Alpha frequency shapes perceptual sensitivity by modulating optimal phase likelihood

Domain: chemistry
Published: 2026-03-03
Editorial outcome: major_revision

Abstract
Whether alpha frequency oscillations orchestrate the pace of sensory sampling is current matter of debate. Using
EEG, we test this hypothesis by investigating whether pre-stimulus instantaneous alpha frequency accounts for
perceptual sensitivity. Our results support the role of alpha frequency in shaping the accuracy of sensory acquisition.
Spontaneous alpha frequency inter-trial fluctuations emerged as a predictor of perceptual decision-making sensitivity
and accuracy, with higher pace accounting for higher sensory precision - an observation supported across
complementary analytical approaches, including Bayesian statistics and computational modelling. Here, we provide
insights into the neural candidate mechanisms through which alpha frequency relates to perceptual decisions.
Specifically, alpha frequency would determine the extent of phase angles covered within the stimulus timeframe. As a
result, higher alpha frequencies may provide more opportunities for the stimulus to coincide with optimal phases for
perception, thereby increasing the likelihood of accurate sensory processing.

Introduction
Converging evidence supports the concept that the human perceptual machinery functions as a discrete system1 that
samples the continuous stream of incoming sensory information into distinct integration windows2. Similar to other
sampling-based processes, the acquisition rate plays a fundamental role in determining the accuracy and fidelity of
the resultant representation. Specifically, higher sampling frequencies are associated with a more precise
representation of the source signal. Alpha rhythms (7-13Hz) have emerged as a prominent neurobiological candidate
involved in governing the parsing of sensory processing into distinct units3. Consequently, we postulated that higher
alpha oscillation frequencies may be connected to heightened perceptual resolution, thereby enhancing the sensitivity
and fidelity of the visual experience4,5. This hypothesis is grounded on several studies6–8, which demonstrated that
both inter-individual and intra-individual differences in Individual alpha frequency (IAF) during task performance play a
significant role in explaining perceptual outcome and the temporal resolution of the visual system. For example, a
higher pre-stimulus alpha frequency was predictive of higher accuracy in discriminating two flashes in the two-flash
fusion (2FF) task9,10 and individuals with a lower occipital IAF had elongated windows of illusion in a sound-induced
flash illusion (SIFI) paradigm11,12. Additionally, we have recently demonstrated that individuals with a higher resting
IAF require less contrast for accurate detection compared to low IAF participants13. Crucially, the reduction vs.
acceleration of alpha frequency by means of neurostimulation enlarged vs. shrunk the sensory integration window,
influencing susceptibility to experiencing the SIFI11,14. Furthermore, transcranial Alternating Current Stimulation
(tACS) applied at different frequencies within the alpha band can alter participants' perceptual switching rate in a
bistable colour-motion binding stimulus, where higher alpha frequencies resulted in accelerated perceptual switches
by shortening perceptual epochs of active binding15. Recent studies16,17 have further underscored that the direction of
IAF modulations varies depending on whether a task requires sensory segregation or integration, with IAF tending to
increase during segregation tasks and decrease during integration tasks.
Despite this recent converging evidence suggesting a role for alpha frequency in sensory sampling, a recent study by
Buerges and Noppeney18 deviated from the prevailing evidence in the field. Indeed, they found, in a sample of 20
individuals within the general population, that neither inter-individual differences nor inter-trial variations in IAF
account for performance outcomes in both a 2FF task and a yes/no detection task. Furthermore, the authors
highlighted that the link between IAF and perceptual performance found in previous studies could be undermined due
to an unaddressed confounding factor associated with participants' response bias. In addition to the limitations
pinpointed by Buerges and Noppeney, other unresolved issues can be identified in the previous literature findings.
For example, most studies examining IAF-related effects on behaviour employed small sample sizes, which is
especially concerning according to a recent meta-analysis suggesting a minimum of 50 participants necessary to
detect robust effects6. Furthermore, the majority of previous studies investigating this scientific question used the
traditional trial-averaging technique. Consequently, there is a lack of robust approaches that enable the exploration of
the finer trial-by-trial relationship between IAF fluctuations and behavioural outcomes. Additionally, the existing body
of evidence has primarily investigated a potential role for IAF in modulating perceptual temporal resolution rather than
delving into its direct impact on shaping sensory accuracy (but see13,19). Here, we posit that alpha rhythm may serve
as a general neural code underpinning perceptual sampling, influencing both temporal and perceptual resolution. The
underlying process can be understood through the classic 2-flash fusion task. In this task, authors have proposed that
when two flashes of light are presented within the same alpha cycle, they are perceived as a single flash because
they are integrated within the same temporal window. Conversely, when the flashes fall in different alpha cycles, they
are perceived as two distinct events, due to separate sampling moments9,11. Building on this concept, we hypothesize
that the same mechanism governs the relationship between IAF and perceptual resolution. For example, if a visual
stimulus is presented in a way that it spans across two different alpha cycles, it will be sampled twice by the brain,
leading to a clearer and more distinct perception. However, if the stimulus falls within a single alpha cycle, it is
processed only once, potentially reducing perceptual performance.
In addition, existing evidence has not investigated the specific mechanism linking higher IAF to more faithful
representations of reality. Here we test the hypothesis that higher vs. lower IAF allows for higher probability of optimal
alpha phase to catch even short-lasting stimuli, leading to successful sensory processing. Specifically, the hypothesis
is based on compelling evidence showing that the accuracy of decision-making relies on the precise phase of alpha
waves during the peri-stimulus period 20–22. The basic idea we propose here is that the mechanism facilitating a more
accurate response with higher (relative to lower) IAF lies in its capacity to span a broader distribution of phase angles
within the same stimulus presentation timeframe. As a consequence, this heightened phase coverage increases the
likelihood of aligning with optimal phase angles, crucial for the accurate perception of stimuli. This proposition, in turn,
leads to another hypothesis closely intertwined: the ability of phase angles in dictating the accuracy of the response is
expected to be more pronounced in participants or trials in which IAF is lower. In such instances, the reduced
likelihood of falling into optimal phases during stimulus processing makes the phase angles an important determinant
of performance (i.e., if the phase is in non-optimal angles, the likelihood of reaching the optimal ones will be reduced).
On the contrary, when alpha is higher, it aids in covering numerous phase angles during stimulus presentation,
thereby increasing the likelihood of processing stimuli within those optimal phases. Our experimental design was
particularly suited to capture this relationship given the presentation time of our stimuli (i.e., 59 ms), where the phase
position becomes more critical due to the limited time available to cover a wide distribution of phase angles.
In this work, we examine how pre-stimulus instantaneous alpha frequency relates to perceptual sensitivity during
visual detection in a contrast detection paradigm. Using EEG, we recorded brain activity non-invasively from
participants while they reported the presence of grey circles embedded in black and white checkerboard patterns.
Each checkerboard was presented for 59 ms and the contrast of the grey circles was titrated for each individual to
their 70% perceptual accuracy. To test for the role of IAF as a candidate sampling mechanism, we have undertaken a
multi-faceted approach. First, we have employed a big-data approach, analysing our hypothesis in a large sample (n
= 125) of individuals within the general population. Secondly, we supplemented the frequentist analyses with
Bayesian statistics (Bayes Factor) to provide an estimate of the effect's strength and direction. Third, we have
exploited computational models such as the signal detection theory (SDT) framework to disentangle potential IAF
effects related to sensitivity vs. bias. Fourth, we analysed a potential relationship between IAF and decisional
parameters extracted using the drift diffusion model (DDM), the most commonly used computational framework for
modelling human perceptual decision-making23–25. Fifth, we have complemented classic binning methods with more
robust analyses that evaluated the trial-by-trial relationship between IAF and behavior26. Subsequently, we delineated
a candidate mechanism enabling a higher IAF to process stimuli more faithfully, demonstrating the inherent interplay
between alpha frequency and phase. To this end, we employed both bin-based and trial-by-trial analyses using alpha
phase, IAF and their interaction as predictors. We find that moment-to-moment variations in alpha frequency are
associated with differences in accuracy and sensitivity. We also observe that alpha phase relates to perceptual
accuracy, and that this phase–accuracy relationship depends on the prevailing alpha frequency. Finally, we outline a
candidate framework in which higher alpha frequency increases the number of phase angles sampled during stimulus
presentation, thereby increasing the chance that the stimulus coincides with phases more favourable for perception.

Results
We assessed the influence of alpha frequency and alpha phase in dictating the accuracy of perceptual reporting in a
visual detection paradigm. Using EEG, we recorded brain activity non-invasively from 125 participants while they
reported the presence of grey circles embedded in black and white checkerboard patterns (Figure 1).
Each checkerboard was presented for 59 ms and the contrast of the grey circles was titrated for each individual to
their 70% perceptual accuracy. In half of the trials, checkerboards did not contain any grey circle embedded in them
(catch trials). Participants were instructed to indicate via the keyboard the presence (key 'k') or absence (key 'm') of
the grey circles inside the checkerboard. No timeout has been set for the response. After collecting the response, the
screen appeared black for 1.9–2.4 s in the inter-trial interval. On average, participants reached 73.5 % accuracy,
testifying to the accuracy of the titration procedure employed. The average RT for the task was 0.88 ± 0.03 s. Correct
responses had faster RT (0.85 ± 0.03 seconds) compared to the slower RT for incorrect responses (0.99 ± 0.04
seconds; see Figure 1B).
We investigated whether fluctuations in prestimulus alpha frequency, alpha phase and their interaction were linked to
perceptual decision-making. To this end, first we computed the first temporal derivative of the phase angle time series
(see methods), representing the instantaneous frequency of a signal within a band-limited range27. We focused our
analysis on a pool of occipito-parietal electrodes located on the midline and right side (i.e., Oz, POz, O2, PO4, PO8),
selecting, for each subject, the electrode that exhibited the maximum alpha power in the prestimulus window. These
electrodes were chosen because the stimulus was consistently presented on the left. Then, we evaluated, using a
diverse range of statistical methods, whether there was an effect of IAF on perceptual outcome. Secondly, we
assessed whether phase angles could distinguish between correct and incorrect decision and, finally, we evaluated
any interaction between IAF and alpha phase able to shed light of the candidate mechanisms underlying their effect
on sensory precision.

Pre-stimulus IAF is linked to variations in perceptual sensitivity
We investigated the influence of pre-stimulus IAF on perceptual sensitivity and bias by employing a binning analysis.
This analysis involved dividing the trials into three terciles based on pre-stimulus IAF18. We time-collapsed the IAF
data by computing the average values from -800ms to -100ms for each participant. First, we showed strong evidence
(BF = 59.79) for a difference in accuracy between the third IAF tercile (mean accuracy = 0.76, SE = 0.01) compared
to the first IAF tercile (mean accuracy = 0.72, SE = 0.01; t115 = -3.71, p < 0.001; Figure 2A). Subsequently, we
computed the signal detection indices from trials within the first and third terciles. The results demonstrated that trials
in the first tercile were associated with reduced sensitivity (mean d’ = 1.46, SE = 0.07) compared to trials within the
third tercile (mean d’ = 1.65, SE = 0.07; t115 = -3.10, p = 0.003; Figure 2B), indicating moderate evidence for a
difference (BF = 9.32). Conversely, the criterion indices estimated in the first (mean c = 0.44, SE = 0.04) vs. third
terciles (mean c = 0.39, SE = 0.04) were not statistically different (t115 = 1.67, p = 0.098, Figure 2C), providing
inconclusive evidence (BF = 0.39). This finding suggests that trials with higher IAF are associated with higher
perceptual sensitivity and accuracy, while the response bias does not appear to be predicted by IAF. The pattern of
results remained consistent when using two or four IAF bins (See supplementary method 1). Furthermore, we found
no statistical differences in behavioural performance when binning trials based on pre-stimulus power (Figure S2).
Moreover, building on our recent finding that individuals with higher resting IAF require less contrast for accurate
detection compared to those with lower IAF13, we further demonstrate that this relationship extends to average pre-
stimulus alpha levels. Specifically, individuals with higher pre-stimulus IAF consistently required lower threshold
contrast to achieve target performance levels (Pearson = -0.21, p = 0.021; Pearson skipped = -0.21, CI = [-0.37, -
0.03]), although the BF indicated only limited evidence for this association (BF = 1.04).

Time-resolved binning analysis revealed that the IAF effect on sensitivity is broadly extended over the pre-
stimulus period.
Having examined the impact of pre-stimulus IAF on perceptual sensitivity using a binning approach, we then
extended our analysis to explore the time dynamics of the effect similar to 18. For this purpose, we re-sorted the trials
into three bins based on the prestimulus instantaneous frequency. However, this time the procedure was iterated for
each pre-stimulus time point separately. Subsequently, we entered sensitivity (d’), criterion (c), and accuracy indices
extracted in each time point into a cluster-based statistic to examine the time-resolved nature of the effect (See
methods). The results indicated that, both accuracy (Figure 2A) and sensitivity (Figure 2B), demonstrated statistically
significant differences between the types of trials, further corroborating the findings highlighted in the previously
conducted time-averaged analysis. Specifically, accuracy was higher in the third than in first IAF terciles within two
temporal clusters: one ranging from -800ms to -390ms (p cluster < 0.001) and another from -310ms to -135ms (p
cluster = 0.009). Furthermore, we observed increased d’ in the third IAF tercile compared to the first IAF tercile in the
time period ranging from -670 ms to -350 ms (p < 0.001) and in the period ranging from -310 ms to -160 ms (p =
0.015). Similarly, when extending the analysis to a broader prestimulus window (–2500 to –100 ms), we found that
IAF predicted sensitivity and accuracy already ~1.5 s before stimulus onset (Figure S1). However, these long-range
effects became stronger and more consistent within the immediate prestimulus interval selected for the main analysis
(–800 to –100 ms). Finally, this time-resolved analysis revealed an effect on the criterion that was not evident when
averaging values across the temporal dimension (Figure 2C). Specifically, in the interval from -635 to -490 ms (p =
0.036), there was a reduction in decisional bias, as indicated by the criterion moving closer to the 0 value, in trials
characterized by higher IAF compared to those with lower IAF. This suggests that more accurate sampling of the
external signal may have indirectly contributed to a reduction in perceptual bias. This is confirmed by a negative
association between perceptual accuracy and bias, with higher sensitivity associated with lower bias (Pearson =
-0.299, p = 0.001; BF = 15.02; Pearson skipped = -0.24, CI = [-0.39 ; -0.06]).

IAF is higher in correct vs. incorrect trials.
Finally, we explored whether the frequency of alpha oscillations before stimulus presentation could also shed light on
the distinction between correct and erroneous responses. Thus, we compared IAF preceding correct vs incorrect
response. The analysis (Figure 3A) yielded a significant effect (t115 = 4.13, p < 0.001), indicating very strong evidence
for a difference (BF = 245.50). Specifically, higher alpha frequency was associated with a correct response (M =
11.35 Hz, SE = 0.1 Hz, 25th percentile = 10.60, 75th percentile = 11.97), while lower alpha frequency was linked to a
higher chance of errors (M = 11.30 Hz, SE = 0.1 Hz, 25th percentile = 10.61, 75th percentile = 11.97). Moreover, we
showed (see Supplementary Method 2) that IAF does not significantly predict the actual choice (i.e., reporting
"present" vs. "absent"). In addition, we demonstrated that there is no statistical relationship between detection
accuracy and alpha power, as alpha power does not differ between correct and incorrect trials (Figure S3).

Trial-by-trial fluctuations in IAF predict the accuracy of perceptual report.
The statistical methods presented so far go in line with the traditional cognitive electrophysiology analyses that rely
on data averaging over trials to maximize the signal-to-noise ratio. In recent times, there has been a growing trend
among researchers to employ single-trial analyses to link EEG activity with metrics of interest defined at the single-
trial level. This approach enables the exploration of the trial-by-trial relationship between brain activity and
behaviour/perception, which is not observable when using traditional averaging-based methods. To this end, we
employed a multiple regression approach using a least-squares solution28 to examine a link between single-trial
estimates of IAF and single-trial accuracy. The conducted analysis demonstrated a positive association between trial-
by-trial variations in IAF and perceptual accuracy (t115 = 4.42, p < 0.001; Figure S3B). This finding implies that trials in
which IAF was higher vs. lower were associated to accurate responses (mean slope = 0.43, SE = 0.1). A Bayesian
paired t-test yielded BF = 766.38, providing very strong evidence for this association. Crucially, we ruled out the
hypothesis that the relationship between IAF and perceptual sensitivity is merely due to spurious changes in alpha
frequency reflecting variations in the power of different alpha generators. Our source analysis demonstrates that all
the observed effects are predicted by the IAF extracted from occipital sources, further confirming the specific role of
occipital areas in accounting for sensory precision29,30 (See supplementary Method 3). While our source-level
reconstruction corroborated the sensor-space findings, the intrinsic limitations of EEG spatial resolution and potential
source mixing constrain the anatomical specificity of these effects.
Moreover, we found no statistical relationship between IAF and RT on a trial-by-trial basis (Supplementary method 4,
mean slope = 0.05, SE = 0.10, t115 = 0.52, p = 0.603), providing strong evidence for the absence of an association (BF
= 0.12). This demonstrates that the impact of IAF on accuracy is unlikely to reflect a speed/accuracy trade-off. If such
trade-off were present, higher IAF would be expected to correlate with slower reaction times, as individuals might
prioritize accuracy over speed; however, the strong evidence for no association between IAF and RT supports the
interpretation that IAF directly enhances response accuracy without affecting response speed. Finally, no statistical
relationship was observed between fluctuations in alpha power and perceptual sensitivity (See figure S4).

