# Hierarchical Bayesian Analyses for Modeling BOLD Time Series Data

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## Abstract

Hierarchical Bayesian analyses have become a popular technique for analyzing complex interactions of important experimental variables. One application where these analyses have great potential is in analyzing neural data. However, estimating parameters for these models can be complicated. Although many software programs facilitate the estimation of parameters within hierarchical Bayesian models, due to some restrictions, complicated workarounds are sometimes necessary to implement a model within the software. One such restriction is convolution, a technique often used in neuroimaging analyses to relate experimental variables to models describing neural activation. Here, we show how to perform convolution within the R programming environment. The strategy here is to pass the convolved neural signal to existing software package for fitting hierarchical Bayesian models to data such as JAGS (Plummer 2003) or Stan (Carpenter et al. 2017). We use the convolution technique as a basis for describing neural time series data and develop five models to describe how subject-, condition-, and brain-area-specific effects interact. To provide a concrete example, we apply these models to fMRI data from a stop-signal task. The models are assessed in terms of model fit, parameter constraint, and generalizability. For these data, our results suggest that while subject and condition constraints are important for both fit and generalization, region of interest constraints did not substantially improve performance.

## Keywords

Stop-signal Hierarchical Bayesian Modeling Functional magnetic resonance imaging## Introduction

Analyzing neural data can be quite complicated as its quality can be affected by a variety of confounding factors, such as motion, technological limitations, and patterns of activation that are unrelated to the experimenter’s interests. As a result, many data analytic strategies have been developed to sieve the important characteristics (i.e., “signal”) from neural data, which are often obscured by sources of variance that are unsystematically related to the independent variables of the experiment (i.e., “noise”). It is particularly challenging to purify the signal within functional magnetic resonance imaging (fMRI) data. For example, preprocessing fMRI data often includes motion correction, slice timing correction, mapping to a standardized space, and spatial smoothing. Even after all of these preprocessing steps, it can be difficult to extract meaningful signal from fMRI data, due to noise and perhaps more importantly, the slow nature of the hemodynamic lag. As a consequence, it can be difficult to quantify statistical evidence or precisely identify the neural basis of cognitive functions.

One convenient way to purify data quality while simultaneously quantifying statistical evidence is through hierarchical Bayesian modeling (Rouder and Lu 2005; Lee 2008; Shiffrin et al. 2008; Ahn et al. 2011; Turner et al. 2013). Within a hierarchical model, experimental effects can be conceptualized as having a common basis, yet still allowing for departures that are intrinsic to particular factors. For example, subjects performing a task in a typical experiment will undoubtedly have some commonalities: they may have similar age ranges, they are all performing the same task, and they may all be mentally healthy. Yet, they may also have individual characteristics that allow them to perform systematically differently from all other subjects in our task. Hierarchical models allow us to balance these different factors in a statistically principled manner. Also, hierarchical Bayesian models improve parameter estimates for a single trial by introducing additional constraints on the estimates and reducing uncertainty. Furthermore, Bayesian statistics allow us to quantify the effects in a way that is exclusively influenced by the data from our current experiment, as well as our prior beliefs about the relative magnitudes of the effects.

Despite the many advantages that hierarchical Bayesian methods provide, they can be complicated to implement. Fortunately, many statistical software packages facilitate Bayesian estimation, enabling Bayesian solutions to otherwise impenetrable analysis questions. However, because many software packages are highly restrictive in the models they can fit to data, sometimes complicated workarounds are necessary to perform more complex analyses. One area where extant software packages have not yet been successfully applied is in the analysis of fMRI data. Although there are potentially a few different reasons (see General Discussion), we believe that a major limiting factor is the lack of accessibility and clarity about how to relate the experimental design to the neural data acquired from the scanner. To accomplish this, a mathematical process called convolution is needed to combine the details of the stimulus presentation with mathematical models that describe hypothesized neurophysiological responses to the stimuli. In many papers, the equations detailing how to perform convolution are somewhat obscured, making it difficult to imagine what the data generating model is and how we can fit the model to data. Although there are software packages available for implementing Bayesian analyses for fMRI data, such as those built into FSL (Chappell et al. 2009) or SPM (Han and Park 2018), these packages are difficult to customize and are unfortunately not hierarchical. All of these complications make it difficult for any aspiring modeler to analyze neural data, and for example, link the parameters of interest to a cognitive model (Turner et al. 2013, 2015, 2016, 2018).

