Unravelling global and selective inhibition utilising a Bayesian hierarchical model approach

University of Amsterdam (UvA)

MSc in Brain and Cognitive Science (MBCS)

Research Project – 12 EC

May 2019 – October 2020

Unravelling global and selective

inhibition utilising a Bayesian

hierarchical model approach

Franck Porteous

1286746 – franck.porteous@pm.me

Supervisor : Scott Isherwood

University of Amsterdam – Department of Psychology

Integrative Model-based Cognitive Neuroscience Research Unit

(IMCN)

~

Co-Assessor : Dr. Dora Erbé-Matzke

University of Amsterdam – Department of Psychology

Table of Contents

ABSTRACT ... 1

GENERAL INTRODUCTION ... 2

INHIBITION ... 2

Inhibition Conceptually ... 2

Why Model Inhibition? ... 4

Chosen Tasks for Studying Inhibition ... 4

RESEARCH QUESTIONS ... 5

GENERAL METHODS ... 6

BAYESIAN ACCUMULATION MODELS ... 6

SAMPLE ... 7

EXPERIMENT 1: MULTI-SOURCES INTERFERENCE TASK ... 8

MATERIALS &METHODS -MSIT ... 8

MSIT ... 8

Conditions ... 9

Sample ... 9

Models ... 10

Linear Ballistic Accumulator (LBA) ... 10

Priors ... 11 Model Comparisons ... 12 Parameter Estimation ... 13 RESULT -MSIT ... 13 Descriptive Statistics ... 13 Model Fit ... 14 Model Selection ... 14

Model Parameter Estimates ... 15

DISCUSSION -MSIT ... 16

EXPERIMENT 2: STOP SIGNAL TASK ... 17

MATERIAL &METHODS -SST ... 18

Experimental Design ... 18

Sample ... 18

Modelling the SST ... 19

The Race Model Framework ... 19

BEESTS ... 20

Parameter Estimation ... 21

RESULTS -SST ... 22

Descriptive Statistics ... 22

Model Parameter Estimates ... 22

DISCUSSION -SST ... 23

CORRELATION IN PARAMETERS OF INHIBITION MODELS ... 24

METHODS ... 24

Data Acquisition ... 24

RESULTS ... 25

Intra-Model Correlation Study ... 25

Inter-Model Correlation Study ... 26

Comparing Measures of Inhibition ... 28

DISCUSSION ... 29 Predicting Inhibition ... 29 Limitations ... 30 GENERAL DISCUSSION ... 31 Future Directions ... 33 REFERENCE: ... 34 APPENDIX ... 37

Abstract

Every day, individuals face situations where the ability to stop their course of action in favour of better-adapted behaviour is essential. This requires inhibition, a crucial component of conflict processing, decision making, and goal-driven conduct in a dynamic environment. Inhibition can be divided into two concepts, global and selective inhibition. Both concepts relate to two tailored neurocomputational methods that enable adequate inhibition performance in different contexts. Here, we investigate these concepts utilising the Stop Signal Task (SST) and a new version of the Multi Sources Interference Task (MSIT), which allows for better control over its sources of interference. We adopt a Bayesian parametric approach to model behavioural data then scrutinise and relate these inhibition concepts. By revealing and correlating parameter values emerging from each model (i.e., BEESTS and LBA), we define two measures of global and selective inhibition, and subsequently, unveil common computational grounds between the driving components of these concepts. While we show a marginal correlation between the models’ parameters, denoting poor performance, we do not find a correlation between our measures of inhibition. Unfortunately, this study suffers from a small sample-size, thus not allowing us to advance robust conclusions.

Keywords: inhibition, global inhibition, selective inhibition, conflict processing, decision making, MSIT, SST, Bayesian modelling, BEESTS, LBA, DMC.

General Introduction

Core to human behaviour, inhibition is indispensable to one’s panel of cognitive functions. In the context of perceptual decision-making (i.e., decisions made based on the integration and accumulation of sensorial evidence over time, until reaching an internal bound), inhibition is the ability to stop proponent responses in favour of better-adapted and relevant behaviours and has earned researchers’ interest and motivation.

In the present study, we explore two forms of inhibition (i.e., selective and global inhibition) with the support of two behavioural tasks and their associated models. We pilot and model a novel version of the Multi Source Inference Task (MSIT; Bush et al., 2003), a simple-to-implement, efficient, and well-known cognitive task measuring selective inhibition in various contexts (Capri et al., 2020; Weigard, Heathcote, & Sripada, 2019). With this, we aim to disentangle sources of interference present in the stimuli, control for them separately and combined, and assess their relative influence on behaviour using the Linear Ballistic Accumulator (LBA; Brown & Heathcote, 2008; Heathcote et al., 2019). We also run the Stop-Signal Task (SST; Lipszyc & Schachar, 2010; Logan & Cowan, 1984; Matzke et al., 2019; Verbruggen et al., 2019; Verbruggen & Logan, 2008, 2009) which measures global inhibition, and models its outcoming behavioural data using an adapted version of the race model (BEESTS; Bayesian

Ex- Gaussian Estimation of STop-Signal RT distributions; Matzke, 2013; Matzke et al., 2013). Then,

we utilise the estimated parameters from both Bayesian hierarchical models to investigate latent variables in both global and selective inhibition mechanisms.

This research aims to gain insights into the nature and mechanisms common to global and selective inhibition by adopting a modelling approach; an approach that permits one to observe and relate underlying variables from those models.

Inhibition

Inhibition Conceptually

Inhibition has remained central to cognitive science, and neuroscience as it is a mechanism that governs our most fundamental behaviours. Considered as one of the five executive functions, inhibition of dominant or prepotent responses allows for the adaptation of one’s behaviours to an ever-changing world (Miyake et al., 2000). It is the ability to suppress a response that is no longer required or appropriate and provides to cognition fundamental tools allowing goal-driven conduct in a dynamic environment (Verbruggen & Logan, 2009). Herein, we focus on two forms of inhibition, selective and global inhibition. We can find selective inhibition in cognitive science literature in a popular task such as the Stroop task, or in ecological everyday-life situations. For example, when grabbing a spice

container on a kitchen shelf, one could gather new sensory information after having initiated an initial action (e.g., reading the label on the container, thus figuring out that this spice is irrelevant for the recipe), and would need to inhibit this movement to let a new and more-relevant action take place (e.g., modifying the hand trajectory toward the correct spice container). Global inhibition, on the other hand, is popular in go/no-go tasks and could be illustrated when one inhibits him/herself from walking on crosswalk to avoid collision with a reckless driver. In these real-life scenarios, both forms of inhibition stop proponent responses, but selective inhibition also considers the content of the stop signal, not just its presence.

Successful inhibition of behaviour is challenging to study as it does, by definition, prevent the production of any behaviour. Thus, observing inhibition using conventional experimental measures (e.g., reaction time or accuracy) is generally impracticable and requires a different methodology. Unravelling key features of inhibition mechanisms can be done by adopting a modelling approach to behavioural data. In the first place, mathematical formalisation (i.e., models) allows to test, in the light of behavioural data, the feasibility of a suggested behavioural mechanism. Secondly, once a model has its structure accepted, the model provides researchers with parameter values that are estimated from participant behaviours. Finally, in fixed frameworks, it allows one to understand changes in behaviour in terms of the models’ parameter value variations. Both global and selective inhibition concepts are investigated using tailored experimental procedures aiming to challenge participants inhibition abilities. We aim to characterise and describe inhibition in terms of the models’ internal parameter values according to different experimental conditions. In other words, our intention is to investigate the variation of latent parameters explaining behavioural changes in two inhibition-related tasks.