Trial-by-trial fluctuations in IAF predict the accuracy of sensory accumulation.
We expanded the previous trial-by-trial analysis, analysing the impact that IAF fluctuations had on DDM parameters.
Unlike SDT, the DDM allows for simultaneous consideration of choice accuracy and full response time distributions,
enabling a better tracing of decision dynamics. From the fitted DDM model, we extracted the drift rate (v),
representing the rate of evidence accumulation, the starting point (z), capturing response bias, the boundary
separation (a), indicating the amount of information required to decide, and the non-decisional time (t), accounting for
the time taken for processes other than perceptual decision-making. Using regression coefficients, we examined the
relationship between trial-to-trial variations in IAF and the DDM parameters. Our analysis revealed a significant
positive association between increased IAF and drift rate parameters (posterior mean IAF ~ v coefficient = 0.049, HDI
= [0.026; 0.073], Figure 3C; q < 0.001). This finding indicates that in trials with relatively higher IAF sensory
information are accumulated in a faster and more precise fashion. Interestingly, we found that IAF was unrelated to
the starting point and the non-decisional time parameter (posterior mean IAF ~ z coefficient = 0.001, HDI = [-0.003; 0.005],
posterior mean IAF ~ t coefficient = -0.001, HDI = [-0.004; 0.002], all q > 0.34), while there was an association
with the boundary separation (posterior mean IAF ~ a coefficient = 0.013, HDI = [-0.001; 0.026], q = 0.031), indicating
that in trials with lower IAF there was a tendency to accumulate less information before deciding. These results
provide valuable insights into the relationship between pre-stimulus IAF and the dynamic process of evidence
accumulation during decision-making.

Instantaneous alpha phase shapes perceptual accuracy
In order to investigate the potential role of IAF phase in explaining the relationship between IAF and perceptual
accuracy, we first investigated whether alpha phase could play an impact on perceptual accuracy by employing a
binning analysis. Therefore, we grouped trials of similar phase together into two bins separately for each participant
and timepoints in the -800 ms to 200 ms window. Each phase bin comprised 180° phase. The first bin contained all
trials with phases between 0° and 180°, while the second one all trials between 180° and 360°. Subsequently, we
entered accuracy indices calculated in the two bins and extracted in each time point into a cluster-based statistic to
investigate the presence of clustered time points in which phase bins impact accuracy. The results indicated that
accuracy exhibited a statistically significant difference between the two bins [time period ranging from -49 ms to -21
ms (p = 0.007), from 5 ms to 25 ms (p = 0.014), from 52 ms to 71 ms (p = 0.027), from 99 ms to 130 ms (p = 0.008)
and from 157 ms to 177 ms (p = 0.046), Figure S6].

Alpha phase effects on perceptual sensitivity are moderated by IAF.
Having demonstrated that stochastic fluctuations in alpha phase are able to shape perceptual accuracy, we
investigated whether IAF could play a moderator role on this relationship. The logic underlying this choice is that an
individual who exhibits a higher alpha frequency covers more phase angles in the same amount of time compared to
an individual with a lower alpha frequency. Therefore, the positioning of the phase at specific time points should have
a greater impact in explaining decision outcomes in individuals with lower (compared to higher) IAF, as they will less
likely show a transition from the favourable to unfavourable bin at critical time points, i.e. during stimulus presentation.
To investigate this hypothesis, we replicated the above-mentioned analysis separately in individuals that exhibited
lower vs. higher pre-stimulus IAF. We found that, in lower prestimulus IAF individuals, the two phase bins were
associated with different levels of accuracy [Figure 4A; time periods ranging from -685 ms to -666 ms (p = 0.048), -
639 ms to -619 ms (p = 0.029), -588 ms to -564 ms (p = 0.030), 5 ms to 17 ms (p = 0.045) and 52 ms to 67 ms (p =
0.046)]. In contrast, no statistically significant phase bin differences were observed in individuals showing higher
prestimulus IAF (Figure 4B). Indeed, in this group, no cluster emerged in which alpha phase was able to intercept
perceptual accuracy. A complementary within-subject, cluster-based permutation test on the interaction contrast
confirmed that this difference reflects a genuine IAF × phase interaction. Specifically, the modulatory effect of alpha
phase on accuracy was significantly stronger in trials with low instantaneous IAF compared to high IAF (Figure S7).

Trial-by-trial fluctuations in alpha phase predict perceptual sensitivity only in lower IAF trials.
In order to corroborate the previous results that assessed the impact of alpha phase in groups of individuals
exhibiting low vs. high prestimulus alpha frequency, we conducted trial-by-trial analyses evaluating, across the entire
sample, whether there was a significant interaction between IAF and alpha phase. This analysis further demonstrated
a positive association between trial-by-trial variations in IAF and perceptual accuracy (mean slope = 0.42, SE = 0.09;
t115 = 4.42, p < 0.001) , providing very strong evidence for an association (BF = 705.60). Moreover, we found that
alpha phase significantly and strongly (BF = 14.64) impacts perceptual accuracy (mean slope = 0.29, SE = 0.09; t115 =
3.26, p = 0.002). Crucially, we found the presence of a significant negative interaction term between IAF and alpha
phase (mean slope = - 0.20, SE = 0.09; t115 = -2.28, p = 0.024), although the corresponding BF indicates limited
evidence for this effect (BF = 1.24). We replicate this pattern of results employing a DDM analysis that included IAF,
alpha phase and the IAF*phase interaction as predictors (Figure 4A). The analysis demonstrated that IAF (posterior
mean IAF ~ v coefficient= 0.071, HDI = [0.037; 0.010], q < 0.001) and instantaneous alpha phase (posterior mean
Alpha phase ~ v coefficient= 0.064, HDI = [0.017; 0.110], q = 0.002) significantly impacted the drift rate parameters.
Again, we found a significant negative interaction between IAF and alpha phase (posterior mean Alpha phase* IAF ~
v coefficient= -0.041, HDI = [-0.087; 0.001], q = 0.043). Crucially, none of these predictors, nor their interactions,
significantly impacted the other DDM parameters (all q > 0.072), except for a significant relationship between alpha
phase and the starting point parameter (posterior mean Alpha phase ~ z coefficient= 0.009, HDI = [0.001; 0.017], q =
0.019), where peak values were associated with a more neutral bias. This finding pointed to a reduced vs. increased
influence of alpha phase in dictating perceptual sensitivity when the prestimulus alpha was higher vs. lower.

Correct vs. incorrect decisions are associated with a different phase angle only in the low IAF group
To further validate our findings beyond the bin-based approach on phase angles, we employed a reverse data-driven
method to examine whether correct versus incorrect responses were associated with distinct phase angles. Watson-
Williams test revealed that, while no significant clusters emerged in either the whole sample or the high IAF group
(Figure 5B), the low IAF group showed a notable cluster spanning from approximately -800 ms to -300 ms (Figure
5A). Within this interval, the phase angles associated with correct and incorrect responses significantly differed (p
cluster = 0.041).

Alpha-phase clustering increases for correct responses only in low IAF individuals.
To further test whether the correctness of the perceptual report was related to alpha phase, we computed intertrial
phase clustering (ITPC) separately for each time point in the -800 ms to 200 ms window. In the whole sample, we
observed a distinction in alpha-clustering between correct and incorrect responses (p = 0.011). However, we aimed to
investigate whether this effect varied as a function of IAF. Given that higher IAF may reduce the impact of phase, we
hypothesized that accounting for IAF could reveal a more nuanced pattern, potentially diversifying the effect based on
the speed of alpha. To address this, we repeated the analysis separately for individuals with low IAF and high IAF.
The results revealed a notable difference: in the high IAF group, there was no significant disparity in ITPC between
correct and incorrect decisions. However, in the low IAF group, ITPC exhibited a significant differentiation (p = 0.045),
suggesting a more pronounced influence of alpha phase on perceptual accuracy in this subgroup (Figure 6A). To
further reinforce the finding, we also computed ITPC across frequencies ranging between 2 and 50 Hz. This analysis
aimed to ascertain that the previously observed effect was specifically tied to the alpha-band. The results of the
cluster-based analysis (Figure 6B) revealed three findings: 1) there was an increase in ITPC in correct vs incorrect
trials, 2) this effect was group-specific, holding true only for individuals with low IAF (p = 0.038) and 3) this effect was
confined to the alpha-low beta band. These findings also held when we directly contrasted the correct–incorrect ITPC
difference between groups: the increase in ITPC for correct versus incorrect perception was greater in Low-IAF than
in High-IAF participants within the high-alpha/low-beta range (Figure S8, p = 0.043).

Methods
Participants
125 individuals (70 female; age range 18–35) participate to the study. All participants signed a written informed
consent prior to taking part in the study, which was conducted in accordance with the Declaration of Helsinki. The
protocol was approved by the Bioethics Committee of the University of Bologna (protocol code 201723, approved on
26 August 2021). All participants had no neurocognitive or psychiatric disorders. The identified results were
aggregated across both genders. Gender was determined based on participants' self-reports. We did not conduct
separate analyses by gender because previous literature did not report a gender effect on the neural markers (i.e.,
IAF), nor on the detection task we focused on. No statistical method was used to predetermine sample size; however,
we aimed to recruit a relatively large sample to ensure stable estimates of inter-individual variability and to increase
the reliability of trial-level analyses.

Experimental procedure
Stimuli were presented on a 18' CRT display (Cathode Ray Tube, CRT, display resolution of 1280 x 1024 pixels,
refresh rate 85 Hz) at a distance of ~ 57 cm in a dimly lit room. Participants sat in a comfortable chair in front of the
monitor. The stimuli were generated and presented using MATLAB (version 2016, The MathWorks Inc., Natick, MA)
and the Psychophysics toolbox. Visual stimuli were checkerboards appearing on the lower left visual field. The
checkerboards presented could contain grey circles within each of the cells (target) or not (catch trials). Stimuli had
spatial frequency of 5.16 cycles/degree, and they were presented only in the lower part of the screen at 4.1°/3.7°
eccentricity (horizontal/vertical). Each black and white checkerboard was flashed for 59 ms. Participants were
instructed to indicate via the keyboard the presence (key 'k') or absence (key 'm') of the grey circles inside the
checkerboard. Following a training phase, each participant underwent a staircase procedure to determine the
contrast level of the grey circles needed to achieve 70% detection accuracy. This procedure was conducted with an
equal number of target-present and target-absent trials. Specifically, participants completed blocks consisting of five
trials with targets present and five trials with targets absent. After each block, the participant's accuracy was
assessed. If the accuracy was 50% or lower, the grey contrast was increased by 4 points. If the accuracy was below
70%, the contrast was increased by 1 point. Conversely, if the accuracy was 90% or higher, the contrast was
decreased by 4 points, and if it was above 70%, it was decreased by 1 point. This procedure was repeated 25 times.
The stimuli used included checkerboards with grey circles embedded, with the lowest contrast being red, green, and
blue (RGB) values of 15/240 (where the first value refers to the contrast of the circles against the white squares and
the second value against the black squares) and the highest contrast being 75/180. The initial contrast for threshold
estimation was set between these two extremes (RGB value: 45/210). The detection threshold was estimated based
on the final three blocks. Including target-absent trials was needed for obtaining a bias-free measure, avoiding
confounding effects related to participants' decision criteria. Without target-absent trials, it would be difficult to
determine if variations in threshold values were due to actual perceptual ability or simply the number of hit rates,
despite equal sensitivity when controlling for false alarms (Green & Swets, 1966). This approach aligns with the
Signal Detection Theory (SDT) principle, where d' is calculated by considering both hit rates and false alarms to
derive a bias-free measure of performance. The bias-free approach means that, assuming equal sensitivity, the
choice of a liberal or conservative decision criterion does not affect the estimation of the contrast threshold, unlike
classical staircase methods (e.g., two-down, one-up methods; Levitt, 1971). Notably, this threshold estimation
method was first implemented in Tarasi et al.24 and in Tarasi et al.25. The second phase comprised 180 trials (Figure
1). Each trial started with the appearance of a neutral cue presented at the centre of the screen. The cue was
presented for 1 s followed by a fixation dot. After a variable delay of 1.2–1.5 s a checkerboard containing (or not) grey
circles at the titrated contrast within it appeared at the bottom left of the monitor for 59 ms. After collecting the
response, the screen appeared black for 1.9–2.4 s in the inter-trial interval. We opted to present the stimulus in only
one hemifield to prevent spontaneous fluctuations in attention between the two hemifields in the prestimulus period
from interfering with the results. Participants had to determine the presence or absence of the grey circles within the
checkerboard and press the button associated with their choice. No timeout has been set for the response.

EEG Acquisition and Analysis
We used a Brain Products actiCHamp EEG amplifier with 64 channels, configured according to the international 10–
10 system. EEG was measured with respect to FCz electrodes, and all impedances were kept below 10 kΩ. EEG
signals were acquired at a rate of 1000 Hz. EEG was processed offline with custom MATLAB scripts (version
R2021a) and with the EEGLAB toolbox68. The EEG recording was filtered in the 0.5–100 Hz band and a notch-filter at
50 Hz was applied. The signals were visually inspected, and noisy channels were spherically interpolated. Then, trials
containing excessive noise, muscle or ocular artefacts was discarded. An average of 149 trials per subject passed
this stage. Next the recording was re-referenced to the average of all electrodes, and the Independent Component
Analysis (ICA) was applied. On average, 6.4 ± 0.38 components were removed, primarily those associated with eye
movements, blinks, muscle artifacts, and noisy electrodes. The signal was downsampled to 256 Hz.

Extract instantaneous alpha frequency and phase.
The instantaneous alpha frequency was extracted following the methodology described in 69 and further extended in
70. First, we have taken the derivative of the time-series data71 in order to remove the aperiodic component in the
data, which could introduce bias in frequency estimation70 (the same IAF–accuracy patterns emerged using
alternative aperiodic removal method, supplementary method 9). Next, for each participant, we selected the occipital
electrode signal (Oz, O2, POz, PO4, PO8) showing the highest alpha power (7-13 Hz) in the pre-stimulus time (from -
800 to -100 ms) computing using the EEGlab spectopo function68. We focused the analysis on the occipito-parietal
electrodes on the midline and right side, given that the stimulus was always presented on the left. We then applied
peak-detection algorithms developed by72 to check the presence of a peak in the chosen electrode. In cases where
no peak could be identified, we selected subsequent electrodes displaying higher alpha power. Participants without
detectable peaks in any of the electrodes considered were excluded from further analysis, as the absence of peak in
the power spectrum could bias frequency sliding to non-frequency shift features70. After these preliminary steps, the
participant count reduced to 116, with 9 individuals removed. The distribution of selected electrodes was as follows:
Oz (11%), POz (25%), O2 (11%), PO4 (22%), and PO8 (31%). Subsequently, we filtered the signal with a zero-
phase, plateau-shaped, band-pass filter with 15% transition zones (MATLAB function filtfilt.m) centred on the
individual oscillation peak extracted in the previous stage (filter range from IAF-2 Hz to IAF+2 Hz. Then, a Hilbert
transform was applied to the filtered data (MATLAB function hilbert.m) to extract phase angle time series. The
instantaneous frequency was obtained by calculating the temporal derivative of the instantaneous phase. We
reduced the noise present in the instantaneous frequency by median filtering ten times the instantaneous frequency
using ten different time windows ranging from 10 to 400 ms. Additionally, we adopted complementary approaches to
examine the role of IAF without relying on the instantaneous frequency method, instead using FFT-based frequency
estimation. First, we computed the power spectrum from the pre-stimulus signal (-800 to -100 ms), matching the time
window used for the instantaneous frequency approach. We then applied Corcoran’s algorithm to the single-trial
power spectra to extract the single-trial IAF. This peak detection method, performed directly on the frequency
spectrum, helps overcome potential limitations of the instantaneous frequency approach70,73. We also implemented a
Gaussian fitting approach: for each trial, we fitted a Gaussian to capture the spectral peak within the alpha band. The
centre frequency of this Gaussian was taken as the IAF. These methods together confirm the relationship between
IAF and perceptual performance.

