Evidence accumulation in the brick task : a model-based cognitive neuroscience study

Universiteit Van Amsterdam

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Evidence Accumulation In The Brick Task:

A Model-Based Cognitive Neuroscience Study

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August 2014
 Author:

Laura FONTANESI Dr. Leendert VAN MAANEN Supervisors:

Abstract

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Perceptual decision making studies usually employ tasks in which the evidence in favour of different alternatives is kept constant within trials. In the brick task (Brown, Steyvers, & Wagenmakers, 2009), however, the evidence presented on the screen varies at discrete time steps within a trial. In this study, we manipulated the duration of the time steps and collected fMRI and behavioural data from 20 participants. In the first part of the study, we fitted reaction time (RT) distributions using two sequential sampling models (SSMs), one of which accounts for differences in the way incoming evidence is weighted within a single trial. In the second part of the study, we implemented a Bayesian model to estimate the evidence presented on the screen when a response was given. We then looked for brain regions sensitive to trial-to-trial variations in these estimates and brain regions sensitive to trial-to-trial variations in RTs. Results from the behavioural analyses support the idea that individual differences play a big role in the way decisions are made in this task, in particular on the way evidence is weighted within a trial. Results from the fMRI analyses suggest that variations in the evidence and variations in RTs predict two non-overlapping brain networks, and highlight different aspects of the decision process.

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Introduction

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Perceptual decision making is the process of choosing a target stimulus among one or more distractors in the presence of noise. Understanding the mechanisms underlying such decisions (i.e. how perception is translated into actions) has been of great interest over the last century in psychology as well as in neuroscience (e.g. Gold & Shadlen, 2007; Smith & Ratcliff, 2004).

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A typical task used to study perceptual decision making is the two-alternative forced-choice (TAFC) task (Figure 1, on the left). In TAFC tasks, participants are presented with two stimuli

displaying different physical features (e.g. a cloud of dots moving in two opposite directions) and are asked to make a choice based on the perceived difference (e.g. choose the direction followed by the largest number of dots). This can be made more or less difficult varying the discriminability of the stimuli and different instructions can be given to stress either accuracy or speed in the answers (e.g. Palmer, Huk, & Shadlen, 2005).

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Sequential sampling models (SSMs) have been successfully used by mathematical psychologists to predict accuracy and reaction times (RT) distributions in TAFC tasks (e.g. Ratcliff & McKoon, 2008). Three basic assumptions are made: evidence in favour of the two alternatives is accumulated over time throughout a trial, the process of evidence accumulation is subject to random fluctuations due to noise in the incoming information, and a choice is made when the accumulated evidence reaches a decision threshold (Bogacz, Brown, Moehlis, Holmes, & Cohen, 2006).

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Cognitive neuroscientists have suggested that perceptual decision making in TAFC tasks is implemented in the brain by a fronto-parietal network, possibly complemented by the basal ganglia (BG) (Keuken et al., 2014). More recently, a new approach has been proposed, known as ‘model-based cognitive neuroscience’ (Corrado & Doya, 2007), in which mathematical modeling is combined with cognitive neuroscience. Following this approach, Forstmann and colleagues (2008) found that stressing speed in the answers in a TAFC task affected the response caution parameter of the formal model used to fit RT distributions, and selectively activated the right anterior striatum and the right pre-supplementary motor area (pre-SMA). Moreover, the estimated caution parameter at the behavioural level was found to be negative

correlated to activation in these areas (i.e. the more the participants tend to make risky decisions, the higher the activity in these areas).

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Decisions in TAFC tasks are quite fast (usually occur within a second). Therefore, TAFC tasks are not very suitable for fMRI analyses when we want to investigate brain activity related to the process of evidence accumulation per se (and not brain activity at the cue level, as for instance in Forstmann and colleagues, 2008) since fMRI temporal resolution is usually about two seconds. Moreover, in TAFC tasks the evidence presented on the screen in favour of the correct alternative is held constant within a trial. In the random dot motion (RDM) task, for example, even though the dots are continuously moving, the percentage of randomly moving dots is the same within a trial. Task difficulty can be manipulated, so that more or less evidence in favour of the correct alternative is presented on the screen, but this is usually done across experimental conditions. However, many perceptual decisions that we make in real life are based on evidence that changes over time. For instance, sailors often need to discriminate between different luminous signals (e.g. indicating the presence of another boat or a lighthouse) during the night, when they are moving and atmospheric conditions continuously change, making the signals more or less visible.

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Expanded judgment tasks (Irwin, Smith, & Mayfield, 1956) differ from other TAFC tasks in that evidence provided to the participants is manipulated at different moments within a trial. This has allowed to study how memory processes interact with evidence accumulation. For instance, Pietsch and Vickers (1997), argue that, in expanded judgment tasks, evidence presented early in the trial influences decisions less than evidence presented later.