While selective inhibition allows cognitive processes to be flexible in a noisy environment by inhibiting non-relevant external interferences, global inhibition relates to the inhibition of an initiated pre-planned behaviour (go-signal) after presentation of a stop signal (e.g., Go/No-go tasks). In this project, global inhibition is better linked to the ability to stop processes already initiated with respect to a new external input. One could assume dichotomy of those systems, however, in vivo imaging studies using technics such as functional Magnetic Resonance Imaging (fMRI) indicate overlapping networks in response inhibition, conflict processing, and adjustments of response thresholds in paradigms pertaining to those concepts (Hollander et al., 2015). As evidence points towards both the existence of shared and specific networks, understanding the implications of neural substrates remains a complex yet crucial question to answer.

Furthermore, inhibition performances are subject to numerous forms of internal and external interferences. Internal interferences, such as cognitive disorders (e.g., Attention-Deficit/Hyperactivity disorder, schizophrenia) represent an additional challenge for some individuals to overcome. On the other hand, external interferences such as visual distractors commonly impair participant performance as they represent new data to integrate and resolve.

Why Model Inhibition?

The behavioural data produced by participants is the consequence of successive and intertwined cognitive mechanisms and thus, represent the response of participants to the challenges presented to them (e.g., an experimental task, or real-life situations). Those behaviours intrinsically hold information about the functions that generate them (Schad et al., 2020), and understanding their nature requires an approach that captures and is capable of mimicking their generation process. To do so, researchers use models that are grounded in theory, such as Bayesian models. Bayesian models are composed with clusters of parameters characterizing the modelled mechanisms, and priors. In the context of Bayesian modelling, priors are initialised to be uninformative and, once the modelling process is achieved, embody the estimated value of their respective parameter. The structure of these models then characterizes the underlying computational principle of the modelled cognitive process and have the power to explain phenomena that originate from mixed sources. Models offer a framework where changes in behaviour can be understood in terms of parameter value variation (Philiastides et al., 2011). Models are, by themselves, non-informative regarding the underlying neural mechanisms concealed in neural data. Once their structure is defined, the parametrisation defines the degree to which a model fits behavioural data. When using models, researchers must be guided by robust theoretical grounds and strong hypothesis concerning the origin of a variations of their models’ parameters. As formulated by Smaldino, “Models are stupid, and we need more of them” (Smaldino, 2017). In this research, we are investigating to what extent our model captures the behavioural data and to what extent our model assists us in understanding how our experimental conditions can lead to behavioural changes.

Chosen Tasks for Studying Inhibition

The formal portrayal of inhibition mechanisms requires straightforward experimental paradigms that allow experimenters to manipulate sources of interferences and most importantly, to target specific behaviours precisely. In this section, we briefly describe the two tasks that we use to test our two forms of inhibition.

The tasks we have chosen to investigate selective inhibition is the Multi-Source Interference Task (MSIT; Bush et al., 2003) and the Stop Signal Task (SST; Logan & Cowan, 1984) for global inhibition. The MSIT is a three-choice decision task where participants have to identify and respond with the identity of a unique digit among two distractors digits, forming a three-digit stimulus. With a panel of distractors and target combinations, we can affect participants’ inhibition performance by inducing two different sources of interference: a Flanker-like effect where the interference is stimulus-relevant (i.e., with the manipulation of the distractors’ identity making them plausible answers) and a Simon-like effect where the interference is task relevant (i.e., manipulating the position of the target to be incongruent with its identity). Herein, we deconstructed the stimuli structure into four conditions: a

congruent condition devoid of interference, two intermediary conditions having each a unique source of interference (i.e., distractor or position effect), and an incongruent condition combining both sources of interference present in the intermediary conditions. These sources of interference cover all possible interferences originating from the stimuli.

The SST is a two-choice go task (e.g. press left when an arrow pointing to the left appears, and vice versa) where the participant occasionally hears a tone, indicating that they have to withhold their response entirely. In both tasks, accuracy and reaction time (RT) will be the central behavioural measures collected as they represent an essential input dimension from which both models will estimate their parameters.

Research Questions

Two research questions motivated this study. While the first one is specific to the paradigm testing the selective inhibition of participants, the other focuses on the possible interaction between selective and global inhibition.

Our first research question will, therefore, focus on both the reproducibility of our experiment with respect to the existing literature and the characterisation of the observed results.

Research question 1 — Can we replicate previously found behavioural differences in participants RT

and accuracy between congruent and incongruent conditions in the MSIT? Moreover, can we discern a distinctive pattern amongst RT and accuracy when sources of interference are uniquely presented or mixed?

Our second research question focuses on comparing each model’s estimated parameter values, both embodying, respectively, our two forms inhibition. Within-subject, we will study the existence of a possible correlation between a selection of LBA parameters varying with the MSIT conditions (selective inhibition with different cognitive charges) and the full set of BEESTS parameters (global inhibition). With this, we aim to gather indications of the possible relationship amongst inhibition models.

Research question 2 — Is participant performance in both inhibition tasks correlated? In other words:

Is performance in one task able to predict performance in the other?

The goal of the present study is to use two tasks, each having a precise approach to inhibition, and two distinct models describing their associated cognitive process, thus, explaining the behavioural responses (e.g., accuracy and RT) that we sought to understand.

This paper is structured such that it will begin with a general method covering the tasks, modelling methods and participant sample, then, present our two experiments with their associated theoretical background, methods, results and discussion, and lastly, the parameter study performed on both models.

General Methods

Bayesian Accumulation Models

Formal mathematisation of behavioural phenomena allows researchers to articulate their methodology towards a mechanistic approach where cognition can be broken down into its essential pieces. Having two concepts of inhibition to investigate, we chose to adopt two parametric Bayesian hierarchical models of evidence accumulation. The hierarchical nature of the models allows us to fit our participant data at both a population-level and participant-level. This approach also allows us to infer and estimate poorly identified parameters of individual subjects from the easily estimated population-parameters. Mathematical frameworks have been developed for years to help answer problems related to parameter estimation in high dimensional spaces. In this research, global and selective inhibition models will be observed with two distinct and tailored models allowing one to examine parameter values with respect to the underlining mechanism elicited by each task. The linear ballistic accumulator (LBA) will let us grasp underlying cognitive parameters involved in decision-making in diverse contexts with multiple response production. For selective inhibition, we will use the MSIT task and model its behavioural data with version of the LBA (Brown & Heathcote, 2008; Heathcote et al., 2019). In this context of conflict between LBAs, we will observe which parametrisation embodies the behavioural data best. Concerning our global inhibition task, the BEESTS will be used to model behavioural data produced by the SST. Heathcote et al. (2018) proposed the dynamical models of choices (DMC) as an elegant, flexible, open-source, R-based framework (R Core Team, 2019) for parameter estimation in decision models such as the Diffusion Decision Model (DDM), the LBA, and BEESTS . Throughout our research, we used this DMC framework to implement our models of choice. Integrated at the DMC core, is the differential evolution population based Markov Chains Monte-Carlo sampling method (pMCMC; Turner et al., 2013). pMCMC enables sampling from the posterior distribution at the group- and individual-levels to make a new estimate of the posterior parameters values (Heathcote et al., 2019; Matzke et al., 2019; van Ravenzwaaij et al., 2018). Thanks to this approach, estimation of parameters requires less computation as they are more straightforward to calculate in large samples than from normal distribution equation directly (van Ravenzwaaij et al., 2018). As Matzke et al. (2019) also phrased it “Bayesian inference is particularly suited for cognitive modelling because it offers a coherent inferential framework, which allows researchers to respect the complexity of the data-generating process and incorporate prior information”. Additionally, such models offer researchers a measure of the level of uncertainty for their parameter estimation. Bayesian hierarchical model thus represents an excellent alternative to classical model as it better embodies behavioural data while being computationally efficient.