IAF binning analysis across pre-stimulus time.
To evaluate the effect of pre-stimulus IAF in directing decisional outcome, we employed a binning analysis using an
approach similar to the one employed by18. Specifically, for each individual, trials were sorted into three terciles
according to the instantaneous frequency (i.e., first-IAF, second-IAF and third-IAF trials), independently for each pre-
stimulus time point. Then, the Signal Detection Theory (SDT) parameters indexing perceptual sensitivity (d’) and bias
(c) were computed separately for the first and the third terciles. d’ reflects standardized measure of discrimination
abilities between the signal and the noise and was calculated as 𝑑=z(HR) − z(FA), where z represents the z-scores
of Hit rate (i.e., HR, the probability of correct response on target trials) and false alarms (i.e., FA, the probability of
incorrect response on catch trials). Conversely, c quantifies a subject’s decision criterion (i.e., c value over 0 implies
the presence of a bias to report the absence of the target) and was calculated as 𝑐= − 𝑧(𝐻𝑅)+𝑧(𝐹𝐴) / 2. To ensure a proper
estimation of d’ and c, we corrected the extreme rates of hit and false alarm. Rates of 0 were replaced with 0.5/n, and
rates of 1 were replaced with (n-0.5)/n, where n is the number of signal or noise trials74 To statistical assess whether the
SDT parameters differed as a function of the bins considered, we entered d’ and c indices extracted across pre-
stimulus time into a cluster-based statistic. The real cluster size was obtained by summing t-values (exceeding an
alpha threshold of 0.05) of two-sided paired t-tests over adjacent time points. The null hypothesis distribution of the
cluster sizes was drawn by randomly shuffling (n = 1000), across participants, the association between SDT
parameters (i.e., d’ and c) and bins (first vs. third IAF terciles) and choosing the maximum cluster value (i.e., summed
t-values in significant adjacent points) per each randomization. From this dummy distribution, we have extracted the
cluster size corresponding to the 95% percentile and we compared the actual cluster size with this statistical
threshold. The same statistical procedure was repeated using accuracy as a dependent variable. We calculated the
p-value for each cluster by determining the position of the observed clusters within the distribution of clusters
generated from the permutation procedure. This approach was applied consistently across all permutation-based
analyses. To ensure the validity of our findings, we conducted additional analyses to verify the consistency of our
results across different bin configurations. Initially, we expanded our primary analysis to include the second IAF bin
and performed ANOVA on accuracy, d', and criterion measures across the three IAF bins. This approach allowed us
to assess whether the differences observed remained significant when including the second bin, rather than just
focusing on comparisons between the first and third bins. In addition, we conducted regression analyses to assess
whether there was a monotonic trend in accuracy and d' across the three bins, which would suggest a gradual
increase in performance with higher IAF (Supplementary Method 1). To further validate our findings, we also explored
the impact of different bin configurations. Specifically, we divided the trials using two and four bins. For the two-bin
configuration, trials were divided into above-median and below-median IAF bins. A paired t-test was used to compare
accuracy and d' between these two trials type. For the four-bin configuration, we divided trials into quartiles and
assessed performance measures across these quartiles.

IAF binning analysis collapsing pre-stimulus time.
In an additional analysis, we averaged IAF across pre-stimulus time (-800 to -100 ms) and we then binned trials into
terciles. This procedure was conducted separately for each individual. Then, we entered d’, c and accuracy extracted
in the first vs. third IAF terciles into both a classic paired t-test and Bayesian paired t-tests to assess whether a
significant difference in SDT parameters emerged depending on the considered bins. The same analysis was then
repeated using the IAF estimated from the single-trial Corcoran-derived measures. Bayesian analysis utilized the
default prior settings: Prior Distribution for Effect Size: Cauchy, with a Scale Parameter of 0.707. We used these
parameters for all the Bayesian test-t performed. To assess the robustness of these results, we repeated Bayesian
test-t using a more conservative prior (scale = 0.5), which yielded Bayes factors that were highly similar to those
obtained with the default prior. We further examined whether RT varied between the first and third terciles of trials
and how these variations related to response accuracy (Supplementary Method 2). Specifically, using paired t-test,
we compared RTs in the first tercile with those in the third tercile, both overall and separated by response accuracy.

IAF binning analysis: correct vs error trials.
To further validate the previous findings, we employed a different approach including IAF as the dependent variable.
In this analysis, we categorized the trials based on their accuracy (correct vs. incorrect trials) and investigated
whether a difference in instantaneous frequency emerged based on the correctness of the response. Correct trials
encompassed both correctly detected target trials (hits) and correctly detected catch trials (correct rejections). On the
other hand, error trials included missing of the target in target-presence trials (miss) and target reporting in catch trials
(false alarms). We then calculated the mean pre-stimulus IAF preceding correct and incorrect trials for each
participant. To assess the statistical difference between IAF for correct vs. incorrect trials, we utilized both classic
paired t-test and Bayesian paired t-test. These complementary statistical approaches allowed us to thoroughly
examine the potential variations in IAF in relation to the accuracy of the response. In order to check that different
trials count between correct and incorrect response inflated the difference between IAF extracted from correct and
incorrect responses, we used a subsampling procedure to match the number of correct and incorrect trials.
Specifically, for each participant, we shuffled the trials with correct responses and matched them with IAF values
count from the incorrect responses. For each permutation, we calculated the average IAF for the correct responses.
This permutation procedure was repeated 1000 times to ensure reliability. We then averaged the surrogate IAF
values obtained from these permutations. Finally, we compared the average IAF values for correct versus incorrect
responses. This comparison corroborated the initial analysis presented in the manuscript, where all trials were
included as correct trials were associated with higher IAF values (IAFcorrect = 11.35 ± 0.01) compared to incorrect trials
(IAFincorrect = 11.30 ± 0.01, t115 = 4.12, p < 0.01). Finally, we employed classical FFT-based approaches to confirm the
presence of IAF differences between correct and incorrect responses. First, we computed the power spectrum
separately for correct and incorrect trials by pooling the pre-stimulus signal within each condition. This method avoids
estimating IAF at the single-trial level, ensuring a more consistent assessment of the frequency spectrum. After
computing the individual power spectra for correct and incorrect responses, we identified the alpha peak using two
approaches. The first, a more traditional method, involved visually detecting the alpha peak, which was not reported if
no clear peak was present. In parallel, we applied a similar analysis using Corcoran’s algorithm, providing a bias-free,
automated, and data-driven estimation of the alpha peak. Finally, we compared whether IAF differed significantly
between correct and incorrect trials.

IAF single-trial regression: Accuracy
To relate single-trial estimates of instantaneous frequency to single-trial accuracy, we used a non-parametric multiple
regression approach similar to that described in 26,28. Specifically, a regression coefficients describing the relationship
between prestimulus IAF and accuracy were estimated according to the following linear model β=(ΧTΧ)−1 ΧTD where Χ represent a design matrix containing one column for the intercept (all 1s), one column of IAF
(each row represented the average IAF observed in the pre-stimulus time in that particular trial), whereas D is the
vector of accuracy scores (dummy coded as 1 or 0). The resulting β coefficients describe the contributions of IAF in
explaining accuracy. First, we averaged the IAF across the pre-stimulus window (-800 to -100 ms) and then applied
z-scoring to reduce the impact of outlier values, and β was estimated using a least-squares solution. β coefficients
were then converted into a z-score relative to a subject-specific null hypothesis distribution obtained by repeatedly
shuffling (n = 2000) the mapping between the accuracy values and the IAF values. The coefficient associated with
the true data mapping was subsequently transformed into a z-statistic, relative to the mean and standard deviation of
the permuted data. According to this approach, it is possible to weight the variability (i.e., noise) present at the
subject-level into the subsequent group-level analyses28. Because the null hypothesis claims that there is no
relationship between IAF and accuracy (i.e., β = 0), we conducted a test-t against 0 to assess the statistical
significance of the observed β coefficients. In order to get information on the strength of the effect, we also investigate
this effect using a Bayesian test-t. We then tested whether the results of this analysis were confirmed when using IAF
estimates derived from an FFT-based approach, such as the one proposed by Corcoran et al. 72, instead of single-
trial IAF estimates obtained through the instantaneous frequency analysis. To corroborate the results, we examine
the relationship between IAF and task accuracy by means of a generalized linear mixed model (GLMM,
supplementary method 10). The dependent variable was task accuracy, coded as a binary variable (0 or 1). The
normalized alpha frequency was used as a continuous predictor. To account for inter-individual variability,
participants were included in the model as a random effect. The specific formula used for the model was: accuracy~
IAF+(1|Subject). MATLAB's fitglme function was used to fit the model to the data, specifying a binomial distribution
and a logit link to reflect the binary nature of the accuracy data. We also analysis whether fluctuations in RT could be
predicted by trial-by-trial IAF modulations. To test this, we repeated the very same regression approach described
above, including RT instead of accuracy as a dependent variable.

IAF single-trial regression: Drift Diffusion Model
We fit participants' decision using the drift diffusion model (DDM), the most widely used computational model of two-
alternative decision-making tasks23,24. The DDM formulation we employed allows for trial-by-trial analysis, enabling us
to examine the influence of individual trial characteristics on decision-making—something not feasible with SDT's
reliance on aggregated data. Additionally, by using both DDM and SDT, we achieve a form of conceptual replication,
strengthening our findings across different analytical frameworks. We used the HDDM package provided by75. Here,
Bayesian inference through Markov chain Monte Carlo (MCMC) sampling is utilized to approximate posterior
distributions for model parameters at both the individual and group levels. This approach is advantageous as it
effectively manages the uncertainty associated with parameter estimates, allowing for robust and precise parameter
estimation. The priors for each parameter in HDDM are informed by a pool of 23 studies, which report the best-fitting
parameters of the Drift Diffusion Model across various decision-making tasks76. By grounding these priors in empirical
data, HDDM ensures that initial parameter assumptions are reliable and informed by a substantial body of existing
research (see the supplement by75 for a visual representations of these priors). Using this package, we estimated
regression coefficients75 to determine the relationship between trial-to-trial variations in IAF and DDM parameters v
(the rate of evidence accumulation, the so-called drift rate), z (the starting point of the accumulation process), a (the
boundary separation between the two decisional thresholds) and t (non-decision time). The resulting coefficients
describe the contribution that trial-by-trial fluctuations in IAF play in modulating the DDM parameters. IAF values were
z-scored to mitigate the influence of outlying trials. We initialized HDDM to draw 5000 posterior samples with the first
500 samples discarded as burn-in. We removed trials with reaction times exceeding 4 seconds and falling below 100
ms before fitting procedure. We inspected traces of model parameters and their autocorrelation to ensure that the
models had properly converged. We examined the overlap of the posterior distributions for the estimated parameters
with zeros, defining significance as less than 5% overlap. Because these are comparisons of Bayesian posterior
distributions, we report the HDDM outcomes as q- rather than p-values.

Assessing the specificity of the relationship between IAF and Perceptual Accuracy relative to Alpha Power
To ensure that the effects observed in relation to IAF were not confounded by oscillatory power, we conducted
several analyses (See Supplementary Figure S1-S5). First, we extracted time-frequency oscillatory amplitude maps
from the aperiodic-free EEG signal. All time–frequency analyses were performed at the subject-specific electrode
used for IAF estimation. Time-frequency decompositions were performed using complex Morlet wavelets for 50
linearly spaced frequencies ranging from 2 Hz to 50 Hz. Wavelets with 3 cycles at the lowest frequency and 11 cycles
at the highest frequency were applied. After obtaining the absolute value of the resultant analytic signal, we
conducted a cluster-based permutation test (1000 permutations, two-sided, p < 0.05) to evaluate whether oscillatory
amplitude varied based on response accuracy (correct vs. incorrect trials) in the prestimulus period (-800 ms, -100
ms). Amplitude was baseline-normalized in dB (-3200ms, -2800ms relative to stimulus onset) for correct and incorrect
trials. Clusters were defined as contiguous suprathreshold samples (p < 0.05) in the 2D time–frequency space,
observed clusters were considered significant if their size exceeded the 95th percentile of the null distribution of
maximum cluster sizes. Next, we extracted trial-by-trial baseline-normalized amplitude from the time-frequency
analysis, averaging it within the -800 ms to -100 ms window, at the frequency bin closest to each participant's
previously determined IAF. We then performed regression analyses, incorporating both trial-by-trial fluctuation in IAF
and alpha power as predictors of perceptual accuracy. Both variables were z-scored within-subject. This analysis
allowed us to determine whether the influence of IAF on performance was independent of alpha power. We also
replicated a bin-based analysis by dividing trials into terciles based on the previously extracted baseline-normalized
trial-by-trial alpha amplitude (averaged over −800 to −100 ms). We compared signal detection theory metrics across
the lowest and highest terciles to confirm that the effects observed with IAF did not hold when considering variations
in alpha power. Lastly, using the same time–frequency parameters and baseline normalization as above, we
compared prestimulus power maps between low and high IAF groups via a cluster-based permutation test (1000
permutations), randomly reassigning participants to groups on each permutation while preserving group sizes.
Clusters were defined as contiguous suprathreshold samples (p < 0.05) in the 2D time–frequency space, observed
clusters were considered significant if their size exceeded the 95th percentile of the null distribution of maximum
cluster sizes.

Alpha phase - binning analysis
To evaluate the effect of alpha phase in directing decisional outcome, we employed a binning analysis (for a similar
approach 77–79). Specifically, for each individual, trials were sorted into two bins according to the instantaneous phase
(i.e., 0°-180° vs. 180°-360°), independently for each time point (-800ms to 200ms). We choose to use two bins in
order to have a sufficient number of trials on which perform the subsequent analysis. To statistical assess whether
the perceptual reports differed as a function of the bins considered, we entered accuracy index extracted across
selected time window into a cluster-based statistic. The real cluster size was obtained by summing t-values
(exceeding an alpha threshold of 0.05) of two-sided paired t-tests over adjacent time points. The null hypothesis
distribution of the cluster sizes was drawn by randomly shuffling (n = 1000), across participants, the association
between accuracy and bins and choosing the maximum cluster value (i.e., summed t-values in significant adjacent
points) per each randomization. From this dummy distribution, we have extracted the cluster size corresponding to
the 95% percentile and we compared the actual cluster size with this statistical threshold. In a follow-up analysis, we
have investigated whether the weight of alpha phase bin in directing the accuracy of the report could be different as a
function of IAF. To this end, we portioned our original sample of 116 participants in terciles based on the mean pre-
stimulus IAF they showed, selecting the first (i.e., low IAF individuals, n = 39) and third (i.e., high IAF individuals, n =
39) terciles. Finally, we compute the above-mentioned analysis separately in the two extracted groups. The two
groups categorized based on IAF did not show differences in oscillatory amplitude, either in alpha or in other
frequency bands (Figure S5).

Alpha phase - single-trial regression
To relate single-trial estimates of instantaneous phase to single-trial accuracy as well as to investigate the interaction
between instantaneous alpha phase and instantaneous alpha speed, we used a non-parametric multiple regression
approach similar to the one used above. Specifically, regression coefficients describing the relationship between 1)
IAF and accuracy, 2) alpha phase and accuracy and 3) the interaction between IAF and alpha phase were estimated
according to the following linear model β=(ΧTΧ)−1 ΧTD. Χ represent a design matrix containing one column for the
intercept (all 1s), one column of IAF (each row represented the average IAF observed in the pre-stimulus time in that
particular trial), one column of phase (dummy coded as 1, if in that trial the phase fell within the 0°-180° space, or 0, if
in that trial the phase fell within the 180°-360° space) and one column representing the interaction between IAF and
alpha phase. D is the vector of accuracy scores (dummy coded as 1 or 0). The resulting β coefficients describe the
contributions of IAF, phase and their interaction in explaining trial-by-trial variation in accuracy. IAF values were z-
scored to mitigate the influence of outlying trials. We choose to extract the phase in the first time-point in which we
found a significant impact on phase on accuracy according to the previously bin-analysis statistic conducted. β was
estimated using a least-squares solution and they were then converted into a z-score relative to a subject-specific null
hypothesis distribution obtained by repeatedly shuffling (n = 1000) the mapping between the accuracy values and the
predictors. We then conducted a test-t against 0 to assess the statistical significance of the observed β coefficients
and we also compute the Bayesian factor.
Furthermore, we have expanded our investigation in order to evaluate whether these relationships held valid in the
DDM framework. To this end, we estimated regression coefficients to determine the relationship between trial-to-trial
variations in IAF, alpha phase and their interaction with DDM parameters v, z, a and t. The resulting coefficients
describe the contribution that trial-by-trial fluctuations in the predictors play in modulating the DDM parameters. We
initialized HDDM to draw 5000 posterior samples with the first 500 samples discarded as burn-in. We removed trials
with reaction times exceeding 4 seconds and falling below 100 ms before fitting procedure. The statistical significance
of the regression coefficients was assessed using a procedure similar to that used in the previous DDM regression
analysis. Specifically, we inspected traces of model parameters and their autocorrelation to ensure that the models
had properly converged, and we examined the overlap of the posterior distributions for the estimated parameters,
defining significance as less than 5% overlap with the zeros value.

