EEG tensor decomposition delineates neurophysiological principles underlying conflict-modulated action restraint and action cancellation

Executive functions are essential for


Introduction
Executive functions (or the often synonymously used term cognitive control) are essential for adaptive behavior (Diamond, 2013), one important aspect of which relates to "interference control" (Diamond, 2013;Friedman and Miyake, 2004;Frings et al., 2015).Yet, "interference control" encompasses many different functions, which have been subject to distinct streams of research in cognitive neuroscience.Interference control processes, for example, have been conceptualized in conflict monitoring theory (Botvinick et al., 2001;Keye et al., 2013).In the forthcoming we use the term conflict monitoring when we refer to interference control processes.Importantly, conflict monitoring theory has not yet often been considered in relation to "response inhibition" (Bari and Robbins, 2013), even though one means to resolve cognitive conflicts is via inhibitory control (Diamond, 2013) and the concepts of inhibitory control and conflict monitoring are closely related.In fact, it has been suggested that inhibitory control processes are needed to manage conflicts; i.e. the unwanted (incompatible/incongruent) response representation (Chmielewski et al., 2014;Klein et al., 2014;Ocklenburg et al., 2011;Stürmer et al., 2000;Verleger et al., 2009;Willemssen et al., 2009).This has been corroborated by several human (Tandonnet et al., 2011;Taylor et al., 2007) and monkey studies (Cisek and Kalaska, 2005).The close links between conflict monitoring and inhibitory control have led to the question in how far both processes interact during cognitive control: Along these lines, it has been shown that conflict monitoring and response inhibition processes interact (Chmielewski et al., 2018;Chmielewski and Beste, 2017;Wendiggensen et al., 2022b) and do thereforeaccording to the additive factor logic (Sternberg, 1969) depend on similar processes and may together reflect the ability to monitor and resolve conflicts/interference.Within the field of response inhibition, different streams of theorizing and research can be distinguished (Bari and Robbins, 2013) such as action restraint or action cancelation (Schachar et al., 2007;Verbruggen et al., 2019).Action restraint stresses proactive inhibitory control processes and is typically examined in Go/Nogo tasks, whereas action cancelation stresses reactive inhibitory control processes and can be investigated in Stop signal tasks (Raud et al., 2020).Yet, it has to be noted that some studies call this distinction into question (Chang et al., 2017;Kaiser et al., 2022), and many conflict tasks involve a combination of proactive and reactive processes (Cooper et al., 2015;Kaiser and Schütz-Bosbach, 2019, p. 2).Regardless of this debate, it has been shown in a Simon task design (Hommel, 2011a) that both action restraint (Chmielewski and Beste, 2017) and action cancelation (Eggert et al., 2023) processes are affected by concomitant stimulus-response conflicts as for example induced by a Simon Task.During a Simon task (Simon and Rudell, 1967;Simon and Small Jr., 1969), individuals are asked to respond to the identity of stimuli (e.g.letters A and B) by a left-or right-hand button press.The important experimental manipulation is that the letters are shown at the left or right on a centrally positioned display.The effect is that whenever the stimulus location and the response effect do not match in their position, a response conflict is induced.Conflicting stimulus-response information affects behavioral performance in action restraint and action cancelation, and there were also modulations in fronto-central and centro-parietal event-related potential (ERP) components (Chmielewski and Beste, 2017;Eggert et al., 2023).These modulations occurred in the N2-P3 complex thought to index conflict monitoring (Folstein et al., 2008;Larson et al., 2014) and different inhibitory control subprocesses depending on modulations in the theta, alpha and beta frequency bands (Huster et al., 2013).
As of now, it is, however, unclear in how far inhibitory control and conflict monitoring really share a common neurophysiological ground.To examine this more closely, it is important to examine response inhibition processes within contexts imposing more less demands on conflict monitoring processes and to examine in how far there are neurophysiological signatures reflecting effects of response inhibition and conflict monitoring.Yet, it is important to also distinguish different forms of inhibitory control processes.Successful goal-directed behavior in everyday life often involves navigating distractors/interfering input, including actions in situations where responses are to be inhibited.Therefore, the previously described conflict effects on both action restraint and action cancelation warrant further investigation.Accordingly, we use the aforementioned Simon Nogo task and Simon Stop task (Chmielewski and Beste, 2017;Eggert et al., 2023), thereby incorporating conflict demands in a Go/Nogo task and in a stop signal task, enabling the examination of the impact of interference/conflicts on both types of response inhibition to reveal underlying shared or distinct mechanisms.The Simon task setup is particularly suitable to investigate conflict processing since, unlike other tasks inducing interference (e.g., the Flanker task or the Stroop task), processes related to stimuli and responses are not confounded (Hommel, 2011a).Interactive effects between conflict monitoring and both action restraint and action cancelation processes suggest that all three aspects may share common mechanistic principles that contribute to interference control, the neurophysiological underpinnings of which remain unknown.This is of conceptual importance because it provides insights into the neural architecture of executive functions, as stressed in older (Diamond, 2013;Friedman and Miyake, 2004), but also newer conceptual framings of action control (Beste et al., 2023;Frings et al., 2020).
Especially newer framings suggest that nominally different action/ response control processes are explainable by a small set of cognitive and neurophysiological processes (Beste et al., 2023;Frings et al., 2024Frings et al., , 2020)).The central step to elucidating these neurophysiological processes is using a suitable method to examine neurophysiological data (e.g., EEG data) from different experiments.Principal component analysis (PCA) and independent component analysis (ICA) have often been used to extract neurophysiological sources and structures using EEG signals (Mike X. Cohen, 2014, p. 20,014;Handy, 2009), but these methods are optimized for two-dimensional data (e.g.covering the spatial and temporal information of EEG data).Crucially, data from typical EEG experiments can yield up to seven dimensions: time, space, frequency, trial, condition, participant, and group/experiment (Cong et al., 2015), the latter of which is of considerable importance when investigating possible overarching neural principles that are relevant across different experimental implementations.These many dimensions can mathematically best be described in tensors.Applying methods optimized for two-dimensional data (i.e., PCA and ICA) is only possible by reducing the data dimension; e.g., by concatenating or stacking the data (Gholamipourbarogh et al., 2022b;N. 2022a;Jonmohamadi et al., 2020).This, inevitably leads to loss of information (Cong et al., 2015).Tensor decomposition techniques, on the other hand, can capture all dimensions of information contained in EEG data (Cong et al., 2015) through exploiting the interrelation between various dimensions of a given dataset, thereby identifying and systematizing a possible core set of neurophysiological processes.In the current study, we therefore use a fourth-order tensor decomposition to look into possible common neurophysiological mechanisms underlying conflict-modulated action restraint and action cancelation.This is an important methodological proof-of-principle step towards a possible overarching conceptual framework of neurophysiological principles underlying multiple domains of action control (Beste et al., 2023).Tensor decomposition yields several components that can be characterized by their frequency spectrum, the time domain and also the combined the time-frequency spectrum information.The latter reflects information that is typically revealed through time-frequency decomposition (wavelet) analysis in more traditional EEG analyses and therefore provides strong connections to research methods more commonly applied in the field.In the current study, we use tensor decomposition methods in explorative analyses to examine processes between conflict-modulated action restraint/action cancelation (i) on a spatial level (scalp topography), (ii) in the time domain, (iii) in the frequency domain, (iv) in the time-frequency domain as provided by the outer product of the time and frequency domain, and (v) on the subject-condition level.
In essence, in the used tensor decomposition analysis the temporal, spectral, and spatial components are consistent across all subjects and the only variations would be in subject and condition features.In our case, this would be the last features in the decomposition process.Consequently, the primary objective of this study is to conduct a comprehensive comparison of these variations among subjects and conditions.The fourth-order tensor decomposition analysis facilitates a direct examination of subject-specific differences within the common components, providing a robust and efficient means of characterizing individual responses without the need for additional source localization techniques typically used to compare distinct brain regions or sources across different subjects.

