Perceptual decision-making in the sound-induced flash illusion: a trial-level hierarchical Bayesian integration of EEG and behavior

Abstract

This study used the sound-induced flash illusion (SIFI) paradigm to investigate how sensory congruency, audiovisual rhythm configurations, and momentary neural states jointly shape perceptual decisions by integrating trial-level EEG and behavioral data within a hierarchical Bayesian regression framework. Thirty healthy adults performed six beep–flash conditions (B0F1, B0F2, B1F1, B1F2, B2F1, and B2F2), encompassing two visual-only baseline conditions, two audiovisually congruent conditions, and two incongruent audiovisual illusion conditions, while response accuracy and reaction time (RT) were recorded together with prestimulus occipital α-band power, auditory and visual N1 amplitudes, and the centro-parietal positivity (CPP) indexing evidence accumulation. We first characterized condition effects and EEG–behavior relationships using repeated-measures ANOVAs and correlations, and then jointly modeled accuracy (Bernoulli likelihood with logit link; effects reported as odds ratios) and RT (Student-t likelihood on logRT; effects interpreted as percentage change in median RT) using trial-level hierarchical Bayesian regression, with model generalizability assessed via Pareto-smoothed importance sampling leave-one-out (PSIS-LOO) cross-validation. Behaviorally, unimodal and congruent conditions yielded faster and more accurate responses, whereas incongruent conditions—particularly the “2 beep–1 flash” configuration—produced elevated illusion rates and markedly prolonged RTs, highlighting the role of causal structure and signal reliability in cross-modal weighting. Neurally, higher trial-level prestimulus α power was reliably associated with reduced odds of a correct response, consistent with lower cortical excitability and a more conservative decision criterion; larger auditory N1 amplitudes predicted faster responses; and CPP slope closely tracked RT in descriptive analyses, consistent with its interpretation as an accumulation-to-bound signal. Model comparisons indicated that experimental conditions accounted for most of the behavioral variance, whereas EEG covariates added limited and non-robust incremental out-of-sample predictive value under the present linear additive specification. Nevertheless, prestimulus α showed a stable negative association with accuracy, and auditory N1 showed a weaker association with response speed, indicating that these neural measures remain theoretically informative even though they did not materially improve overall model generalizability.

Supplementary Information

The online version contains supplementary material available at https://doi.org/10.1186/s12993-026-00344-y.

Introduction

How multisensory integration shapes perceptual decisions under noise and conflict is a central question in cognitive neuroscience. In natural environments, combining signals across modalities can improve detection, discrimination, and temporal precision, whereas temporal or numerical discrepancies can also produce systematic biases, perceptual illusions, and performance costs [19, 21, 33, 37, 38]. The sound-induced flash illusion (SIFI) has become a canonical testbed for this problem: when a single visual flash is paired with two brief auditory beeps, observers frequently report “seeing” two flashes (fission). In the fusion configuration, two visual flashes paired with a single beep can bias observers toward reporting that only one flash was seen [21, 37, 38]. Because SIFI manipulations are tightly controlled and highly reproducible, they offer a useful paradigm for quantifying how visual and auditory evidence are weighted when inferring the numerosity of brief events.

Converging behavioral, neuroimaging, and computational work suggests that multisensory integration is conditional rather than obligatory. Within approximately Bayesian causal-inference accounts, vision and audition each provide noisy sensory evidence (likelihoods) whose influence is weighted by their precision, but integration is contingent on an inferred causal structure: the system must decide whether signals arise from a common cause (C = 1) or from separate causes (C = 2), and only when the posterior for a common cause is sufficiently high does it favor integration [32–35]. In SIFI, brief audiovisual stimulus-onset asynchronies and rhythm manipulations systematically alter both the likelihood structure and the inferred probability of a common cause, shifting cross-modal weighting and thus perceived flash numerosity [19, 21, 37, 38].

At the level of condition means, this interplay between causal inference and reliability-based integration yields a characteristic behavioral pattern. Unimodal and audiovisually congruent configurations typically produce higher accuracy and faster responses, whereas incongruent configurations—most notably “2 beeps–1 flash (B2F1)”—increase illusion rates and impose reaction-time (RT) slowing [19, 37, 38]. However, such mean-level comparisons offer limited insight into why a given trial is perceived as illusory, or why responses speed up or slow down on a particular trial. To address these questions, it is necessary to move from condition averages to a trial-level approach that links momentary neural states to single-trial choices and RTs.

Scalp EEG provides a suitable window onto these fast dynamics. Prestimulus α-band power and phase index transient changes in cortical excitability and channel gain, biasing subjective visibility and confidence without necessarily improving objective discrimination [2, 17, 36]. The aperiodic 1/f component of the power spectrum further parametrizes the background excitation–inhibition balance, shaping the “precision field” in which sensory evidence is processed [5]. Early sensory event-related potentials (ERPs), including visual and auditory N1, provide indices of momentary sensory encoding that are shaped by attention, prediction, and cross-modal interactions [10, 16], van den [39]. Importantly, veridical and illusory percepts are not necessarily neurally equivalent. In the sound-induced flash illusion and related audiovisual illusion paradigms, altered early occipital and multisensory ERP responses have been reported on illusion trials, suggesting that these components should be interpreted as context-sensitive neural correlates of percept formation rather than as invariant markers that map one-to-one onto physical stimulus structure [18, 21, 27, 28]. At the decision-readout level, the centro-parietal positivity (CPP) has been established as a supramodal “accumulation-to-bound” signal whose slope and amplitude closely track RT and subjective confidence across tasks [6, 22, 29]. Additional topographical signatures, such as the auditory-evoked contralateral occipital positivity (ACOP), reflect cross-modal orienting that facilitates visual-cortical processing at contralateral sites [9, 15, 20]. Together, these markers provide candidate neural mechanisms linking causal inference, attention, and evidence accumulation to trial-by-trial behavior in SIFI.

Methodologically, however, work on SIFI and multisensory decision-making more broadly still faces several structural limitations. First, within-person (trial-to-trial) and between-person (between-subject) effects are often conflated. Without explicit decomposition and appropriate centering (e.g., group-mean centering), slope estimates can mix these levels, leading to ecological fallacies and biased inferences about trial-level neural predictors [4, 7]. Second, RT distributions are typically skewed with heavy tails and occasional outliers,Gaussian-based mean comparisons are therefore fragile, whereas robust distributional approaches (e.g., modeling logRT with heavy-tailed likelihoods) more faithfully capture processing differences [14, 25, 26, 31]. Third, statistical practice in this literature remains heavily anchored to null-hypothesis significance testing (NHST) and single p-value thresholds, despite clear calls to move beyond dichotomous “significant/non-significant” decisions toward richer effect-size and uncertainty quantification [1, 43].

At the level of model assessment, posterior predictive checks and cross-model predictive evidence remain underused, limiting transparent evaluation of model–data fit and generalizability. Bayesian predictive criteria such as the widely applicable information criterion (WAIC) and Pareto-smoothed importance sampling leave-one-out (PSIS-LOO) cross-validation provide principled tools for comparing hierarchical models, robust likelihoods, and link functions on a common scale while quantifying uncertainty and model separability [40–42]. Posterior predictive checks and visualization are central elements of the Bayesian workflow, allowing researchers to test whether fitted models can reproduce substantively important distributional features [11–13]. In parallel, modeling multiple neural markers in isolation raises multiple-comparison and comparability issues. This motivates hierarchical joint modeling frameworks in which accuracy is expressed on the log-odds/odds-ratio scale, RT on the log scale as percentage change, and multiple EEG predictors (e.g., CPP slope, prestimulus α, visual and auditory N1) enter a single coherent model [30, 44].