Watson-Williams test for phase difference between correct and incorrect choices
To investigate whether accurate and inaccurate perceptual decisions are associated with different phase angles, we
analysed phase data across timepoints ranging from -800 to 200 ms. Specifically, we aimed to determine if the phase
angles differed as a function of decision accuracy using the Watson-Williams test. This test, the circular analogue of
the t-test, evaluates whether two samples of phase angles (corresponding to correct vs. incorrect decisions) are
drawn from different distributions80. The Watson-Williams test accounts for both the mean phase angle and the
circular variance within each sample, making it well-suited for analysing phase differences in electrophysiological
data81–84.To assess the significance of phase differences, we employed a permutation-based analysis. For each
timepoint where the Watson-Williams test indicated a significant difference in phase angles between correct and
incorrect decisions, we calculated the corresponding F-value and summed these values across contiguous significant
timepoints. We then compared the observed clusters of F-values with a null distribution of clusters generated by
permuting the association between phase angles and decision outcomes 1000 times (i.e., shuffling the phase angle-
accuracy relationship). This approach allowed us to determine whether the observed clusters were significantly larger
than those expected under the null hypothesis. The analysis was performed on the entire sample, as well as
separately for the low and high IAF groups, to explore potential differences between these groups.

Intertrial phase clustering (ITPC) analysis – instantaneous alpha phase
We computed ITPC separately for correct trials and for incorrect trials using the same instantaneous alpha phase
data that were used in all other analyses. ITPC is a measure of the phase consistency across trials at a given time
point, and it varies from 0 to 1, where 0 corresponds to no phase clustering and 1 to perfect clustering across trials.
For each subject, ITPC values were calculated separately for correct and incorrect trials. To ensure that our
observation will not be biased by the different number of correct vs incorrect trials, we conducted a resampling
analysis in which trials were randomly sampled from each condition so that the minimum number of trials in each
condition was matched within each subject (similar to 82). This procedure was iterated 500 times to ensure that all
trials were used and that reliable estimates were obtained. To statistical assess whether there was a significance
difference in phase clustering between correct vs. incorrect choices, we entered ITPC index extracted across the -
800 ms to 200 ms time window into a cluster-based statistic. The real cluster size was obtained by summing t-values
(exceeding an alpha threshold of 0.05) of two-sided paired t-tests over adjacent time points. The null hypothesis
distribution of the cluster sizes was drawn by randomly shuffling (n = 1000), for each participants, the association
between accuracy and ITPC and choosing the maximum cluster value (i.e., summed t-values in significant adjacent
points) per each randomization deriving from two-sided paired t-tests run on the permuted data. From this dummy
distribution, we have extracted the cluster size corresponding to the 95% percentile and we compared the actual
cluster size with this statistical threshold. After conduction this analysis considering the whole sample, we repeat this
analysis separately in the low vs. high IAF group.