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The brick task (Brown et al., 2009) (Figure 1, on the right) is an expanded judgment task in

which evidence in favour of the two alternatives is gradually presented within a trial, and the speed at which it is presented can be directly manipulated. However, unlike other expanded judgment tasks (e.g. Smith & Vickers, 1989; Usher & McClelland, 2001), the evidence presented since the beginning of the trial remains on the screen until a response is given, allowing to investigate the process of evidence accumulation while limiting the memory requirements of the task.

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Figure 1. Example of a TAFC task, the RDM task (on the left), and the brick task (on the

right). In the RDM task, a cloud of moving dots is presented at the beginning of the trial and participants are asked in which direction the majority of dots is moving (e.g. left or right). Stimuli are kept on the screen until a response is given, and decisions in this task usually take around 1 second. In the brick task, participants are presented with an empty screen at the beginning of the trial, and are asked which of the columns accumulates faster (e.g. column A or column B). During the trial, bricks accumulate at discrete time steps. The choice of the duration of each time steps affects how long participants will take to make a decision.

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In this study, 20 participants performed the brick task in an MRI scanner. Two experimental conditions were presented, varying the speed at which evidence was accumulating on the screen, but providing the same instructions (neither speed nor accuracy were stressed) and keeping the discriminability of the two stimuli constant across conditions.

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To describe the process of evidence accumulation in the brick task, Brown and colleagues (2009) proposed two Bayesian ideal observer models and a simple heuristic model. In the first part of the study, we wanted to investigate individual differences in how participants make decisions in this task and whether different components of the decision process were affected by just varying the speed of evidence accumulation on the screen. Since ideal observer models are not very suitable for this purposes, we chose to fit two SSMs, the Drift Diffusion Model (DDM; Ratcliff, 1978) and the more complex Ornstein-Uhlenbeck (O-U; Busemeyer & Townsend, 1993) model to behavioural data.These two models make different assumptions on the accumulation process: in the DDM, the same weight is given to evidence that is accumulated in different moments within a trial, while in the O-U model an additional parameter is added, allowing for variations in the evidence weights within a trial.

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In the second part of the study, we wanted to investigate which brain areas are involved in the process of evidence accumulation. Given the nature of the task, it was possible to know the height of the two columns at each time step during each trial. We used this additional

information to estimate the evidence on the screen at the moment a response was given. To do so, we defined a Bayesian model in which the two accumulation rates are estimated at each time step based on the number of fallen bricks. Using these estimates, we looked for brain areas in which the activity was modulated by trial-to-trial variations of the evidence on the screen and compared the results with brain activity modulated by trial-to-trial variations of the RTs.


Materials and Methods

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Participants

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Twenty participants (eleven female; mean age, 24; SD age, 7) took part in the study. An informed consent was given before the experiment started and the study was approved by the University of Amsterdam Ethics Committee. All participants had normal or corrected-to-normal vision. 

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Behavioural Task

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At the beginning of each trial a fixation cross was presented in the middle of the screen. After a fixed time step (0.217 sec in the first, and 0.417 sec in the second condition) there was 80% chance for a brick to fall in one column and 60% chance for a brick to fall in the other column. In each condition, there was a maximum of 25 time steps per trial (corresponding to 5.425 sec in the first, and 10.425 sec in the second condition), and trials ended as soon as a response was given or the maximum number of time steps was reached. At the end of each trial, a visual feedback was given, indicating the accuracy of the given answer. Participants were instructed to press a button (left or right) corresponding to the column in which bricks were accumulating faster (i.e. the column in which a brick had 80% chance to fall in each time step). Neither accuracy nor speed were stressed. Since the difference in accumulation rate between the faster and the slower column was the same in the two conditions, task difficulty has not been explicitly manipulated. Conditions consisted of 100 trials each and were presented in two mixed blocks, for a total of 200 trials per participant (of which 20 were null trials), so a maximum of 90 trials per condition was presented. The target column had 50% chance of being on the left or on the right side of the fixation cross, so there was no overall bias for either left or right responses.

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Drift Diffusion Model and Ornstein-Uhlenbeck Model

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To fit RT distributions for correct and incorrect trials, we chose two SSMs, the DDM and the O-U model, and estimated parameters for each participant in each condition. Both the DDM and the O-U model are random-walk models: they assume that evidence in favour of the two alternatives is accumulated in a single total (i.e. the difference in evidence for one and the

other alternative) and a choice is made when this single variable reaches a decision threshold (i.e. one for the correct and one for the incorrect response). In the DDM (Figure 2, top-right),

evidence accumulates in each trial following:

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The variable x indicates the difference between the evidence in favour of the correct alternative and the evidence in favour of the incorrect alternative at time t, and dx is the change in x in each small time interval dt. A is the drift rate (i.e. amount of evidence that is accumulated in each dt), assumed to have a positive mean but to vary across trials, due to variance in the way evidence is encoded. cdW is Gaussian noise with mean 0 and variance

c2_{dt, and indicates variance in the drift rate within trials (i.e. noise in the incoming}

information). Every trial is assumed to start at x=0, meaning that there is no initial bias for one of the two alternatives. Two additional parameters are the decision threshold, assumed to be constant within a trial (meaning that the decision criterion does not decay as time passes), and the non-decision time (i.e. part of the RT that is not related to the decision process itself), assumed to be constant across trials. In the full DDMboth non-decision time and starting point are left free to vary across trials (Ratcliff, & Tuerlinckx, 2002).

