lowpass filter of with 45 Hz cut-off and 5 Hz transition bandwidth, and high-pass filtering at 0.1 Hz cut-off with 1 Hz bandwidth. Following [13], trials with pronounced ocular artefacts were rejected by the vertical EOG difference exceeding 100 μV, excluding 796 trials (15.9%). 395 of these trials were from three specific participants, who therefore failed to reach a minimum of at least 30 trials in each looming condition. These three participants were excluded from further analysis. Further ocular artefacts were identified and removed from the EEG data per participant, using EEGLAB’s independent component analysis (ICA) functionality. The final dataset included 22 participants with a total of 4013 looming trials, i.e., an average of 46 trials per participant and looming condition. For event-related potential (ERP) analysis, the EEG data of looming trials were divided into epochs from 1 s before to 8 s after the looming onset in each trial (sufficient to include all responses in all conditions given the 0.03 rad/s trial expiry threshold mentioned above), and the EEG data for each epoch were baseline-corrected, using the average of the last 200 ms before the looming onset as baseline. Then, the five channels centered on Pz mentioned above were averaged to yield the final signal used in the CPP analyses (per-condition averages across participants shown in Fig 3B and 3C; per-condition averages per participants shown in Fig G of S1 Appendix). The analysis illustrated in Fig 3C was also rerun after disabling the 0.1 Hz EEG high-pass filter and the ICA ocular artefact removal, confirming that neither of these two preprocessing steps substantially altered the obtained CPP signatures; see Fig 3D and Fig F in S1 Appendix.

For the analyses which focused specifically on the onset of the CPP, we increased the signal to noise ratio by sorting trials on response time per participant and looming condition, separating the sorted trials into groups of five trials, and taking the average response-locked ERP within each such group. We then identified the CPP onset for each averaged trial as the last sample where the averaged response-locked ERP was less than 30% of its value at the overt response. We excluded averaged trials where this did not occur within 1 s before the response, or where the ERP at response was less than +2 μV, resulting in a total exclusion of 284 (37.4%) out of the 760 averaged trials. 134 of these exclusions were due to five participants, who distinguished themselves from the rest either by having no clear ERP peak at response (Cohen’s d < 0.3 when comparing ERP between 0.5 s before response and at response) or by the at-response peak occurring at close to zero voltage; see Fig G in S1 Appendix. These five participants were excluded from the CPP onset analyses, leaving a data set of 17 participants and 445 averaged trials (an average of 26 per participant). Since our CPP onset estimation method was novel, and not originally planned for, we also conducted sensitivity analyses. As illustrated in Figs F and H of S1 Appendix, these analyses showed that the obtained CPP onset estimates were robust to variations in the parameters of our method, but also that our method was not suitable for use on non high-pass filtered ERP data, which would have been desirable since the high-pass filtering we used subtly altered the CPP signal (cf. Fig 3C and 3D). Future work should improve on these onset estimation methods, for example along the lines of the approach in [90].

Statistical analysis

To test for effects of the kinematical looming conditions on the optical expansion rate at behavioral detection response, we carried out a repeated-measures ANOVA, implemented using MATLAB’s anovan function with participant as a random factor, independent variables initial car distance (2 levels) × acceleration magnitude (2 levels) × experimental block (5 levels) × pre-looming wait time (5 levels), limited to first-order interactions only, and as the dependent variable; see Table B in S1 Appendix for full results. To study separation between response-locked ERPs for different looming conditions (Fig 3C and 3D), we performed ANOVAs at 20 ms intervals, with the response-locked ERP as the dependent variable, participant as a random factor, and looming condition (4 levels) as independent variable. We also performed this type of ANOVA to test for an effect of looming condition on CPP onset relative response (Fig 3F). The 95% confidence intervals in Fig 3F and 3H were obtained by 100,000 sample bootstraps from the empirical data in question.