Intertrial phase clustering (ITPC) analysis – Time-frequency decomposition
To corroborate the frequency-specificity of the effect, we have also computed ITPC at frequencies between 2 and 50
Hz, separately for the low vs. high IAF groups. Time-frequency decompositions were performed for 50 linearly spaced
frequencies between 2 Hz and 50 Hz, extracted from epochs spanning −100 to 200 ms centred on target onset and
using wavelets of 3 cycles at the lowest frequency to 11 cycles at the highest frequency. After extracting the angle of
the resultant analytic signal, we computed ITPC for each frequency and time points. As before, we equate the
number of trials when computing ITPC for correct and incorrect trials per each participant. Then, we investigated,
separately for the low vs. high IAF group, whether there was a significant difference in the ITPC value between the
two conditions by mean of cluster-based statistics.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors conclude that alpha frequency influences sensory precision by modulating the distribution over phases.\nThere are a couple of conceptual and metholodological issues:\n1. Can the hypothesized mechanism explain variation in observers’ behaviour?\nThe authors suggest that higher alpha frequency increases the probability that the optimal phase will be present during stimulus presentation. Let’s think this through: stimulus duration = 60ms. The authors observed an alpha frequency of 11.15 Hz (correct trials) vs. 11.12 Hz (wrong trials) i.e. a difference in alpha frequency of 0.03 Hz. Hence, the length of the alpha cycle decreases from 89.928 ms for wrong trials to 89.686 ms for correct trials. The coverage of alpha phases for wrong trials about 66.7% and for correct trials about 66.9% , i.e. a tiny increase in the coverage of alpha phases of 0.2%.\nCan such a tiny increase in the coverage of alpha phases really explain a measureable behavioural effect? Can it explain a behavioural effect that can be measured reliably with only 150 trials per participant?\n2. Is it possible to measure a change in alpha frequency of 0.03 Hz reliably given the constraints imposed by the experiment?\nTo estimate a 0.03 Hz difference reliably and obtain such an extremely high spectral resolution, one requires very long data segments. However, the segments are limited to only brief 700 ms prestimulus baseline.\nMoreover, the estimation of the instantaneous phase with the Hilbert transform depends on the filter parameters. No data are shown that confirm the appropriateness of a filter range of IAF +/- 2 Hz.\nThe authors take the first temporal derivative to remove an aperiodic component. However, this is often not sufficient to remove 1/f noise as an important confound.\nIn addition to these two fundamental problems, there are a couple of other issues:\n1. Changes in power which are a key confound are not reported.\n2. No eye tracking data are reported. Eye movements are crucial confounds that need to be excluded.\n3. The authors report only results at the sensor level. This makes it impossible to draw decisive conclusions about the underlying neural mechanisms. It is very likely that alpha frequency as measured on the scalp arises from multiple neural generators e.g. parietal and occipital sources which may have different frequencies. Hence, putative changes in alpha frequency may reflect changes in power of different alpha generators (rather than a change in alpha frequency).\n4. The experimental paradigm involves crowding, detection etc. – possible effects of alpha frequency may act through various mechanisms making strong conclusions difficult.\n5. It seems that there are only 150 trials per participant. How many trials fell into different categories (e.g. correct vs. wrong)? What is the impact of the correction procedure applied to the d-prime (for small number of trials, biases can be substantial)\n6. Quite a few methodological details are missing e.g. visual angle, luminance, stimulus duration = 60 ms with a refresh rate of 85Hz - shouldn't that be a multiple integer? Parameters for Bayesian t-test? Statistical analysis of accuracy? Generalized linear mixed effects models are recommended. Fitting procedures and parameter recovery of HDDM?", "reviewer_2": "This paper extends current proposal regarding perceptual sampling by the human alpha rhythm by proposing that the relevance of alpha phase can be captured only at slower frequencies. This is based on the idea that with faster rhythms, it is more likely that a stimulus will fall under a ‘good’ phase of the alpha rhythm. The mechanism is elegant and consistent with previous proposals. The methods are sound and overall, the paper is well written, albeit with some typos. I have the following comments.\nAbstract and line 388: “Bayesian statistics, and advanced computational models” – I think that the authors are referring to using the drift diffusion model, and the word “advanced” is not necessary or useful (DDM was described by Ratfcliff and colleagues in 1978). Why not just write “drift diffusion model” which is the same number of words?\nAbstract, I would tone down the “unprecedented” from the “we provide… insights into the neural mechanisms through which IAF influences perceptual decisions” – I think that the seeds of the idea have been in some of the writings by VanRullen, Busch and others, as well as in some papers from the 1930's onwards, and we expect most things published in this journal to be “unprecedented”, so it is fine without the hype words.\nAbstract:\nthis sentence seems to be missing a word: “Higher probability for covering optimal alpha phases during same stimulus presentation, crucial for accurate perception, is intrinsic to faster than slower IAF.” Perhaps “is intrinsic to faster rather than slower IAF”…. \nMethods:\nConcerning the methods. It is established that alpha power, more consistently than frequency or phase, has an effect on visual detection. Additionally, peak alpha frequency often covaries with alpha power. Given the small magnitude of the effect reported it is imperative that the authors perform the same analysis they conducted by binning alpha frequency but by binning alpha power. In other words, the author should clearly show that their results are not driven by an effect in alpha power.\nThe manuscript seems to be a reanalysis of a previous dataset used in another paper published by the same authors (https://doi.org/10.1162/jocn_a_02026). If this is the case, the authors should clearly acknowledge this through the manuscript and summarize the previous analyses and findings.\nDiscussion:\nResults\nCriterion – perhaps there is some weak effect at specific time points that is missed by correcting over the whole time period? Other studies have reported criterion changes with alpha power, so it would be good to be careful about how this is worded.\nFigure 4C – It is interesting to note that the significant bins for ITPC in slow IAF participants, which are not found for the fast IAF individuals, is in the high alpha to low beta range (appears to be 10-18 Hz approximately from the figure, at least). It would be good to point this out, rather than simply describing it as “confined to the alpha band” (line 294), which does not seem quite accurate.\nI agree that drawing a clear connection between studies showing a role of alpha in detection (cited in the paper) and studies showing a role of alpha in temporal integration is a valuable theoretical contribution. In addition, the ideas put forward could be tested, and I think that it would be useful to make these implications clear with a few examples. For example, the role of alpha phase and IAF on detection should vary depending on stimulus duration (see Figure 5). Second, increasing IAF externally via entrainment/neurostimulation could improve detection while also decreasing the role of phase. Given that IAF varies naturally between people and as a function of age and neural disorders (such as SCZ as mentioned by the authors), \nIt would be good to be more precise with the use of the word “precision”. For example: “In alignment with these findings, our results strongly contribute to this body of research as they indicated a significant role of IAF in dictating the precision of sensory acquisition, as spontaneous inter-trials fluctuations in IAF were able to strongly predict the accuracy of perceptual decision.” As the authors note, they only measured accuracy and not precision of sensory representations since it was a simple detection (present/absent) task. It may indeed be the case that IAF might influence the amount of time that a brief stimulus is sampled during optimal phase, but the authors did not show that this is the case or that this decreased time necessarily reduces the precision of the neural response or perceptual interpretation. Again, this could be tested with a different paradigm, leading to a strong hypothesis that a task in which duration of sensory sampling improves precision is influenced by IAF and alpha phase.\nThe authors suggest that their model could explain null findings based on the fact that previous research has overlooked the importance of peak alpha frequency when looking at the effect of phase. Similar proposals have been advanced. For example, it has been shown that other intervening variables could explain the effect (or lack thereof) of alpha phase/frequency. Particularly, consideration of alpha power (Fakche et al. 2022 eNeuro) and/or aperiodic properties of the EEG (Deodato & Melcher, 2024, biorkiv) has been shown to affect result. These alternative viewpoints could enrich the discussion.\nGeneral: please use spellcheck for minor errors such as “temporal integratrion” and so on. I think that simply using Word to check the text would help.\nOther comments:\nLines 56-57 if the authors were the first to postulate such hypothesis, they should add an appropriate reference. Otherwise appropriate references could be:\nValera et al. 1981\nKristofferson 1967\nLines 62-69 present an oversimplified view of the effects of neurostimulation on temporal integration windows. Many researches using entrainment or neurostimulation reported effects that are not entirely supporting this view. This should be acknowledged.\nLines 83-87 “This leads to the intriguing hypothesis that variations in alpha frequency play a more basic role in visual processing beyond its temporal aspects, influencing visual sensitivity and extending even to low-level stimulus features.”\nIt’s not clear how one thing would lead to another, the authors need to make a better point at this. This seems to be one of the key points in the paper. However, it is presented in an approximate way.\nLines 97-99. “This proposition, in turn, leads to another hypothesis closely intertwined: the ability of phase angles in dictating the accuracy of the response is expected to be more pronounced in participants or trials in which IAF is slower.”\nThis is true only for short lived stimuli. Markedly shorter than the length of an alpha cycle but not too short. The authors should specify in introduction and discussion the importance of stimulus length on the proposed mechanism.\nLine 145 “perceptual decision-making perception” it’s likely a typo.\nLines 173-174. There seems to be an inconsistency between what the authors reported (-800 to -450) and the figure 1.\nLines 193-194, The effect is very small (it depends on the decimal places), this is concerning.\nLines 203-204. There is a statistic report but no indication over which channel/time-point or cluster.\nLine 337. Integratrion is a typo\nLines 358-368. Again, the mechanism proposed is sound and consistent with previous evidences. However, the author should point that within the mechanism they propose these results should be expected only for stimuli with a specific length.\nFigure 3 is missing the letter identifiers (A,B, and C).\nLine 446: “and further ampliated” (while ampliated is indeed a word historically it is rare to find it in regular discourse)", "reviewer_3": "This study presents a novel and interesting view on the mechanisms underlying alpha oscillations’ effect on visual perception. They use a visual detection task and investigate how instantaneous and individual alpha frequency and alpha phase impact perceptual performance. The study demonstrates that individual alpha frequency (IAF) significantly influences perceptual accuracy, with faster IAF predicting higher sensitivity and accuracy in sensory acquisition. This relationship is moderated by the phase of alpha oscillations, where individuals with slower IAF exhibit more pronounced phase-related variations in perceptual accuracy, whereas those with faster IAF cover a broader range of phase angles, reducing phase influence on decision accuracy. These findings provide critical insights into previously reported conflicting results regarding alpha phase and perception. The authors used state-of-the-art analysis methods and computational modelling of the behavioral performance. They clearly present the current state of the field and the shortcomings of current theoretical models of alpha oscillations in perception.\nI consider that this study is of very high quality and could have an important impact in the field. I have some concerns with the clarity, level of details reported and some of the analytical choices and controls. I list my comments below, roughly in order of importance. I believe that most of my remarks and questions will not change the main conclusions of the paper but hopefully improve its impact and quality.\n1. The DDM analysis is interesting but some aspects should be explained in more details and better motivated:\na. The task's temporal constraints (none in this study) should be indicated before presenting the DDM results as such constraints heavily impacts DDM fitting, e.g. a relatively strict response deadline will impair DDM fit as the RT distribution is truncated. The authors explain how RTs were selected for this analysis in the methods (and their choices seem valid), but providing more information on the task structure (at the beginning of the result section or in the DDM section) would help readers interpret the findings.\nb. The authors mention that it takes into account RTs but without elaborating on how RTs are integrated in their proposed mechanistic view of how IAF and phase shape perception. See also my other comment below on the lack of information on RTs in this study.\nc. The authors should elaborate on what the DDM analysis adds on top of the SDT analysis as the two indices reported from the DDM (drift and bias) are conceptually very close to the SDT ones (sensitivity and criterion). Or whether the DDM analysis was used as a conceptual replication of the SDT findings.\nd. The DDM commonly allows to estimate two other parameters not reported in the current study: bound and non-decision time. These parameters are not reported in the current study. Moreover, in the original paper presenting the DDM model used in the current study, Wiecki et al. (2013) explain that one should to allow all DDM parameters to vary according to conditions (here IAF) and perform model comparisons to determine which parameters should be fixed and which ones should be allowed to vary with conditions. I do not believe that this procedure was followed in the current study.\ne. The authors do not describe how the diagram representing the DDM model (Fig. 2C) was obtained. Is it a generic graphical representation of the DDM or were the distributions and drift arrows simulated based on the model fit to the data? Also, the diagram should be described in more details in the caption.\n2. The behavioral task is not described in sufficient details:\na. The stimulus composition is unclear. If I understand correctly, every stimulus was a checkerboard composed of black and white cells in catch trials, and with superimposed grey circles in target trials. Was a target composed of the checkerboard with a circle in every cell or only a subset of cells? It seems like it was in every cell but this is not explicitly mentioned in the methods.\nb. What was the size of the entire stimulus (in degrees of visual angle)? What was the size of each square/what was the spatial frequency?\nc. In the only depiction of the stimuli in Fig. 5, it seems like circles placed on white versus black cells had different shades of grey, what was the difference between these circles' colors? What specific feature of the circles was titrated in the staircase?\nd. The staircase procedure is not explained, please provide more information on the type of procedure used (1-up-2-down? Quest method?), how many trials were included in this staircase, etc.\ne. No information is provided on reaction times in this task (neither descriptive statistics nor any statistical test) although this performance index is used in the DDM analysis. This information is essential to evaluate the paradigm in general and some of the findings, and should thus be presented at the beginning of the results section (at least descriptive stats and/or graphical representation).\n3. In general, I think the task should be explained in more details at the beginning of the results section and accompanied with graphical depiction of the trials time course, and together with descriptive statistics (and/or plots) of accuracy and RTs.\n4. Because (one of) the main axe(s) of the study is individual differences in IAF, the authors should document this further.\na. It would be informative to know how much IAF varies within an individual and whether this variability is correlated with the individual's average IAF (e.g. is prestimulus IAF more/less variable in individuals with higher average IAF?). One inspiration for representing this variability could be Fig. 2 of Grabot & van Wassenhove (2017) (https://journals.sagepub.com/doi/full/10.1177/0956797616689369?journalCode=pssa) investigating interindividual differences in temporal-order-judgments. Of course, this is just a suggestion for graphically representing this variability.\nb. Moreover, since objective target discriminability was titrated for each participant, and according to the authors' claims, there should be a correlation between participants' average IAF and their titrated contrast reached during the staircase. Showing this correlation would strengthen the authors claim on the impact of IAF on perceptual performance.\n5. Some details are missing in the EEG methods:\na. I guess the authors only inspected occipito-parietal electrodes on the midline and right side because the stimuli were always displayed on the left visual hemifield. This should be explicitly stated in the methods section.\nb. The authors omit important details on the EEG recording, namely: how many electrodes were used and what system/brand/device was used?\nc. In the ICA preprocessing step: how many components were the data decomposed in? How many components were discarded per participant on average? The author mention that \"components containing artifacts that could be clearly distinguished from brain-driven EEG signals\", does that mean that all \"non-brain-driven\" component were discarded? In a typical 64 electrodes EEG dataset ICA decomposition, if the ICA decomposes the signal into 64 components, most components are not brain-driven, therefore it would mean that in this study a large number of components were discarded. Or did the author discard a specific and restricted type of components (e.g. blinks, lateral eye-movements, as is commonly done)?\nd. The filtering should be described in more details (type of filter, cut-off frequencies, etc.) as this could potentially have an impact on oscillatory measures and would be necessary to replicate this study.\n6. For the analyses described in the \"Instantaneous alpha phase shapes perceptual accuracy\" section, the authors use 2 phase bins and argue that this number of bins was chosen to have sufficient number of trials per bin. I understand aiming to retain enough trials per bin but I don't understand how the two bins were chosen. I am not an expert on phase analyses but since absolute phase of neural oscillations measured with EEG is meaningless (peaks and throughs depend on cortical folding and local or distant orientation of the dipoles contributing to the signal), wouldn't it be more sensitive/statistically powerful to choose phase bins individually using a data driven method? For instance, computing a phase opposition index between correct and incorrect trials? I might be missing a critical aspect of the analysis that would not be possible with another method, but I would like the author to better motivate this analytical choice.\n7. The authors used 3 phase bins in the IAF analysis. They mention using the same procedure as another study but they should provide some evidence that the results do not depend on this specific setting. Do the result hold (at least qualitatively) when using 2 or 4 frequency bins? Moreover, the authors only compare the 1st and 3rd bin, I think this is a valid analytical choice, but the effect in the 2nd bin should be presented at least qualitatively or graphically. It would be valuable to know whether the effects observed in the 2nd bin are in-between the 1st and 3rd, or whether there is a non-linear relationship between IAF and perceptual performance.\n8. The authors do not seem to control for alpha power in any of the analyses. Although the effect of alpha power on perceptual processes is still debated in the literature, this could be an important confound in the reported analyses, e.g. alpha power in trials grouped in the 1st and 3rd bins according to IAF values could be different and thus put the authors’ claims into question. This should also be investigated at the interindividual level: is alpha power different in low versus high IAF participants?\n9. The authors should at least mention which sensors were used in the analyses at the beginning of the results section. I understand that the details of sensor selection belong to the methods but knowing which electrodes were used for the analyses is fundamental to interpret the results. Moreover, in the results section they describe how electrodes were selected but it would be informative to provide the final distribution of selected electrodes across participants (i.e. how many times was each electrode selected). Also, if I understand correctly, only one electrode was used for each participant? This could be more clearly stated.\n10. The authors do not discuss potential origins of individual differences in IAF. Although there might not be much to elaborate on in the literature (one potential factor I know of is white matter structure (Minami et al., 2020)), this is a central aspect of their study and I believe it would be interesting to at least propose future avenues for research on this question.\nMinami, S., Oishi, H., Takemura, H. & Amano, K. Inter-individual differences in occipital alpha oscillations correlate with white matter tissue properties of the optic radiation. eNeuro 7, 2 (2020).\n11. The 3rd IAF bin in Fig1 A & B seems to fluctuate rhythmically and the two significant clusters in Fig. 1A are roughly separated by a theta cycle (300-250ms). Could the authors comment on this? More generally, the authors do not elaborate on potential cross-frequency coupling although numerous studies have linked theta-alpha coupling in visual perception. I do not believe that this is mandatory but it would be interesting to discuss these potential interactions.\n12. It would be interesting to discuss how the present findings relate to computational models of (alpha) oscillations (e.g. Karvat & Landau, 2024, although this specific study is concerned with temporal integration).\nKarvat, G., & Landau, A. N. (2024). A role for bottom–up alpha oscillations in temporal integration. Journal of Cognitive Neuroscience, 36(4), 632-639.\n13. In the \"Instantaneous alpha phase shapes perceptual accuracy\" section, no statistic or p-value is given. The authors should provide some information on the statistical results there.\n14. There are a number of typos and errors across the manuscript (e.g. L.249 “invstgate”, the citation of the DDM package used is Wiecki, not Wiekli, and it is not included in the references, etc.)\n15. A perceptually homogenous color-scale should be used in Fig.4C\n(https://www.mathworks.com/matlabcentral/fileexchange/51986-perceptually-uniform-colormaps)\n16. The title of section \"Alpha-Phase clustering increases following correct responses only in slow IAF individuals.\" is confusing as the effect that is shown is post-stimulus presentation but not necessarily post-response (or at least we cannot know since RTs are not reported)."}, "author_response": "We would like to thank Reviewer 1 for their insightful comments, which have enabled us to further detail the nature of our effect and clarify that the methodology employed for peak estimation is state-of-the-art, allowing for highly precise frequency resolution. Additionally, these observations have helped us contextualize our findings within the existing literature on the relationship between alpha oscillations and perceptual sensitivity. Below, we have provided a point-by-point response to all comments received.\n1. Can the hypothesized mechanism explain variation in observers’ behaviour?\nThe authors suggest that higher alpha frequency increases the probability that the optimal phase will be present during stimulus presentation. Let’s think this through: stimulus duration = 60ms. The authors observed an alpha frequency of 11.15 Hz (correct trials) vs. 11.12 Hz (wrong trials) i.e. a difference in alpha frequency of 0.03 Hz. Hence, the length of the alpha cycle decreases from 89.928 ms for wrong trials to 89.686 ms for correct trials. The coverage of alpha phases for wrong trials about 66.7% and for correct trials about 66.9% , i.e. a tiny increase in the coverage of alpha phases of 0.2%. Can such a tiny increase in the coverage of alpha phases really explain a measureable behavioural effect? Can it explain a behavioural effect that can be measured reliably with only 150 trials per participant?