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The O-U model is similar to the DDM but an additional parameter is added, λ, that is linear to x:

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The amount of evidence that is accumulated at each dt, depends on the drift rate A and on the current state of accumulated evidence x, which is multiplied by λ. Depending on the sign of λ, the O-U process can be either stable or unstable (Bogacz et al., 2006), and as λ tends to 0, the O-U model reduces to the DDM.

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When λ < 0, the O-U process is stable (Figure2, top-left): if the current state of accumulated

evidence x is bigger than the fixed point -A/λ (usually between 0 and the correct threshold), x tends to decrease; whereas if x is smaller than -A/λ, x tends to increase. This has two main

dx = Adt + cdW,

x(0) = 0.

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Figure 2. Representation of the stable O-U (top-left), the DDM (top-right), and the

unstable O-U (bottom-left). Frequency distributions for correct and incorrect trials are represented respectively at the top and the bottom of each plot, and the paths of evidence accumulation of five trials are represented in the middle. Data were generated from three sets of parameter values, that differed only in the parameter λ (λ = -2 in the stable O-U, λ = 0 in the DDM, and λ = 2 in the unstable O-U). The fixed-point of the O-U process is represented as a red line, and (since the mean drift rate was set at 0.05) was equal to -A/λ = 0.025 (stable O-U) and -A/λ = -0.025 (unstable O-U).

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consequences on the decision process: (a) evidence accumulated early in the trial weights less than evidence accumulated later (recency effect), and (b) the state of preference for one or the other alternative oscillates back and forth before crossing the decision threshold, so that more time is required to make a decision (as in choices between two undesirable alternatives).

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When λ > 0, the O-U process is unstable (Figure 2, bottom-left): if the current state of

accumulated evidence x is bigger than the fixed point -A/λ (usually between the incorrect

correct (81%) − 0.10 0.00 0.05 0.10 incorrect (19%) correct (74%) − 0.10 0.00 0.05 0.10 incorrect (26%) correct (68%) − 0.10 0.00 0.05 0.10 incorrect (32%)

threshold and 0), x tends to increase; whereas if x is smaller than -A/λ, x tends to decrease. This has two main consequences on the decision process: (a) evidence accumulated later in the trial weights less than evidence accumulated earlier (primacy effect), and (b) the state of preference for one or the other alternative accelerates towards the decision threshold, so that less time is required to make a decision (as in choices between two desirable alternatives).

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Model fit and model comparison

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We estimated four parameters for the DDM and five for the O-U model, separately for each participant and each condition. The variance in the drift rate within trials was fixed at 1 for both models (this is usually fixed in parameter estimation as a scaling parameter). For the DDM, we estimated the mean drift rate and the variance in the drift rate across trials, the non-decision time and the decision threshold. For the O-U, we estimated the same parameters and the additional parameter λ. Given the complexity of the likelihood function of the DDM and the O-U model, we used an approximation of the likelihood function that is based on kernel density estimation (as in Turner & Sederberg, 2014, see also Van Zandt, 2000). Kernel density estimates were based on 20000 simulated trials. We therefore performed maximum-likelihood estimation in a two stages procedure. In the first stage, we selected 100 best fitting set of parameter values (80 for the DDM), from a pool of 1000 sets of parameter values (800 for the DDM). Parameter boundaries were set relatively wide and were constrained based on assumptions in the models (i.e. drift rate, variance in the drift rate, threshold and non-decision time are strictly positive). In the second stage, we performed differential evolution optimisation as implemented in the R package ‘DEoptim’ (Ardia, Mullen, Peterson, & Ulrich, 2013) using the results from the first stage as initial population, and for 250 iterations.

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To assess the goodness of this procedure, we ran a small parameter recovery study before fitting the data from the experiment. We chose a set of parameter values for the O-U model (mean RT, 1.9 s; SD RT, 0.5; accuracy, 79%) and generated 100 data sets of 100 trials each, of which 2% were outliers (sampled from a uniform distribution). Mean and variance of the recovered parameters were then computed to assess unbiasedness and variance of the estimates (Ratcliff, & Tuerlinckx, 2002).

To compare the goodness-of-fit of the O-U model and the DDM, Akaike Information Criterion (AIC; Akaike, 1973) weights (Wagenmakers, & Farrell, 2004) and Bayesian Information Criterion (BIC) weights (Schwarz, 1978) were computed, for each subject and for each condition.

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Once determined which of the two models gave a better account of the data, we tested if the estimated parameters differed across conditions using Bayesian paired t-test (Rouder, Speckman, Sun, Morey, & Iverson, 2009).