Participants
The sample recruited for the current study consisted of N = 37 righthanded participants.As determined by a telephone screening, only participants who reported no psychiatric or neurological illnesses were invited to take part in the study.In order to detect psychiatric morbidity, all participating subjects completed the Adult Self-Report for ages 18-59 (ASR; Achenbach, 2015) via an online-questionnaire on SoSci.de(Leiner, 2019) prior to the in-person appointment.Furthermore, the participants completed the Mehrfachwahl-Wortschatz-Intelligenztest (Version B, MWT-B; Lehrl, 2005); all participants had an average IQ >91.Altogether, 9 participants were not included in the final sample due to scores above the cut-off values in the ASR, outliers in the behavioral data, or an insufficient number of trials for the analysis of the EEG data.The final sample consisted of N = 28 (19 females, age range 20 to 33, mean age 25.75 ± 3.94).We planned our study with standard Simon effects in mind that have been reported with large effect sizes (dz > 0.8).Thus, using an alpha of α = 0.05 we aimed for a power of 1-β = 0.99, to reliably replicate the Simon effect, and accordingly needed a sample of at least N = 27 participants as determined by G*Power (Faul et al., 2007) with F tests as the test family, "ANOVA: Repeated measures, within factors" as the statistical test and an a priori type of power analysis.The participants signed an informed consent form and received either financial reimbursement or course credit upon the completion of their study appointment.The ethics committee of the Faculty of Medicine of TU Dresden approved the study.The participants performed both tasks (see below) in a counterbalanced order.

Task -Simon Stop (conflict-modulated action cancelation)
The task paradigm combined a Simon task (Simon and Rudell, 1967;Simon and Small Jr., 1969) with a stop signal setting (Logan et al., 1984) as recently published by our group (Eggert et al., 2023).In the center of a blue screen, a fixation point was shown during the entire duration of the experiment.To both sides of the fixation point, a white frame box was shown.Each trial constituted the presentation of either the letter "A" or the letter "B" in one of the two white frame boxes.The letter stimuli were presented in normal font and yellow color.The participants were instructed to press the left control key with the left index finger when an "A" was shown and to press the right control key with the right index finger when a "B" was presented.The participants were asked to disregard the spatial location of the letter stimulus.Hence, two conditions essential to a Simon task were created: A response of the hand on the same side as the location of the presented letter stimulus established a congruent trial, while a response of the hand on the opposite side as the location of the presented letter stimulus established an incongruent trial.
The task consisted of Go trials and Stop trials in order to incorporate the stop signal element.In Go trials, the letter stimulus remained unchanged for 1700 ms or until a response was given, which was determined to be either "correct" or "incorrect".The lack of a response in Go trials was classified as a "miss".Crucially, the Stop trials included a stop signal, i.e. the respective letter stimulus turned red after a variable delay.Here, the participants were instructed to stop any response that might have already been initiated.The stop signal was shown for 1700 ms.In line with Verbruggen et al. (Verbruggen et al., 2019), we aimed to achieve a stopping probability of 0.5.Accordingly, the time period between the onset of the presentation of the letter stimulus and the letter turning redi.e. the stop signal delay (SSD) -was varied, depending on the response accuracy of each participant in the preceding trial.At the beginning of the task, the SSD was 250 ms.In the event of a "correct rejection" (i.e. the participant correctly stopped a response in a stop trial), the SSD was increased by 50 ms in the subsequent trial with an upper limit of 1000 ms.In the event of a "failure to stop" (i.e. the participant was not able to stop the response in a stop trial), the SSD was decreased by 50 ms in the subsequent trial with a lower limit of 50 ms.When the mean reaction time of the preceding 50 trials was larger than the set reaction time of 450 ms, the prompt "Bitte versuchen Sie, schneller zu drücken" (German for "Please try to respond faster") was displayed in the center of the screen for 2000 ms.The duration of the intertrial interval was jittered between 1300 and 1700 ms.
Before the start of the experiment, all participants performed a standardized exercise consisting of 36 trials.Here, direct feedback regarding the accuracy of responses was presented to the participants after every trial.The main task comprised of a total of 936 trials, with 77% Go trials (720 trials) and 23% Stop trials (216 trials).Half of the Go trials and half of the Stop trials were congruent and incongruent, respectively (360 congruent/incongruent trial in the Go conditions, 108 congruent/incongruent trials in Stop conditions).The task consisted of nine blocks, each of which included 80 Go trials and 24 Stop trials.Between blocks, it was possible for the participants to take a break.The trials were presented in a randomized order.The total task duration was approx.40 min.