\nThank you for your comment. We believe this perspective stems from viewing the mean observer as an “actual” observer. However, it is important to clarify that the mean observer represents an arithmetic aggregate of mean values both within and across participants. Indeed, the variability reported by looking at the “mean observer” does not represent the range of variability within and between participants, and as such it hampers the impact that instead bin-based and trial-by-trial analysis provided. The mean observer effect was reported not as a main finding but as a confirmatory finding supported by strong frequentist and Bayesian effect statistics. It just means that intraindividual and interindividual variability are important mediators of this effect as shown in the set of analyses provided. Indeed, despite the intra- and inter- individual variability, the small difference noted in this analysis between correct and incorrect responses, as highlighted by Reviewer 1, does not diminish the findings. On the contrary, this result is exactly what one might expect given the nature of the phenomenon, it is therefore highly plausible and reflects previous effect sizes by seminal works (see below). Incidentally, following the improvement of the filter prompted by the reviewer’s useful comment, we have demonstrated that this difference increases to 0.05 Hz, and we will refer to this value henceforth. Thus, we believe this does not undermine the plausibility of an interaction between individual alpha frequency (IAF) and phase; rather, it reinforces the replicability and robustness of these findings, which have been similarly reported by several independent laboratories. Below we present 5 key points to support this statement:\n1) Within-Individual Variability: The observed difference in IAF between correct and incorrect trials might be small at the arithmetical level because the phase coverage is quite similar within the same individual. To illustrate this, we calculated the standard deviation of IAF across time, trials, and participants, resulting in a value of 0.89 Hz (strikingly close to the value of 0.9Hz highlighted by the seminal work of Haegens et al., NeuroImage, 2014). Therefore, for a participant with an IAF of 10 Hz, the range of IAF values will fall between 8.2 and 11.8 Hz in 95% of cases (MEAN ± 2 STD). Comparing these two extreme cases, the stimulus coverage would be 49% for an IAF of 8.2 Hz and 71% for an IAF of 11.8 Hz. This important difference in coverage is in line with the hypothesis that a higher IAF significantly increases the likelihood of accurate responses compared to a lower IAF due to this differential phase coverage. However, in other scenarios where the within-subject IAF difference is less pronounced between different trials, the coverage would show less variation, as the reviewer correctly pointed out. For example, with a 0.2 standard deviation difference (10 Hz - 0.2 * 0.89 = 9.82 Hz; 10 Hz + 0.2 * 0.89 = 10.18 Hz), the coverage would be 58.9% for an IAF of 9.82 Hz and 61.1% for an IAF of 10.18 Hz. Therefore, in this case, the likelihood of the stimulus falling within the optimal phase is similar.\nConnected to this point, it has to be considered the strong impact of intra-participant averaging procedure we employed. When averaging IAF within each participant across all time points (800 points) and trials (150 trials), this process smooths out inter-time and inter-trials fluctuations, reducing the variability of IAF values. Differences that might be significant at the single trial and timepoint become less pronounced when averaged across numerous points and trials. This step alone reduces the impact of any extreme IAF values by bringing them closer to the participant's mean IAF.\nCrucially, we have evidence validating this hypothesized scenario empirically. In fact, recognizing the significant impact of the averaging procedure, we incorporated a bin-based analysis in the initial version of the manuscript (see image attached below). This analysis divided trials into the 1st and 3rd bins based on individual alpha frequency (IAF) values on a subject-by-subject basis. Crucially, this analysis reduces the smoothing effect of averaging at the intra-individual level, allowing us to consider the trials in which the deviation from the mean was consistent (on average, above (3rd bin) vs. below (1st bin) 1 SD from the mean). This is proved by the simple observation that in the first bin, the IAF average across participants was 10.90 Hz, while in the third bin, it was 12.19 Hz. So, by smoothing out the effect of the mean, the difference is around 1.3 Hz between the first and the third terciles. This value is about 26 times larger than the difference of 0.05 Hz. In this case, on average, in the first bin the alpha cycle covers approximately 65.4% of the stimulus, while in the 3rd bin the alpha cycle covers approximately 73.2% of the stimulus. Considering 116 individuals with 150 trials each, the difference created a notable impact. Indeed, the difference in phase coverage is compatible with the highlighted behavioral effect, in which the average accuracy in the third bin is 3 points higher than the accuracy measured in the first bin.\n2) Moreover, another crucial step to consider is the impact of inter-participant averaging. After averaging IAF values within each participant, these averaged values have been further averaged across all 116 participants. This inter-participant averaging further reduces variability. For all these reasons, differences in IAF between correct and incorrect responses that were apparent within individuals are less likely to stand out in the grand average. The averaging process integrates a wide range of individual IAF differences, encompassing varying degrees of response accuracy and inherent IAF variability. Consequently, the differences between correct and incorrect responses observed at the individual level tend to converge towards the mean when averaged over a large number of participants. Indeed, this process exemplifies regression toward the mean, a statistical phenomenon where extreme values tend to move closer to the average upon repeated measurements or averaging. By averaging IAF values within and across participants, extreme differences are moderated. As a result, the final estimate shows smaller differences in IAF between correct and incorrect responses than might be seen in individual trials. Consequently, the overall IAF differences are less pronounced in this type of analysis, as the repeated averaging process obscures the more extreme values and highlights those closer to the mean IAF. This is precisely the reason that led us to also adopt trial-by-trial analysis that are able to weigh the single-trials impact, avoiding averaging procedure (indeed, this procedure allowed us to find an even greater Bayes Factor in the trial-by-trial variability that was however already high in the mean value analysis).\nMoreover, in acknowledging the influence of inter-individual difference, we incorporated an additional analysis in the initial version of the paper that we find compelling in illustrating the relationship between IAF and phase. Specifically, we replicated the binning-based approach, this time binning alpha frequencies not within participants but based on their inter-individual IAF. Specifically, we separated participants into two groups: one with high alpha frequency (average 12.38 Hz) and another with low alpha frequency (average 10.29 Hz). Once again, by avoiding averaging across the entire sample, the differentiation between individuals becomes evident — a difference of 2.09 Hz, which is approximately 42 times greater than 0.05 Hz.\nWe hypothesized that individuals with higher IAF would sufficiently cover the stimulus, regardless of the specific phase angle at stimulus onset. Faster alpha frequencies naturally provide greater coverage, allowing individuals to adapt even if the stimulus begins in an unfavourable phase. This is also supported by the greater variability characterizing individuals with faster IAF, as empirically demonstrated in the revised manuscript (Fig. 1C). Therefore, if variations in alpha are unable to bring significant differences in the covered phase angle, the role of phase should be less crucial. Conversely, we expected individuals with slower IAF to be more phase-dependent, unable to effectively compensate for starting from an unfavorable phase angle. Our results exactly confirmed this hypothesis: individuals with higher alpha frequencies showed a lesser impact of phase on performance, while those with lower alpha frequencies were significantly influenced by phase. The difference in frequency between these groups (i.e., 2.09 Hz) supports this disparity. For example, individuals with faster alpha cycles (80.77 ms) would cover approximately 74% of an alpha cycle within the 60 ms stimulus window, whereas those with slower alpha cycles (97.18 ms) would cover about 62%. Therefore, even though there is inter-trial and inter-time variability as described in point 1, this has less impact on individuals with a high IAF because, even in the \"worst\" cases (i.e., 2DS below the mean), a high IAF still allows for sufficient coverage of the stimulus. So, the magnitude of the different phase coverage between the two sub-populations is significant and thus explains the interaction IAF*Phase we robustly demonstrated.\nIn summary, we repeat again that, while the observed differences in IAF may appear small at the grand average level, they are meaningful when considering the broader distribution and variability within individuals. These differences, even if minor in their mean absolute value, can significantly influence behavioral outcomes, supporting the conclusion that alpha frequency influences perceptual sensitivity by modulating the distribution over phases.\n3) To provide a broader perspective beyond our own research group, we present 3 figures from highly relevant studies published in top-tier journals (including one in Nature Communications) investigating the relationship between IAF and perceptual outcomes (Nelli et al., Nat. Comm. 2017; Samaha et al., Curr. Biol., 2015; Wutz et al., PNAS, 2018). As it can be clearly seen, the magnitude of differences reported between correct and incorrect responses strictly mirrors what is reported in our findings. This underscores that even subtle differences, which appear subtle only in terms of their mean absolute value for the reasons discussed earlier and which apply equally in these studies, significantly influence perception when averaged across numerous trials and participants.\n4) Another very relevant point concerns the statistical interpretation of the findings. A fundamental principle in statistics is that relying on arithmetic or absolute values to assess significance and impact can be misleading. This approach is similar to basing the presence of significance on a figure that represents the average of two measurements. It is important to note that both the analysis highlighted contrasting IAF in correct vs. incorrect trials and the bin-based analysis are statistically compelling, as shown by the Bayes Factor above 50, which strongly supports the hypothesis linking IAF to accuracy.\n5) Regarding the possibility of detecting this effect with just 150 trials per participant, we would like to point out that, while it is true that we have approximately this number of trials per participants, we tested 125 participants [116 included in the final analyses after removing 9 as prescribed by the guidelines indicated by (Samaha and Cohen, 2022)], which we believe is the largest sample size ever used to investigate the relationship between IAF and perception. To put this into perspective, consider that a recent paper that has attracted significant attention by Buerges and Noppeney (Nat. Hum. Beh., 2022), which concluded no relationship between IAF and perceptual sensitivity, included only 20 participants—a huge contrast to our significantly larger sample size (more than 100 additional participants). Furthermore, even if one considers that 150 trials may introduce a low signal-to-noise ratio by increasing the noise component due to their relatively limited number, our observed results demonstrate that the effect is substantial enough to effectively overcome any potential noise introduced.\nIn conclusion, we hope the reviewer will now convene with us on the methodological and statistical strength of our observed effects which indicate their robustness and significance in the relationship between IAF and perceptual sensitivity.\n2. Is it possible to measure a change in alpha frequency of 0.03 Hz reliably given the constraints imposed by the experiment?\nTo estimate a 0.03 Hz difference reliably and obtain such an extremely high spectral resolution, one requires very long data segments. However, the segments are limited to only brief 700 ms prestimulus baseline.\nMoreover, the estimation of the instantaneous phase with the Hilbert transform depends on the filter parameters. No data are shown that confirm the appropriateness of a filter range of IAF +/- 2 Hz.\nThe authors take the first temporal derivative to remove an aperiodic component. However, this is often not sufficient to remove 1/f noise as an important confound.\nWe completely agree that detecting a difference of 0.03 Hz (as well as 0.05Hz) requires extremely high spectral resolution. However, to avoid the limitations of classic FFT analysis, which would have hindered our ability to detect this subtle difference, we utilized instantaneous frequency analysis to overcome this constraint (Cohen, MIT press, 2014).\nImportantly, the vast majority of studies published in top-tier journals investigating the role of IAF in perception (Benwell et al., NeuroImage, 2022; Buccellato et al., NeuroImage, 2023; Buerges and Noppeney, Nat. Hum. Beh., 2022; Drewes et al., Cerebral Cortex, 2022; Nelli et al., Nat. Comm. 2017; Samaha et al., Curr. Biol., 2015; Sharp et al., J. Neuroscience, 2022; Shen et al., Plos Biology, 2019; Wutz et al., PNAS, 2018) have utilized the very same data-analysis approach specifically to detect these subtle effects (see previously attached figures). Furthermore, we have employed an even more robust approach than previous papers by individually selecting the filter, centered around the IAF, on a subject-by-subject basis, as recommended by relevant methodological work on this topic (Samaha and Cohen, NeuroImage, 2022).\nMoreover, as previously shown (see previously attached figures), the average difference observed in our study (0.05 Hz) dramatically aligns with findings from these influential and pivotal studies examining differences between correct and incorrect responses using different experimental protocols. Therefore, suggesting that such differences cannot be reliably traced implies fundamental flaws in both our study and these seminal studies. If such flaws existed, it would necessitate a significant reassessment of the field. However, we believe that asserting that the effect cannot be measured is a very strong and unjustified claim, potentially undermining a decade of research that has been corroborated and widely replicated.\nPlease note that our study, however, went well over replicating this mean average effect, instructing on the mechanisms determining the how alpha frequency impacts sensory sampling.\nLast but not least, related to the previous points, this relationship is notably robust, as supported by the Bayes Factor (BF) being above 30 across the majority of analysis (again, indicating strong evidence in favor of H1). We believe that it is very unlikely that such a strong effect arose purely by chance, if it cannot even be measured properly.\nMoreover, we agree with the reviewer about the importance of designing an appropriate filter for extracting the instantaneous alpha frequency. Upon review, we found that the initial filter employed slightly different parameters compared to those commonly used in the literature (London et al., 2022), so we adjusted it accordingly. Specifically, we filtered the signal using a zero-phase, plateau-shaped, band-pass filter with 15% transition zones (Matlab function filtfilt.m). It is important to highlight that the overall pattern of results remains fully consistent with the initial version. Attached are some figures showing the power spectrum obtained using the spectopo function after the filtering process for selected subjects. As observed, the filter effectively captures the alpha band and crucially adapts to the precise range based on each subject’s individual alpha frequency, while frequencies outside this range are attenuated.\nFinally, we are happy to present a comparison of the power spectrum averaged across subjects (see attached figure), both with and without the removal of the 1/f aperiodic component. As clearly shown, our technique effectively isolates and removes the aperiodic noise, while preserving the essential periodic features of the signal. A key indicator of this is the alpha peak, which remains relatively consistent between the two plots, demonstrating that alpha activity was effectively preserved throughout the procedure.\nFurther evidence supporting the efficacy of our approach is observed in the beta band: in the left plot (without 1/f removal), no distinct peak can be extracted, whereas the right plot (with 1/f removal) reveals a prominent beta peak, clearly visible due to the successful attenuation of aperiodic noise.\n1. Changes in power which are a key confound are not reported.\nWe noted that not only Reviewer 1, but also Reviewers 2 and 3 requested these control analyses on alpha power, and we fully agree that their inclusion would strengthen the proposed framework.\nWe fully agree with them on the importance of clarifying the specificity of the effects linking IAF to perceptual performance, ruling out the possibility that the same relationship could be driven by alpha power. To address this concern, we conducted a series of analyses that consistently demonstrated that alpha power does not significantly influence perceptual performance. Below is a summary of the analyses, which are detailed in the supplementary materials.\nFirst, we extracted time-frequency amplitude maps and conducted a cluster-based permutation test to assess whether oscillatory amplitude differed based on response accuracy (correct vs. incorrect trials). The statistical analysis revealed no significant clusters distinguishing correct from incorrect responses (p = 0.43).\nNext, we performed regression analyses using a least-squares approach, adding alpha power as an additional predictor alongside IAF. The results indicated that IAF remained a strong predictor of performance (slope = 0.37, SEM = 0.09; t = 4.04, p < 0.001, BF = 179.63), while alpha power (mean slope = -0.14, SEM = 0.09; t = -1.57, p = 0.12, BF = 0.34) did not demonstrate any predictive value. A similar pattern emerged when using a GLM model (mean slope for alpha power = -0.03, p = 0.13; mean slope for IAF = 0.08, p < 0.001).\nWe also conducted bin analyses by dividing trials into terciles based on alpha amplitude. The results showed no significant differences in sensitivity and criterion between trials in the first and third terciles (all p > 0.09, all BF < 0.41).\nFinally, we conducted another time-frequency analysis comparing oscillatory amplitude between participants in the slow and fast IAF groups. No significant clusters were found between the two groups (p = 0.14).\nIn conclusion, these analyses consistently demonstrate that alpha power does not contribute to explaining perceptual performance, in contrast to the robust influence of IAF.\nHere we report the new section added in the methods section:\nAssessing the specificity of the relationship between IAF and Perceptual Accuracy relative to Alpha Power\nTo ensure that the effects observed in relation to IAF were not confounded by oscillatory power, we conducted several analyses (See Supplementary Materials). First, we extracted time-frequency oscillatory amplitude maps for each participant from the specific electrodes used to compute IAF. Time-frequency decompositions were performed for 50 linearly spaced frequencies ranging from 2 Hz to 50 Hz. Wavelets with 3 cycles at the lowest frequency and 11 cycles at the highest frequency were applied. After obtaining the absolute value of the resultant analytic signal, we conducted a cluster-based permutation test to evaluate whether oscillatory amplitude varied based on response accuracy (correct vs. incorrect trials). Next, we extracted the trial-by-trial amplitude from the time-frequency analysis, averaging it within the -800 ms to -100 ms window, at the frequency closest to each participant's previously determined IAF. We performed regression analyses using a least-squares approach, incorporating both trial-by-trial fluctuation in IAF and alpha power as predictors of perceptual accuracy. This analysis allowed us to determine whether the influence of IAF on performance was independent of alpha power. We also replicated a bin-based analysis by dividing trials into terciles based on the previously extracted trial-by-trial alpha amplitude. We compared signal detection theory metrics across the lowest and highest terciles to confirm that the effects observed with IAF did not hold when considering variations in alpha power. Lastly, we conducted a time-frequency analysis using the same parameters as those applied in the comparison of correct and incorrect responses. This analysis aimed to assess potential differences in oscillatory amplitude between participants categorized into low and high IAF groups.\nHere we include the new part and figure included in the supplementary materials:\nFigure S1. No significant differences observed in behavioral performance between first and third power terciles\nWe replicated the initial analysis included in the main text by binning trials based on pre-stimulus power in the first and third terciles. The analysis revealed that this binning approach did not dictate behavioural performance (all t < 1.68, all p > 0.09, all BF < 0.41).\nFigure S2. Correct and Incorrect trials are not characterized by difference in oscillatory amplitude.\nWe conducted a time-frequency analysis to assess whether correct responses exhibited different amplitudes compared to incorrect responses. The cluster-based analysis revealed no significant differences between the two types of trials, indicating that power, in contrast to IAF, is not capable of differentiating correctness in the trials.\nFigure S3. Trial-by-trial fluctuations in alpha power do not account for perceptual accuracy.\nWe conducted an analysis to evaluate whether trial-by-trial fluctuations in power, alongside those of IAF, influenced the accuracy of the responses provided. We extracted the power calculated from the time-frequency analysis by taking the alpha frequency closest to the IAF, normalizing it through a z-score transformation. Subsequently, we created a design matrix that included the intercept, IAF, and normalized alpha power. The regression analysis confirmed the significant predictive ability of IAF fluctuations on perceptual accuracy (mean slope = 0.37 p < 0.001). In contrast, fluctuations in power did not modulate the participant's ability to accurately retain the stimulus (mean slope = -0.15 p = 0.12).\nFigure S4. Low and Fast IAF groups do not exhibit significant differences in oscillatory amplitude.\nWe conducted a time-frequency analysis to assess whether participants in the low IAF frequency group exhibited different amplitude profiles compared to those in the high IAF frequency group. The cluster-based analysis revealed no significant differences between the two populations, indicating that power does not effectively differentiate between the two groups.\n2. No eye tracking data are reported. Eye movements are crucial confounds that need to be excluded.\nWe agree that eye-related artifacts could potentially act as a significant confound in the EEG analysis we conducted. However, it is important to note that we have already addressed potential confounds related to eye movements by employing Independent Component Analysis (ICA), which is a widely recognized technique for offline artifact removal, in the first version of the manuscript. This method is the gold-standard to eliminate artifacts associated with eye movements from our data. Finally, it is noteworthy that the seminal studies described in the above comments (Nelli et al., Nat. Comm. 2017; Samaha et al., Curr. Biol., 2015; Wutz et al., PNAS, 2018) also employed the same artifact removal techniques (i.e., ICA) in order to remove this type of artifact.\n3. The authors report only results at the sensor level. This makes it impossible to draw decisive conclusions about the underlying neural mechanisms. It is very likely that alpha frequency as measured on the scalp arises from multiple neural generators e.g. parietal and occipital sources which may have different frequencies. Hence, putative changes in alpha frequency may reflect changes in power of different alpha generators (rather than a change in alpha frequency).