Here, we present a more accessible introduction to modeling neural activation by providing concrete convolution equations, and connecting these equations with computer code written in R. The benefit of this approach is that once the design matrix for the experimental data is computed, it can easily be passed to user-friendly Bayesian software programs such as Just Another Gibbs Sampler (Plummer 2003, JAGS;) or Stan (Carpenter et al. 2017). These programs make Bayesian inference quite convenient, as several algorithms facilitate efficient and accurate estimation of the model parameters (e.g., Markov chain Monte Carlo). We then develop five models of neural time series data that increase in complexity in terms of the number of effects that they are able to accommodate. Although the models are intended to be generally applicable, we apply them to data from a stop-signal task to provide a concrete example of implementation. Within our task, we assess the models’ ability to both fit data and to generalize to new data through out-of-sample prediction analyses. Hence, the present goal of this article is to show the importance of condition- and subject-level constraints in the stop-signal task, while providing the tools necessary for others to carry out hierarchical Bayesian analyses of fMRI time series data in their own research.

The outline of the article is as follows. First, we give a brief overview of the literature on Bayesian analysis of fMRI data, and of the literature on response inhibition. Second, we explain how the general linear model analysis used in fMRI can be implemented in a Bayesian framework. Here, we explain how convolution is done effectively for two types of experimental designs. Third, we develop five models of neural time series data. An advantage of these models, and a deviation from typical fMRI analyses, is that they extract estimates of neural activation for each trial/ stimulus, as opposed to a single estimate averaged across trials for each condition. Fourth, we describe our stop-signal task and the fMRI procedures. Fifth, we present model fitting results from one run of the stop-signal task. Sixth, we use an out-of-sample prediction analysis to test the generalizability of the models across different runs of the stop-signal task. We conclude with a summary of the results and a discussion of limitations and future directions.

### Bayesian Analyses of Neural Activation

One of the most standard methods in fMRI data analysis is the general linear model (GLM). The GLM can be considered a multiple linear regression model applied to fMRI time series data, where the data are modeled as a linear combination of factors such as the condition, stimulus, and baseline level of activation. In addition, task-irrelevant factors such as scanner drift, physiological noise, and autocorrelation due to hemodynamic properties are sometimes modeled (Friston et al. 1995). Although there are scenarios where the linear modeling assumptions necessary for the GLM do not hold for fMRI data (Monti 2011; Poline and Brett 2012), the GLM approach still remains the most frequently used data analysis method due to its simplicity and approachability. However, fMRI analysis—particularly the GLM—could be improved using Bayesian statistics, where the uncertainty of model parameters must be quantified in terms of posterior distributions. Although fMRI analysis has been predominantly frequentist, we certainly are not the first to suggest using Bayesian statistics to improve fMRI analysis. Bayesian techniques such as spatial priors, adaptive priors, and model comparisons have been applied to fMRI research in various ways over the past few decades (for a review, see Zhang et al. (2015)).

A natural extension of a Bayesian GLM applied to a single subject is to introduce a hierarchical structure. Although it is a technical point, extensions to hierarchical models are relatively easy in the Bayesian framework, compared to frequentist statistics. This is because Bayesian statistics factorize multi-dimensional model parameters into a series of conditional probabilities that depend on the structure of the model (Kruschke 2014; Woolrich 2012). By building a hierarchical layer, the information extracted from one subject’s data helps to constrain the inference process for other subjects. This information sharing allows subject-specific effect parameters to be less affected by random noise, because each parameter is affected by both the subject’s data and the information learned across the group (Kruschke 2014). Given these properties, one can view the addition of a hierarchical structure for an fMRI time series as an extension of the standard GLM; whereas the standard GLM may only estimate condition-level effects, adding a hierarchy enables the model to capture both condition-level effects and stimulus-specific effects.

We are also not the first to use *hierarchical* Bayesian techniques to analyze fMRI data. For example, hierarchical Bayesian analyses have been used to impose spatial constraints on voxel-level analyses (Bowman et al. 2008). Although the analyses presented in this article are based on clusters of voxels that comprise regions of interest (ROIs), the methods presented here can easily be integrated into existing pipelines for voxel-based fMRI analyses (see the General Discussion).

Unfortunately, Bayesian approaches to neuroscience are complex. With such complicated methods, it is difficult to find a balance between ease of implementation and ability to customize. Many Bayesian estimation programs are home-brewed, which are complicated to write. Others are deeply imbedded within software packages, and do not allow the user to tailor the models to their experimental design and analytic goals. Another barrier of using Bayesian statistics is the computational burden. Programs such as JAGS and Stan have helped ease this burden, and also increased accessibility by creating a user-friendly working environment that can be managed within R. In this paper, we show how one can first perform convolution in R (which be translated to other programming languages such as Matlab or Python) and then pass the resulting convolved signal to your preferred Bayesian software package.