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Estimate of the evidence on the screen at each time step

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To built trial-to-trial estimates of the state of accumulated evidence on the screen (to be later used in the fMRI parametric analyses) we implemented a Bayesian model. This model takes into account knowledge about the distribution from which the presented evidence is sampled at each time step, and knowledge about the height of the two column at each time step of the presented trials. Contrary to the ideal-observer model proposed by Brown and colleagues (2009), this model does not assume that the participants know or vaguely know the two rates of the columns. Moreover, in this model we look at the probability that one column is ‘faster’ than the other, whereas in the ideal observer model the probability that the rate of a certain column is the known target rate is computed.

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During the brick task, participants are asked to say which of the two columns accumulates faster (i.e. which of the two column is higher on average). Bricks fall in the two columns following two independent binomial distributions with rates θt = 0.8 (target column) and θd =

0.6 (distractor column). At the beginning of the trial, the two rates are not known , and have 1

to be estimated based on the number of falling bricks. Therefore, after n time steps:

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✓

t

⇠ Beta(1 + s

t

, 1 + f

t

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✓

d

⇠ Beta(1 + s

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= ✓

t

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Here we assume that, at the beginning of each trial, the prior knowledge of the two rates is: θt~

1

where θt is the posterior distribution of the rate of the target column, which follows a beta

distribution with parameters α = 1 + st (st being thenumber of bricks that fell in the target

column) and β = 1 + ft (ft = n - st). θd is the posterior distribution of the rate of the distractor

column, which follows a beta distribution with parameters α = 1 + sd (sd being thenumber of

bricks that fell in the distractor column) and β = 1 + fd (fd = n - sd). The variable δ is the

difference between the two rates after n time steps. The probability of δ being positive (i.e. probability that θt is higher than θd) can be computed as the area underlying the posterior

distribution of δ for δ > 0 (Figure 3).

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Figure 3. Posterior distribution of δ (i.e. difference between the rates of the two columns) after 5 (on the left) and 20 (on the right) time steps since the beginning of a simulated trial. After 5 time steps, 3 bricks have fallen in both columns, so that st = sd = 3, and ft = fd =

2. Therefore, the evidence in favour of the correct alternative (i.e. δ being positive) is p = 0.50 (area shaded in red). After 20 time steps, 13 bricks have fallen in the target column and 9 in the distractor column, so that st = 13, sd = 9, ft = 7, and fd = 11. Therefore, the

evidence in favour of the correct alternative (i.e. δ being positive) is p = 0.89 (area shaded in red). As n increases, the estimate of δ is more precise, and p also increases.

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fMRI data acquisition

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Imaging data were acquired in two scanning sessions (i.e. one for each block of the experiment) using a Philips 3T Achieva scanner. At the beginning of the experiment, T1 anatomical scans were acquired for each subject (220 slices; TR, 8.2 s; TE, 3.8 ms; flip angle, 8°; FOV, 240 mm; voxel size, 1 * 1 * 1 mm). For functional imaging, EPI scans were acquired in transverse orientation (slice thickness, 3 mm; 37 slices; TR, 2 s; TE, 27.63 ms; flip angle, 76.1°; FOV, 240 mm; voxel size, 3 × 3 × 3 mm). Trials started every 12 s: at the beginning of

n = 5 δ Density −1.0 −0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 n = 20 δ Density −1.0 −0.5 0.0 0.5 1.0 0.0 1.0 2.0 3.0

each trial a fixation cross was presented for either 0, 500, 1000, or 1500 ms, so that the time in between trials was not always the same.

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fMRI preprocessing

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fMRI analyses were performed using FEAT (FMRI Expert Analysis Tool), version 6.00, part of FSL (FMRIB’s Software Library, www.fmrib.ox.ac.uk/fsl). Images were realigned to correct for small head movements using MCFLIRT motion correction tool (Jenkinson, Bannister, Brady, & Smith, 2002). Data were spatially smoothed using a 5-mm FWHM Gaussian kernel, temporally filtered using a nonlinear high-pass filter (100-s cutoff), and pre-whitened. All functional images were registered using the subjects’ individual high-resolution anatomical images acquired at the beginning of the experiment and normalised into MNI space by linear scaling.

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fMRI data analysis

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The statistical analyses were based on the general linear model (GLM). The design matrix was convolved using a double gamma haemodynamic response function (HRF) and its temporal derivative. Two separate analyses were performed, following a parametric (or variable impulse) approach (as described in Grinband, Wager, Lindquist, Ferrera, & Hirsch, 2008).

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In the first analysis, a total of nine regressors were included: response hand (left and right), null trials, responses around or below chance (in favour of the opposite alternative), trials in the first, and trials in the second condition. The remaining two regressors were added modulating the amplitude of the HRF based on differences in RTs (one for the first and one for the second condition). The second analysis was similar to the first but the last two regressors were added modulating differences in the evidence in favour of the given response (estimated with the Bayesian model described above).

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For each subject and each session we built four contrasts: one for trials in the first condition (non-modulated), one for trials in the second condition (non-modulated), one for trials in the first condition (modulated based on evidence or RTs), and one for trials in the second condition (modulated based on evidence or RTs). Second-level analyses were performed to

combine the two runs from each participant for each condition (fixed effect). A group-level analysis was then performed for each condition (mixed effect).