\nThis is a great comment. We agree that addressing the possibility of spurious effects resulting from an artificial \"merging\" of alpha activity generated by different sources would be a valuable enhancement to the paper. In response to the Reviewer observations, we would like to clarify that we have conducted source-level analyses that provided more detailed insights into the underlying neural mechanisms.\nOur source-level analysis results clearly demonstrate that alpha activity originating from the occipital lobe serves as a strong predictor of the observed behavior. In contrast, while alpha activity generated from the right parietal lobe shows some significant associations with behavioral measures, these associations do not withstand a Bayesian approach, which reveals a higher probability for the null hypothesis (H0) compared to the alternative hypothesis (H1) in many cases. This suggests that the influence of the parietal lobe may be less robust, further supporting the notion that the association between interindividual alpha frequency (IAF) and perceptual sensitivity is more directly related to occipital activity. We believe these source-level results not only clarify the roles of different brain regions in alpha activity but also provide a deeper understanding of the neural mechanisms involved in the studied behavior. It is important to highlight that, given the limited spatial resolution of EEG, we cannot definitively conclude that our occipital and parietal signals were fully separated; there may have been residual mixing of signals even after source reconstruction. Nevertheless, we observed a divergence between occipital and parietal alpha peak frequencies in shaping perceptual performance. Overall, we are confident that this integration of source-level analyses strengthens the conclusions of our work.\nSpecifically, we have added in the main text that:\nwe also ruled out the hypothesis that the relationship between IAF and perceptual sensitivity is merely due to spurious changes in alpha frequency reflecting variations in the power of different alpha generators. Our source analysis demonstrates that all observed effects are robustly predicted by the IAF extracted from occipital sources.\nAnd we have added in SI the following analyses:\nWe addressed the concern that the observed relationship between IAF and behavior might be an artifact of merging alpha activity originating from distinct sources by performing a source-level analysis, inspired by the approach in Haegens et al. (2014). Specifically, we used the Desikan-Killiany atlas to select occipital (Pericalcarine sulcus and lateral occipital cortex) and parietal regions (Inferior Parietal and Superior Parietal cortex). For each region, we identified the sources with the highest alpha power and extracted the IAF separately for occipital and parietal sources, thereby mirroring the procedure used in our sensor-level analysis. Then, we investigated the influence of pre-stimulus IAF on perceptual sensitivity and bias by employing a binning analysis using the IAF extracted for each individual and separately for the two sources (i.e., occipital and parietal). This analysis involved dividing the trials into three terciles based on pre-stimulus IAF. Firstly, we time-collapsed the IAF data by computing the average values from -800ms to -100ms for each participant. Subsequently, we computed the signal detection indices from trials within the first and third terciles. The results demonstrated that trials in the first tercile were associated with reduced sensitivity compared to trials within the third tercile both in the right occipital (mean d’ = 1.42, SE = 0.08; mean d’ first tercile third = 1.61, SE = 0.06; t = -3.56, p < 0.01; BF = 37.44; Fig. 1A) and in the right parietal sources (mean d’ = 1.46, SE = 0.08; mean d’ = 1.61, SE = 0.06; t = -2.33, p = 0.02; BF = 1.43).\nSimilarly, accuracy was higher in the third compared to the first terciles when considering the occipital (mean accuracy = 0.71, SE = 0.01; mean accuracy first tercile = 0.75, SE = 0.01; t = -3.95, p < 0.01; BF = 129.50) as well as parietal sources (mean accuracy = 0.72, SE = 0.01; mean accuracy = 0.75, first tercile third tercile SE = 0.01; t = -2.78, p < 0.01; BF = 4.11). Conversely, the criterion indices estimated in the first vs. third terciles were not statistically different (Occipital source: mean c = 0.42, SE = 0.05; mean c = 0.40, SE = 0.05; t = 0.94, p = 0.35; BF = 0.17; Parietal source: mean c = 0.44, SE = 0.05, mean c = 0.43, SE = 0.04, t = 0.25, p = 0.80, BF = 0.11).\nIt is important to note that the strength of the relationship between IAF and perceptual performance is significantly more pronounced when considering occipital IAF compared to parietal IAF, suggesting that the alpha activity generated in the occipital lobe has a more critical impact on performance. This aspect becomes even more pronounced when we examined the differentiation of IAF in correct and incorrect responses, as well as the trial-by-trial analyses linking accuracy fluctuations to IAF variations. In this context, only the IAF extracted from the occipital source reached full statistical significance when comparing IAF in correct (IAF = 11.27, SE = 0.1) and in incorrect trials (IAF = 11.24, SE = 0.11, correct incorrect t = 2.48, p = 0.01, BF = 2.02) as well as in the trial-by-trial analysis (mean slope = 0.31, SE = 0.1, t = 3.22¸ p < 0.01, BF = 13.25), while the IAF extracted at the parietal level showed only non-significant trends, further weakened by Bayesian factors indicating a higher probability for the null compared to the alternative hypothesis (IAF = 11.24, SE = 0.1; IAF = 11.21, SE = 0.11, t = 1.86, p correct incorrect 106 = 0.07, BF = 0.56; mean slope = 0.20, SE = 0.1, t = 1.89¸ p = 0.06, BF = 0.60).\n4. The experimental paradigm involves crowding, detection etc. – possible effects of alpha frequency may act through various mechanisms making strong conclusions difficult.\nAll potential confounding effects were carefully considered during stimulus conceptualization and calibration. Furthermore, any potential confounds (e.g., crowding) would be consistent across conditions and individuals, as stimuli vary solely by their contrast, which is the only parameter that could explain the results. Therefore, there are no random variations that could explain the observed effect.\n5. It seems that there are only 150 trials per participant. How many trials fell into different categories (e.g. correct vs. wrong)? What is the impact of the correction procedure applied to the d-prime (for small number of trials, biases can be substantial)\nAs reported in the paper, the average detection accuracy was 73%. Therefore, on average, 41 trials were incorrect, which are a fair number to extract alpha-related measures. Nevertheless, we implemented a subsampling procedure, matching the number of correct and incorrect responses at the individual level, in order to demonstrate that the trial count does not influence the outcome. Specifically, we included this analysis in the main text:\nIn order to check that different trials count between correct and incorrect response inflated the difference between IAF extracted from correct and incorrect responses, we used a subsampling procedure to match the number of correct and incorrect trials. Specifically, for each participant, we shuffled the trials with correct responses and matched them with IAF values count from the incorrect responses. For each permutation, we calculated the average IAF for the correct responses. This permutation procedure was repeated 500 times to ensure reliability. We then averaged the surrogate IAF values obtained from these permutations. Finally, we compared the average IAF values for correct versus incorrect responses. This comparison corroborated the initial analysis presented in the manuscript, where all trials were included as correct trials were associated with higher IAF values (IAF = 11.35 ± 0.01) compared to incorrect trials (IAF = 11.30 ± 0.01, t = 4.15, p < 0.01).\nFurthermore, we calculated the Signal Detection Theory (SDT) index d' and criterion by employing a different corrections strategy to ensure that the loglinear approach has not impacted the obtained results. To this end, we have used the classical approach of adjusting only the extreme rates of hit and false alarm. Rates of 0 were replaced with 0.5/n, and rates of 1 were replaced with (n - 0.5)/n, where n is the number of signal or noise trials (Macmillan & Kaplan, 1985). In the new version of the paper, we have now employed this correction method (described in the methods, see below) and changed the text and statistics accordingly.\nTo ensure a proper estimation of d’ and c, we corrected the extreme rates of hit and false alarm. Rates of 0 were replaced with 0.5/n, and rates of 1 were replaced with (n - 0.5)/n, where n is the number of signal or noise trials (Macmillan and Kaplan, 1985)\n6. Quite a few methodological details are missing e.g. visual angle, luminance, stimulus duration = 60 ms with a refresh rate of 85Hz - shouldn't that be a multiple integer? Parameters for Bayesian t-test? Statistical analysis of accuracy? Generalized linear mixed effects models are recommended. Fitting procedures and parameter recovery of HDDM?\nThank you for the comment. We have added this information in the new version of the paper.\n1. Regarding the information about the presented stimuli, we have added the following information in the methods section:\nStimuli had spatial frequency of 5.16 cycles/degree, and they were presented only in the lower part of the screen at 4.1°/3.7° eccentricity (horizontal/vertical). Luminance was maintained constant in all the level of the stimuli (mean 146 RGB).\n2. We have also amended the stimulus duration in the new version of the manuscript.\nAs stimuli lasted for 5 frames, the presentation time was 59 ms.\n3. Regarding the parameters for Bayesian t-test, we integrated that:\nThe analysis utilized the default prior settings: Prior Distribution for Effect Size: Cauchy, with a Scale Parameter of 0.707. We used these parameters for all the Bayesian t-test performed.\n4. Regarding the statistical analysis employed to investigate the relationship between IAF and accuracy, we used a non-parametric multiple regression approach similar to that described in Cohen and Cavanagh (2011) and Samaha et al. (2017), which shared our very same aim of analyzing how trial-by-trail fluctuations of a neural marker is able to intercept trial-by-trial variability in decisional outcomes. However, we agree that complementing this analysis with generalized linear mixed effects models is a valuable addition. To this end, we fitted this type of model and confirmed the same pattern of results.\nIn the new version of the methods, we added that:\nTo corroborate the results, we examine the relationship between IAF and task accuracy by means of a generalized linear mixed model (GLMM). The dependent variable was task accuracy, coded as a binary variable (0 or 1). The normalized alpha frequency was used as a continuous predictor. To account for inter-individual variability, participants were included in the model as a random effect. The specific formula used for the model was:𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 ∼ 𝐼𝐴𝐹 +(1 ∣ 𝑆𝑢𝑏𝑗𝑒𝑐𝑡). MATLAB's fitglme function was used to fit the model to the data, specifying a binomial distribution and a logit link to reflect the binary nature of the accuracy data.\nAnd in the supplementary material:\nGLMM analysis corroborated the IAF-sensitivity finding derived from the non-parametric multiple regression analysis (see main text) as IAF was a significant predictor of trial-by-trial accuracy (slope = 0.13, p < 0.01). This effect remained significant when accounting for trial-by-trial fluctuations in alpha power (slope IAF= 0.07, p < 0.01; slope alpha power = -0.03, p = 0.14).\n5. We appreciate the opportunity to provide further clarification regarding the fitting procedures and parameter recovery of HDDM.\nSpecifically, we included in the new version of the manuscript that:\nWe used the HDDM package provided by (Wiecki et al., 2013). Here, Bayesian inference through Markov chain Monte Carlo (MCMC) sampling is utilized to approximate posterior distributions for model parameters at both the individual and group levels. This approach is advantageous as it effectively manages the uncertainty associated with parameter estimates, allowing for robust and precise parameter estimation. The priors for each parameter in HDDM are informed by a pool of 23 studies, which report the best-fitting parameters of the Drift Diffusion Model across various decision-making tasks (Matzke and Wagenmakers, 2009). By grounding these priors in empirical data, HDDM ensures that initial parameter assumptions are reliable and informed by a substantial body of existing research (see the supplement by (Wiecki et al., 2013) for a visual representations of these priors)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have made significant efforts to revise their manuscript. However, several methodological concerns raised previously remain valid and require further attention. Below, I briefly outline the key issues along with recommendations for improvement.\n1. The main conclusions still rely on computing a so-called ‘instantaneous alpha frequency’ by applying the Hilbert transform to obtain phase and then computing the derivative. However, this approach is flawed. The term ‘instantaneous frequency’ is misleading in the sense that frequency can accurately be computed only over finite time windows. Instantaneous frequency estimates are not consistent with fundamental principles in physics, as greater frequency resolution requires longer time windows. Michael Cohen who initially proposed this method showed in subsequently published research that this approach leads to biased estimates that depend strongly on 1/f noise and SNR. While the authors attempted to remove 1/f noise, their approach appears to be problematic. As shown on p.12 in their rebuttal, the PSD after 1/f noise removal shows substantially lower alpha power (an order of magnitude lower), which is rather surprising given the original PSD. To ensure unbiased frequency estimates, it is crucial to adhere to established methods such as FFT over longer time windows. This approach is widely validated and avoids the pitfalls associated with ‘instantaneous frequency’ calculations.\n2. The authors suggest that despite the minute differences in IAF within each participant they observe substantial variability when computing variance pooled over time, trials and participants (p.2 in rebuttal). However, this approach conflates within trial variability across time, within-participant variability across trials and between-subject variability. Because the statistical inference (i.e. comparison: correct vs incorrect) is across trials within each participant, this conflation of variability is not appropriate.\n3. In support of their own findings, the authors cite previous studies in high impact journals that reported comparable differences in alpha frequency. However, these older studies relied on the instantaneous alpha frequency estimation approach which has since then been shown to be flawed.\n4. Alpha power is another important confound. The authors show that there is not a significant difference in alpha power using classical statistics. However, there is non-decisive evidence using Bayesian analysis (in fact, even weak evidence for a power difference). Hence, alpha power confounds cannot be completely excluded.\nDespite these concerns, the study may contribute to the field by sparking further research into the role of alpha frequency. However, ideally (and as previously suggested), the authors should address the methodological concerns and employ techniques known to provide unbiased alpha frequency estimates (i.e. FFT over finite timewindows).", "reviewer_3": "The authors have addressed all my comments and suggestions in great details. I commend the authors on the work and care invested in responding to each comment or concern I had.\nI believe the manuscript was greatly improved and that this is a well-designed and very impactful study.", "reviewer_4": "I was invited to substitute for Reviewer 2 and assess whether the concerns raised have been appropriately addressed. First and foremost, I share the highly positive impression of the manuscript and would like to congratulate the authors on their outstanding work. The study represents an ambitious effort to scrutinize the relationship between IAF and visual perception, featuring a large sample size, multiple analytical approaches that converge on a main conclusion, and a conceptual integration of distinct research strands (perceptual and temporal sampling). Moreover, it generates testable research questions that will undoubtedly inspire future studies. Taken together, these strengths make the manuscript a strong candidate for publication in Nature Communications, which I strongly endorse.\nA significant portion of Reviewer 2's comments focused on wording, typos, overlooked literature, and the need for clarifying remarks (e.g., regarding the strength of effects and the scope of the IAF-perception relationship, i.e., restriction to only very short stimuli). I believe the authors have done an excellent job addressing these concerns both in their response and in the revised manuscript.\nThe most critical issue raised by Reviewer 2 concerns the potential influence of alpha power on instantaneous frequency. The authors employ multiple approaches to rule out a statistically significant impact on the IAF effect, though one could argue that the statistical approach in Figure S2 is overly conservative, as it tests a very broad frequency and time range. However, my concern lies more with the reliability of extracting IAF at the single-trial level, which is crucial for several of the presented analyses. The methods section states that individuals without a clear IAF were excluded (ll. 582–584), but this does not fully account for various ways in which the power spectrum might influence the findings. For instance, changes in non-alpha frequency bands associated with more accurate perception could “leak” into the narrow-band filtered data, thereby affecting the estimated peak frequencies.\nTo address this, it would be helpful to present Figure S2 in power spectral form over a narrower range (e.g., up to 20 Hz) for both conditions, overlaid with an estimate of variability. Another potential issue is that in trials where participants are less accurate, noisier data could make alpha peak estimation less reliable. One way to test this would be perhaps to use the “aperiodic-free” single-trial power spectra and fit Gaussians to estimate peak frequency and Goodness-of-Fit per trial. If the effect is genuinely driven by IAF, Goodness-of-Fit should not differ between conditions.\nThat said, compared to the field addressing similar questions, the authors have already conducted extensive analyses. Given that I am joining the review process at a late stage, I wouldn’t insist on these additional checks but suggest them as potential refinements or discussion points.\nAlthough addressing this issue is not strictly within my role as a substitute for Reviewer 2, I found the response to Reviewer 1’s concern about the potential influence of eye movements to be overly defensive. While ICA correction removes a significant portion of instantaneous volume conduction effects, it does not necessarily eliminate the neural activity associated with eye movements. There is exciting ongoing research exploring the relationship between oculomotor behavior and alpha activity (see, for example, work by Ole Jensen’s lab). If the authors wish to further investigate this, they could consider analyzing the ‘blink component’ (acknowledging that this primarily captures vertical movements) or defining a frontal electrode group for analysis using the non-eye-movement-corrected data. However, I do not insist on these additional analyses."}, "author_response": "We appreciate the reviewers’ detailed comments and agree that it is useful to confirm that our findings linking alpha frequency with perceptual accuracy are robust and not based on a single method of extraction, i.e., the instantaneous frequency approach. To address these concerns, we implemented new set of analyses that confirmed the reliability of the finding derived from the instantaneous frequency calculation:\nALTERNATIVE ANALYSIS 1: As an initial complementary analysis, we pooled trials from each condition (correct versus incorrect) to circumvent the limitations of single-trial estimates, which can be vulnerable to noise due to the brief prestimulus period. This approach enables us to obtain a more stable, high signal-to-noise estimate of the alpha peak, thereby mitigating the potential imprecision inherent in single-trial analyses. Therefore, for each subject we visually inspected the pooled power spectra for correct and incorrect trials separately and only included those subjects for whom the alpha peak was clearly identifiable in both conditions (excluded participants N = 15 out of 116). t-tests comparing the extracted IAF values revealed that correct responses were associated with a significantly higher IAF, underscoring the relationship between elevated alpha frequency and task performance.\nALTERNATIVE ANALYSIS 2: Moreover, we extended our analysis by applying an automated, data-driven algorithm—following the approach of Corcoran et al. (2018)—to extract IAF from the same pooled power spectra. This approach first smooths the power spectrum using a Savitzky-Golay filter, which is critical for reducing random noise and revealing the underlying spectral structure. Following smoothing, the algorithm computes the first derivative of the spectrum with respect to frequency, identifying the local maximum in the alpha band as the IAF. This objective method reduces the subjectivity inherent in visual inspection. Importantly, the algorithm is designed to be highly conservative, providing an estimate only when the signal is robust, thereby preventing unreliable peaks from being forced into the analysis (N = 34 out of 116). Despite the strictness of this approach, the resulting estimates still revealed a significantly higher IAF in correct trials, confirming the robustness of our findings independently of the extraction method adopted. Notably, alternative analyses 3 and 4 (see below) using a trial-by-trial FFT-based estimation approach yielded the same results without requiring the exclusion of any participant.\nALTERNATIVE ANALYSIS 1 and 2 have been added in the new version of the supplementary materials:\nIAF extracted from the pooled correct trials was higher than that extracted from the pooled incorrect trials.\nBeyond single-trial analyses, we also examined whether IAF differences between correct and incorrect trials were evident when estimated from pooled data. This approach provides a complementary perspective, as it avoids trial-by-trial variability and instead focuses on broader spectral properties of the prestimulus signal. Specifically, we computed the power spectrum by pooling all trials within each response category (correct vs. incorrect) and identified the alpha peak separately for each condition. Our results revealed a systematic difference: IAF extracted from the pooled correct trials was significantly higher than that extracted from the pooled incorrect trials. This finding was consistent across subjects and was confirmed using both visual inspection of the power spectrum (t100 = 3.11, p < 0.01, BF = 9.80) of the spectra and the Corcoran et al. (2018) algorithm for automated peak detection applied to the aperiodic-free power spectrum (t81 = 2.77, p < 0.01, BF = 3.73). Importantly, this pattern replicated the results obtained using the single-trial approach, reinforcing the conclusion that prestimulus IAF is a key predictor of perceptual accuracy. The consistency across different estimation methods highlights the robustness of the observed effect, ruling out the possibility that it arises from noise-related fluctuations in single-trial analyses.\nALTERNATIVE ANALYSIS 3. Secondly, also following the insights from reviewer 4, we verified the single-trial IAF estimates using complementary FFT-based approaches as included in the new version of the methods section:\nAdditionally, we adopted complementary approaches to examine the role of IAF without relying on the instantaneous frequency method, instead using FFT-based frequency estimation. First, we computed the power spectrum from the pre-stimulus signal (-800 to -100 ms), matching the time window used for the instantaneous frequency approach. We then applied Corcoran’s algorithm to the single-trial power spectra to extract the single-trial IAF. This peak detection method, performed directly on the frequency spectrum, helps overcome potential limitations of the instantaneous frequency approach (Samaha and Cohen, 2022; Schoffelen et al., 2024).\nSpecifically, we adapted the Corcoran algorithm and applied it separately to each single trial. Subsequentially, we replicated both the tercile-based and trial-by-trial analyses using these single-trial IAF estimates. For the tercile-based analysis, trials were divided into three groups based on the prestimulus IAF, and performance metrics were compared between the lowest and highest terciles. Consistent with our previous results, while no effect was observed on criterion, both sensitivity and accuracy were significantly higher in the third IAF tercile. In the trial-by-trial analysis, fluctuations in IAF on individual trials were used to predict perceptual performance, and the results confirmed that higher IAF on a given trial corresponded to an increased probability of correctly detecting the presence/absence of the target.