Against this backdrop, the present study uses a SIFI paradigm to formulate a trial-level integration problem that links Bayesian integration, causal inference, and evidence accumulation theories to quantitative EEG and behavioral data. Thirty healthy adults performed beep–flash combinations spanning unimodal, audiovisually congruent, and incongruent (fusion/fission) conditions while response accuracy (ACC) and RT were recorded. Time-locked EEG was used to extract trial-level prestimulus occipital α power, visual and auditory N1 amplitudes, and CPP slope. We first characterized condition effects and EEG–behavior relationships using repeated-measures ANOVAs and correlations, and then jointly modeled ACC (Bernoulli likelihood with logit link; effects reported as odds ratios) and RT (Student-t likelihood on logRT; effects interpreted as percentage change in median RT) using hierarchical Bayesian regression. Model generalizability was assessed via PSIS-LOO cross-validation within a Bayesian workflow [11, 13, 41, 45].

Guided by prior empirical and theoretical work on prestimulus alpha oscillations, early sensory ERPs, and centro-parietal accumulation signals in perceptual and multisensory decision-making [17, 21, 22, 29, 36], yet retaining an exploratory scope, we formulated the following hypotheses regarding trial-level EEG predictors while controlling for paradigm manipulations (congruency and illusion vs. non-illusion):

H1 (Prestimulus α and performance). Higher prestimulus occipital α power, reflecting lower cortical excitability and reduced sensory gain, will be associated with lower odds of a correct response and slower responses at the single-trial level, particularly increasing illusion probability under incongruent configurations [2, 17, 36].

H2 (CPP slope and evidence accumulation). Steeper CPP build-up, indexing faster evidence accumulation, will be associated with shorter RTs and higher accuracy across conditions, consistent with accumulation-to-bound accounts and recent links between CPP dynamics and subjective confidence [6, 22, 29].

H3 (Sensory N1 amplitudes and context dependence). Larger auditory and visual N1 amplitudes, as readouts of early sensory evidence and attentional gain, will covary with perceptual judgments and RT. We expect auditory N1, which directly reflects encoding of beep numerosity and rhythm, to show stronger associations with RT and to exhibit context dependence (interactions with congruency/illusion) than visual N1, whose effects may be more weakly expressed in the present bilateral display [10, 16],van den [39].

By embedding these EEG predictors and condition manipulations within a unified hierarchical Bayesian framework, we aim to quantify their relative contributions to single-trial perceptual decisions in SIFI and to evaluate how much incremental predictive value neural covariates add beyond experimental factors alone.

Methods

Participants

Sample size planning was conducted prior to data collection. Because the study also included conventional repeated-measures analyses of behavioral outcomes, we used a repeated-measures ANOVA power analysis in G*Power 3.1.9.2 as a pragmatic lower-bound planning tool for the 3 × 2 within-subject design (flash condition: 1F vs. 2F; sound condition: 0B vs. 1B vs. 2B), assuming an expected effect size of f = 0.20, α = 0.05, and power = 0.80, which yielded a target sample of 24 participants [8]. We emphasize that this calculation was used for study planning rather than as a justification of Bayesian evidential thresholds. The primary inferential framework of the present study is Bayesian, and model evaluation was based on posterior uncertainty and out-of-sample predictive performance rather than null-hypothesis significance testing alone [24].

Thirty healthy adults (mean age 21.6 ± 5.0 years; 19 women) were recruited via online advertisements and poster announcements at Beijing Sport University and in the surrounding community. All participants were right-handed and reported no history of psychiatric disorders. To ensure sufficient task engagement and perceptual reliability, participants with mean accuracy below 50% in the congruent baseline conditions (B1F1 and B2F2) were excluded (cf. [3]), resulting in the exclusion of five individuals. Data from the remaining 25 participants were included in the final analyses. All participants provided written informed consent prior to the experiment and agreed to the use of their anonymized data for publication. Each participant received 100 RMB as compensation upon completion of the study. Ethical approval was obtained from Beijing Sport University (2024349H) and all procedures conformed to the Declaration of Helsinki.

Design and stimuli

The experiment was programmed using MATLAB 2021 with Psychtoolbox 3.1.7 [23]. Visual stimuli were displayed on a Lenovo IdeaCentre AIO 3 all-in-one computer with a screen resolution of 1024 × 768 pixels and a refresh rate of 60 Hz. The visual flashes consisted of white disks (RGB: 255, 255, 255,radius = 2°) presented simultaneously on both the left and right sides, each positioned 7° from the central fixation point against a black background (RGB: 0, 0, 0). Each flash was displayed for 17 ms.

Auditory stimuli were pure tones (beeps) of 3.5 kHz presented at 75 dB SPL for 17 ms, delivered binaurally through headphones. The experiment followed a 3 (sound: 0, 1, or 2 beeps) × 2 (vision: 1 or 2 flashes) within-subject design, yielding six conditions in total: B0F1, B0F2, B1F1, B1F2, B2F1, and B2F2. Here, “B” denotes the number of auditory beeps and “F” denotes the number of visual flashes. B0F1 and B0F2 served as visual-only baseline conditions, whereas B1F1, B2F2, B1F2, and B2F1 were the four audiovisual combinations shown in Fig. 1b. For example, B2F1 indicates one simultaneous visual flash accompanied by two auditory beeps. The inter-beep and inter-flash intervals were both 66 ms. To improve consistency, we use uppercase condition codes throughout the manuscript: B0F1 and B0F2 denote the visual-only baseline conditions, B1F1 and B2F2 denote the audiovisually congruent conditions, B1F2 denotes the fusion configuration, and B2F1 denotes the fission configuration.

Fig. 1.

Experimental paradigm of the sound-induced flash illusion. a In the revised paradigm, white disks were presented simultaneously and symmetrically 7° to the left and right of a central fixation point. b The four audiovisual beep–flash combinations (B1F1, B2F1, B1F2, B2F2) are illustrated. The two visual-only baseline conditions (B0F1 and B0F2) were also included in the experiment and analyses but are not displayed in this panel for visual simplicity. B1F1 and B2F2 represent congruent conditions, whereas B1F2 and B2F1 correspond to incongruent conditions. c Electrode clusters used for regional analyses. Blue dots indicate the visual region of interest (ROI: Oz, POz, O1, O2, PO7, PO8), and yellow dots indicate the auditory region of interest (ROI: Fz, FCz, Cz, CPz)

Procedure

Participants were instructed to maintain fixation on the central point throughout the experiment and to judge whether they perceived one or two visual flashes while ignoring the auditory beeps. To counterbalance response mapping, half of the participants pressed the left mouse button for “one flash” and the right button for “two flashes,” while the mapping was reversed for the other half. Each participant completed 400 trials. The inter-trial interval (ITI) varied randomly between 400 and 700 ms in 100-ms steps. Trials within each block were presented in pseudo-random order to prevent sequence predictability.

EEG recording

EEG data were recorded using a portable multimodal amplifier (Smarting Pro + , mBrainTrain, Belgrade, Serbia) and a standard elastic cap arranged according to the extended international 10–10/10–05 system. Scalp landmarks (nasion, inion, and left/right preauricular points) were measured and marked prior to cap placement. The cap was then positioned with Cz at the vertex to ensure consistent coverage across participants. The montage included frontal, central, parietal, occipital, and temporal sites (e.g., Fp1/2, F3/4, Fz, FC series, C series, CP series, P series, O1/2, T7/8), enabling subsequent topographical and regional analyses.