Sample

Our two tasks were embedded in a behavioural multi-task study as a part of a larger project conducted at the Integrated Model-based & Cognitive Neuroscience research unit. Despite the reduced amount of participant data, we still observe a counter-balanced order within our five tasks. In an effort of establishing internal validity, a matrix for counterbalancing among tasks was determined prior to testing. In order to achieve statistical power and equilibrium in task order, it was determined that 72 participants were required for this study. Moreover, in order to get a measure of intra-subject variability, two experimental sessions featuring all five tasks were required for each participant.

Participants were recruited using an online platform provided by the University of Amsterdam (UvA) faculty of psychology without any pre-screening criteria. Unfortunately, following the SARS-COV-19 outbreak and subsequent cessation of activity from the testing facilities, only a portion of the expected participants could perform the experiments (first session n=23, second session: n=8). Due to the unbalanced sample sizes across sessions, we discarded the second session and were therefore unable to conduct an intra-subject study. The MSIT and SST were always performed successively. The order participants completed the MSIT and SST were counterbalanced, such that 11 participants performed the tasks in the order MSIT-SST and 10 in the opposite order. Moreover, eight participants started the multi-task with the MSIT/SST pair, five after one to two tasks, and seven after all other tasks. Thus, we do not report any pattern of task-priming (e.g., a counterbalanced amount of each task was performed before the MSIT/SST pair) in our dataset. All 23 participants were right-handed English-speaking subjects, with normal or corrected-to-normal vision. After a brief introduction to the nature of each individual task and the multi-task format, the participants gave their informed consent. Also, if anything came to the participant’s attention, an experimenter could be called at any point during the testing period. The full length of the experiment was approximately an hour and twenty minutes.

All experimental procedures were approved by a local institutional review board at the University of Amsterdam (UvA) and were under the ethical standards of the Declaration of Helsinki. The experiments were conducted at the UvA testing facilities in individual and private testing rooms.

Experiment 1: Multi-Sources Interference Task

In this project, we will pilot a new version of the MSIT, providing four different conditions that enable us to detangle sources of interference previously undistinguishable in the original task. Our version of the MSIT makes use of two distinct sources of interference capable being presented separately, or in combination. These are a Flanker-like effect (FE; Bush et al., 2003; Eriksen & Eriksen, 1974) and Simon-like effect (SE; Bush et al., 2003; Simon & Berbaum, 1990). As previously mentioned, developing a systematic control for the source of inference across the MSIT trials allows robust identification of the condition impact on behaviour and model’s parameters values.

Materials & Methods - MSIT

MSIT

The MSIT consists of a presentation of a 3-digit stimulus containing one unique number where participants have to discriminate the unique digit by pressing a key corresponding to its identity (e.g., press “3” for stimulus “030”). Our version of the MSIT was modelled after a typical and well-accepted experimental design commonly used in the literature (Bush et al., 2003; Weigard, Heathcote, & Sripada, 2019). The task offers the opportunity to generate different classes of stimuli (i.e., conditions that separate multiple sources of interferences) and, subsequently, modulate the associated cognitive load required for the production of their respective correct responses. Figure 1 shows the structure of a typical MSIT trial. Stimuli were presented on a black background at the centre of a white fixation circle. The fixation circle was presented for 1 second, then contained the 3-digit stimuli (until response or for a maximum of 1.6 s), a speed-related feedback lasting for 400 ms, and was finally emptied for an inter-trial-interval (ITI) of 2 seconds.

Figure 1. Classical sequence of event in a MSIT trial. Duration between each phase is described along the

arrowed lines. This figure is also available at https://bit.ly/2J2M8FD under a Creative Commons CCBY licence https://creativecommons.org/licenses/by/2.0/ 1 sec Until response or 1.6 sec 400 ms ITI 2 sec Or

Before performing 240 trials divided in two experimental blocks of 120 trials, subjects could train with 25 practice trials. In this practice block only, two successive feedbacks were given after each trial, thus urging participants to remain on the edge of their individual performances and increase their error rate (which is essential to the models). The feedback consisted of two messages (lasting 400 ms each) within the fixation circle indicating if the subject was on time (“IN TIME” in green coloured letters, or “TOO SLOW” in red) and if the produced response was correct (“CORRECT” in green coloured letters, or “INCORRECT” in red).

During each trial, participants responded using the keyboard numerical key “1”, “2” or “3”. Participants were instructed to indicate the identity of the unique number, not its index (e.g., for the stimulus “002”, pressing “2” is a correct response while “3” would answer for the index and will be considered as an incorrect response).

Conditions

For each of the 240 trials of the MSIT, one of four different conditions can be presented. In a congruent condition (CON) no distractors are shown (i.e., empty spaces are filled with “0”, which is not a plausible answer) and the target-digit identity is congruent with its position (e.g., “100”, “020” and “003”). In the

FE condition (FLA), the target-digit identity and position stay congruent, but the distractors’ identity

is now a feasible option (e.g., “113” or “323”; Eriksen & Eriksen, 1974). In the SE condition (SIM), the position and index of the target-digit are incongruent (e.g., “100” is congruent and “300” is not; Hedge & Marsh, 1975) while distractors are not feasible options (i.e., filled with zeros). Finally, in the

incongruent condition (INC), both SE and FE are presented in the stimuli making the identity and

position of the target-digit incongruent and employing distractors that are feasible options (e.g., “221”, “311”, “323”). Disentangling SE and FE allow us to gain insights into the relative influence of each of those interference sources. As these effects are structurally diverse, it is plausible that each effect would affect behavioural outcomes differently.

Sample

Among the original participant pool that performed the MSIT (N=23), two subjects did not respond using the designated keys, rendering their data unusable. We required participants to have an accuracy across condition above chance level (i.e., here, 33%). One participant did not fulfil that condition by having near-zero accuracy in two of four conditions and was thus rejected. Ultimately, 20 participants (mean age: 20.85 ± 15.19, age-range: [17-34], 15 females) were included in the analysis.

Trials with an RT lower than 0.150 sec and without responses were removed as we considered them to be respectively, sub-conscious and too slow or not regarding the instructions (e.g., “respond as fast as possible”). With this criteria, we were left with the final dataset (n = 4 762, RT: 0.5677 ± 0.0307 sec).

Models

Linear Ballistic Accumulator (LBA)

The subjects’ MSIT behavioural data are modelled using the LBA (Brown & Heathcote, 2008). This model describes the process of sensorial evidence gathering, guiding toward the expression of behavioural responses (see Figure 2 for a visual depiction of the LBA). In the framework of the LBA, cognitive constructs such as “evidence accumulation”, “decision threshold” and “visual integration” are characterizable and quantifiable using the models’ parameters and their values (Heathcote et al., 2015).