\nThese new analyses have been added in the new version of the supplementary materials:\nFFT-based method for estimating single-trial IAF confirmed the pattern of results observed with the instantaneous frequency approach.\nTo further validate the reliability of single-trial IAF estimation, we compared the results obtained using the FFT-based method with those derived from the instantaneous frequency approach. Specifically, we applied the Corcoran et al. (2018) algorithm to the single-trial power spectra, allowing us to extract IAF values based on a robust peak detection method rather than phase-based instantaneous frequency calculations. The key advantage of this approach is its reliance on spectral features rather than time-domain fluctuations, reducing the influence of transient noise. Our analyses revealed a strong correspondence between the two methods. Specifically, trials in which the FFT-based IAF estimates were in the third terciles showed higher accuracy (t115 = 3.03, p < 0.01, BF = 7.64) and sensitivity (t115 = 2.77, p < 0.01, BF = 3.85), while maintaining the same level of criterion (t115 = 1.46, p = 0.15, BF = 0.29). Moreover, fluctuation in trial-by-trial FFT-bases IAF estimate impacted the probability of making a correct detection of the target (t115 =, p < 0.01, BF = 40.31). Therefore, both methods consistently indicated that higher IAF was associated with better perceptual performance, confirming the robustness of our original findings. These results suggest that different analytical frameworks can reliably capture the same underlying neural dynamics, reinforcing the conclusion that prestimulus IAF fluctuations play a crucial role in shaping perceptual outcomes.\nALTERNATIVE ANALYSIS 4. Following the request of reviewer 4, we further extended our analysis by implementing a Gaussian function-based approach to estimate the IAF on a trial-by-trial basis. In this method, a Gaussian curve is fitted to the single-trial power spectrum. The peak of this fitted Gaussian is taken as the IAF for that trial. Building on these single-trial estimates, we conducted complementary analyses. In the tercile-based analysis, trials were sorted into three groups based on their prestimulus IAF values. Performance metrics were then compared between the lowest and highest terciles; consistent with our previous results, while criterion remained unaffected, both sensitivity and accuracy were significantly higher in the highest IAF tercile. In the trial-by-trial analysis, we examined how fluctuations in IAF predicted perceptual performance. The results confirmed that a higher IAF on any given trial corresponded to an increased probability of correctly detecting the presence or absence of the target. Moreover, correct trials were associated with higher IAF compared to incorrect trials. These new analyses have been added in the new version of the supplementary materials:\nSingle-Trial IAF Estimation via Aperiodic-Free Gaussian Fitting\nTo further validate the reliability of our single-trial IAF estimation, we conducted an additional analysis using an aperiodic-free Gaussian fitting approach. In this procedure, the power spectrum was computed over the prestimulus window using Welch's method per each trial separately, focusing on the alpha band. An initial peak estimate was determined by identifying the frequency with the highest power within this band. Around this peak, we defined a narrow fitting window extending 3 Hz on either side to isolate the region of interest. Within this window, a Gaussian function—characterized by its amplitude, centre frequency, and width—was fitted to the data, and the centre frequency of the fitted Gaussian was taken as the IAF for that trial. An adjusted R² was calculated to assess the goodness-of-fit, and trials were excluded if the adjusted R² was below 0.9 or if the IAF was estimated at the boundaries of the fitting window, suggesting that the fit was constrained rather than reflecting a true spectral peak. First, we also compared the goodness-of-fit between correct and incorrect trials and observed no significant differences (Adjuster r2 correct = 0.98± 0.01; Adjuster r2 incorrect = 0.98 ± 0.01; t115 = -0.33, p = 0.74). Moreover, the IAF estimates obtained with this method successfully predicted participants' performance, replicating the results presented in the main text. Specifically, terciles analysis confirmed that trials included in the third vs. first terciles are characterized by higher sensitivity and accuracy (all t115 > 2.96, all p < 0.01; all BF > 6.46). Moreover, correct responses were associated with higher IAF compared to incorrect responses (t115 = 2.65, p < 0.01; BF = 2.91) and trial-by-trial fluctuations in IAF predicts variations in perceptual accuracy (t115 = 2.69, p < 0.01; BF = 3.24)\nAlternative analysis 5. Regarding the comment relative to the appropriateness of removal of 1/f component, we would like to emphasize that, although the absolute alpha power appears reduced following 1/f noise removal, this decrease does not reflect a lower signal-to-noise ratio (SNR). We computed the relative alpha power, which compares the alpha power to the total power across the spectrum. This ratio inherently offers a more precise indices of the alpha signal's prominence relative to the background. Our analysis revealed that after removing the 1/f component, the relative alpha power increased (38%) compared to the spectra where the 1/f component was retained (33%), indicating that the alpha signal is more clearly distinguished from the background noise. Moreover, we employed an alternative approach to remove 1/f, by applying a polynomial fitting procedure to the power spectrum. First, for each trial, we computed the FFT to obtain both the power spectrum and the phase information, then converted the power spectrum into a logarithmic scale. In this log-transformed space, we fit a second-degree polynomial to capture the 1/f component. By subtracting this polynomial from the log power spectrum, we effectively removed the 1/f noise (see figure below).\nFinally, we recombined the corrected power spectrum with the original phase information and performed an inverse FFT to reconstruct the time-domain signal without the 1/f contribution. We computed the relative alpha power and, even using this alternative approach, it is superior compared to the original signal (40%). Moreover, we replicated the original analysis linking IAF to perceptual precision when extracting IAF from this aperiodic-free signal, as detailed in the supplementary materials:\nConfirmation of the Robust IAF–Sensory Precision Relationship Using an Alternative 1/f Removal Method\nWe conducted supplementary investigations to assess whether the results linking IAF and sensory accuracy are robust to different methods for removing the 1/f component. Specifically, we applied an alternative approach that uses a polynomial fitting procedure to remove 1/f noise. For each trial, we computed the FFT using Welch’s method to obtain both the power spectrum and the phase information over the prestimulus window. The power spectrum was then converted into a logarithmic scale, and a second-degree polynomial was fitted to capture the 1/f component. By subtracting this polynomial from the log-transformed power spectrum and converting the result back to the original scale, we effectively removed the 1/f noise. The corrected power spectrum was then recombined with the original phase information and transformed back into the time domain using an inverse FFT.\nWe subsequently replicated our original analysis linking IAF to perceptual precision using the IAF estimates derived from this aperiodic-free signal. The results confirm that our findings are robust and consistent regardless of the specific method used for 1/f removal. Specifically, tercile analysis confirmed that trials in the third versus the first tercile are characterized by higher sensitivity and accuracy (all t115 > 2.80, all p < 0.01; all BF > 4.16). Moreover, correct responses were associated with higher IAF compared to incorrect responses (t115 = 4.29, p < 0.01; BF = 439.41), and trial-by-trial fluctuations in IAF predicted variations in perceptual accuracy (t115 = 4.42, p < 0.01; BF = 703.19).\nCollectively, these multiple lines of evidence demonstrate that the relationship between IAF and perceptual accuracy is robust and not an artifact of the instantaneous frequency method. By validating our initial results using both classical and data-driven spectral analyses, ensuring the reliability of single-trial estimates, and replicating our findings with independent analytical frameworks (FFT), we are confident that these additional analyses convincingly establish the validity of our original conclusions and underscore the genuine neurophysiological basis of the observed effects."}, {"round": 3, "reviewer_comments": {"reviewer_4": "The authors have done a great job in addresing my concerns. This is a great paper. Congratulations.", "reviewer_5": "In an EEG study of the role of alpha oscillations in visual perceptioin, the authors investigate how indivdiual and instantaneous alpha frequency (IAF), together with alpha phase, influence visual accuracy and sensitivity in a visual detection task. Their key findins is that IAF and phase jointly modulate visual perception, and the authors provide data and derive a qualitative model that lower IAF makes the impact of phase more likely. The study is well designed, uses a large sample (n =125) and sophisticated analysis methods including SDT and DDMs. Overall, the work appears as an important contribution in the ongoing debate of whether IAF modulates visual perception, including a novel view how IAF may link to alpha phase effects on visua perception.\nI have been added as a reviewer after first rounds of revision, and have been asked to especially focus on a previous reviewer (#1) comment regarding the computation of instantaneous IAF, potentially being biased by 1/f noise and SNR, power confounds, and eye motion confounds. So I mainly focus on the author’s reply to these concerns:\n• Inst. IAF approach: The authors provided four additional analyses where they avoided computing instantanous IAF, but computed and compared IAF from power spectra of trials with correct vs. incorrect response using establised methods (Corcoran protocol, FFT methods in a fixed prestimulus interval). Specifically, they could reproduce their main findings (higher IAF has better visual accuracy/sensitivity) when estimating alpha peaks from single-trial data using the Corcoran approach. Further, the authors provide an alternative approach to remove 1/f noise using polynomial fitting. In all cases, the author could replicate their main findings. While I share the concerns of reviewer #1 that the instantaneous IAF may be partially a misnomer because frequency cannot be estimated instantenously (i.e., it needs extensive time windows for computation to achive sufficient frequency resolution to distinguish tiny frequency differences between conditions), the author provide convincing complementary analysis their their results are robust using other methods, avoiding the inst. IAF method’s potential pitfalls.\n• Power confounds: The authors provide additional analyses showing that power differences between correct and incorrect decision are absent or minimal (also using Bayes factors), esp. compared to the IAF effects. (Minor: the supplemental fig. seems to lack a color bar description and a figure legend). So power confounds of the IAF effects appear unlikely.\n• Previous reviewers also mentioned potential eye motion confounds, which were adressed previously. I do not think that eye motion confounds are specifically problematic in this study because at prestimulus time points, before any stimulus, participants have no information to alter their eye motion behavior. In principle, a link between differentiial eye motion (e.g. blinks) and IAF is conceivable, where IAF then arose from the eye motion, but this appears rather unlikely. A simple control analysis would be to compare e.g. blink frequency before stimulus onset between correct and incorrect trials.\nOverall, I want to emphasize that the study is very well conducted and an important contribution. When reading the manuscripts, some additional points (none major) came to my attention, that may help to further improve the work:\nIntro:\n• The theoretical link between IAF and alpha phase appears plausible on first sight, because faster IAF leads to faster cycling trough optimal and non-optimal phases, so that the optimal phase is more likely hit by the stimulus. On the other hand, faster cycling also means that the oszillator stays only for a shorter period within the optimal phase, compared to a slower IAF. So the ‘positive’ effect of faster phase cycling could be well offset by a shorter duration within the optimal phase. This could be formalized/visualized by a phase-over-time plot.\nResults:\n• Time-resolved analyses: Strikingly, sensitivity and accuracy already diverge (even though non-significantly) much earlier than the stimulus onset, already at -800ms. One would expect that at least for very early prestimulus time points (e.g. -1.5 s), IAF is not predictive of sensitivity/accuracy. Analyzing such a temporal emergence over longer time periods would serve as an important control, enhancing the specifity of a main finding.\n• Alpha phase analysis: The author report alpha phase effects that occur around stimulus onset (starting at -49ms) or after stimulus onset. Thus, it appears that the phase effect is related to stimulus processing, because the prestimulus effect could well arise from temporal smearing of the time-frequency analyses into the the peri-stimulus window. So the phase effect may not arise from ongoing alpha oscillations. Further, the authors may show a (supplemental) figure of the time course of the effect.\n• IAF x phase interaction analysis: This analysis mixes within- (phase) and between (IAF) subject levels: Showing that a phase-effect exists in low IAF individuals, but not in high IAF individuals does not allow the conclusion of a significant difference of phase effects between IAF groups (i.e. absence of evidence is not evidence of absence). This could be formally assessed with an interaction effect in this analysis. In principle, this analysis could also be computed purely within individuals, e.g. using a 2 x 2 binning approach on IAF x phase bin, showing an interaction effect. The trial-by-trial analysis of interaction effects shows that this is in principle the case, but a 2x2 approach could well visualize the effect.\n• ITPC analysis: Again, the question is whether correct-incorrect contrast differs between high and low IAFs. This interaction effect is not tested directly.\nDiscussion:\n• Fig. 7: Even though the model is convincing at first sight, it makes a critical assumption: For accurate visual perception, the optimal phase must be hit, independent for how long stimulus processing falls into this phase. Under this assumption, higher IAF will indeed increase the likelihood that a stimulus at least briefly touches the optimal phase due to rapid cycling. On the other hand, the amount of time a stimulus is processed during the optimal phase could also be critical for accurate perception. Here, a lower IAF may increase the probability for a high optimal-phase duration. Thus, these two effects might cancel each other. Only a quantitative model could distinguish these cases.\nMinor\n• Line 335: The polar plot should only be presented describtively, but phases should not be tested again because they arise from the already significant cluster (double dipping).\n• Line 434: Reference to Fig. 7 may make more sense?"}, "author_response": "We thank the reviewer for this valuable feedback. Moreover, we have now added a color bar description to the figure, specifying what the color represents, as this information was missing in the previous version.\n• Previous reviewers also mentioned potential eye motion confounds, which were adressed previously. I do not think that eye motion confounds are specifically problematic in this study because at prestimulus time points, before any stimulus, participants have no information to alter their eye motion behavior. In principle, a link between differentiial eye motion (e.g. blinks) and IAF is conceivable, where IAF then arose from the eye motion, but this appears rather unlikely. A simple control analysis would be to compare e.g. blink frequency before stimulus onset between correct and incorrect trials.\nWe thank the reviewer for bringing up this point. As correctly noted, eye motion confounds were already considered in previous versions of the manuscript. We would remark here that we performed ICA-based artifact correction on all EEG datasets, carefully identifying and removing components associated with blinks and eye movements based on their characteristic scalp topographies and time courses. This procedure is a well-established approach to eliminate ocular artifacts from EEG data (Cohen, MIT Press, 2014). Moreover, as you stated, since the effects of interest concern prestimulus time points, when no visual information is yet available to drive systematic eye movements, it is highly unlikely that residual eye activity could explain the observed differences. Taken together, these steps ensure that our findings cannot be attributed to eye motion confounds.\nOverall, I want to emphasize that the study is very well conducted and an important contribution.\nThank you for your thoughtful evaluation and for recognizing the rigor of our work and the effectiveness of our responses to prior concerns.\nWhen reading the manuscripts, some additional points (none major) came to my attention, that may help to further improve the work:\nIntro:\n• The theoretical link between IAF and alpha phase appears plausible on first sight, because faster IAF leads to faster cycling trough optimal and non-optimal phases, so that the optimal phase is more likely hit by the stimulus. On the other hand, faster cycling also means that the oszillator stays only for a shorter period within the optimal phase, compared to a slower IAF. So the ‘positive’ effect of faster phase cycling could be well offset by a shorter duration within the optimal phase. This could be formalized/visualized by a phase-over-time plot.\nWe thank the reviewer for this insightful comment. We fully agree that faster alpha rhythms entail a trade-off between more frequent oscillatory cycles, providing increased opportunities for a stimulus to coincide with an optimal excitability phase, and shorter durations spent within each instantaneous phase. This is indeed a key conceptual point that we address more systematically in our response to your final comment, which relates to the same issue (see below). We also have now explicitly acknowledged this conceptual balance in our theoretical reasoning in the revised Discussion (see below). Briefly, we consider it unlikely that perceptual accuracy depends merely on how long the system remains within a single optimal phase. A longer single optimal phase represents an integration window of a single sampling unit, like a snapshot, where a longer duration allows for just one sample rather than multiple successive ones. Empirical evidence indicates that slower alpha rhythms favor temporal integration at the expense of segregation and temporal precision (Wutz et al., 2018, PNAS; Samaha et al., 2015, Current Biology; Cecere et al., 2015, Current Biology; Frisoni et al., 2025 Cortex). Such prolonged integration may be advantageous in contexts requiring stable perceptual grouping (Wutz et al., PNAS, 2018), but in tasks demanding rapid updating or precise temporal parsing, faster alpha cycling — promoting segregation and dynamic sampling — provides a clear advantage. Faster alpha rhythms may therefore enhance perception by enabling multiple rapid “checks” of the sensory input, allowing the brain to re-sample and refine information within the same temporal window. This interpretation anticipates the theoretical reasoning expanded in the revised Discussion, where we integrate this trade-off into a broader framework of recurrent sensory sampling and dynamic updating (see also Samaha et al., Current Biology, 2015).\nImportantly, this reasoning also suggests that what matters may not be the absolute duration spent within an optimal phase, but rather the number of distinct opportunities the system has to align with such a phase. In other words, repeated sampling may confer a stronger benefit than prolonged single exposures, as each new cycle provides an independent chance to validate or revise the initial sensory evidence. A similar principle has been empirically demonstrated in the two-flash fusion paradigm8, where temporal discrimination was not enhanced by longer single perceptual episodes, but instead by the opportunity to generate two separate sampling events. Consistent evidence comes from Cecere et al. 10, who demonstrated that individual alpha frequency determines the temporal window of perceptual integration, with slower alpha rhythms leading to longer integration windows and, consequently, greater temporal fusion across sensory events. Similarly, Wutz et al. 45 found that the peak frequency of alpha oscillations increased when visual task demands required temporal segregation compared with integration. Thus, a faster IAF increases not only the likelihood of “hitting” the optimal phase at least once, but also the number of distinct samples that can be accrued within a given interval. By contrast, slower rhythms, although extending the duration of each phase, risk both missing the optimal phase altogether and relying excessively on a single, potentially misleading shot of information.\nTaken together, this framework reinforces the view that perceptual accuracy is optimized through multiple brief samplings rather than prolonged single exposures.\nWe have also amended the caption of Figure 7 to clarify this reasoning, emphasizing that the duration spent within a single, even optimal, phase is not the key driver of perceptual accuracy Rather, accuracy depends on how many opportunities the oscillatory cycle has to align with an optimal phase across recurrent stimulus presentations. This clarification highlights that faster alpha frequencies enhance perceptual performance by increasing the likelihood of such optimal phase alignments within a given time window.\nResults:\n• Time-resolved analyses: Strikingly, sensitivity and accuracy already diverge (even though non-significantly) much earlier than the stimulus onset, already at -800ms. One would expect that at least for very early prestimulus time points (e.g. -1.5 s), IAF is not predictive of sensitivity/accuracy. Analyzing such a temporal emergence over longer time periods would serve as an important control, enhancing the specifity of a main finding.\nWe thank the reviewer for raising this important point. To address it, we extended our analyses to an earlier prestimulus window (–2500 to –100 ms). These additional results show that IAF predicts sensitivity and accuracy even at ~1.5 s before stimulus onset. This suggests that IAF captures an oscillatory state that can influence perceptual performance well in advance of stimulus onset. Importantly, however, when considering very remote periods (–2500 to –1500 ms), performance is indistinguishable across the considered terciles. Moreover, the predictive effect emerges more clearly as the stimulus approaches, and becomes markedly stronger and more consistent in the immediate prestimulus interval. Finally, such long-range predictive effects are not unprecedented. For example, Samaha et al. (PNAS, 2015, Fig. 4B, see below) observed that IAF predicted performance already about one second before target onset. Our findings are thus consistent with prior work showing that alpha dynamics can convey predispositional signals well before the stimulus, while gaining functional specificity in the critical prestimulus period.\nWe included the presented analysis in the new version of the supplementary materials\n• Alpha phase analysis: The author report alpha phase effects that occur around stimulus onset (starting at -49ms) or after stimulus onset. Thus, it appears that the phase effect is related to stimulus processing, because the prestimulus effect could well arise from temporal smearing of the time-frequency analyses into the the peri-stimulus window. So the phase effect may not arise from ongoing alpha oscillations. Further, the authors may show a (supplemental) figure of the time course of the effect.\nThank you for this useful comment. We have now included an additional figure (new Supplementary Figure S6) showing the full-time course of the alpha phase effect in the whole sample. Regarding the concern about potential temporal smearing, we would like to clarify that the reported phase effect reflects a phase-dependent modulation of perceptual accuracy - that is, behavioural performance differs as a function of the instantaneous alpha phase in the peristimulus time. We agree that temporal smearing in the time-frequency decomposition may blur the exact temporal boundaries of the estimate, but it cannot artificially produce a systematic relationship between phase and behaviour. Smearing would affect temporal precision, not the existence or direction of a phase-behaviour coupling. Importantly, when comparing the phase-related effect in participants binned by their individual alpha frequency (i.e., high and low IAF individuals), we observed significant phase effects also at time poin"}]

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