During acquisition, EEG was referenced online to the common average of all active scalp electrodes for all participants. The ground electrode was placed according to the manufacturer’s recommendations, and electrode impedance and baseline noise levels were checked prior to recording.

EEG signals were sampled at 500 Hz with the system’s built-in analog anti-alias filter. After rechecking the raw data and preprocessing records, we confirmed that all datasets included in the present analyses were recorded at this sampling rate; therefore, no offline resampling step was required. No additional online band-pass or notch filters were applied beyond the hardware anti-alias filtering; all spectral and temporal filtering was performed offline.

Before recording, the electrode–skin interface was prepared using standard abrasive gel and conductive paste to reduce impedance. Impedances for all channels were kept below 10 kΩ wherever possible. Channels that could not reach this threshold after reasonable adjustment were flagged for later inspection and, if necessary, handled in subsequent data-quality control.

The experiment was conducted in a quiet, dimly lit, electrically shielded room. The EEG amplifier, experiment computer, and display/audio hardware shared a common electrical ground, and power and signal cables were routed separately to minimize electromagnetic interference. Participants were seated comfortably with their forearms supported and completed practice trials before the formal task. Event triggers marking stimulus onset and other task events were sent from the stimulus computer to the EEG amplifier via a hardware trigger interface and recorded in the continuous EEG data stream for precise temporal alignment between EEG and behavioral data.

EEG data preprocessing and feature extraction

EEG preprocessing was performed in MATLAB using EEGLAB (v2023.0). Raw continuous data were first visually inspected to identify gross artifacts and malfunctioning channels. Bad channels were either interpolated using spherical splines based on neighboring electrodes or, if unrecoverable, marked for exclusion from regional averages but retained for ICA decomposition to preserve spatial information. All datasets were then re-referenced to the common average if not already recorded in that configuration.

Continuous EEG data were notch-filtered at 48–52 Hz to remove line noise, high-pass filtered at 0.1 Hz, and low-pass filtered at 40 Hz using zero-phase finite impulse response filters. When Fp1/Fp2 were present, surrogate vertical and horizontal electrooculogram (EOG) channels were constructed from these electrodes to aid artifact identification. Independent component analysis (ICA) was performed on a copy of the data high-pass filtered at 1 Hz to improve component separation. Components were classified using ICLabel, and those predominantly reflecting ocular, muscular, cardiac, or channel-noise artifacts were removed. The resulting ICA weights were then back-projected to the main (0.1–40 Hz) dataset.

Data were epoched time-locked to stimulus onset from − 1000 to 2000 ms. Each epoch was baseline-corrected using the  − 200 to 0 ms interval. Epochs with amplitudes exceeding ± 200 µV at any scalp channel were rejected automatically, and additional residual artifacts were removed based on visual inspection.

For event-related potentials (ERPs), a posterior occipital ROI (Oz, POz, O1, O2) was used to extract visual components. Within this ROI, mean amplitudes were computed for C1 (50–90 ms), P1 (80–130 ms), N1 (130–190 ms), and later visual components. For the auditory ROI (Fz, FCz, Cz, CPz), amplitudes of P50 (40–70 ms), N1 (80–120 ms), and P2 (150–200 ms) were calculated, and P50/N1 peak latencies were derived. To index evidence accumulation, a centro-parietal ROI (CPz, Pz, CP1, CP2) was used to extract the centro-parietal positivity (CPP). In response-locked epochs (− 300 to 0 ms relative to response), a linear regression was fit to the CPP waveform in a predefined time window to estimate CPP slope, and CPP onset latency was defined as the first time point after 100 ms post-stimulus at which the waveform exceeded a data-driven threshold for a sustained duration. When response–hand mapping was specified, the lateralized readiness potential (LRP) was computed from C3/C4 as the contralateral-minus-ipsilateral difference, and its onset latency was derived using the same criterion.

In the frequency domain, Welch’s method was used to estimate prestimulus α-band (8–12 Hz) power at occipital sites over a − 1000 to 0 ms window. Instantaneous α phase in the  − 100 to 0 ms interval was obtained via band-pass filtering (8–12 Hz) followed by a Hilbert transform. To characterize aperiodic activity, power spectra between 2 and 40 Hz (excluding 7–13 Hz and 48–52 Hz) were fit with a parametric model to estimate the 1/f exponent and offset [5]. In addition, α and β (13–30 Hz) power and their lateralization indices were computed over the − 1000 to 0 ms interval.

Behavioral measures included response accuracy (defined as a match between reported and physical number of flashes, given the instruction to report visual numerosity) and RT (in ms). Trial-level EEG features (e.g., prestimulus occipital α power, CPP slope, visual N1 amplitude, auditory N1 amplitude) were aligned with single-trial behavioral data and served as predictors for both classical analyses and hierarchical Bayesian modeling.

Statistical analysis

Exploratory trial-level associations between EEG markers and behavior were first examined using classical analyses. For these purposes, trial-level EEG predictors included prestimulus α power, CPP slope, visual N1 amplitude, and auditory N1 amplitude. All EEG variables were standardized within participant (z-scored across trials) to remove between-subject mean differences and place measures on a comparable scale. Trials with RT < 50 ms were excluded, and remaining RTs were log-transformed (logRT) to reduce skewness and stabilize variance. For repeated-measures ANOVAs, accuracy and logRT were averaged within each subject × condition cell.

Experimental factors were defined at two complementary levels. For the primary behavioral analyses, we used a six-level within-subject Condition factor corresponding to the six experimental conditions: B0F1, B0F2, B1F1, B1F2, B2F1, and B2F2. For theoretically motivated follow-up analyses, we additionally defined a two-level Audio factor (no sound vs. sound-present) and, within sound-present trials, a two-level Congruency factor (congruent vs. incongruent). To probe potential interactions between EEG indices and task conditions, each continuous EEG predictor was discretized into within-subject tertiles, yielding a three-level factor (Ter1–Ter3) representing low, medium, and high values for that participant.

The statistical workflow was as follows. First, one-way repeated-measures ANOVAs with the six-level factor Condition were conducted on mean accuracy and mean logRT to characterize the overall behavioral pattern across the six experimental conditions. Second, a 2 (Audio: no sound vs. sound) × 3 (CPP_ter: Ter1–Ter3) repeated-measures ANOVA was used to examine how CPP tertiles modulated performance as a function of sound presence. Third, within sound-present trials, separate 2 (Congruency: congruent vs. incongruent) × 3 (EEG_ter: Ter1–Ter3) repeated-measures ANOVAs were run for each EEG predictor to assess whether trial-wise EEG fluctuations differentially influenced behavior in congruent versus incongruent contexts. For factors with more than two levels, sphericity was assessed and Greenhouse–Geisser corrections were applied where appropriate. Main effects and interactions were followed by paired-samples t tests; for each contrast we report t, degrees of freedom, 95% confidence intervals, and the corresponding p value. Where needed, p values were adjusted for multiple comparisons using Bonferroni correction within each family of tests.