### Measuring Response Inhibition

One domain where hierarchical Bayesian analysis can be helpful is when measuring response inhibition. Response inhibition is considered an important component of cognitive control (Miyake et al. 2000; Aron 2007; Logan 1985) and has interesting applications to individual differences (Miyake and Friedman 2012), attention deficit hyperactivity disorder (Nigg 2001; Schachar and Logan 1990), obsessive-compulsive disorder (Bannon et al. 2002; Penadés et al. 2007), and substance use disorders (Monterosso et al. 2005; Nigg et al. 2006). Additionally, tasks designed to measure response inhibition often produce many missing behavioral observations. For example, if a subject was instructed to withhold their response, a correct action would result in no behavioral data. To gain a better perspective on the cognitive processes that allowed for successful response inhibition, we can look to patterns of neural measures during the trial. In this sense, the issue of sparse or missing data can be circumvented by thorough analysis of brain data.

Response inhibition has been studied extensively in neuroscience and mathematical psychology. Considerable research has led to well-developed theories of response inhibition, in both of these fields. It is important to note that most of this research, with a few exceptions, is either focused entirely on neuroscience or modeling, without much overlap between the two fields. We begin a brief review of the response inhibition theories in neuroscience, primarily within the context of fMRI experiments.

Within neuroscience, cognitive control theories, in general, are based on the idea that fronto-parietal connectivity allows for cognitively regulatory abilities (Miller and Cohen 2001; Jung and Haier 2007). Additionally, individual differences found in these tasks arise from differences in fronto-parietal connectivity, or whole-brain connectivity to the prefrontal cortex (Cole et al. 2012). Response inhibition, including both stopping and not going, has been found to involve the right inferior frontal gyrus, presupplementary motor area, and the basal ganglia, although the distinct role of each area in the network is contested (Aron et al. 2014; Chikazoe et al. 2009; Sebastian et al. 2016; Verbruggen and Logan 2008).

In addition to these neural theories, response inhibition theories have been extensively modeled to examine the mechanisms of the behavior. Behavioral models of the stop-signal task are generally based on a race process, where go processes and stop processes are different “racing” accumulators (Logan and Cowan 1984). More recent models add features such as stochastic accumulators (Logan et al. 2014) and the ability to estimate entire stop signal reaction time distributions (Matzke et al. 2013). While many of these models focus on purely behavioral data, some have incorporated neuroscience by using single-unit neurophysiology in experiments involving nonhuman primates to constrain behavioral models and differentiate between competing theories (Logan et al. 2015; Boucher et al. 2007).

Two tasks commonly used to measure response inhibition are the go/no-go and stop-signal tasks. In the go/no-go task, subjects are instructed to respond to one stimulus (or set of stimuli), often by invoking a motor response (i.e., pressing a button), and not to respond to a different stimulus or set of stimuli. The stop-signal task extends this basic paradigm by adding a stopping condition, where a go signal is presented, but after a set delay, a “stop-signal” is presented. While the tasks both look at response inhibition, not going (in the go/no-go task) and stopping (in the stop-signal task) have overlapping, but not identical networks (Rubia et al. 2001; Swick et al. 2011). The development of a task that has both stopping and not going components extended upon this theory by confirming that both types of inhibition share a common network, but stopping and not going have distinct subprocesses within that network (Sebastian et al. 2013). The task we use in this paper has components of a go/no-go task built into the stop-signal task to further analyze these nuances in response inhibition. In this next section, we begin by detailing how to construct a time series of neural activity that can later be constrained by task-specific details.

## Bayesian Implementation of the General Linear Model Analysis

Unlike many applications of Bayesian statistics for cognitive models, to perform Bayesian inference on neural activation, we must relate the stimulus effects to an entire time series worth of neural activity throughout the experiment. The reason for this is due to what is called *hemodynamic lag*, where the effects of an individual stimulus may linger for up to 30 s after stimulus presentation. These lingering effects can have a major impact on our ability to understand the systematic relationship that our independent variables have on brain activity for most realistic experimental designs where stimuli are presented within 30 s of one another. Hence, to properly estimate stimulus effects, we must carefully consider the specific sequence of stimulus presentations.