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Finally, four conjunction analyses (Nichols, Brett, Andersson, Wager, & Poline, 2005) were performed: first, we looked at common areas modulated by evidence and RTs separately for each condition. Then, we looked for common areas modulated by RTs between the two conditions and common areas modulated by evidence between the two conditions.


Results

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One participant was excluded from both behavioural and fMRI analyses since the participant did not complete the experiment. One participant was excluded from the fRMI analyses due to excessive head motion. Two other participants were excluded from the fMRI analyses due to excessive noise in the data.

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Behavioural data

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Figure 4illustrates the behavioural data. As the duration of the time steps increases, mean

RTs also increase (BF = 1403202.0), and the number of time steps (BF = 19608.8) decreases. However, as the duration of the time steps increases, accuracy (BF = 1.3) does not change. The same holds for the estimate of evidence on the screen at the moment a response is given (based on the Bayesian model described in the Methods section) (BF = 0.4).

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Figure 4. From the left: mean RTs, mean accuracy, mean number of time steps before a

response is given, and mean probability that the given response is the correct one, at two different time steps durations. Error bars represent 95% confidence intervals.

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217 ms 417 ms

Time steps duration

Mean R Ts (s) 0 1 2 3 4 5 217 ms 417 ms

Time steps duration

Mean Accur acy 0.0 0.2 0.4 0.6 0.8 1.0 217 ms 417 ms

Time steps duration

Mean n time steps

0 2 4 6 8 10 12 14 217 ms 417 ms

Time steps duration

Mean p f or the giv en response 0.0 0.2 0.4 0.6 0.8 1.0

Parameter recovery

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The results of the parameter recovery study are presented in Table 1 and Figure 5. We

obtained fairly accurate estimates of the mean drift rate and the non-decision time. Both the λ and the decision threshold seem to be slightly underestimated, but not dramatically biased (especially considering the range of possible values of λ). However, the variability of the drift rate across trials does not seem to be recovered well (values are spread all over the parameter space).

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Table 1. Mean and SD of recovered parameters from 100 data sets of 100 trials with 2% of

contaminated data.

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Figure 5. Recovered parameters from 100 data sets with 2% of contaminated data. The

red lines in the scatter plots represent the true values and the range on the x-axes represent the parameter boundaries.

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Mean Drift Rate SD Drift Rate _{Across Trials} λ _ThresholdDecision Non-decision _{time (s)}

True value 0.080 0.010 2.00 0.90 0.300 Mean estimates 0.080 0.026 1.77 0.76 0.291 SD estimates 0.013 0.018 0.12 0.13 0.074 0.0 0.1 0.2 0.3 0.4 0.5 − 140 − 120 − 100

Mean Drift Rate

Log Lik elihood 0.00 0.02 0.04 0.06 0.08 0.10 − 140 − 120 − 100 SD Drift Rate Log Lik elihood −4 −2 0 2 4 − 140 − 120 − 100 λ Log Lik elihood 0.0 0.5 1.0 1.5 2.0 2.5 − 140 − 120 − 100 Threshold Log Lik elihood 0.3 0.4 0.5 0.6 − 140 − 120 − 100 Non−decision time (s) Log Lik elihood

Model comparison

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Results of model comparison are reported in Table 2. AIC indicates the O-U model as the

preferred model for most participants in both conditions (15 participants in the first and 13 in the second), while BIC favours the O-U model in the first (12 participants) and the DDM in the second condition (10 participants). Figure 6 shows differences between AIC and BIC

weights, and how strong is the evidence for one or the other model. Even though AIC tends to prefer the O-U model, the probability of the O-U being the best model is not always strong (i.e. weights are not always close to 1). However, there seem to be no cases in which there is high probability that the DDM is better than the O-U (i.e. weights that are close to 0).

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Table 2. Differences in AIC/BIC with respect to the AIC of the best candidate model, AIC/

BIC weights in favour of the O-U model, per participant per condition. When either Δ