Given the skewed distributions and potential outliers in trial-wise measures, EEG–behavior correlations were quantified using Spearman’s rank correlation coefficient (ρ). Ninety-five percent confidence intervals for ρ were obtained from Fisher-z–transformed values. Within each analysis family (e.g., correlations across multiple EEG markers), p values were adjusted using the Benjamini–Hochberg false discovery rate (FDR) procedure, and statistical significance was evaluated on the FDR-corrected values. Prior to hierarchical Bayesian modeling of accuracy and RT, variance inflation factors (VIFs) were computed for the four EEG regressors to assess potential multicollinearity. Consistent with contemporary Bayesian recommendations, the Bayesian analyses in the present study were interpreted in terms of posterior distributions, uncertainty intervals, and predictive performance, rather than by reference to frequentist power criteria alone [24].

Trial-level Hierarchical Bayesian regression

We fitted trial-level hierarchical Bayesian regression models for both accuracy (ACC) and reaction time (RT). ACC was modeled with a Bernoulli likelihood and logit link, whereas RT was modeled on the log scale using a Student-t likelihood. All models included participant-specific random (varying) intercepts. Trial-level (within-subject) EEG predictors entered the models as random slopes that were allowed to vary across participants, whereas between-subject EEG components and experimental condition dummies were treated as fixed effects.

Fixed-effect coefficients were assigned weakly informative Normal(0, 1) priors. Standard deviations of random effects followed Half-Normal priors. For the Student-t RT model, the degrees-of-freedom parameter was specified as ν = 1 + Exponential(1/30), allowing for heavy-tailed but finite-variance distributions. Sampling was performed in PyMC using the No-U-Turn Sampler (NUTS) with the same general configuration (number of chains, warm-up and retained iterations, target acceptance rate, and random seeds) as for the other models in this study.

Prior to modeling, predictors were decomposed into within- and between-subject components and standardized. Trials with missing values on any modeling variable were removed using listwise deletion within the analysis subset.

Notation and data structure

Let s = 1,…,S index subjects and t = 1,…, T_s index trials for subject s.

∈{0,1}: correctness.

> 0: reaction time in seconds.

∈: within-subject standardized EEG components.

∈: between-subject standardized EEG components (subject means).

∈: condition dummies (baseline omitted, so K − 1 are included).

Within–between decomposition and standardization were implemented as follows. For a given EEG predictor (e.g., CPP slope, prestimulus α power, visual N1 amplitude, auditory N1 amplitude), we first computed each subject’s mean over trials,

and then defined:

Within-subject component (trial-level z-score):

Between-subject component (subject mean, standardized across subjects):

The within-subject coefficient quantifies the instantaneous effect of a 1-SD deviation of a given predictor from an individual’s own mean on the outcome. The between-subject coefficient captures how between-participant differences of 1 SD in the same predictor (i.e., baseline level differences) relate to overall performance. Including both components in the model helps to avoid ecological fallacies and bias arising from conflating within- and between-subject variation.

Model specification and comparison

ACC model (Bernoulli-logit)

With logit(p) = log{p/(1 − p)}, the linear predictor is

Likelihood and link:

Hierarchy

denotes the group-level mean slope for the within-subject component of predictor. captures its between-participant variability. represents the between-subject fixed effect for predictor. indexes the condition effects, with the omitted category serving as the baseline..

For any predictor x, the corresponding logit coefficient β can be expressed on the probability scale as an odds ratio (OR) given by exp(β). For within-subject predictors, exp() quantifies the multiplicative change in the odds of a correct response associated with a 1-SD increase in the EEG marker relative to that participant’s own mean. For between-subject predictors, exp() analogously reflects the odds ratio associated with a 1-SD difference in the participant-level mean of the same marker across individuals.

RT model (log-RT Student-t)

To address heavy tails, we model log-RT with a Student-t likelihood:

Hierarchy and interpretation.

The subject-specific intercepts and within-subject random slopes followed the same hierarchical structure as specified in Eq. (5). On the log scale, exponentiating the coefficient β yields an interpretable percentage change: a 1-SD increase in a given predictor is associated with an approximate change of (e^β − 1) × 100% in the median RT.

Prior specification

We employ weakly informative priors:

For RT:

ν =1+Exponential(1/30).

Inference and computation

We use PyMC’s NUTS for MCMC sampling (4 chains; 2000 warmup; 2000 saved draws; target_accept = 0.90; seed = 2024). Convergence is assessed via , effective sample sizes, trace plots, and energy/step-size diagnostics. Model adequacy is evaluated using PSIS-LOO and WAIC [41], as well as posterior predictive checks: calibration and Brier score for ACC; 50%/90% predictive interval coverage, observed–predicted plots, and RMSE/MAE for RT. Bayesian R² is computed per Eqs.

ACC (Bernoulli-logit),

RT(log-RT t-likelihood), ,

ACC: Report posterior means and 95% HDIs for within and between effects; provide OR = exp(β) with intervals for interpretability.

RT: Report percent changes (e^β − 1) × 100% with 95% HDIs; optionally back-transform to seconds for conditional predictions.

Hierarchical variance: report σ_α and σ_w,j, along with ICC via Eqs.

, ν > 2.

Model comparison: present PSIS-LOO/WAIC elpd (and SE) and Pareto-k proportions > 0.7.

Predictive performance: AUC, Brier, and calibration for ACC; R², coverage, RMSE/MAE for RT.

Model family and comparison

We compared three hierarchical Bayesian regression models for both ACC and RT: M0: included only experimental condition (d_st) and participant-specific random intercepts, providing a baseline in which behavior is driven solely by paradigm manipulations.

M1: extended M0 by adding EEG main effects—prestimulus α power, CPP slope, visual N1 amplitude, and auditory N1 amplitude—each decomposed into within- and between-subject components.

M2: further extended M1 by including an interaction between trial-level auditory N1 (within-subject component) and sound presence (Audio factor), allowing the effect of auditory N1 on behavior to depend on whether a sound was present or absent.

Model selection focused on predictive performance rather than in-sample fit. We used Pareto-smoothed importance sampling leave-one-out cross-validation (PSIS-LOO) to estimate the expected log predictive density (ELPD-LOO) and its standard error. For ease of comparison, we also report LOOIC (− 2 × ELPD-LOO). Models with higher ELPD-LOO (or lower LOOIC) were preferred; when differences in ELPD-LOO were small relative to their uncertainty (|ΔELPD-LOO|< 1–2 × SE), models were treated as predictively indistinguishable, in which case we favored the more parsimonious specification and, where appropriate, report stacking weights for model averaging. PSIS-LOO was chosen over WAIC and DIC because it is generally more robust in hierarchical models with finite samples and potential high-leverage observations, and it provides both standard errors for model comparisons and Pareto k diagnostics to assess the reliability of the importance-sampling approximation.

Results

Behavioral performance analyses

Condition significantly affected both accuracy and RT. As shown in Fig. 2a, left, congruent and visual-only conditions generally yielded higher accuracy than incongruent conditions, with B1F2 showing the lowest accuracy. A one-way repeated-measures ANOVA with the six-level factor Condition confirmed a significant main effect on accuracy, F(5, 120) = 9.72, p < 0.001, partial η² = 0.288.

Fig. 2.