Figure 2. Graphical representation of the LBA. This figure is also available at https://bit.ly/34oR8wl under a Creative Commons CCBY licence https://creativecommons.org/licenses/by/2.0/

Here, the MSIT stimuli contain both relevant attributes, which help the participant in determining the correct response, and irrelevant attributes which impair the participant’s performance by being associated with incorrect responses (i.e., a distractor). The MSIT digits-stimuli represents multiple sources of information that participants have to comprehend in order to produce a correct answer (i.e., each element-digit of the stimuli has its accumulation framed by one LBA). A competition between LBAs can be seen as a race between accumulators gathering evidence from a noisy signal until one “wins” by reaching a critical threshold (Smith & Ratcliff, 2004). Under this framework, both response accuracy and RTs can be explained in terms of the model’s internal parameters and the cognitive constructs implicated in solving the task can be understood.

The LBA describes the competition between three accumulators, each representing the accumulation process occurring for each element of the stimulus. For all accumulators on a given trial, the LBAs respective rate of evidence accumulation is constant and is drawn from a normal distribution parameterized with a mean and a standard deviation. These previously mentioned parameters are commonly estimated separately with regard to the accumulator match/mismatch (i.e., accumulator

Decision threshold

Starting point

Accumulation rate

and standard deviation (sd_v) (B)

(A)

associated to correct and incorrect answer for a specific trial), with the response it accumulates evidence for, and the trial-specific correct response. This race among accumulators ends when one of them reaches a decision boundary (denoted by a “B” parameter) triggering the behavioural response. The models also account for the delay that sensory integration (t0) yields — also referenced as “non-decision time” — and form a uniform distribution of noisiness at the start point (A), for each accumulator, across trials.

Thanks to the relatively low error rate of the participant in each cell, we sought to considerer a limited amount of models (see Table 1).

Table 1. Parametrisation of the LBA models.

Model Model Parameter

A B mean_v sd_v t0 st0

Model 1 Fixed P * R M C * M Fixed Fixed

Model 2 Fixed P * R C C * M Fixed Fixed

Model 3 Fixed P * R C * M C * M Fixed Fixed

Model 4 Fixed P * R M C * M C Fixed

Model 5 Fixed P * R C * M M Fixed Fixed

Model 6 Fixed P * R * C M M Fixed Fixed

Model 7 Fixed P * R * C C * M M Fixed Fixed

Model 8 Fixed Fixed Fixed Fixed Fixed Fixed

For example, as shown in Table 1, “Model 1” has A (starting point), t0 (non-decision time) and st0 (inter-trial variability) fixed. At the same time, B (threshold) varies with P (the index of the target) and R (response given by the participant), “mean_v” (the mean drift rate annotated DR) varies with match/mismatch (M), “sd_v” varies with C (condition) and M. In “Model 5” we also leave A, t0 and st0 fixed. In contrast, B varies with P and R, “mean_v” varies with C and M, and “sd_v” varies with M. “Model 6” is a control where all parameters are fixed. The rationale behind fixing the A parameter in all models is that B already reflects the quantity of evidence required from an accumulator to induce a behavioural response. Also, we assume that at any trial, participants are not exposed to any meddling (e.g., drugs, visual interferences) that may disrupt their sensory integration (t0) nor their inter-trial variation (st0).

Priors

For all models, priors were initialised to be broad and non-informative. Refined priors are necessary when the amount of available data updating the posterior is sparse (Turner et al., 2013). Priors were crafted from a truncated normal distribution with centralised mean, full standard deviation, and ample parameter space to explore at the participant level.

The priors were truncated normal distribution (TND) that can be found below with their respective scale, lower bound, and upper bound:

Aμ ~ TND(μ = 1, σ = 0,5 , lower bound = 0, upper bound =

∞

Bμ ~ TND(μ = 1, σ = 0,5 , lower bound = 0, upper bound =

∞

mean_vμ ~ TND(μ = 2, σ = 1 , lower bound = 0, upper bound =

∞

sd_vμ ~ TND(μ = 1, σ = 1 , lower bound = 0, upper bound =

∞

t0μ ~ TND(μ = 0,5, σ = 0,5 , lower bound = 0, upper bound =

∞

At the hyper level the parameters of each LBA were estimated using location prior (mu-prior) and scale prior (sigma-prior).

Model Comparisons

The LBA, like many cognitive models, shows highly correlated parameters that render its estimation difficult at the subject-level when data is sparse (Kolossa & Kopp, 2018; Weigard, Heathcote, Matzke, et al., 2019). Furthermore, limiting the number of estimated parameters to the bare minimum is essential to reduce parameter space to explore (i.e., to be allowed to vary, a parameter must be relevant for the model in both ecological and theoretical grounds). In order to choose judiciously among our six Bayesian hierarchical models, we will make use of two information criterion (ICs). Namely, the Deviance Information Criterion (DIC; Berg et al., 2004) and the Bayesian Predictive Information Criterion (BPIC; Ando, 2007). These permit the estimation of the goodness of fit of a model populated by MCMC simulation while penalising parameter setups that are not parsimonious.

The DIC is an adaptation of the Akaike information criterion (AIC; Cavanaugh & Neath, 2019) to hierarchical Bayesian models. This criterion provides a measure of goodness of fit based mainly on two factors: a model’s suitability to behavioural data and its internal complexity. Similarly, the BPIC allows estimating the posterior mean of an expected loglikelihood from a predictive distribution. Additionally, the BPIC allows circumventing an assumption requested by the DIC declaring that the “true model” must be incorporated in future observations generated by the target models’ parametric family of distribution. In other words, the BPIC allows a robust asymptotic estimation when the family of this distribution may not contain the true distribution of the parameter (Ando, 2007). While the DIC has a tendency to prefer over-fitted models, the more conservative BPIC criterion tend penalise model-complexity more effectively.

Finding which parametrisation gives a model a superior understanding of the behavioural data enables us to trace back which cognitive constructs are involved in the conflict resolution presented by the task. With respect to the model comparison, we aim to characterize better the mechanism behind conflict resolution in a situation with multiple sources of interference.

Parameter Estimation

Each of the parameter simulations were performed using 65 chains, an initial burn-in period of 400 samples, and a thinning of 10 (i.e., retain one in every ten samples to avoid auto-correlation and save file space). We performed a visual inspection of the chains and computed the Gelman-Rubin statistic (Gelman & Rubin, 1992) to confirm the end of the burnin period. In order to pass the visual inspection, the chains had to form a stable and overlapping group across the range of iterations (comparable to a “hairy caterpillar”). Shadowing previous research making use of the Gelman-Rubin measure, we considered that participant chains had converged if the Gelman-Rubin statistic value was equal or under 1.10 (Matzke et al., 2019; Weigard, Heathcote, Matzke, et al., 2019; Weigard, Heathcote, & Sripada, 2019).

From the starting point taken of each participant’s behavioural data, we determined the starting point of the hyperparameter sampling process. Once made, we ran 800 iterations to get a proper sample from the prior. Based on the previous distributions, we ran an additional 800 iterations to obtain a better likelihood. Finally, we ran the final 400 iterations. Prior to calculating an estimation of parameters, we reiterated the computation of the Gelman-Rubin statistic to gauge convergence and stability of the latter chains.

Result - MSIT

Descriptive Statistics

Across all 4 conditions, we observed significant changes in RTs (F(3, 79)= 7.88, p < 0.01) and accuracy (F(3, 79)= 23.32, p<0.01). Figure 3.A and 3.B depict accuracy and RT of participants across conditions, respectively. In Figure 3.A, we observe that both CON and FLA accuracy remain high with respect to SIM and INC. Figure 3.B shows that RTs are increasingly longer for conditions with more associated cognitive load, thus, forming a staircase pattern of increasingly RTs across conditions in the following order: CON, SIM, FLA and INC.