Computational models

The optical expansion rate accumulator models investigated here can be described by the following discrete update equation for the accumulated evidence E at time step i: (1) where is an accumulation gain parameter either drawn at random per trial from a normal distribution with σ_K as a free parameter (model AV), or kept constant per participant (σ_K = 0; model A), Δt is the discrete simulation time step length, and the final term is a discrete implementation of a Wiener process with noise intensity σ, with ν(i) drawn at random per time step from a standard normal distribution . Note that we included a reflecting lower boundary at zero (the max function), as often done for evidence accumulators with a single decision boundary [7, 20, 91, 92]. The optical expansion rate is the time derivative of the optical size of the collision obstacle (here, the lead vehicle): (2) where W and D(i) are the width of and momentary distance to the collision obstacle, and V(i) is the momentary speed at which it is coming closer [28]. The accumulator model makes a detection decision once E(i)≥1 (E is in arbitrary units, so one of decision threshold, K, or σ can be fixed without loss of generality) overtly responding a non-decision time T_ND later. A fraction α_ND = 0.3 of T_ND was assumed to occur before the evidence accumulation, based on [29, p. 189]; this value did not affect the behavioral model fits or comparisons, only model evidence visualizations like the one in Fig 3E. For a description of the other accumulator model variants that were also tested, see the Supporting information.

The stochastic threshold model investigated here (model T) can be written on a similar form: (3) again with overt detection response a time T_ND after E(i)≥1, i.e., the parameter is the looming detection threshold parameter. Note that the estimated value of σ for this model will depend on the discrete simulation time step length; we used Δt = 0.02 s across all models. The models were simulated from the start of each trial, i.e., the pre-looming wait time was also simulated, and the accumulator models were initialised at E = 0 for each trial.

Model fitting by approximate Bayesian computation

Our main goal of ABC was parameter estimation rather than model selection, so we used relatively broad, non-informative priors. To make the obtained Bayes Factors reasonably meaningful, and for increased computational efficiency, we identified approximate limits for the model priors by means of a first ABC fit to the information available from the previous test track experiment [52]; see Supplementary material for details. We then fitted each model variant (T, A, AV, etc) to each participant separately. To implement ABC rejection sampling [93, 94], for each parameterization sampled at random from the prior distribution, we generated a simulated data set of the exact same nature and size as the data obtained from one human participant, (same number of repetitions, pre-looming wait times, etc.) and calculated and stored a set of twenty summary statistics; the response time quantiles {0.1, 0.3, 0.5, 0.7, 0.9} for each of the four looming conditions separately. We then compared the simulated RT quantiles to each human participant’s data, for each participant retaining only parameter samples where all twenty absolute differences between simulated and observed RT quantiles were below a rejection threshold ϵ_RT. These retained samples provide an approximation of the posterior parameter distribution for the participant [93, 94]. There are several more advanced versions of ABC than this basic rejection sampling algorithm, but this method was preferred here because it allowed us to obtain individual per-participant fits without extra simulations of the model (which is the computationally costly step), and because it made computationally feasible the investigations illustrated in Fig E of S1 Appendix, showing that the ABC model comparisons were robust to the choice of the hyperparameter ϵ_RT. In the Supporting information we also describe how we used ABC to jointly fit both our behavioral and neural data, finding that excessive model flexibility was not the reason for the mismatch between our CPP observations and our behavioral models’ evidence traces.

Model fitting by maximum likelihood estimation

Maximum likelihood estimation of model parameters was carried out using exhaustive grid search over the model parameter ranges identified by the abovementioned initial ABC fits to the previous test track experiment, with each parameter’s range uniformly divided into 20 searched grid values. For each parameterization in this grid, as in the ABC fits a replica of the real experiment was simulated, but upscaled by a factor 20 to 1000 trials per looming condition, yielding a numerical response time distribution per condition, estimated at a bin size of 0.25 s. For these fits, all models were also extended by assuming a probability P_C = 0.01 (our conclusions were robust to variations in this value) per trial of ‘contaminant’ responses poorly described by our model, e.g., due to temporary lapses in participant attention [95], modeled as a uniform distribution across the time range from looming onset to trial expiry. (Whereas our ABC fitting method is robust to such contaminant responses, they can disrupt the MLE fits if they fall in low probability response time bins, which may occasionally be numerically estimated to zero probability by the contaminant-free model.) Likelihoods were then estimated from the resulting numerical probability distributions, per participant and model parameterization. This model fitting method was computationally feasible for models with up to four free parameters (T, A, AV, AG, AL); based on the results from the ABC fits we would not expect to see substantial further improvements with the more complex models.