Sub ΔAIC DDM ΔAIC O-U AIC weights ΔBIC DDM ΔBIC O-U BIC weights

217 ms 417 ms 217 ms 417 ms 217 ms 417 ms 217 ms 417 ms 217 ms 417 ms 217 ms 417 ms 1 17.32 5.55 0.00 0.00 1.000 0.941 15.06 3.30 0.00 0.00 0.999 0.839 2 4.73 0.02 0.00 0.00 0.914 0.503 2.53 0.00 0.00 2.26 0.780 0.244 3 0.00 0.00 0.96 2.26 0.382 0.244 0.00 0.00 3.13 4.60 0.173 0.091 4 13.07 2.01 0.00 0.00 0.999 0.732 10.85 0.00 0.00 0.26 0.996 0.468 5 18.72 36.47 0.00 0.00 1.000 1.000 16.42 34.26 0.00 0.00 1.000 1.000 6 1.39 24.27 0.00 0.00 0.667 1.000 0.00 21.90 0.68 0.00 0.416 1.000 7 3.57 1.03 0.00 0.00 0.856 0.627 1.30 0.00 0.00 1.18 0.657 0.357 8 4.01 1.23 0.00 0.00 0.881 0.649 1.74 0.00 0.00 0.98 0.704 0.380 9 5.64 6.14 0.00 0.00 0.944 0.956 3.31 3.97 0.00 0.00 0.840 0.879 10 9.44 27.44 0.00 0.00 0.991 1.000 7.25 25.14 0.00 0.00 0.974 1.000 11 7.06 17.68 0.00 0.00 0.972 1.000 4.82 15.41 0.00 0.00 0.917 1.000 12 0.00 0.00 0.80 2.18 0.401 0.252 0.00 0.00 3.00 4.48 0.182 0.096 13 1.87 0.00 0.00 0.63 0.718 0.422 0.00 0.00 0.35 2.87 0.456 0.192 14 24.94 32.77 0.00 0.00 1.000 1.000 22.66 30.54 0.00 0.00 1.000 1.000 15 3.44 3.83 0.00 0.00 0.848 0.872 1.22 1.56 0.00 0.00 0.648 0.686 16 0.00 0.00 1.90 1.73 0.279 0.296 0.00 0.00 4.17 3.97 0.111 0.121 17 1.36 0.00 0.00 2.07 0.663 0.262 0.00 0.00 0.81 4.35 0.400 0.102 18 7.94 21.58 0.00 0.00 0.981 1.000 5.73 19.31 0.00 0.00 0.946 1.000 19 0.00 0.00 2.39 0.85 0.232 0.395 0.00 0.00 4.58 3.04 0.092 0.180

AIC or Δ BIC are 0 for a particular model, it means that it is the preferred model. AIC/ BIC weights can be interpreted as the probability that the O-U model is the best model.

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Figure 6. AIC and BIC weights comparing the O-U model fit with the DDM fit for each

participant in the two conditions (left and right). The thick dotted line in the middle of the graph represents the value (0.5) indicating equal support for the two models. Values above 0.5 indicate support for the O-U model, while values below 0.5 indicate support for the DDM.

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Ornstein-Uhlenbeck model parameters

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Figure 7 shows the results from fitting the O-U model to behavioural data. We performed six

paired Bayesian t-tests to look at differences between model parameters across conditions. The Bayes Factor for the mean drift rate was 1966.9, 0.2 for the the standard deviation of the drift rate across trials, 224.4 for the λ parameter, 3.6 for the threshold, and 0.3 for the non-decision time. Moreover, BF = 0.2 for the fixed-point (-A/λ, where A is the mean drift rate).

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Figure 8 shows the relationship between the estimated λ parameter and the difference in

AIC between the O-U model and the DDM. As the λ parameter approaches 0, the O-U model reduces to the DDM and therefore the difference in model fit becomes smaller. Since the two models make similar predictions, the more complexity of the O-U model weights more in the model comparison. 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 217 ms time steps Subject O − U vs DDM fit 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 417 ms time steps Subject O − U vs DDM fit AIC weights BIC weights

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Figure 7. Boxplots of the estimated parameters of the O-U model, in 20 participants and

at two time steps durations.

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Figure 8. The estimated λ parameter is plotted against difference between AIC for the O-U model and the DDM. The dotted line represents the fitted regression line between the two variables.

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● ● ● ● 0.04 0.06 0.08

Time step duration

mA 217 ms 417 ms ● ● 0.00 0.01 0.02 0.03 0.04

Time step duration

sdA 217 ms 417 ms ● ● ● − 2 − 1 0 1 2

Time step duration

lambda 217 ms 417 ms ● 0.2 0.4 0.6 0.8

Time step duration

Thr 217 ms 417 ms 0.0 0.1 0.2 0.3 0.4

Time step duration

NDT 217 ms 417 ms ● ● ● − 0.6 − 0.2 0.2

Time step duration

Fix ed point 217 ms 417 ms 0 10 20 30 − 2 − 1 0 1 2 First Condition Delta AIC lambda − 0.6 − 0.2 0.0 0.2 fix ed point ( − A/lambda) 0 10 20 30 − 2 − 1 0 1 2 Second Condition Delta AIC lambda − 0.6 − 0.2 0.0 0.2 fix ed point ( − A/lambda) Page ! of !19 30

Brain activity modulated by variations in reaction times

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Table 3 and Figure 9 show the results from the conjunction analysis investigating common

clusters between brain regions sensitive to trial-to-trial variations in RTs in the two conditions. Results are reported at an uncorrected threshold of z > 2.3 and cluster corrected at p > 0.05.

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Table 3. Results from the conjunction analysis (brain regions sensitive to trial-to-trial variations in RTs in the two conditions). The reported areas are based on the Harvard-Oxford cortical structural atlas in FSLview. A z-threshold of 2.3 and a p-cluster threshold of 0.05 was applied. Max refers to the maximum z-value in each cluster and max x, max y, and max z are the voxel coordinates of the relative peak.

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Fig 9. Results from the conjunction analysis (brain regions sensitive to trial-to-trial variations in RTs in the two conditions).