Model Fit

Model Selection

In order to determine which models combined the best fit to data and the most relevant regarding fit and parsimony, both DIC and BPIC were performed for each model. Table 2 reports the summed minimum deviance (SMD), DIC and BPIC values and Gelman-Rubin statistic for all models. The SMD relates the goodness of fit of a given model to behavioural data regardless of its parameterisation while our ICs proposed their values factorising models’ parameterisation as a penalty. Both BIC and BPIC are considered as being robust measures for Bayesian hierarchical model comparisons, but the BPIC penalises complex parameterisation even more.

Table 2. LBA model comparison. For each of the eight models this table show the summed minimum

deviance (SMD), the DIC, the BPIC and Gelman-Rubon statistic.

Model SMD DIC BPIC Gelman-Rubin

1 -3148.124 -2536.638 -2201.633 1.04 2 -1911.364 -1223.621 -843.679 1.11 3 -4151.179 -3425.465 -3034.339 1.06 4 -3993.402 -3302.202 -2930.571 1.05 5 -3996.876 -3385.716 -3052.986 1.05 6 -4464.408 -3517.294 -3030.026 1.16 7 -4636.189 -3618.516 -3099.007 1.39 8 3862.517 4005.643 4097.422 1.01 Figure 3.

MSIT descriptive plots of participants’ performance (i.e., Accuracy and RT) A. Boxplot of the participant accuracy in function of the stimuli condition. B. Boxplot representing the reaction time (RT) of participant in function of the conditions.

Here we can observe that “Model 7” obtained the lowest DIC and BPIC value, but also score a Gelman-Rubin statistic above 1.10 (1.39) indicating that the convergence of the chains is not satisfactory. Therefore, we retained “Model 5” for further investigation as it obtained the second lowest DIC and BPIC value and indicating a Gelman-Rubin statistic value inferior to 1.10. Investigation of the participants individual ICs values between “model 5” and the second-best model (“model 3”) show that “Model 5” is chosen over “Model 3” for respectively 60% and 70% of the participants individual DIC and BPIC values. Additional observation of the goodness of fit was perform using cumulative distribution functions reveals that the chosen model fit correctly to behavioural data (see Appendix 1).

Model Parameter Estimates

In our selected model, the focus is brought on the LBAs accumulation rate according to the conditions. This rate is described by the mean of the normal distribution modelling it. Table 3 shows the mean values of the parameter estimate across participants and Appendix 2 offers the estimated value of each parameters for all participants. Difference in DR values amongst conditions were significant for true accumulators (f(3) = 177.87, p<.05) and false accumulators (f(3) = 21.99, p<.05).

Table 3. Mean parameter values for the winning model ("Model 5"). "Uno", "Dos", and "Tres" is the target

position within the stimulus, "true" and "false" depend on the correctness of the accumulator, and "ONE", "TWO", and "THREE" are the participants' responses.

Parameters Mean Value

A 0.78 B.uno.ONE 1.15 B.dos.ONE 1.68 B.tres.ONE 1.53 B.uno.TWO 1.77 B.dos.TWO 1.16 B.tres.TWO 1.81 B.uno.THREE 1.64 B.dos.THREE 1.66 B.tres.THREE 1.17 mean_v.CON.true 4.56 mean_v.FLA.true 3.50 mean_v.INC.true 3.77 mean_v.SIM.true 4.86 mean_v.CON.false 0.32 mean_v.FLA.false 0.71 mean_v.INC.false 0.97 mean_v.SIM.false 1.03 sd_v.true 0.91 sd_v.false 1.91 t0 0.14

Discussion - MSIT

In regard to our first research question, we can confirm with results previously found in the literature. As the cognitive charge (e.g., amount of interference present in the stimuli) increases, overall RTs increase and accuracy decreases (Bush et al., 2003; Kawai et al., 2012). Between our two conditions with unique sources of interference (SIM and FLA), we observe a greater variability of the accuracy in SIM, and higher overall accuracy compared to FLA. Trials of the SIM condition provide stimuli that appear to be easy to answer because of the amount of relevant visual elements is at its lowest (e.g., only one target digit and two zeros). Possibly, as the SE interfere with the task rules (i.e., identity of the target is incongruent with its position, thus engaging with the “responding to identity and not position” rule) participants would respond faster and often incorrectly in the SIM condition. Subsequently, accuracy in trials with SE was lower compared to FE, and RTs in trial with SE were faster than those containing FE.

Our selected MSIT model (“Model 5”) focuses on the variation of the accumulation rate parameter according to condition. Generally, LBA models that allow variations in accumulation rate aim to capture changes in cognitive processing of perceptual data, while LBA models allowing thresholds to vary are better indicators of cautious behaviours amongst participants (Bogacz et al., 2010; Mulder et al., 2013). In our LBA, the threshold parameter was allowed to vary with the position of the target-digit and with the response-key given by the participant, and the accumulation rate varied with conditions and match/mismatch (i.e., whether the accumulator match the correct response or not). With the model parametrisation allowing DR to vary with conditions, we observe significant difference of DR rate for matched and unmatched accumulators thus showing the models’ reliability in characterizing participants’ behaviours.

Nevertheless, our experiment still presents two properties for which we did not control. Amongst stimuli of the INC condition, we noticed a small deviation in the stimuli validity such as “311” could be considered trickier than “322”. Indeed, in “311” and “322” both FE and SE are existing. However, the FE distractors are not only viable options, but they also designate the position (not the identity) of the target-stimulus (while in “322”, distractors do not), thus representing a possible source of additional interference and within condition variation. Secondly, a drawback to the utilisation of the FE comes from the difference between the shapes of a target-distractors pair in term of basic visual discrimination (e.g., “223” could be considered more challenging than “113”). Unfortunately, due to our low sample size and experimental design, we could not assess such differences. Future research could investigate if performance could be affected by such deviance in the stimuli structure.

Altogether, the model selection highlights that according to condition in the MSIT the measured behavioural changes is driven by changes in accumulation rates.

Experiment 2: Stop Signal Task

The SST is an efficient two-choice response time experimental paradigm which enables the testing of global inhibition. Using the SST, stop-signal reaction time (SSRT) can be estimated and utilised as a measure of inhibition. In this study, we consider the SSRT to be, specifically, a measure of global inhibition.

SSRT estimation is essential to various domains in cognitive science and neuroscience as it is frequently used as a measure of inhibition. Through a meta-analysis, Lipszyc et al. (2010) provides evidence that changes in inhibition performance in the SST is present across participants with common disorders (i.e., attention-deficit hyperactivity disorder, obsessive-compulsive disorder and schizophrenia), supporting the relevance of the task in a clinical context.

Participants were presented with arrows shaped stimuli (e.g., a white arrow on a black background) and had to indicate the direction of the arrow using two designated direction keys (e.g., arrow keys “←” and “→”). Figure 4 illustrates a sequence of two experimental trials (with and without stop-signal) and Appendix 3 presents a sequence of practice trials (without and with a stop-signal).