Consistent with other Bayesian cognitive modeling applications (Turner et al. 2017, 2018), we first define a generative model that describes how stimuli provoke neural activation for a given region of interest. This generative model is used to describe the effects of each stimulus in the experimental design through a set of activation parameters. Once each individual stimulus is described, we must combine these descriptions to form a *convolved* neural signal. The convolved neural signal is a prediction from our generative model, and so it can be compared to data observed from an experiment to assess how well our model prediction matches the observed data. Close matches indicate that the set of activation parameters are accurate, whereas poor matches indicate that at least a subset of activation parameters need to be adjusted. The process of inference is to adjust all of the activation parameters such that close matches can be obtained for all neural time series of interest.

Before providing all the details of our hierarchical models, we felt it is necessary to detail the core component of these models that relates each individual stimulus to a predicted time series of neural activity. While we have provided similar descriptions elsewhere (Palestro et al. 2018), the experimental design and purpose of the analyses below are both quite different. Furthermore, despite its simplicity, we find that most descriptions of convolution are somewhat difficult to apply generally. Thus, we provide a description of convolution in our specific task so that other researchers wishing to use these tools may have them readily accessible.

In this section, we describe how to construct a time series of neural activity in three parts. First, we describe the generative model of neural activity for a single stimulus. Second, we describe how critical experimental design variables should be collected and organized, to allow for formal mathematical descriptions of neural activity. Third, we describe how the experimental design variables can be used with the generative model for each specific stimulus to produce a predicted time series for neural activity. This operation is known as convolution, and we discuss two cases of convolution: finite impulse and the more general boxcar convolution. To aid in this discussion, we provide code in the Appendices that can implement the analyses we report here. Appendix A provides the means to calculate the canonical hemodynamic response function, Appendix B enables the user to specify the stimulus presentation for a specific experimental design, Appendix C provides the function for performing convolution, and finally, Appendix D provides the JAGS code for one of the five hierarchical models we investigate here.

### Hemodynamic Responses and Convolution

Unfortunately, scientists have yet to develop a way of directly observing neural activity within the brain in a noninvasive way that is safe for humans. One of the current best approaches for measuring said neural activity is through indirect observation of metabolic changes in blood flow so that neurons consuming oxygen and glucose can be replenished. The onset of neural activity leads to a systematic series of local physiological changes, most importantly the change in concentration of oxyhemoglobin and deoxyhemoglobin (Poldrack et al. 2011). Ultimately, fMRI experiments measure these concentrations through the blood-oxygenation-level-dependent (BOLD) response. The BOLD response is intended to measure changes in brain activity as a dependent variable in our experiments: if the BOLD response in a given brain area is systematically related to the independent variable in our experiment, then it follows that the area is somehow being recruited to process the stimuli.

*t*is time,

*β*is the amplitude of the response, and Γ(

*x*) = (

*x*− 1)! denotes the gamma function. Many of the parameters in Eq. 1 are fixed to specific values by convention:

*a*

_{1}= 6,

*a*

_{2}= 16,

*b*

_{1}= 1,

*b*

_{2}= 1, and

*c*= 1/6. These parameters define the shape of the HRF. While we could freely estimate these to reflect their variability across the brain or with stimulus types (Aguirre et al. 1998; Buckner 1998), they are often assumed to have a canonical form by using specific parameter values for simplicity in computation. Hence, the key parameter in Eq. 1 is

*β*, as it defines how active a given voxel or region of interest is following a stimulus presentation. Appendix A provides R code that can be used to implement Eq. 1.

### Construction of a Design Matrix

To construct a prediction of how neural activity changes as a consequence of the stimulus variables, we must first organize variables that define the experimental environment. Specifically, it is important that we know (1) which stimulus was presented at what point in time and (2) how that particular stimulus should affect the neural activity. As experimenters, we easily have access to (1), and we can organize this information within a “design” matrix **X**. The design matrix **X** must be constructed by setting individual regressors for each level of the independent variable, and some specification must be made for each trial (Rissman et al. 2004; Mumford et al. 2012). **X** has a row for each point in time a neural measure is collected (i.e., repetition time), a column for each stimulus effect, and a column for baseline activation (analogous to a *y*-intercept term in linear regression). Within **X**, each column is constructed by shifting the template HRF in Eq. 1 to the point in time at which a stimulus was presented. More technically, the HRF must be convolved with what is called a an impulse (response) function that defines the points in time a stimulus was presented.