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Brain activity modulated by variations in estimated evidence

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There were no significantly activated cluster in the conjunction analysis investigating common clusters between brain regions sensitive to trial-to-trial variations in evidence in

Cluster Index Area Voxels P MAX MAX X (vox) MAX Y (vox) MAX Z (vox)

5

Lateral Occipital Cortex,

inferior division 8839 1.08e-15 9.16 19 29 36

4 Insular Cortex 4690 6.02e-10 6.38 28 73 37

3 Lateral Occipital _Cortex 4323 2.28e-09 7.56 69 26 39

2 Superior Parietal _Lobule 1367 0.000802 5.13 63 40 61

the two conditions. Table 4 and 5, and Figure 10 and 11 summarise the group-level contrasts

for the first and second condition. Results are reported at an uncorrected threshold of z > 2.3, with a cluster extent of k > 20.

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Table 4. Brain regions sensitive to trial-to-trial variations in evidence in the first condition. The reported areas are based on the Harvard-Oxford cortical structural atlas in FSLview. A z-threshold of 2.3 and was applied and only clusters of more than 20 contiguous voxels are reported. Max refers to the maximum z-value in each cluster and max x, max y, and max z are the voxel coordinates of the relative peak.

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Fig 10. Brain regions sensitive to trial-to-trial variations in evidence in the first condition.

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Cluster Index Area Voxels MAX MAX X (mm) MAX Y (mm) MAX Z (mm)

6 _{Cortex, superior division}Lateral Occipital 78 2.87 -34 -76 38

5 _{Cortex, superior division}Lateral Occipital 50 2.82 46 -72 40

4 Precuneus Cortex 41 2.72 -18 -56 20 3 Inferior Temporal Gyrus, temporooccipital part 32 3.03 -52 -46 -16 2 Occipital Pole 29 3.15 4 -96 10

1 _{Cortex, superior division}Lateral Occipital 28 2.7 -52 -70 22

Cluster Index Area Voxels MAX MAX X (mm) MAX Y (mm) MAX Z (mm)

17 _{Cortex, superior division}Lateral Occipital 1165 4.37 -48 -70 28

16 Frontal Pole 817 3.77 6 64 8

15 Middle Frontal Gyrus 412 4.75 -24 16 38

14 Precuneus Cortex 411 3.7 -8 -56 2

13 Lateral Occipital Cortex, _{superior division} 272 3.41 48 -58 24

12 Occipital Pole 257 3.78 4 -96 0

11 _{Gyrus, pars triangularis}Inferior Frontal 212 3.87 -52 22 16

Table 5. Brain regions sensitive to trial-to-trial variations in evidence in the second condition. The reported areas are based on the Harvard-Oxford cortical structural atlas in FSLview. A z-threshold of 2.3 and was applied and only clusters of more than 20 contiguous voxels are reported. Max refers to the maximum z-value in each cluster and max x, max y, and max z are the voxel coordinates of the relative peak.

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Fig 11. Brain regions sensitive to trial-to-trial variations in evidence in the second condition.

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9 Occipital Fusiform Gyrus 104 3.51 -18 -84 -16

8 56 3.44 42 -70 -46

7 Parahippocampal Gyrus, _{posterior division} 53 3.14 -24 -22 -24

6 51 3.37 16 -12 22

5 Middle Frontal Gyrus 37 3.32 38 18 38

4 Frontal Pole 34 2.73 32 50 -14

3 Frontal Orbital Cortex 26 2.93 -30 34 -18

2 Superior Frontal Gyrus 24 2.94 2 48 34

Discussion

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Results from model comparison suggest that the O-U model is overall better than the DDM to fit RT distributions in the brick task. The O-U model is very similar to the DDM, but accounts for differences in the way the same amount of evidence weights, depending on the current state of accumulated evidence (and, therefore, on the moment it is presented within a trial). These differences are determined by a value, the ‘fixed-point’ of the O-U process, that depends on both the drift rate and the additional λ parameter. The closest this fixed-point is to zero, the more the O-U model prediction differs from the DDM. Moreover, depending on wether the fixed-point is positive (negative λ) or negative (positive λ), the O-U process is respectively stable or unstable, and results in different behavioural patterns. Fitting the O-U model to RT distributions, we obtained positive λ parameter estimates for most participants (all but one). In this case, the O-U is an unstable process: evidence presented earlier weights more in the decision process and the behavioural pattern is similar to choices between two desirable alternatives and more liberal decisions (Busemeyer, & Townsend, 1993). This can be interpreted in the light of the fact that trials in the brick task are relatively long and that only a written feedback is presented (no penalty nor rewards).

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Moreover, we found strong evidence that the mean drift rate and the λ parameter are higher in the first (time step duration of .217 s) compared to the second condition (time step duration of .417 s) but that the fixed-point is the same in the two conditions. Differences in the mean drift rate reflect the difference in evidence accumulation rate, which was explicitly manipulated varying the duration of the time steps: the higher the duration, the slower the accumulation rate. Interpreting differences in the λ parameter per se, however, might be misleading: as stated above, different behavioural patterns are obtained depending on the position (sign and distance from zero) of the fixed-point, that is determined dividing the drift rate by the λ parameter. Therefore, our results suggest that the observed individual differences do not depend on the duration of the time steps: participants that make more liberal decisions in the first condition also make more liberal decisions in the second condition.