Figure 4. A classical sequence of event in the SST. The first four slides depict a stop-trial while the three

last one shows a go-trial. The diagonal axis under the slides represent the duration of each phase. This figure is also available at https://bit.ly/3ojLSCa under a Creative Commons CCBY licence

https://creativecommons.org/licenses/by/2.0/

For a limited amount of trials, participants heard a salient tone (stop-signal; SS) indicating that they had to withhold their response entirely (i.e., global inhibition of any ongoing response). In order to keep participants on the edge of individual performance level (i.e., ~50% chance of producing a correct

500 ms SSD Until response or 2 sec - SSD ITI 1 sec Tone 500 ms Until response or 2 sec ITI 1 sec Stop-trial Go-trial

response on any given stop-signal trials), we used a staircase system which allows for modulation of the delay between the presentation of the arrow and the SS (Stop Signal Delay; SSD). Practically, longer SSDs represent trickier trials as participants have more time post-go-signal to engage in executing a response before the interruption of the SS.

Material & Methods - SST

Experimental Design

We instructed participants that responses to go-signals had to be completed immediately, regardless of the possibility of a SS intervention, and forewarned that inhibition following SS and a response to the go-signal should be treated equally (Verbruggen et al., 2019). Participants had a twenty-five-trial practice session with correctness-related feedback after each trial. Following the practice session, participants performed two blocks of 160 trials, accounting for a total of 320 trials per participant, thus complying to the minimum of 200 trials proposed by Verbruggen et al., (2019). As Figure 4 shows above, each trial consisted of the presentation of a fixation circle for 500 ms followed by the stimulus, which remained until the participant responded or until there was a period of two seconds minus the current SSD. During the practice session, correctness-related feedback was presented for 500 ms. Frustration generated by the failure to stop on a stop-trial can encourage participants to violate an essential rule of the SST, stipulating that the involvement in producing a response to the go-signal must be equal to the involvement in inhibiting the response after the presentation of a stop-signal. To avoid having participants implement a waiting strategy, we fixed the number of stop-trials to a quarter of the total amount of trials across the experimental design (i.e., 80 trials, Verbruggen et al., 2019).

The SST allows for measurement of two main dependent variables, RTs and SSRTs. The experimental paradigm is constructed such that the SSD is increased by 50ms after each unsuccessful stop-trial. Conversely, when the subject responds during a stop trial, the SSD is decreased by 50 ms. The initial value of the SSD was 200 ms with a maximal value range of 0s to 900 ms. In the effort of gathering enough data for each condition, we aimed to use this approach to fix the participant’s response probability (i.e., error rate) in stop-trials to 0.5 (i.e., p(response | SS) ≈ 0.5). Participants accuracy on stop trials was thus required to be within a range of 25-75% of correct responses to be considered for modelling (Congdon et al., 2012; Verbruggen et al., 2019).

Sample

Amongst the 23 participants that performed the MSIT, six were excluded. Two used incorrect response keys and four had a stop-trial accuracy above our inclusion criteria, leaving us with 17 participants (age 20.53 ± 5.51 years, range 17-25, 13 females) for analysis. Participants excluded due to high accuracy on stop-trials were believably not compliant with the “involvement rule”, and subsequently

implemented a waiting strategy leading to a pattern of growing RTs in the earlier trials. Also, an internal clock failure from the experimental software uncalibrated the RT measurement of the first trial of each block for all participants, and thus led to the loss of those RT measures.

All go-trial data points with a RT under 0.150 sec were too short to be considered as a conscious action and were excluded. At the same time, no datapoints had to be excluded due for above reasonable RTs since our response time window was of 2 sec. Go-omissions (i.e., participant did not respond during a go-trials) had to be under 5% of the total amount of go-trials. Also, go-accuracy (i.e., participant discriminated the direction of the stimulus correctly) had to be equal or above 95% for go-trials.

Stop-accuracy (i.e., the ratio of correct inhibition against failed inhibition on stop-trials) had to be 50 ± 25%

correct.

Modelling the SST

Here, we describe the modelling framework under which we studied the SST. Then, we present our choice of model, associated priors, and methods for the modelling of the SST behavioural data. And finally, the method that we employ.

The Race Model Framework

The race-model is defining a framework for a race among two processes occurring in an SST stop-trial (Eagle et al., 2008; Heathcote et al., 2019; Logan & Cowan, 1984; Matzke et al., 2019; Verbruggen et al., 2019; Verbruggen & Logan, 2008, 2009; Weigard, Heathcote, Matzke, et al., 2019). The race model describes the competition between the response to a go-stimulus (i.e., go-runner) and the inhibition of this response after stop-signal onset (i.e., stop-runner). See Figure 5 for a visual depiction. The stop signal-delay (SSD) represents the difference between the trigger time of the go-runner and the stop-runner. The first runner to finish its race can express its associated behaviour. In other words, when the stop-runner wins the race, the response is inhibited, and when the go-runner wins the race, the response is produced.

Figure 5. The two-racer ex-Gaussian stop-signal race model. The horizontal axis represents time with, at its origin, the presentation of the go-signal. Following the period of SSD, the SS is presented. At this point in time, both stop and go processes are engaged and in competition. The grey distribution at the top of the figure represents the RT for failed inhibition (i.e., observed RTs on stop-trials). The top line represents the Ex-Gaussian go-runner RT distribution and three parameters defining it (mean, standard deviation and exponential component). Defined by the same three parameters, the bottom line shows the Ex-Gaussian SSRT distribution (the stop-runner). This figure is also available at https://tinyurl.com/ydezqk3p under a Creative Commons CCBY licence https://creativecommons.org/licenses/by/2.0/

Each participant’s RT for go-trials and failed stop-trials were inspected under the race-models’ assumption that RT in go-trials are shorter than in stop-trials, even when the participant fails to inhibit his response.

Practically, direct observation of the finishing time of a stop-runner is unrealisable since successful inhibitions do not produce any form of behavioural responses (i.e., the participant does not act in any way). Moreover, since our focus lays on the inhibition processes, alternative instructions such as requesting the participant to press another button would be irrelevant as it would consist of a new form of go-run. The present methods aim to estimate the distribution of the SSRT by inferring its parameters from the SSD values and the difference between RT distribution for stop-failure and go-success.

BEESTS

To model the behavioural data gathered using the SST, we choose to utilise the BEESTS approach. Under the assumption of a complete race model (i.e., both set of parameters describing RTs and SSRTs distribution are both independents), modelling with BEESTS allows us to estimate parameters of the SSRT distribution (i.e., mean, standard deviation, exponential coefficient), but also, to account for both trigger and go failures (designated by “tf” and “gf” respectively).

Priors

Priors for BEESTS were initialised similarly to LBA priors, TND with centralised mean and full standard deviation, broad, and non-informative. Upper and lower bound are described below:

mu ~ TND(μ = 0.50, σ = 1, lower bound = 0, upper bound = 2) muS ~ TND(μ = 0.20, σ = 1, lower bound = 0, upper bound = 2) sigma ~ TND(μ = 0.05, σ = 1, lower bound = 0, upper bound = 2) sigmaS ~ TND(μ = 0.03, σ = 1, lower bound = 0, upper bound = 2) tau ~ TND(μ = 0.08, σ = 1, lower bound = 0, upper bound = 2) tauS ~ TND(μ = 0.05, σ = 1, lower bound = 0, upper bound = 2) tf ~ TND(μ = 0.10, σ = 1, lower bound = 0, upper bound = 2) gf ~ TND(μ = 0.10, σ = 1, lower bound = 0, upper bound = 2)

Parameter Estimation

BEESTS assumes that a convolution between a Gaussian (i.e., normal) and an exponential distribution describes the SSRT and RT distribution, respectively. Each of these distributions is defined by three parameters: a mean (mu), a standard deviation (sigma), and an exponential component (tau).