Task -Simon Nogo (conflict-modulated action restraint)
The task combined a Simon task (Simon and Rudell, 1967;Simon and Small Jr., 1969) with a Go/Nogo task (Chmielewski et al., 2018;Ghin et al., 2022).Throughout the entire experiment, a white fixation cross was shown in the center of a black screen.To both the right and left side of the fixation cross a white box was displayed.Every trial commenced with the presentation of a stimulus in the form of a letter (A or B) for 200 ms in one of the two boxes, while a distractor stimulus in the form of three vertically aligned bars was shown in the respective other box.The task consisted of Go trials and Nogo trials: In Go trials, the letter stimulus was shown in normal font, while it was shown in cursive and bold font in Nogo trials.In Go trials, participants were asked to respond as quickly as possible by pressing the left control key with the left index finger upon the presentation of an "A" and by pressing the right control key with the right index finger upon the presentation of a "B".The response was to be given irrespective of the location of the letter stimulus, thereby generating congruent trials when the letter stimulus was presented on the same side as the responding hand (A on the left or B on the right) and incongruent trials when the letter stimulus was presented on the opposite side from the responding hand (A on the right or B on the left).In the Nogo trials, the participants were asked to withhold any response.
The experiment consisted of 720 trials in 6 blocks of equal length, which were separated by short breaks.Every block comprised of the same ratio of Go trials to Nogo trials.Half of the Go trials and half of the Nogo trials were congruent.All conditions were distributed equally and randomized across all blocks of the task.In Go trials, a reaction in the time window of 250-1000 ms after the onset of the presentation of the stimulus was classified as a valid response.In the event of an erroneous reaction, the response was classified as "incorrect", while the omission of a response was classified as a "miss".If no response had been given after 500 ms, the prompt "Faster!" was shown on above the fixation cross on the screen.In Nogo trials, any response in the time window of 250-1000 ms after stimulus presentation constituted a "false alarm".Every trial was concluded after 1700 ms.The intertrial interval had a duration between 1300 and 1700 ms.The total task duration was approx.30 min.
Fig. 1 shows the different kinds of trials inherent to both tasks.Both tasks were presented using the Presentation software.All participants carried out both the Simon Stop task and the Simon Nogo task in the same session.The tasks were presented in a randomized order.
In order to investigate both proactive and reactive cognitive control processes (i.e.action restraint and action cancelation) in the same context/experimental procedure, we also combined both tasks in a unified experimental design ("Simon Inhibition").Here, we used a cue to distinguish between the two task elements and thereby provided the same preparatory time window before the stimulus onset for both tasks.Further information regarding the task design, data collection and analysis of this experiment examined in an unrelated sample of participants is provided in the supplementary material (supplementary note 2).

EEG recording and pre-processing
60 Ag/AgCl electrodes in an elastic cap (EasyCap) were fitted to the participants for the recording of the EEG, using BrainAmp amplifier and the Brain Vision Recorder 1.2 software (Brain Products).The coordinates θ = 58, ϕ = 78 and θ = 90, ϕ = 90 were determined as locations for the ground and the reference electrodes, respectively.Impedances were maintained at below 5kΩ and the sampling rate was set to 500 Hz.Automagic (Pedroni et al., 2019) and EEGLAB (Delorme and Makeig, 2004) were used for the preprocessing, which entailed the resampling of the data to 256 Hz as a first step, followed by the removal N. Gholamipourbarogh et al. of flat channels and the re-referencing of the EEG data to an average reference.Subsequently, two different pipelines were applied; the PREP preprocessing pipeline (Bigdely-Shamlo et al., 2015) and the EEGLAB clean_rawdata() pipeline.By applying a multitaper algorithm, the PREP pipeline eliminated line noise at 50 Hz and succeeding the removal of interference caused by bad channels, another average reference was employed.Using a FIR high-pass filter of 0.5 Hz (order 1286, stop-band attenuation 80 dB, transition band 0.25 -0.75 Hz), the clean_rawdata pipeline first identified and discarded noisy or outlier channels, thereby detrending the EEG data.After the subsequent reconstruction of epochs displaying particularly strong power (i.e., >15 SDs in relation to calibration data) by applying Artifact Subspace Reconstruction (ASR; burst criterion: 15; (Mullen et al., 2013) and the ensuing removal of time windows, which could not be reconstructed, a lowpass filter of 40 Hz (sinc FIR filter; order: 86; Widmann et al., 2015) was implemented.The following step entailed the removal of electrooculography artifacts by employing a subtraction method (Parra et al., 2005) as well as the classification and removal of other remaining artifacts (e.g., artifacts related to muscle movement or loose electrodes) based on an independent component analysis (ICA) using the Multiple Artifact Rejection Algorithm (MARA; Winkler et al., 2014Winkler et al., , 2011)).After the detection and removal of components including cardiac artifacts using ICLabel (Pion-Tonachini et al., 2019), an interpolation of the previously removed channels was carried out using a spherical method.
During the subsequent segmentation of the data of both tasks, segments locked to the target stimulus (− 2000 to 2000 ms) were established for all conditions (correct Go trials and correct rejections in Nogo/ Stop trials in both the congruent and incongruent conditions).A baseline correction was carried out based on the time window of − 200 ms to 0 ms before stimulus presentation.The choice of the time range (− 200 to 0 ms) for baseline correction was influenced by a previous study that examined a similar task (Eggert et al., 2023) and to keep the methodological approach with this earlier work.We conducted baseline correction as a preliminary step for further investigation into ERP amplitudes.Although time-domain baseline correction is not required for subsequent time-frequency analysis, it does not negatively impact the data.

Tensor decomposition
We used the tensor decomposition method to investigate data from the Simon Nogo (action restraint-related task) and the Simon Stop (action cancelation-related task) task (with different congruency and response conditions) and to extract multi-domain features.
We utilized a four-dimensional tensor that incorporated information about frequency, time, channel, and subject-condition-task relationships.This allowed us to capture spectral, spatial, and temporal information, as well as their variations across different subjects, conditions, and tasks through the extracted components.To construct the tensor, we obtained a total of 40 trials (epochs) for each subject in each condition, spanning a time duration of − 200 to 1000 ms.We obtained average trial data by averaging the epochs within each subject and across different conditions and tasks.The averaged epochs were then transformed into a time-frequency representation (TFR) using the continuous wavelet transform (CWT), enabling us to examine frequencies ranging from 0.5 Hz to 20 Hz.The TFR data from all subjects, conditions, and tasks were used to generate the fourth-order tensor.The tensor had dimensions of frequency × time × channel × subject-condition-task, specifically 32 × 154 × 60 × 224.Here, 32 frequency points represented the spectrum from 0.5 to 20 Hz, 154 time points represented a duration of 1.2 s, 60 channel points represented the scalp electrodes, and 224 indicated data from 28 subjects across 4 different conditions (congruency and response), and 2 tasks (Simon Nogo and Simon Stop).