Behavioral performance, EEG tertiles, and cross-measure correlations. a Mean accuracy and mean RT by condition (error bars =  ± SE of subject means). b Panel plots for Audio × CPP tertile (Ter1/Ter2/Ter3) on ACC and logRT across all trials. c Within sound-present trials, Congruency (congruent vs. incongruent) crossed with EEG tertiles shown separately for CPP slope, prestimulus α-power, visual N1, and auditory N1; lines depict subject-mean trajectories with ± SE. d Spearman correlation matrix among α, CPP slope, visual N1, auditory N1, ACC, and logRT; asterisks mark cells tested for FDR-corrected significance. logRT = natural log of RT after excluding responses < 50 ms. Tertiles are computed within participant

As shown in Fig. 2a, right, RT exhibited a parallel but not identical pattern: B0F1 was the fastest condition, whereas B2F1 produced the greatest slowing. A one-way repeated-measures ANOVA with the six-level factor Condition confirmed a significant main effect on logRT, F(5, 120) = 16.75, p < 0.001, partial η² = 0.411. Detailed descriptive statistics are presented in Table 1, selected theoretically central pairwise comparisons are summarized in Table 2, and the full set of Bonferroni-corrected pairwise comparisons is reported in Table S1.

Table 1.

Descriptive statistics for behavioral performance across the six experimental conditions

Condition	Accuracy, M	Accuracy, SE	RT (ms), M	RT (ms), SE	logRT, M	logRT, SE
B0F1	0.952	0.011	659.730	16.804	6.460	0.023
B0F2	0.847	0.032	680.123	19.192	6.484	0.027
B1F1	0.963	0.010	672.023	21.785	6.479	0.028
B1F2	0.731	0.058	743.781	28.749	6.557	0.037
B2F1	0.808	0.056	788.145	28.716	6.624	0.035
B2F2	0.962	0.013	698.728	19.955	6.513	0.027

Table 2.

Key Bonferroni-corrected pairwise comparisons for behavioral performance across conditions

Outcome	Contrast	Mean difference	95% CI	t(24)	Bonferroni-p	Cohen’s dz	Interpretation
Accuracy	B1F1 vs. B1F2	0.233	[0.113, 0.353]	3.993	0.008	0.799	Congruent > incongruent
Accuracy	B2F2 vs. B1F2	0.231	[0.119, 0.344]	4.243	0.004	0.849	Congruent > incongruent
Accuracy	B0F1 vs. B1F2	0.221	[0.097, 0.345]	3.679	0.018	0.736	Visual-only > incongruent
RT (logRT)	B2F1 vs. B0F1	0.164	[0.114, 0.214]	6.785	<0 .001	1.357	Incongruent slower
RT (logRT)	B2F1 vs. B0F2	0.140	[0.090, 0.191]	5.753	< 0.001	1.151	Incongruent slower
RT (logRT)	B2F1 vs. B1F1	0.146	[0.100, 0.192]	6.523	< 0.001	1.305	Incongruent slower
RT (logRT)	B2F1 vs. B2F2	0.112	[0.072, 0.151]	5.849	< 0.001	1.170	Incongruent slower
RT (logRT)	B2F1 vs. B1F2	0.067	[0.025, 0.109]	3.316	0.043	0.663	B2F1 slowest

Only theoretically central contrasts are shown in the main text. The full set of Bonferroni-corrected pairwise comparisons is reported in Table S1

Accuracy and logRT were first averaged within each subject × condition cell. Descriptive RT values are presented in milliseconds for readability, whereas inferential tests for response latency were conducted on logRT.

Detailed descriptive statistics for ACC and RT are presented in Table 1, and Bonferroni-corrected pairwise comparisons are reported in Table 2. In brief, B1F2 yielded the lowest response accuracy, whereas B2F1 produced the longest response times, indicating that audiovisual incongruency impaired both choice accuracy and processing efficiency. More specifically, B1F1, B2F2, and B0F1 were among the highest-accuracy conditions, whereas B2F1 and especially B1F2 were associated with reduced performance. For RT, B0F1 was the fastest condition, while B2F1 showed the greatest slowing. These pairwise contrasts are reported in full in Table 2 and are therefore not repeated exhaustively in the main text.

Classical EEG analyses

To improve readability, the principal repeated-measures ANOVA results for the EEG-tertile analyses are summarized in Table 3, with full numerical details reported in the corresponding text. Figure 2b and c illustrate the tertile-based performance patterns, whereas Fig. 2d summarizes the correlation structure among behavioral and EEG measures.

Table 3.

Summary of repeated-measures ANOVA results for EEG-tertile analyses

Analysis block	EEG predictor	ACC: main pattern	logRT: main pattern
Audio × EEG tertile	CPP tertile	Audio main effect, F(1, 25) = 4.79, p = 0.038, ηp2 = 0.161; no CPP or interaction effect	Audio main effect, F(1, 25) = 23.41, p < 0.001, ηp2 = 0.484; no CPP or interaction effect
Congruency × EEG tertile	CPP tertile	Congruency main effect, F(1, 25) = 17.49, p < 0.001, ηp2 = 0.412	Congruency main effect, F(1, 25) = 28.22, p < 0.001, ηp2 = .530; CPP tertile main effect, F(2, 50) = 4.07, p = 0.023, ηp2 = .140
Congruency × EEG tertile	Prestimulus α tertile	Congruency main effect, F(1, 25) = 18.44, p < 0.001, ηp2 = .424; α tertile main effect, F(2, 50) = 4.33, p = 0.0185, ηp2 = 0.148; interaction, F(2, 50) = 4.09, p = 0.0226, ηp2 = 0.141	Congruency main effect, F(1, 25) = 31.94, p < 0.001, ηp2 = 0.561
Congruency × EEG tertile	Visual N1 tertile	Congruency main effect, F(1, 25) = 17.82, p = 0.00028, ηp2 = 0.416	Congruency main effect, F(1, 25) = 31.93, p < 0.001, ηp2 = 0.561
Congruency × EEG tertile	Auditory N1 tertile	Congruency main effect, F(1, 25) = 17.85, p = 0.00028, ηp2 = 0.417	Congruency main effect, F(1, 25) = 32.64, p < 0.001, ηp2 = 0.566; auditory N1 tertile main effect, F(2, 50) = 5.10, p = 0.00968, ηp2 = 0.169; interaction, F(2, 50) = 3.88, p = 0.0271, ηp2 = 0.134

A two-way repeated-measures ANOVA with factors Audio (present vs. absent) and CPP tertile (Ter1/Ter2/Ter3) on all trials showed a significant main effect of Audio on both accuracy and speed (ACC: F(1,25) = 4.79, p = 0.038; logRT: F(1,25) = 23.41, p < 0.001), indicating overall faster and more accurate performance when sound was present (Fig. 2b). The main effect of CPP tertile was not significant (ACC: F(2,50) = 2.38, p = 0.103; logRT: F(2,50) = 2.65, p = 0.080), and the Audio × CPP interaction was also not significant (ACC: F(2,50) = 0.93, p = 0.402; logRT: F(2,50) = 0.61, p = 0.547) (Fig. 2b; Table 3).

Restricting analyses to audio trials, we ran 2 × 3 repeated-measures ANOVAs with factors Congruency (congruent vs. incongruent) and EEG tertile (Ter1/Ter2/Ter3), using in turn the within-subject tertiles of CPP slope, prestimulus α-power, visual N1, and auditory N1 as the second factor (Fig. 2c; Table 3).