In hierarchical Bayesian modelling, population-level parameters describe the global behaviours of the model parameter values. Here, we employ two sets of hyperparameters to describe the distribution, one for the mean (i.e., a location parameter), and one for the standard deviation (i.e., a scale parameter) of said distributions. We estimated parameter values using DMC utilising the pMCMC. Each parameter was estimated using 33 chains (DMC set by default using three times the amount of parameters) and a thinning of 10. The convergence of the chains was assessed using the Gelman-Rubin statistic and visual inspection under the same set of verifications used for the LBA (i.e., value below the recommended criterion 1.10 and having a “hairy caterpillar” silhouette). We initiated a period of 100 samples to create a confident starting point for higher-up parameter estimations. Then, we ran an additional 100 samples to start the converging procedure appropriately before running a final 250 samples for robust parameter estimations.

Results - SST

Descriptive Statistics

Overall cognitive performance in the SST was high and showed little inter-individual variation. Stop accuracy (i.e., participant inhibit response at the SS presentation) had a mean percentage of 51.2 ± 6.70%; thus, in alignment with our staircase system. Go omissions (i.e., participant failed to respond during a trial) were low across participants (0.493 ± 0.77%, mean and standard deviation) while go-accuracy (i.e., correctly indicating the direction of the arrow) was high (0.995 ± 0.008%). Modelling with BEESTS works under the assumption that the participant validates the horse-race model. At this point of the analysis, we do not have access to the SSRT on successful stop-trial but can subtract RTs in go-trial and RTs on unsuccessful stop-trial to validate the horse-race model. Using this non-parametric computation, we validate the race model for all participants (mean difference = 0.161 ± 0.100 ms).

Model Parameter Estimates

Parameters’ estimated values can be found in Table 4.

Table 4. Estimated parameter value for all participants and mean parameter value.

Participants mu.true sigma.true tau.true muS sigmaS tauS tf gf

1 0.34 0.04 0.31 0.13 0.02 0.02 0.12 0.01 2 0.63 0.21 0.21 0.13 0.02 0.01 0.02 0.01 3 0.39 0.08 0.10 0.12 0.03 0.02 0.04 0.00 4 0.87 0.35 0.14 0.17 0.03 0.03 0.19 0.01 5 0.31 0.04 0.13 0.14 0.02 0.01 0.07 0.01 6 0.36 0.06 0.15 0.13 0.03 0.02 0.06 0.00 7 0.30 0.04 0.07 0.14 0.02 0.02 0.04 0.00 8 0.31 0.04 0.12 0.16 0.05 0.03 0.13 0.00 9 1.09 0.32 0.11 0.15 0.02 0.02 0.03 0.01 10 0.39 0.08 0.16 0.13 0.03 0.02 0.07 0.00 11 0.32 0.04 0.17 0.20 0.03 0.03 0.05 0.00 12 1.00 0.31 0.10 0.14 0.02 0.02 0.02 0.01 13 0.59 0.16 0.25 0.12 0.03 0.02 0.03 0.00 14 0.37 0.06 0.18 0.14 0.02 0.02 0.05 0.00 15 0.87 0.23 0.14 0.13 0.02 0.01 0.02 0.00 16 0.55 0.13 0.11 0.13 0.02 0.01 0.02 0.00 17 0.47 0.10 0.17 0.15 0.02 0.01 0.02 0.00 Mean 0.54 0.13 0.15 0.14 0.02 0.02 0.06 0.01

Additional observation of the goodness of fit was perform using cumulative distribution functions (see Figure 6).

Figure 6. Cumulative Distribution Functions (CDF) for BEESTS. Participant data is shown using thick lines,

with 10th, 30th, 50th, 70th, and 90th percentiles marked with open points. The models’ prediction is shown using thin lines and filled points. Grey dot clusters show the uncertainty around each percentile (100 random samples from the posterior).

Discussion - SST

In this experiment, we examined the concept of global inhibition through the lens of the race model. To do so, we modelled the SSTs’ behavioural data using BEESTS, and estimated parameter values that will be used to compute a global inhibition score.

The models’ goodness of fit is illustrated by the cumulative distribution functions shown in Figure 6. This graphical representation of the model’s goodness of fit shows a random selection of samples from the posterior distribution (i.e., the estimated parameter values distribution) alongside participants’ behavioural data. Here, we can see that the model has embodied the cognitive events occurring during the SST, thus offering robust estimation of its parameter values.

0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 left GO RT (s) Probability

Average Data LEFT

0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 right GO RT (s) Probability

Average Data RIGHT

0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 left SS RT (s) Probability

Average Data LEFT

0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 right SS RT (s) Probability

Correlation in Parameters of Inhibition Models

Our interest lies in the dynamics behind the driving components implicated in both global and selective inhibition. To investigate evidence of cofounding variable between these concepts of inhibition, we compare parameters from both LBA and BEESTS models. With the LBA parameter values, we describe participant behavioural changes across the MSIT conditions and define our measure for selective inhibition. We construct our global inhibition measure (i.e., SSRT) with the parameters from BEESTS. In the following section, we explore inter-parameter correlations between models (LBA and BEESTS), in function of the MSIT conditions and amongst participants.

Both model frameworks possess their own specific set of parameters, each modelling a specific aspect of inhibition. Being parsimonious in the construction of the model and its parametrisation allows limiting autocorrelation of said parameters and redundancy in their function (e.g., limits parameters trade-off). This conciseness is essential to the final step of this research as our between-model comparison is based on relevant data.

Here, we aim to investigate whether or not the estimated parameter values from a model can predict those for another model within the same participant. We will focus specifically on variation in parameter values between both models according to condition. Using the mean SSRT from the BEESTS modelling approach and the LBAs’ mean differences of accumulation rates with respect to conditions, we will observe how predictive parameter values from a model are in regard to the other model.

Methods

Data Acquisition

Parameter values were retrieved from both BEESTS and LBA models at the subject level. Selective inhibition was computed within subject as the difference between DR (parameter “mean_v”) in CON and INC conditions. Amongst the four MSIT conditions, we considered that in terms of cognitive load CON and INC were representative of the least and most demanding condition, respectively. Thus, the difference between the INC and CON mean DR parameter values shows each participant’s ability to change their internal parameter values according to the condition (selective inhibition). Global inhibition was estimated using participants’ SSRTs. The SSRTs were computed by summing the mean (muS) of the Gaussian component and the exponential parameter reflecting the skewed aspect of the ex-Gaussian distribution (tauS) of the SSRTs distribution.

The dataset was constituted with only fifteen participants. These participants have successfully completed both tasks and had their model parameter values estimated in both models. Amongst the eight participants that had to be excluded, one was already excluded from both tasks, two were excluded

from the MSIT, and five from the SST. Although this sample size is of low statistical power, we investigated the presence of linear associations between parameters values using the recommended Pearson coefficient (Kirch, 2008). Additionally, the method employed by DMC enables the parameter estimation process to feed into both group- and participant-level alternatively. One could say that each individual participant possesses, in their estimated parameter value, a trace of the group parameters and of other participants as well. Some participants were excluded from this analysis because they did not have their behavioural data modelled in both models but still left a “trace” in other participants’ parameters values. This exclusion does not spoil the validity of their data or the validity of the selected participants for this step.