Tensor decomposition algorithm
One commonly used model for tensor decomposition that generalizes singular value decomposition to higher-order tensors is the CANDE-COMP/PARAFAC (CP) model (Kolda and Bader, 2009).The CP model can be used to extract low-dimensional, multi-connected descriptions from multidimensional data by breaking it down into a sum of rank-1 tensors of smaller dimensions.This decomposition factorizes a tensor into a sum of component rank-1 tensors, each of which represents a unique pattern in the data.In our study, we employed Nonnegative CP (Canonical Polyadic) tensor decomposition using a hierarchical alternating least squares algorithm with an inexact block strategy (https://g ithub.com/wangdeqing/Nonnegative_Tensor_Decomposition)using MATLAB based Tensor toolbox (Brett W. Bader, Tamara G. Kolda and others, n.d.).This approach allows us to estimate nonnegative and sparse components.The algorithm was to incorporate sparsity as a criterion for component estimation due to its relevance in EEG tensor decomposition.Specifically, the extracted spectral components in EEG data often exhibit sparsity, indicating the presence of narrow-band frequencies that are associated with specific brain activities.By leveraging sparsity as a criterion, we aim to capture these distinct spectral patterns in the decomposition process.Additional details regarding the algorithm will be provided.
In this paper, operator For an N-mode tensor X in ℝ I1×⋅⋅⋅×IN , the rank-R CP decomposition of X is represented by the approximation equation shown, where a r (n) ∈ ℝ In are unit vectors with weight vector λ∈ℝ R , and • denotes the outer product.
In the NCP tensor decomposition, the non-negativity constraint is added to the CP decomposition, which means a 1 0 for all r.

Optimization
Given an Nth-order nonnegative tensor X∈ ℝ I1×⋅⋅⋅×IN and a positive number R, the NCP aims to solve the following minimization problem: N are the estimated factor matrices in different modes, I n is the size in mode-n, and R is the selected number of components.In the Kruskal operator, the estimated factor matrices can be expressed as the sum of R rank-1 tensors in outer product form: where a r (n) represents the rth column of A (n) .
To address rank deficiency and convergence issues, the objective function in Eq. ( 2) includes sparse regularization terms using a proximal algorithm.This incorporates a squared Frobenius norm as a proximal regularization item, which promotes sparsity in the feature matrices.To further improve the efficiency of the decomposition process, an inexact block coordinate descent (IBCD) scheme is incorporated.The IBCD scheme optimizes the features by iteratively updating one feature while keeping the other fixed.It alternates between updating features until convergence.The IBCD scheme is designed to handle large-scale tensors and significantly reduces the computational burden.The algorithm also converges to a stationary point of the optimization problem, ensuring the accuracy and reliability of the decomposition results (Wang et al., 2021).

Component number selection
When processing and analyzing EEG signals using tensor decomposition, it is crucial to determine the appropriate number of extracted components for each mode.Determining the tensor rank R can be challenging in tensor decomposition.We used stability as a criterion for selecting the number of components.For the stability analysis of extracted components, an algorithm called tensor spectral clustering (TSC) was implemented.This method allows us to co-cluster and assess the stability information from different modes simultaneously.The procedure of TSC for clustering samples with multiple modalities are as follows: 1. Formation of weighted adjacency matrices for each mode.2. Definition of transition matrices for each mode, leading to the formation of a transition tensor.3. Extraction of the top k eigenvectors corresponding to the largest eigenvalues of the last mode of the transition tensor using Higher Order Singular Value Decomposition. 4. Normalization of the extracted eigenvectors.5. Clustering of the normalized vectors using Hierarchical clustering.6. Assignment of original samples to clusters based on the clustering results.
We performed the tensor decomposition on a given dataset multiple times (K times) using the same algorithm and parameters, but with different initial conditions for each run.The algorithm functions based on this reasoning that a stable component should always appear in each execution, demonstrating the reliability and effectiveness of it.Given, the model order R, for each mode (spectral, temporal, spatial, subjectcondition), there were R × K components.Then the correlation between each pair of components was calculated for each mode to create similarity matrices.These similarity matrices (for each mode W spectral , W temporal , W spatial and W subject-condition ∈ ℝ RK×RK ) were then used as input for clustering.The number of clusters was defined as the same as the number of extracted components R, with stable components producing a tight cluster.The stability index is quantified with the average intra-cluster similarities: where 〈L i , L j〉 denotes the amount of similarity between the i th and j-th components and S k represents the set of indices that jointly make up the k-th cluster.In a stable component extraction, the inner similarity of the corresponding cluster should be close to 1, and the stability index of unstable components would be close to 0. The average of the component stability indices is used to define algorithm stability.The algorithm is stable when the chosen model order is appropriate for the tensor being decomposed.As a result, hyperparameters like the model order can be chosen based on the algorithm's stability.A detailed explanation of the tensor spectral clustering method has been provided elsewhere (Hu et al., 2021).All analyses were implemented in MATLAB 2020b.Fig. 2 displays the analysis steps regarding the neurophysiological data.

Behavioral data -Simon stop
All statistical analyses were conducted using SPSS Version 29.The behavioral measures were analyzed separately for the Go trials and Stop trials by performing repeated-measures ANOVAs with the factors "position" (left vs. right) and "congruency" (congruent vs. incongruent).In the event of a lack of normal distribution of any variable (based on the results of Kolmogorov-Smirnov tests), the respective analyses were conducted using Wilcoxon tests.

Go trials (Simon stop)
Overall, the results showed an average accuracy rate of 95.74% in the Go trials.The repeated-measures ANOVA for the accuracy revealed a main effect of "congruency", with participants showing a higher accuracy rate in the congruent condition (97.29% ± 2.37) compared to the incongruent condition (93.87% ± 4.84; F(1,27) = 26.83,p < .001,ƞ p 2 = 0.50).No other main or interaction effect was found to be significant (all F < 2.49, all p > .126).In respect to the reaction times, the results showed a main effect of "congruency", revealing faster responses in the congruent trials (486 ms ± 72) than in the incongruent trials (514 ms ± 68; F(1,27) = 83.22,p < .001,ƞ p 2 = 0.76).No other main or interaction effect was found to be significant (all F < 3.76, all p > .063).