Restricting analyses to sound-present trials, 2 × 3 repeated-measures ANOVAs with factors Congruency (congruent vs. incongruent) and EEG tertile showed the following pattern. For CPP_ter, Congruency yielded significant main effects on both ACC and logRT (ACC: F(1,25) = 17.49, p < 0.001; logRT: F(1,25) = 28.22, p < 0.001), and CPP_ter exerted an additional main effect on logRT (F(2,50) = 4.07, p = 0.023), with no interaction (F(2,50) = 0.02, p = 0.983). For α_ter, Congruency again showed significant main effects (ACC: F(1,25) = 18.44, p < 0.001; logRT: F(1,25) = 31.94, p < 0.001); α_ter had a significant main effect on ACC (F(2,50) = 4.33, p = 0.0185) and a significant Congruency × α_ter interaction (F(2,50) = 4.09, p = 0.0226), whereas for logRT neither the α_ter main effect nor the interaction reached significance (p = 0.129 and 0.694). For visual N1_ter, Congruency was significant for both ACC and logRT (ACC: F(1,25) = 17.82, p = 0.00028; logRT: F(1,25) = 31.93, p < 0.001), but neither the visual N1_ter main effect nor the interaction was significant. Finally, for auditory N1_ter, Congruency was significant for both ACC and logRT (ACC: F(1,25) = 17.85, p = 0.00028; logRT: F(1,25) = 32.64, p < 0.001); auditory N1_ter also showed a significant main effect on logRT (F(2,50) = 5.10, p = 0.00968) and a significant Congruency × aN1_ter interaction (F(2,50) = 3.88, p = 0.0271), while for ACC neither the main effect nor the interaction was significant (p = 0.077 and 0.273). In sum, Congruency robustly modulated both accuracy and speed across EEG partitions; CPP_ter and aN1_ter contributed additional variance to response speed, α_ter influenced accuracy and interacted with Congruency, and visual N1_ter showed no independent or interactive effects in this discretized framework.

Taken together (Fig. 2c), these classical analyses indicate that the primary behavioral contrast is between congruent and incongruent beep–flash conditions, with unimodal trials generally performing at least as well as congruent trials. Trial-wise EEG tertiles exerted, at best, modest and inconsistent effects on performance. This motivates moving beyond tertile-based ANOVAs to trial-level regression, where continuous EEG covariates and condition dummies can be modeled jointly and more sensitively.

Finally, three correlations were most robust (Fig. 2d). First, accuracy and response time were negatively associated (ρ =  − 0.238, 95% CI [− 0.265,  − 0.211]), indicating a “more accurate, faster” pattern at the aggregate level. Second, visual and auditory N1 amplitudes were negatively associated at the trial level (ρ =  −  0.208, 95% CI [− 0.235,  − 0.180]), hinting at cross-modal anticovariance in early evoked responses or metric sign differences. Third, CPP slope correlated negatively with RT (ρ = − 0.096, 95% CI [− 0.124, − 0.068]), consistent with faster evidence accumulation yielding shorter latencies. Several small effects reached significance: prestimulus α correlated positively with visual N1 (ρ = 0.08***), negatively with auditory N1 (ρ =  − 0.04*), and negatively with ACC (ρ =  − 0.04*); auditory N1 was also weakly negatively correlated with ACC (ρ =  − 0.04*). Visual N1–ACC and auditory N1–RT associations were near zero or non-significant overall. Variance-inflation factors for the four EEG predictors were 1.00–1.06, indicating negligible multicollinearity and supporting concurrent inclusion of α, CPP, visual N1, and auditory N1 in the same hierarchical models. In light of these results, visual N1 shows no strong overall main effect on ACC/RT, yet its stable negative relationship with auditory N1 and potential modality- or condition-dependent interactions remain theoretically informative; its retention in modeling will be adjudicated by model comparison.

In congruent condition (B1F1 vs. B2F2), prestimulus α power, CPP slope, and visual N1 showed no significant effects of Condition, Accuracy, or their interaction (p > 0.05). Only auditory N1 yielded a significant Accuracy main effect—larger (more positive) amplitudes on error trials, F(1,46) = 6.82, p = 0.012, partial η² = 0.129—whereas Condition and the interaction were non-significant (p > 0.05). Bonferroni-corrected simple effects were also non-significant (p > 0.05) (Fig. 3).

Fig. 3.

ERPs and prestimulus α in congruent conditions (B1F1, B2F2). Top row: event-related potentials (ERPs). A waveforms at the visual ROI. B waveforms at the auditory ROI. Bottom row: prestimulus α. C band-limited power. D power spectra within the prestimulus window. Blue/orange denote B1F1/B2F2, solid/dashed denote correct/error trials, and shaded ribbons indicate variability. Overall, early visual components (P1/N1) exhibit similar morphology across the two congruent conditions. The auditory composite shows larger amplitudes in the two-tone condition. Prestimulus α is slightly higher on error than on correct trials, whereas the α peak frequency remains stable

In incongruent condition (B1F2 vs. B2F1), prestimulus α, CPP slope, visual N1, and auditory N1, the effects of Condition, Accuracy, and their interaction were uniformly non-significant (p > 0.05) (Fig. 4). Taken together, mean-window ERP/α differences primarily emerge as a correct–error separation in the congruent context and do not reliably appear under incongruency.

Fig. 4.

ERPs and prestimulus α in incongruent conditions (B1F2, B2F1). A Visual ROI ERPs. B Auditory ROI ERPs. C prestimulus α power. D prestimulus power spectra. Blue/orange denote B1F2/B2F1, solid/dashed denote correct/error trials, and shaded ribbons indicate variability. Under incongruency, visual waveforms diverge more clearly after ~ 200 ms, with B2F1 showing a deeper late negativity. The auditory composite exhibits larger amplitudes in the “two-tone” condition. Prestimulus α is again slightly higher on error than on correct trials, and spectral shapes strongly overlap across conditions

As shown in Fig. 5, correct trials exhibited canonical distributions for all components. The visual N1 (130–200 ms) showed a posterior occipito-parietal negativity with clear contralateral predominance relative to visual stimulation. The auditory N1 (80–130 ms) displayed a fronto-central negative maximum with a stable spatial configuration across conditions; compared with the single-tone conditions, the two-tone conditions produced larger auditory N1 amplitudes. Within the CPP window (300–600 ms), a broad centro-parietal positivity was evident, with an expanded scalp extent on correct trials.

Fig. 5.

Scalp topographies for correct trials: N1 (visual/auditory) and CPP. Each row corresponds to a condition (from top to bottom: B1F1, B2F2, B1F2, B2F1). Each column corresponds to a time window/component (from left to right: visual N1, 130–200 ms; auditory N1, 80–130 ms; CPP, 300–600 ms). Color bars indicate average voltage (μV; warm = positive, cool = negative). The visual N1 shows a contralateral occipito-parietal negativity; the auditory N1 exhibits a fronto-central negative maximum; the CPP displays a centro-parietal positivity. Two-tone conditions yield larger auditory N1 amplitudes, while overall spatial patterns remain consistent across conditions

As shown in Fig. 6, error trials were characterized by globally reduced peak amplitudes for both N1 components and the CPP. The centro-parietal positivity in the CPP window appeared spatially contracted, although the overall topographical patterns remained consistent with the canonical morphology.

Fig. 6.