On first analysis we observed possible intra-models’ autocorrelation of parameters. Secondly, we adopted an inter-model parameter study all-the-while respecting the MSIT condition effects over LBA parameters values. To do so, only parameters that we let vary with the conditions were considered for this step. Finally, a third analysis evaluates the relationship amongst inhibition scores within participants. As mentioned above, the sum of muS and tauS represent our global inhibition coefficient while the difference between the mean of the DR distribution in INC and CON condition represents selective inhibition.

Results

Intra-Model Correlation Study

Figure 7.A and Figure 7.B show the correlations found in parameter values across participants for BEESTS and LBA, respectively. In Figure 7.A we can see two strong correlations amongst the model parameters “mu.true” and “sigma.true”, and muS tauS. In both occurrences the parameters are describing a similar aspect of the distribution shape (mean and standard deviation), hence, potentially explaining a large amount of trade-off which leads to high correlation scores. Besides those two instances, the amount of autocorrelation amongst BEESTS parameters remains non-significant. Correlation amongst LBA parameters does not show unusual patterns. Strong correlation values remain in-between parameters describing similar aspects of the model, thus denoting possible trade-off between the latter.

Figure 7. Colour scale represents the Pearson coefficient of correlation among parameters values across

participants (with 1 indicating a perfect positive correlation, 0 the absence of correlation and -1 a negative correlation). A. Correlation matrix for BEESTS parameters. B. Correlation matrix for LBA parameters.

Inter-Model Correlation Study

For this analysis, we selected LBA parameters that only varied in function of the condition (i.e., mean DR for false and true accumulator in each four conditions). All parameters from BEESTS were included in the analysis. Figure 8 shows all correlations between parameters across models, their distribution and associated scatterplot. Pearson Coeff A s s o s s o s W s W o W r r r r f f f f r f m r r r m A. B

.

Corr:

Corr: Corr:

Corr: Corr: Corr:

Corr: Corr: Corr: Corr:

Corr: Corr: Corr: Corr: Corr:

Corr: Corr: Corr: Corr: Corr: Corr:

Corr: Corr: Corr: Corr: Corr: Corr: Corr:

Corr: Corr: Corr: Corr: Corr: Corr: Corr: Corr:

Corr: Corr: Corr: Corr: Corr: Corr: Corr: Corr: Corr:

mean_v.Con. false mean_v.Con.t rue mean_v.Fla. false mean_v.Fla.t rue mean_v.Inc. false mean_v.Inc.t rue mean_v.Sim. false mean_v.Sim.t rue m uS tauS mean_v.Con. false mean_v.Con.t rue mean_v.Fla. false mean_v.Fla.t rue mean_v.Inc. false mean_v.Inc.t rue mean_v.Sim. false mean_v.Sim.t rue m uS tauS 0.25 0.30 0.35 4.4 4.5 4.6 4.7 0.5 1.0 3.25 3.50 3.75 4.00 0.4 0.6 0.8 1.0 1.2 1.4 3.00 3.25 3.50 3.75 4.00 4.25 0.4 0.8 1.2 1.6 4.6 4.8 5.0 5.2 0.12 0.14 0.16 0.18 0.015 0.020 0.025 0.030 0 2 4 6 4.4 4.5 4.6 4.7 1.0 0.5 _{4.00 3.75 3.50 3.25} 1.4 1.2 1.0 0.8 0.6 0.4 _{4.25 4.00 3.75 3.50 3.25 3.00} 1.6 1.2 0.8 0.4 5.2 5.0 4.8 4.6 _{0.18 0.16 0.14 0.12} 0.015 0.020 0.025 0.030 F igu re 8 . C or re la tion m at ri x for p ar am et er s of i nt er es t in L B A a nd B E E ST S. T he c el ls c ons tit ut ing the di agon al ( top -l ef t to bot tom -r ight ) re pr es ent t he di st ri but ion of t he da ta poi nt s. U nd er t hi s di agona l, ea ch c el l c ont ai ns a s ca tte r pl ot t he pa ra m et er v al ue s fo r al l pa rt ic ipa nt s. C el ls a t th e top of t he di ag ona l hol d t he P ea rs on co ef fi ci ent va lu e c al cul at ed f or thi s s pe ci fi c s ubs et of d at a (a st er ixi s indi ca te s igni fi ca tiv e c or re la tion s).

Two sets of correlations show significance. Both correlations implicate the “mean_v.INC.false”, a parameter that depicts the accumulation of evidence for the false accumulator in the INC. The BEESTS parameters correlating with “mean_v.INC.false” are both implicated in the construction of the SSRT distribution and therefore of the global inhibition measure that we sought to explore.

The relationship between “mean_v.INC.false” and SSRT (sum of “muS” and “tauS”) amongst subjects is found to be moderately correlated (Figure 9, r(14) = .54, p =.035).

Figure 9. For all selected participants, a scatterplot of the global inhibition score (SSRT) against the

mean_v.INC.false parameter.

Comparing Measures of Inhibition

As previously mentioned, the core of this research is to investigate the connection between two forms of inhibition. A measure for each form has been defined beforehand (i.e., SSRT for global inhibition, and the difference between DR in the MSIT conditions CON and INC for selective inhibition). Here, we correlate global and selective inhibition values across participants. Figure 10 shows the initial scatterplot of the participant scores for both global and selective inhibition.

Figure 10. For all selected participants, a scatterplot of the global inhibition score (SSRT) against selective

We do not observe any significant correlation (r(14)= -0.044, p = .874) in-between the global and selective inhibition measures.

Discussion

Predicting Inhibition

In this section, we used two models and their previously estimated parameters to determine the measure of global and selective inhibition. To do so, we employed the SSRT distribution described by BEESTS to produce a global inhibition measure. Similarly, the LBAs’ DR parameters assisted in creating a selective inhibition measure. With this, we conducted correlation studies intra-model, inter-model and finally, with those two measures, between concepts of inhibition.

Besides two instances, the intra-model correlation study showed marginal correlations in the BEESTS parameters. The two instances reveal to be issued from parameter trade-off as both pairs of parameters describe, respectively, the same modelled object. The LBA contained a considerable amount of strong correlation in-between its parameters. This presence of firm correlations can be due to numerous factors such as lousy parametrisation, trade-off, or redundancy in the parameters’ task in modelling behavioural data. Upon closer inspection, it does not appear to result in any concerning patterns.

In the second analysis, we observe that the parameter describing the accumulation of evidence for the false accumulator (i.e., accumulator for incorrect responses) in the INC condition of the MSIT was correlating positively with the two parameters constituting the SSRT distribution. The first element in that correlation shows a lack of performance from the participants when its values increases. Indeed, as a participant accumulates more evidence for the wrong answer, they are more likely to reach the accumulator threshold and respond incorrectly. The second element of the correlation is the pair of parameters describing the SSRT distribution (i.e., “muS” and “tauS”). Longer SSRTs denote that the stop-process is slower than the go-process, thus resulting in low inhibition after the presentation of a SS. As SSRT values increase, the performance of the participant in global inhibition worsens. Moreover, we see that this relationship only holds for false accumulators of the INC condition where interferences are mixed and the cognitive load is high. Overall, this second analysis shows evidence that SSRT values could predict when participants set their accumulation rate for the false accumulator in the INC condition erroneously. Unfortunately, this tendency appears to be driven by two datapoints. Also, the weak sample size of the dataset is impairing our analysis with lacks statistical power thus preventing any robust conclusion.

Altogether, this show hints of a relationship between markers of poor inhibition performances. This could be understood, if shown to remain with a larger sample size, as an evidence that both models could find common grounds amongst participants’ behavioural data.