Stop trials (Simon stop)
The mean error rate (i.e. the probability of responding in a Stop trial) was estimated at 50.66% (± 1.85), with a mean SSD of 227 ms.There was a main effect of "congruency" with regard to the error rate, with participants showing a higher error rate in the congruent trials (51.12% ± 2.20) than in the incongruent trials (50.20% ± 1.60; F(1,27) = 19.90, p < .001,ƞ p 2 = 0.42).Furthermore, there was an interaction effect of the factors "congruency" and "position" (F(1,27) = 12.71, p = .001,ƞ p 2 = 0.32).Post-hoc analyses with Bonferroni-correction (now with α = 0.025) revealed that there was no difference in the accuracy rate between positions in the incongruent trials (Z = − 1.41, p = .157).In the congruent trials, on the other hand, the error rate was higher when the stimulus was shown on the right side (51.65% ± 2.73) than when the stimulus was shown on the left side (50.60% ± 2.02; Z = − 2.56, p = .010).The main effect of "position" was not significant (F = 1.77, p = .194).
With regard to the reaction times of the "failures to stop" (i.e.Stop trials where a response was given erroneously), the analysis found a main effect of "congruency", with participants responding faster in the congruent condition (427 ms ± 49) than in the incongruent condition (457 ms ± 51; F(1,27) = 70.01,p < .001,ƞ p 2 = 0.72).None of the other effects in respect to the reaction times reached significance level (all F < 2.12, all p > .157).
The mean estimation method was implemented in order to estimate the stop signal reaction time (SSRT).The results showed a shorter SSRT in the congruent trials (269 ms ± 28) than in the incongruent trials (278 ms ± 33; F(1,27) = 5.26, p = .030,ƞ p 2 = 0.16).No other effect was found to be significant for the SSRT (all F〈 0.58, all p〉 0.451).The distribution of the SSRT is shown in Fig. 3.

Behavioral data -Simon Nogo
Regarding the accuracy in the Go trials, a Wilcoxon test showed a higher accuracy rate in the congruent trials (94.47% ± 4.08) than in the incongruent trials (89.12% ± 8.21; Z = − 3.47, p < .001).In the analysis of the Nogo trials, a paired-samples t-test revealed a higher error rate in the congruent condition (26.79% ± 12.25) than in the incongruent condition (19.78% ± 10.27; t(27) = 5.48, p < .001).
In the analysis of reaction times in the Go trials, a paired-samples ttest showed faster reaction times in the congruent trials (450 ms ± 46) than in the incongruent trials (477 ms ± 45; t(27) = − 9.60, p < .001).Finally, the analysis of the reaction times of the false alarms also revealed faster reaction times in the congruent trials (398 ms ± 40) than in the incongruent trials (425 ± 44; t(27) = − 4.33, p < .001).The distribution of the Nogo false alarm rates is shown in Fig. 4.

Component number selection
We applied the NCP algorithm on a fourth-order tensor dataset, progressively increasing the number of components (R) from 3 to 60.For each R-value, we repeated the NCP algorithm 30 times with different initial values to assess the reproducibility of the estimated components.The stability index was then computed using tensor spectral clustering.To evaluate the performance of our framework, we calculated the average stability index of the estimated components for each model order.Fig. 5 shows the average stability indices for model orders between 3 and 60.
From there on, we pursued a data-driven analysis approach with a data-driven criterion on how to select the tensor solutions to be analyzed in more detail.As outlined, the important aspect for the study question is a three-way interaction of the factors "task", "trial type" and "congruency".The reason is that the tensor decomposition was conducted to examine whether a neurophysiological signature can be isolated showing an effect of "task", "trial type" and "congruency".This is important, because at present it unclear how a conflict modulated action restraint (Simon Nogo task implementation) shares similarities with a conflict modulated action cancelation (Simon Stop task implementation.The main question of the current study thus involved the concomitant consideration of 3 experimental factors.Therefore, especially interaction effects are important to consider.Main effects are shown in the Supplemental Material; see Supplementary Note 1).A series of repeatedmeasures ANOVAs with the factors "task" (Simon Nogo task vs. Simon   Stop task), "trial type" (Go trials vs. Nogo/Stop trials, depending on the type of task) and "congruency" (congruent vs. incongruent) was carried out to statistically analyze the results of the tensor decomposition.In the event that the assumption of normality of a variable was violated, Wilcoxon tests were used in the post-hoc analyses.We started the statistical data analysis for the solution with the highest average stability index (0.947).This was evident for the model order of 7. Importantly, since no three-way interaction was obtained, we then analyzed the solution with the second highest average stability index (0.947), which was evident for the model order of 36 (with the average stability index 0.939).Since a three-way interaction was evident for this model order, data analysis was stopped after this analysis.The features of the extracted components and the way the factors involved interact with each other have been explained in the following.

Tensor decomposition with 7 component numbers.
When applying the tensor decomposition algorithm with 7 component numbers, we identified three components (2, 4 and 6) that demonstrated interaction effects between different factors.All the statistical results of the component analysis as well as the post-hoc analysis can be found in the supplemental material (Supplementary Note 1).Since no three-way interaction of the factors "task", "trial type" and "congruency" was obtained, we analyzed the next solution of the tensor decomposition with the second highest stability index of the estimated components.This was evident for 36 component numbers.

Tensor decomposition with 36 component numbers.
The average stability index of this solution was 0.939.Among the tensor decomposition results obtained with 36 component numbers, nine components (6, 8, 14, 15, 17, 19, 31, 34 and 35) demonstrated interaction effects between different factors.All the statistical results of the post-hoc analyses are shown in Table 1 for the three-way interactions and Table 2 for the two-way interactions.Importantly, three components revealed a three-way interaction of the factors "task", "trial type" and "congruency".Fig. 6 displays the features of the components showing three-way interactions while Fig. 7 shows the features of the components showing two-way interactions.