Scalp topographies for error trials: N1 (visual/auditory) and CPP. Layout follows Fig. 5 (rows: B1F1, B2F2, B1F2, B2F1; columns: visual N1 130–200 ms, auditory N1 80–130 ms, CPP 300–600 ms; color bars in μV). Relative to correct trials, peak amplitudes are globally reduced for both N1 components and the CPP, and the centro-parietal positivity in the CPP window appears spatially contracted. Nonetheless, each component retains its canonical topographical configuration

By visual inspection, correct trials revealed a contralateral occipital positivity emerging after auditory cueing—i.e., a subtle positive shift over occipito-parietal sites contralateral to the cued side during ~ 130–200 ms, which appeared to propagate into a wider posterior positivity as the CPP developed. This contralateral posterior pattern aligns with the topographical signature of the auditory-evoked contralateral occipital positivity (ACOP). In contrast, error trials showed a less pronounced contralateral posterior positivity and a more restricted centro-parietal CPP field. These descriptive topographies suggest that cross-modal orienting/weighting associated with ACOP may facilitate subsequent evidence accumulation indexed by the CPP on trials that ultimately yield correct responses.

Hierarchical Bayesian model comparison

For ACC, ELPD-LOO values were: m0 =  − 1359.94 (SE = 41.83; LOOIC = 2719.87; p_loo = 38.79), m1 =  − 1358.52 (SE = 41.74; LOOIC = 2717.04; p_loo = 43.25), and m2 =  − 1341.09 (SE = 42.11; LOOIC = 2682.18; p_loo = 55.71). Using m2 as the reference, ΔELPD(m0 − m2) =  − 18.84, SE(Δ) = 59.36, |z|= 0.32; ΔELPD(m1 − m2) = − 17.43, SE(Δ) = 59.29, |z|= 0.29. Thus, although m2 is best on ELPD-LOO/LOOIC, its advantage over m0/m1 is not robust (|Δ|< 2 × SE), and the higher p_loo indicates increased complexity.

For RT, ELPD-LOO values were: m0 = 949.73 (SE = 55.72; LOOIC = − 1899.46; p_loo = 71.45), m1 = 950.03 (SE = 55.67; LOOIC =  − 1900.06; p_loo = 75.38), and m2 = 950.02 (SE = 55.68; LOOIC = − 1900.05; p_loo = 79.60). With m1 as reference, ΔELPD(m0 − m1) =   − 0.30, SE(Δ) = 78.77, |z| ≈ 0.00; ΔELPD(m2 − m1) =  − 0.00, SE(Δ) = 78.74, |z| ≈ 0.00. The three models are practically equivalent for RT; m1 is marginally better on ELPD-LOO/LOOIC, but differences are far from robust, and p_loo increases from m0 to m2.

The model family was: M0 (within: CPP, α; between: CPP, α; plus condition dummies), M1 (M0 + trial-level Auditory N1), and M2 (M1 + Auditory N1 × Audio interaction). Diagnostics indicated 4 chains × 2000 draws, and adequate ESS. Effect patterns were: RT (log-Student-t; multiplicative (exp(β)): relative to the congruent baseline, fission and fusion robustly increased RT (~ + 11.7% and + 3.8%), whereas visual-only slightly decreased RT (~ − 2.5%). Trial-level EEG predictors and the Audio interaction were close to unity with intervals crossing 1; between-subject terms showed limited trial-wise explanatory power. ACC (logit; OR): non-congruent conditions markedly reduced accuracy (ORs < 1); trial-level α was stably negative (OR≈0.88), Auditory N1 was borderline negative; CPP and between terms largely crossed 1; the interaction did not alter conclusions. Directions and magnitudes were consistent across m0/m1/m2, indicating specification-robust findings.

Across outcomes, the “best” models were m2 (ACC) and m1 (RT), yet their ΔELPD-LOO over alternatives were not robust. Given the prediction–complexity trade-off and interpretability, m1 is a defensible unified model (best for RT and near-equivalent to m2; slightly worse for ACC but not robustly so and with lower p_loo).

Parameter estimates of the unified model (m1)

As the unified reporting model, m1 augments M0 by adding trial-level Auditory N1 amplitude (z-score), retaining between-subject CPP and α, and including condition dummies. For RT, m1’s condition effects mirror M0/M2 with negligible differences; the trial-level Auditory N1 effect on RT is 1.005 (95% HDI [0.999, 1.011]), with the interval crossing 1. Trial-level CPP/α and between-subject CPP/α estimates are likewise near unity, indicating that under the current linear/additive specification EEG predictors contribute little at the single-trial level, while RT variation is primarily driven by experimental conditions.

For ACC, condition main effects are strong and stable (non-congruent/illusion conditions with OR < 1; congruent-like levels with OR > 1), consistent with M0/M2; trial-level α power shows a robust negative association (OR = 0.878, 95% HDI [0.792, 0.972]), Auditory N1 is borderline negative (OR = 0.890, 95% HDI [0.796, 0.995]), while CPP (within) and between-subject CPP/α mostly cross 1. Considering PSIS-LOO and the parameter estimates together, m1 was retained as a unified reporting model for parsimony rather than because EEG covariates provided a clear predictive advantage. RT was dominated by condition effects; for ACC, trial-level prestimulus α showed a robust negative association and auditory N1 a weaker negative association, whereas the remaining EEG terms were unstable and the model-level improvements over condition-only formulations were not robust.

Discussion

The present study examined how stimulus congruency, audiovisual numerosity configurations, and momentary neural states jointly shape perceptual decisions in the sound-induced flash illusion at the single-trial level. Across classical and hierarchical Bayesian analyses, three findings were most robust. First, task condition was the primary determinant of behavior, with congruent and visual-only trials yielding better performance than incongruent trials, especially B2F1. Second, among the EEG markers, prestimulus alpha showed the clearest negative association with accuracy, whereas auditory N1 was more closely related to response speed and CPP primarily tracked RT at the descriptive level. Third, model comparison indicated that adding linear EEG covariates did not produce a robust out-of-sample predictive advantage over condition-only models. This pattern frames the discussion below, in which we interpret the behavioral dominance of condition effects together with the more limited but theoretically informative role of trial-level neural fluctuations.

Congruency effects in the present SIFI paradigm followed the expected pattern: audiovisually congruent and unimodal configurations yielded faster and more accurate responses, whereas incongruent configurations impaired performance, with the “2 beep–1 flash” (B2F1) condition producing particularly pronounced RT slowing. This pattern is consistent with sound-induced flash illusion accounts in which additional beeps perturb perceived flash numerosity, reduce the posterior probability of a common cause, and increase the cost of resolving cross-modal conflict [19, 37, 38]. In our implementation, auditory rhythms likely facilitated temporal prediction and arousal under congruency, but under incongruency they instead promoted causal segregation and illusory numerosity, leading to the marked slowdown in B2F1 relative to B1F2. Thus, the behavioral data support a perspective in which causal inference and reliability-based weighting jointly shape multisensory decisions in SIFI.

The negative association between prestimulus α power and accuracy fits the view that α indexes transient cortical excitability and biases reporting thresholds. Prior work shows that higher α does not improve objective discrimination but is linked to lower subjective visibility/confidence and more conservative reports [2, 17, 36]. Under cross-modal conflict and causal uncertainty, elevated α implies reduced sensory gain and a weaker bias toward common-cause integration, yielding lower accuracy and more conservative responding.

Larger auditory N1 amplitudes predicted faster responses, indicating that early sensory evidence strength/feature-specific gain shortens decision latency. This dovetails with evidence that attention/expectation amplifies N1 and facilitates subsequent judgments, and it suggests that stronger auditory encoding can partly offset conflict-related slowing in incongruent trials [10].