While **X** can be defined from the experimental design and a template HRF, we now must specify the magnitude of each stimulus effect on neural activity. To do this, we can define an amplitude coefficient vector *β* such that each stimulus can be scaled according to Eq. 1. In general, we will not know the values of *β* that will scale each HRF appropriate to perfectly match our data. Instead, we must estimate the the values in *β* using, in our applications, Bayesian inference. In particular, we will show how the single-stimulus estimates can be constrained by building up hierarchical structures to explain neural activity.

**X**, and the location of each bar with respect to time corresponds to the row within

**X**where an individual HRF would eventually be placed. The height of each bar corresponds to the values of the parameters in

*β*. For this example,

*β*includes stimulus-wise activation coefficients:

*β*). The middle panel of Fig. 1 shows that the individual HRFs for this particular stimulus sequence—which was actually taken from our experimental design below—clearly overlap through the experimental session. Therefore, any attempt to estimate individual

*β*parameters would be hopeless without considering the entire time series.

Fortunately, the BOLD response exhibits a linear time invariant (LTI) property that enables us to isolate the effects of each individual stimulus presentation on the observed neural time series (Boynton et al. 1996). The first result implied by the LTI property is that if neural activation follows a stimulus presentation but peak activation is delayed by some arbitrary amount, the BOLD response will also start at the time of the stimulus presentation and be delayed by the same amount. This implies that if an appropriate function is specified for neural activation, it can be shifted to the time at which a given stimulus is presented. This property is illustrated in the middle panel of Fig. 1 where each HRF is shifted to the stimulus onset. The second result implied by the LTI property is that the scale parameter *β* (height of each HRF in the second row) is linearly related to amplitude of the neural activation (height of each line in the first row). In other words, trials with larger neural activation will also have larger *β* values. The final implication of the LTI property is in regards to how the individual HRFs are combined to form one convolved signal, which we now discuss.

### Convolution

*h*(

*t*), and the other defines the times at which the stimuli are presented

*f*(

*t*). The shape of the HRF

*h*(

*t*) was defined in Eq. 1. Conceptually, it is easy to define a function

*f*(

*t*) as an indicator function that specifies when a stimulus is present as

*f*(

*t*) with

*h*(

*t*) is defined as

*τ*allows us to slide the HRF along the time axis until we reach the value of

*τ*such that

*f*(

*τ*) = 1. In other words, the variable

*τ*is completely inconsequential as it is integrated out over the temporal axis. At this location, (

*f*∗

*h*)(

*t*) becomes the HRF shifted by

*τ*, as shown for each individual HRF in the middle panel of Fig. 1.

Equation 3 considers the case of any generic function for *f*(*t*) and all continuous values of time *t*. However, there are a few things we can do to simplify Eq. 3 for the purposes of pedagogy and pragmatic applications. First, we can assume a discrete representation for time, rather than a continuous one. This assumption is justified because fMRI scanners are incapable of producing measures of brain activity that are practically continuous through time (i.e., they usually provide a set of measures from the whole brain every 1.5 or 2 s). With a discrete representation in place, we can use summation rather than integration in Eq. 3. Second, depending on the duration of stimulus presentation, *f*(*t*) may take different forms. The simplest case is when *f*(*t*) = 1 for only one point in time *t* for a given stimulus, which is called an impulse function (e.g., the top row of Fig. 1). The more general case is when *f*(*t*) = 1 for a period of time, such as a few seconds. In this case, the function is often called a boxcar function. We begin with a description of convolution using an impulse function because it is conceptually simpler, and then move to the more general form of boxcar convolution.

### Impulse Convolution

*f*(

*t*), where stimuli are shown and removed immediately, such that

*f*(

*t*) = 1 for a single point in time

*t*for each stimulus. Because of the impulse function, the convolution operation in Eq. 3 is equivalent to simply shifting the starting point of the HRF

*h*(

*t*), without additional scaling by

*β*. For convenience, we define

*h*

_{0,i}(

*t*) as the HRF corresponding to the

*i*th stimulus presentation; that is,

*h*

_{0,i}(

*t*) contains the same information in

*h*(

*t*), but is shifted in time to correspond to the

*i*th stimulus presentation. Using a discrete representation of time and an impulse function, we can perform convolution by combining these individual HRFs

*h*

_{0,i}(

*t*) in the following way:

*R*is the total number of stimulus presentations, and

*β*

^{0}denotes an intercept term that shifts the time series throughout the experiment. The bottom panel of Fig. 1 shows the predicted convolved neural activity

**N**(

*t*) from the model across the 30-s time window (solid black line), along with the individual HRFs from the middle panel as a reference. Because the third result of the LTI property suggests that the aggregated neural signal is a linear combination of the individual stimulus effects, we can use Eq. 4 to generate the convolved signal. Here, each individual effect is described by a separate HRF, and these individual HRFs from the middle panel are integrated together (i.e., through summation) to form the convolved signal.