Components showing three-way interactions. Component 6
Component 6 exhibited positive activity with a peak occurring around 200 ms.Frequency analysis revealed activity between 2 and 4 Hz.Regarding the topography, positive activity was observed in the central-parietal region.The statistical results revealed a three-way interaction of the factors "task", "trial type" and "congruency" for component 6 (F(1,27) = 4.64, p = .040,ƞ p 2 = 0.147).However, on the task level, the interaction of "trial type" and "congruency" was not significant for neither the Simon Nogo task nor the Simon Stop task.Component 14 Component 14 demonstrated sustained positive activity with a peak at around 200 ms.In the spectral domain, positive activity was observed within the frequency range of 2 to 8 Hz, with a peak around 5 Hz.The topography revealed right parietal activity.The statistical results revealed a three-way interaction of the factors "task", "trial type" and "congruency" (F(1,27) = 6.55, p = .016,ƞ p 2 = 0.195).In the Simon Nogo task the results showed a significant "trial type x congruency" interaction, with post-hoc comparisons revealing higher values for the incongruent trials (0.056 ± 0.052) than for the congruent trials (0.038 ± 0.042) in the Nogo trial type, while there was no significant difference between congruency conditions in the Go trial type.On the other hand, the interaction of the factors "trial type" and "congruency" was not significant in the Simon Stop task.Component 15 Component 15 exhibited positive temporal activity, with a peak occurring around 300 ms.In the frequency domain, activity was observed between 2 and 4 Hz with a peak at 3 Hz followed by a smaller activity between 8 and 14 Hz.Spatially, a left parietal-occipital activation was observed.The statistical results revealed a three-way interaction of the factors "task", "trial type" and "congruency" (F(1,27) = 4.60, p = .041,ƞ p 2 = 0.146).On the task level, the interaction of "trial type" and "congruency" was significant in the Simon Nogo task.Here, further post-hoc tests showed higher values in the Nogo trials (0.098 ± 0.111) than in the Go trials (0.066 ± 0.078) in the congruent condition, while there was no significant difference between trials types in the incongruent condition.On the level of trial type, there was no difference between the congruency conditions in the both Go and Nogo trials.In the Simon Stop task, the interaction of "trial type" and "congruency" did not reach significance.

Components showing two-way interactions. Component 8
Temporal analysis of component 8 uncovered a distinct positive waveform in the time period of 200 to 1000 ms after the stimulus onset, with a peak at around 600 ms.Moreover, the spectral analysis indicated activity within the frequency range of 3 to 6 Hz, with a peak at around 4 Hz.Regarding spatial analysis, the feature showed activity distributed prominently in the frontal-central and posterior regions.The statistical analysis showed an interaction effect of the factors "task" and "trial type" (F(1,27) = 10.03,p = .004,ƞ p 2 = 0.271).There were higher values in the Go trials (0.451 ± 0.045) than in the Nogo trials (0.026 ± 0.033) in the Simon Nogo task.There was no significant difference based on trial type in the Simon Stop task.Component 14 The features of component 14 have been explained in the 3.2.2.2.1 section.The statistical analysis showed an interaction effect of the factors "trial type" and "congruency" (F(1,27) = 4.67, p = .040,ƞ p 2 = 0.147).Here, there was no significant difference for any of the post-hoc comparisons for either of the trial types.
Component 17 Component 17 displayed a temporal feature characterized by a distinct peak at 200 ms.In the frequency domain, a peak at 10 Hz was observed with the waveform in the time range of 4 to 16 Hz.The spatial analysis showed a significant positive activation in the parietal-occipital region.A significant interaction effect of the factors "trial type" and "congruency" was revealed (F(1,27) = 7.98, p = .009,ƞ p 2 = 0.228).The post-hoc analysis revealed higher values in the congruent trials (0.115 ± 0.099) compared to the incongruent trials (0.093 ± 0.088) in the Nogo trials, while there was no difference between congruency conditions in the Go trials.
Component 19 Temporal analysis of component 19 showed a prolonged positive peak around 400 ms, while the spectral analysis indicated a positive peak ranging at around 3 Hz.With respect to the spatial feature, positive activations were observed in the central region.The statistical analysis showed an interaction effect of the factors "task" and "trial type" (F (1,27) = 6.56, p = .016,ƞ p 2 = 0.195).There were higher values in the Nogo trials (0.116 ± 0.083) than in the Go trials (0.053 ± 0.045) in the Simon Nogo task.The post-hoc analysis also revealed higher values in the Nogo trials (0.054 ± 0.052) compared to the Go trials (0.029 ± 0.029) in the Simon Stop task.An additional interaction effect between the factors "trial type" and "congruency" was revealed by the statistical analysis (F(1,27) = 6.74, p = .015,ƞ p 2 = 0.200).There were higher values in the Nogo trials (0.093 ± 0.060) than in the Go trials (0.036 ± 0.032) in the congruent trials.Similarly, the post-hoc analysis indicated that the Nogo trials had higher values (0.077 ± 0.058) than the Go trials (0.047 ± 0.039) in the incongruent trials.
Component 31 Examination of component 31 in the temporal domain showed consistent positive activity across the entire duration, with a notable peak at 400 ms.Spectral analysis revealed a positive peak around 4 Hz.Regarding spatial characteristics, positive activations were evident in frontal-central and superior-parietal regions.The statistical analysis showed an interaction effect of the factors "task" and "trial type" (F (1,27) = 5.01, p = .034,ƞ p 2 = 0.156).There were higher values in the Nogo trials (0.109 ± 0.080) than in the Go trials (0.060 ± 0.062) in the Simon Nogo task.However, there was no significant difference observed based on trial type in the Simon Stop task.
Component 34 Temporal analysis of component 34 revealed positive activity peaking around 400 ms.Spectral analysis indicated positive activity within the 2 to 4 Hz frequency range, with a peak at around 3 Hz,

Table 1
Summary of the post-hoc test statistics of the significant three-way interaction effects of the factors "task", "trial type" and "congruency" for all respective components in the model order of 36 (components 6, 14 and 15).Separate repeated-measures ANOVAs were conducted for each task level.In the event of a significant result, Wilcoxon tests were performed.Observed differences in these post-hoc comparisons are presented, along the with respective Z-values and p-values.Significant results are highlighted in bold font.

Table 2
Summary of the post-hoc analyses of the significant two-way interaction effects for all respective components in the model order of 36 (components 8, 14, 17, 19, 31, 34 and 35).In the event of a significant result in the post-hoc comparisons, the observed differences are presented, along the with respective Z-values and p-values.Significant results are highlighted in bold font.followed by smaller activity between 8 and 12 Hz.Regarding topography, significant activity was evident in the right parietal region.Statistical analysis indicated an interaction effect of the factors "trial type" and "congruency" for component 34 (F(1,27) = 5.53, p = .026,ƞ p 2 = 0.170).However, none of the post-hoc comparisons for either trial type showed significant differences.

Component 35
Component 35 displayed a temporal feature characterized by a distinct peak at 200 ms.In the frequency domain, a peak at 10 Hz was noted.The spatial analysis showed a significant positive activation in the left temporal-parietal region.A significant interaction effect of the factors "task" and "congruency" was revealed (F(1,27) = 4.68, p = .040,ƞ p 2 = 0.148).However, none of the post-hoc comparisons for neither task yielded significant differences.
The features of components with significant main effects for factors "task," "trial type," and "congruency" are available in the supplementary material (supplementary note 1).Additionally, the analysis of the components that did not show any statistical effect can also be found in the same supplementary material (Supplementary Note 1).