CPP tightly tracked response time; its slope/accumulation rate varied with evidence quality and approached a stable bound at response, replicating key properties of a supramodal accumulation-to-bound signal. Trial-to-trial variations in CPP rate predicted RT even at matched physical evidence [22, 29]. In light of our behavioral pattern, this implies that the mapping from sensory evidence to the decision variable is jointly modulated by causal structure (common vs. separate causes) and transient excitability (α).

Topographically, we observed the auditory-evoked contralateral occipital positivity (ACOP) following auditory cues: a contralateral occipital positivity within ~ 200–450 ms and a broader centro-parietal positivity. ACOP indexes reflexive capture of visual-cortical resources by peripheral sounds, facilitating processing of co-localized visual targets [9, 15, 20]. This mechanism provides a neural account for faster responses under congruency: cross-modal orienting and sensory gain increase effective input to the accumulator.

The absence of robust effects for visual N1 may reflect (1) reduced lateralized sensitivity due to bilateral symmetric display; (2) a weaker bottleneck role of visual N1, with its impact mediated by higher-level accumulation (CPP); and/or (3) task-specific functional dissociations whereby auditory N1 more directly drives temporalization and causal adjudication in the present paradigm.

We used hierarchical Bayesian regression to (1) quantify trial-level (within-person) neural fluctuations while controlling between-person variability and (2) exploit partial pooling to stabilize estimates and reduce overfitting in modest samples. Within–between decomposition via group-mean centering clarifies the interpretation of slopes and avoids conflating within- with between-level effects [7]. This strategy aligns with the contemporary Bayesian workflow [13]. Following the Bayesian workflow, we used (1) prior and prior-predictive checks, (2) convergence diagnostics, and (3) posterior predictive checks plus PSIS-LOO to support interpretability and transportability [13].

We assessed out-of-sample predictive performance using PSIS-LOO, interpreting model differences via ΔELPD-LOO and its standard error SE(Δ) (a common rule of thumb is ∣Δ∣ > 2 × SE to claim a clear advantage). We also reported p_loo (effective complexity) and LOOIC. Compared with WAIC, PSIS-LOO is more robust under weak priors or influential observations, and its reliability is diagnosed by Pareto-k with a recommended threshold of k < 0.7 [41]. In our case, RT models (m0/m1/m2) were nearly equivalent in ΔELPD-LOO, whereas m2’s slight advantage for ACC did not exceed 2 × SE; thus, the added complexity did not translate into stable predictive gains, motivating m1 as the unified reporting model.

Without high-order interactions, m1 already captured the dominant effects of task conditions on RT/ACC and translated trial-level EEG covariates (α, aN1, CPP) into comparable effects. Adding interaction terms in m2 did not yield stable generalization gains. When the true data-generating process is not among candidates, elaborating structure often fails to deliver replicable predictive improvement [45].

These findings have two broader theoretical implications. First, in SIFI, transient neural-state measures may act less as strong standalone determinants of single-trial behavior than as modulators that tune the susceptibility of the perceptual system to experimentally imposed causal structure. In other words, stimulus configuration appears to set the dominant decision context, whereas prestimulus excitability and early sensory encoding may bias how strongly that context is expressed on a given trial. Second, the dissociation between robust descriptive EEG–behavior associations and limited model-level predictive gains suggests that neural markers can remain mechanistically informative even when their incremental predictive contribution is small under a linear additive model. This distinction between theoretical relevance and predictive leverage implies that future work may need nonlinear, latency-sensitive, or state-by-condition formulations to more fully capture neurobehavioral coupling in multisensory perception.

Our study has several limitations. First, if sensory energy and salience are not fully equated across modalities and conditions, mismatch trials may induce weight reallocation that becomes intertwined with causal inference, potentially exaggerating or masking multisensory integration effects. Future studies should better equate sound level, duration, visual contrast, and noise, and report loudness/visibility matching to disentangle energy from causal structure. Although CPP is widely interpreted as a supramodal accumulation-to-bound signal, its sensor-level spatiotemporal profile may overlap with late attention or motor preparation. Amplitude/slope alone cannot fully disambiguate accumulation from concurrent processes; combining CPP with source analyses, generative decision modeling, or causal perturbations would strengthen this interpretation. We observed contralateral occipital positivity compatible with ACOP, yet its functional meaning remains debated: ACOP often reflects enhancement (not suppression) of visual cortex via cross-modal exogenous orienting and can be sensitive to task demands and cue settings. Thus, ACOP should not be uncritically equated with inevitable boosts to subsequent visual evidence. The sample size and composition (e.g., age/sex/cognitive traits) may constrain generalizability. Symmetric bilateral flashes with fixed-intensity beeps control confounds but reduce ecological validity relative to real-world contexts. Future studies should expand and stratify samples and incorporate more ecologically grounded manipulations while preserving internal validity.

Conclusions

Using a sound-induced flash illusion paradigm, this study integrated trial-level EEG measures with behavioral performance in a hierarchical Bayesian regression framework that jointly modeled accuracy and reaction time, thereby characterizing how sensory congruency, audiovisual rhythm configurations, and momentary neural states shape perceptual decisions. Behaviorally, unimodal and audiovisually congruent conditions yielded faster and more accurate responses, whereas incongruent configurations—particularly the “2 beep–1 flash” condition—produced markedly elevated illusion rates and prolonged reaction times, indicating that causal structure and signal reliability jointly govern cross-modal weighting and conflict costs. Neurally, after accounting for condition effects, higher prestimulus α power was reliably associated with reduced odds of a correct response, auditory N1 amplitude showed a modest trade-off of slightly lower accuracy but faster responses, and CPP slope closely tracked reaction time at the descriptive level; scalp topographies further revealed that correct trials were characterized by stronger N1 and CPP components together with a contralateral occipital positivity compatible with the ACOP, consistent with a cross-modal orienting process that facilitates subsequent evidence accumulation. Taken together, the findings establish a reusable trial-level neurobehavioral integration pipeline linking Bayesian integration, causal inference, and evidence accumulation to EEG and behavior. However, model comparison showed that task conditions accounted for most of the out-of-sample behavioral variance, whereas EEG covariates contributed limited and non-robust additional predictive value under the current linear additive specification. Accordingly, the primary contribution of the neural measures in the present study is mechanistic and interpretive rather than a strong incremental gain in single-trial prediction.

Author contributions

HY, DD, and HQ contributed to conceptualization. HY developed the methodology. HY, DD, and HM developed and implemented the software. HY and DD curated the data. DD, HQ, and HM conducted the investigation. HQ, HM, CW, LC, and BW performed validation. HQ, HM, CW, YZ, and BW conducted the formal analysis. HM, CW, LC, and BW provided supervision. LC and BW acquired funding. HY and DD prepared the visualizations. YZ, LC, and BW managed project administration. LC and B.W. provided resources. H.Y. and D.D. wrote the original draft. HY, DD, HQ, HM, CW, YZ, LC, and BW reviewed and edited the manuscript. All authors reviewed and approved the final manuscript.

Funding

This study was supported by the National Key Research and Development Program of China (Grant No. 2023YFF1203704) and the Top Graduate Innovative Research Program of Beijing Sport University (Grant No. 2025027).

Data availability

The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Ethics approval

Ethical approval was obtained from Beijing Sport University (2024349H) and all procedures conformed to the Declaration of Helsinki.’

Consent to participate

All participants provided written informed consent prior to the experiment and agreed to the use of their anonymized data for publication.

Consent for publication

All participants consented to the publication of anonymized data.

Competing interests

The authors declare no competing interests.