Results from the fMRI analyses suggest that variations in the RTs and variations in the estimated evidence for the given response reflect activation in different brain areas. Variations in RTs seems to be more related to task-related areas (i.e. visual and pre-motor areas), while variations in evidence seems to be more related to activation in frontal, middle-frontal areas and with activation in the precuneus. However, since we found no significant clusters in common between the first and the second condition, results for evidence modulation are more difficult to interpret.

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As a conclusion, in the first part of the project we observed that accounting for individual differences is very important in this task. Not only RTs are longer compared to other TAFC tasks, but since neither speed nor accuracy were stressed, mean RTs and the shape of RT distributions vary significantly between participants. The λ parameter of the O-U model describes how different participants behave in this task: the higher the λ, the riskier the behaviour. In the second part of the project we observed how trial-to-trial estimates can be exploited to highlight different aspects of the decision process.

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Future Developments

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So far, we used two models, the O-U model and a Bayesian model, to fit respectively behavioural and fMRI data. We showed how the first model accounts for individual differences in the way participants behave in the brick task, while the second model accounts for the additional information available in this task, and can be used to modulate the trial-to-trial amplitude of the impulse function in the GLM. Figure 12 and 14 illustrates RT

distributions and O-U model fit in the first and second condition for two participants. Participant in Figure 12 makes more liberal (i.e. shorter and less accurate) decisions than

participant in Figure 14, and the estimated fixed-point in the O-U process is closer to zero.

O-U model fit is based solely on RTs and accuracy in the trials and does not take into account the knowledge about what is happening on the screen. Figure 13 and 15 show the

Bayesian estimates of the evidence for the given response for the same participants as a function of RTs in the two conditions. These plots also show that the first participant makes more liberal decisions: in some trials decisions are made when the probability that the chosen column is the correct one is at the chance level or even less than chance level (i.e. there is more evidence for the non-chosen column).

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Compared to other TAFC task, in the brick task we know the distribution from which the accumulating evidence on the screen is sampled (i.e. two independent binomial distributions) and the state of evidence on the screen at each time step (i.e. the height of the two columns). Moreover, evidence on the screen accumulates at discrete time steps. Both the DDM and the O-U model, however, assume that evidence in favour of the correct alternative is sampled continuously in time from a normal distribution with mean the drift rate. Accounting for the knowledge about evidence accumulation on the screen might improve model fit and interpretability of the results, giving a better account of the investigated decision process.

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When λ is positive, the O-U model describes a process in which more weight is given to evidence as a function of its distance from the fixed-point, that is somewhere close to the starting point. As a consequence, more weight will be given to evidence accumulated earlier in the trial, leading to more liberal decisions (i.e. faster and less accurate). This behavioural pattern has some similarities with the one observed in models in which the decision

threshold is not kept constant, but decreases as a function of time within a trial. However, the two models make different assumptions about the decision processes.

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We are therefore going to fit a model that incorporates the main aspects of the models we used so far: accounts for evidence integration over time, individual differences, and knowledge about the accumulating evidence on the screen. Having such a model, we will investigate two main aspects of decision making in the brick task: (a) if a model that accounts for collapsing boundaries is better than a model that accounts for different evidence weights within a trial; (b) which brain areas are related to individual differences in the task.


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Fig 12. RT distributions and O-U model prediction in one participant in the first (on the left) and second (on the right) condition.

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Fig 13. Bayesian estimates of the evidence in favour of the given response as a function of time in one participant (same as Figure 12) in the first (on the left) and second (on the right) conditions.

The black crosses indicate incorrect trials.

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correct (data) incorrect (data) correct (model) incorrect (model) − 0.4 − 0.2 0.0 0.2 0.4 fixed point correct (data) incorrect (data) correct (model) incorrect (model) − 0.6 − 0.4 − 0.2 0.0 0.2 0.4 0.6 fixed point 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 RT Evidence f or the giv en response 0 2 4 6 8 10 12 0.0 0.2 0.4 0.6 0.8 1.0 RT Evidence f or the giv en response

Fig 14. RT distributions and O-U model prediction in one participant in the first (on the left) and second (on the right) condition.

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Fig 15. Bayesian estimates of the evidence in favour of the given response as a function of time in one participant (same as Figure 14) in the first (on the left) and second (on the right) conditions.

The black crosses indicate incorrect trials.


− 0.3 − 0.2 − 0.1 0.0 0.1 0.2 0.3 fixed point − 0.4 − 0.2 0.0 0.2 0.4 fixed point 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 RT Evidence f or the giv en response 0 2 4 6 8 10 12 0.0 0.2 0.4 0.6 0.8 1.0 RT Evidence f or the giv en response

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