Secondary experiment
It may be argued that the Stop task related and the Go/Nogo task related implementation of the Simon task differ in their experimental procedure making it difficult to compare the tasks.For example, the Simon-stop version includes a variable time delay before the target occurrence (turning red or not), which induces an important proactive effect of the cognitive control.Therefore, we designed an additional task incorporating both the Simon Stop and the Simon Nogo trials.This additional experiment is shown in the supplemental material see Supplementary Note 2).Here, the trials pertaining to each task type are preceded by a respective cue (instead of using a letter, we implemented a triangle and a circle, to avoid any overlap with the letters A and B that are used as stimuli).This design enables the participant to prepare for the upcoming trial based on the knowledge that it will either be a Go or Stop trial (pertaining to the Simon Stop element of the task, preceded by a circle as a cue), or a Go or Nogo trial (pertaining to the Simon Nogo element of the task, preceded by a triangle as a cue).Thus, for both types of tasks the subject now engages in a preparatory process.The results for this task (i.e. the task incorporating both the Simon Nogo and the Simon Stop tasks in the same experimental design) task can be found in the supplementary material (see Supplementary Note 2).Overall, the main findings here constitute interaction effects between the factors "trial type" and "congruency" for 3 of the components (out of 14 components), showing higher condition feature values in the incongruent trials compared to the congruent trials in the Go condition, with no differences between the congruency conditions in the Nogo trials.Please refer to the supplementary material for further details Supplementary Note 2).

Discussion
The focus of the current study is to examine overarching neurophysiological processes underlying conflict-modulated action restraint and action cancelation processes as processes recruited by interference control.This question is of conceptual relevance because it provides insights into the neural architecture of executive functions, as highlighted in older (Diamond, 2013;Friedman and Miyake, 2004), and more recent research on action control (Beste et al., 2023;Frings et al., 2020).Previous findings suggest that the brain implements different types of conflicts resolution processes via different cognitive and neural mechanisms (Kaiser et al., 2019;Kaiser and Schütz-Bosbach, 2021;Verbruggen et al., 2014).On the other hand, particularly more recently, evidence is accumulating that neural reactivity during conflict processing can be explained by motor slowing and/or sensory processing speed, i.e. general cognitive processes that are not related to any specific subset of cognitive control mechanisms (Beldzik and Ullsperger, 2024;Kaiser et al., 2023;Weigard and Sripada, 2021).Along similar lines of overarching processes, especially newer conceptual framings suggest that nominally different action/response control processes are explainable by a small set of cognitive and neurophysiological processes (Beste et al., 2023;Frings et al., 2020).However, as of now, direct evidence for such overarching neurophysiological processesas evident in different tasks tapping into overlapping cognitive functionshas been elusive because certain available methodological approaches (i.e., tensor decomposition) have not been applied.This study provides proof of principle on the usefulness of tensor decomposition methods to Fig. 6.The spatial features, temporal features, spectral features and outer products of all components in the model order of 36 that revealed a three-way interaction effect, along with illustrations of these interactions.For the latter, * denotes a significant interaction in the post-hoc analysis.
delineate neurophysiological processes underlying conflict-modulated response inhibition processes.The behavioral data replicate previous findings using the tasks on action restraint (Chmielewski et al., 2018;Chmielewski and Beste, 2017;Yu et al., 2022) and action cancelation processes (Eggert et al., 2023), showing that the implementation of the tasks was successful and the tasks used measured distinct aspects of action restraint and action cancelation.
Most important are the results from the tensor decomposition analysis.The tensor decomposition algorithm reached the highest stability for components when 36 components were extracted (i.e., an average stability 0.939).There were three-way interactions between the factors for 3 components (i.e., component numbers 6, 14 and 15).It was shown that only for components 14 and 15 reliable effects were evident.For component 6, the post-hoc tests were not significant.Components 14 and 15 revealed delta/theta band activity at parieto-occipital electrode sites with a maximum activity between 200 and 400 ms.The entire pattern of results reflected by component 14 and 15 (i.e., the only components revealing a substantial three-way interaction of the factors "task", "trial type" and "congruency") suggest that conflict-modulated action restraint reveals different activity patterns depending on different factors (i.e.trial type and congruency).The effect of these different factors can only be derived by applying multi-dimensional analyses like tensor decomposition.Below, we discuss components 14 and 15 in detail: For both components 14 and 15 (showing delta/theta band activity at parieto-occipital electrode sites with a maximum activity between 200 and 400 ms), activity reflected by these components was attributable to the conflict modulations of action restraint (i.e., Nogo task implementation).There were also important differences with regard to the time domain and frequency signatures clearly reflected by the outer product of the time and frequency domain (i.e., the time-Fig.7. The spatial features, temporal features, spectral features and outer products of all components in the model order of 36 that revealed a two-way interaction effect, along with illustrations of these interactions.For the latter, * denotes a significant difference in the post-hoc analysis.frequency representation).Conflict modulations of action restraint in component 14 revealed activity in the high theta band, while the conflict modulations of action restraint in component 15 showed activity in the delta\low theta band.While component 14 showed a difference between the congruency conditions only in the trials where inhibitory processes were required, component 15 did not show any differences between congruency conditions for neither the Go nor Nogo trial types.However, component 15 revealed a difference between trial types (i.e.Go and Nogo) in the congruent condition.These results indicate that inhibitory processes as indexed by action restraint demands rely on early onset activity in the high theta frequency band when modulated by conflict processing, whereas they show a later-onset P3 time window-like activity in the low theta band for the most demanding condition (i.e. the Nogo trials in the congruent condition).During congruent trials, response selection has been found to run via more automated routes of processing (De Jong et al., 1994;Keye et al., 2013;Kornblum et al., 1990;Mückschel et al., 2016) (but see (Hommel and Wiers, 2017) for critique on this account), making inhibitory control particularly demanding.Indeed, the feature value of component 15 (and hence low theta band activity) was highest during stopping in congruent Simon task trials suggesting that low theta band-related inhibitory control processes is strongest in situations where action restraint processes are most demanding.The spatial information modulations of components 14 and 15 (i.e., at parieto-occipital electrode sites) are similar in terms of an importance of processes reflected by parieto-occipital electrodes, suggesting attentional selection process play a role in inhibitory processes modulated by conflict processing.Notably, results from a study examining the importance of cognitive-neurophysiological processes for understanding performance in conflict monitoring (as examined in a Simon task) by applying deep learning and explainable artificial intelligence methods (Vahid et al., 2020) provide converging evidence.Here, it was shown that activity at parieto-occipital electrode sites in the same time windows as found in the current study were most informative to understand the neurophysiological dynamics underlying conflict monitoring in Simon task stimulus-response conflicts (Vahid et al., 2020).The current tensor decomposition findings stress the importance of theta band activity for cognitive control processes.At present, especially medial frontal theta band activity has been suggested to be important because its biophysical nature allows an integration of information across distinct brain regions (Cavanagh and Frank, 2014;Michael X. Cohen, 2014).While the current findings can for methodological reasons not provide insights into the functional neuroanatomy, the findings suggest that not only frontal theta band activity, but also posterior theta band activity is able to do so.This may be the case because especially parietal structures have been implicated in connecting perception and action (Geng and Vossel, 2013;Gottlieb, 2007;Gottlieb and Snyder, 2010).Even though it cannot be excluded that the involvement of posterior theta band activity may mainly reflect the role of spatial attentional processes in the task applied (Hommel, 2011b;Leuthold, 2011), it should be noted that cognitive control processes in the Simon task can well be explained by ideomotor theory inspired concepts (e.g.event file coding) (Hommel, 2011a).For this, theta band activity also in parietal structures plays an important role (Beste et al., 2023;Rawish et al., 2024;Wendiggensen et al., 2023;P. 2022a).This suggest that there is a general role of theta band activity (possibly unrelated to the cortical region of origin) for cognitive control processes.
When looking at the components with an interaction between task and trial type, it became evident that trial type (i.e.Go or Nogo/Stop trial) modulated response inhibition processes (components 8, 19 and 31), particularly activity in action restraint (as can be seen in components 8 and 31).Component 8 indicated activity within the frequency range of 3 to 6 Hz (peak at around 4 Hz) with a peak around 600 ms (i.e.around button press in Go trials) and at frontal-central and posterior regions.Component 31 revealed a peak at 400 ms mainly in the theta frequency band (i.e. 4 Hz) at frontal-central and superior-parietal regions.Thus, there was a different activity pattern with regard to timing and frequency band depending on the response type: Higher values in the Go trials than in the inhibition trials was related to a later onset (approx.500-600 ms post-stimulus) and activity in the high theta frequency band (component 8), while higher values in the inhibition trials than in the Go trials was associated with a slightly earlier onset (approx.400 ms post-stimulus) and activity in the delta/low theta frequency band (components 19 and 31).The fact that we observed activities in the low theta frequency band might be due to the time-frequency transformation algorithm with the different scaling for the different frequency bands, which made tensor decomposition algorithm more sensitive to the lower frequency bands.
In addition to delta/theta band activity, the analysis also extracted components with activity of the alpha frequency bandmost importantly, in component 17.During conflict-modulated response inhibition (as indexed by the interaction between "trial type" and "congruency"), the activity of component 17 including alpha band activity in the first 200 ms after stimulus presentation at occipital electrode sites was stronger during inhibition and responding particularly in congruent trials.Of note, alpha band activity has not only been implicated in early attentional orienting (Herrmann and Knight, 2001;Luck and Kappenman, 2012), which may play a role due to the lateralized presentation of stimulus information in the current task (Leuthold, 2011;Schneider et al., 2012), but is also of importance in implementing inhibitory control (Beste et al., 2023;Klimesch, 2012;Konjusha et al., 2023;Yu et al., 2024).It has been suggested that through alpha band activity, top-down inhibitory control can be exerted (Beste et al., 2023) and that alpha band activity modulates how previously associated stimulus-response mappings are retrieved to allow the selection of the appropriate action given a specific stimulus input (Beste et al., 2023).In line with component 17 showing occipital alpha band activity shortly after stimulus presentation, the interaction effect of the factors "trial type" and "congruency" was also found in the results for the Simon Inhibition task (new task design described in the supplementary material).However, these results from the additional experiment showed no modulation of response inhibition by conflict information, thereby suggesting that tensor decomposition is sensitive to the manner of task implementation (see Supplementary Note 2) (block-by-block, as in the original experimental design, vs. trial-by-trial, as in the new task design), thereby suggesting that switch probability contexts and transitional probabilities should be taken into consideration when choosing an analysis method (such as tensor decomposition).This is of relevance since many theorists suggest that the contextual embeddings under which adaptive behavior has to unfold has an impact on this (Arbula et al., 2017;Braem and Egner, 2018;Capizzi et al., 2020;Frings et al., 2020;Tarantino et al., 2016).Along similar lines, there is mounting evidence that inter-trial periods (previously most used for baseline correction procedures) reflect brain states affecting cognitive processes shortly thereafter (Wainio-Theberge et al., 2021;Wolff et al., 2021Wolff et al., , 2019)), particularly during inhibitory control (Prochnow et al., 2022;Pscherer et al., 2023;Wendiggensen et al., 2022b).

Conclusion
In summary, the study provides proof of principle that tensor decomposition methods are powerful to delineate the interplay of different action control processes on a neurophysiological levelin the case of the current study the interplay of conflict monitoring and action restraint/ action cancelation (aka response inhibition).Using tensor decomposition, we showed how conflicts modulate action restraint and action cancelation processes and delineate common and distinct neural processes underlying this interplay.Spatial information modulations were similar with processes reflected by parieto-occipital electrodes being important, suggesting attentional selection processes playing a role.The data also show that tensor decomposition is sensitive to the manner of task implementation (block-by-block, as in the original experimental design, vs. trial-by-trial, as in the new task design), thereby suggesting that switch probability/transitional probabilities should be taken into consideration when choosing an analysis method (such as tensor decomposition).The study provides a blueprint of how to use tensor decomposition methods to delineate common and distinct neural mechanisms underlying action control functions using EEG data.

Data and code availability statement
The anonymized data reported in the study and the associated data analysis codes can be made available for other researchers upon request sent to the corresponding authors.This procedure complies with the requirements of our funding agents and the institutional ethics approval.

Declaration of competing interest
There are no conflicts of interest

Fig. 1 .
Fig. 1.All trial types inherent to both tasks: Go trials (both Simon Nogo and Simon Stop), Nogo trials (Simon Nogo) and Stop trials (Simon Stop) for both congruency conditions.

Fig. 3 .
Fig. 3. Distribution of SSRT in the Simon Stop task for both congruency conditions.

Fig. 4 .
Fig. 4. Distribution of Nogo false alarms in the Simon Nogo task for both congruency conditions.

Fig. 5 .
Fig. 5. Average stability indices for the model orders between 3 and 60, highlighting the two model orders with the highest average stability indices (7 and 36) which were chosen for further analysis.