*β*with software packages such as JAGS or Stan, one strategy is to first construct the design matrix

**X**in our home environment (e.g., R or MATLAB) so that we can pass the constructed variable to the Bayesian software package of our choice. Because the design matrix only needs to be computed once, we can use advanced techniques (e.g., the fast Fourier transform) to aid in the computation. Although we do not investigate these methods here, this may be an important consideration for different problems. Once the functions in Appendices A-C are loaded into our R working environment, the following block of code shows how we can perform convolution with an impulse response function:

The function hrf.conv takes the HRF with a unit-level amplitude (i.e., *h*_{0}(*x*); Eq. 1, see Appendix A) and returns convolved HRFs using design variables such as stimulus onset times (t.start), stimulus duration (duration), the number of measurements (measurement), and TR (TR). The last argument resolution is only necessary for boxcar convolution as the function hrf.conv was intended to subsume impulse convolution as a special case of boxcar convolution. Accordingly, to perform convolution with an impulse function, simply set duration to zero. We also decided to exclude the baseline activation column from **X** and add the baseline activation separately to simplify computation.

*t*= 4,7,12, and 20. Here, we assume that we acquired 30 measurements with

*T*

*R*= 2. The resulting convolution used in line 10 is shown in the left panel of Fig. 2. For more in-depth details of how hrf.conv works, we refer the reader to Appendix C where the full code is provided.

#### Boxcar Convolution

Although using impulse functions makes the conceptualization of convolution significantly easier, it may not serve some readers well. For example, impulse functions would not be appropriate in experiments that use block designs, or slow event-related designs where stimuli are presented for many seconds. In these cases, we could not simply shift the common HRF template to one specific time. Instead, boxcar convolution would be needed.

*t*= 0.5. If we define a boxcar function with extremely high resolution (red circles), the true stimulus presentation details are well approximated. However, if we use a grid with low temporal resolution, such as the same resolution as our scan acquisition (blue squares), our boxcar function would consist of a single point at

*t*= 2. This low resolution would not approximate the true stimulus presentation details, and so the ensuing convolution is unlikely to allow us to investigate the systematic effects the stimuli have on the neural time series.

The left panel of Fig. 3 illustrates that one potential strategy for solving the resolution problem is to implement an “upsampling-convolving-downsampling” (UCD) cycle. First, we could increase the temporal resolution of the boxcar function (i.e., upsampling) within R, so that the stimulus presentation details can be approximated well. Second, we could use the high-resolution boxcar function to convolve it with the HRF template to produce a prediction for the neural time series. Using a discrete, albeit high-resolution, grid would still enable us to approximate the integral in Eq. 3 with the summation in Eq. 4. Third, we could decrease the temporal resolution of the convolved signal (i.e., downsampling) to match the fMRI acquisition. This last step would allow for an easy comparison to the observed neural time series through the GLM procedure described above, preserving our ability to simply pass the convolved design matrix X to any software package we wish to use (e.g., JAGS, Stan).

At this point, one may wonder for which stimulus durations a UCD process would be necessary. To evaluate the effects of resolution on the accuracy of the impulse function, the right panel of Fig. 3 shows the correlation (i.e., *y*-axis) between the predicted neural time series using impulse convolution (i.e., left panel of Fig. 2) and boxcar convolution (i.e., right panel of Fig. 2) for a single stimulus as a function of stimulus duration (i.e., *x*-axis), using the canonical HRF. The dark gray-shaded area represents the stimulus duration and the light gray-shaded area represents the acquisition time for our particular experiment below. The right panel shows that the correlation remains high (e.g., greater than 0.9) for stimulus presentations that are shorter than 2 s. This indicates that even for somewhat slow stimulus presentations, impulse convolution is nearly as effective as the UCD cycle with a high-resolution grid (red lines). The yellow lines show that downsampling from the high-resolution grid has no effect on the correlations, which is expected given that downsampling occurs after the convolution step.

As mentioned above, the function hrf.conv is made to subsume impulse convolution. For boxcar convolution, we need to specify a stimulus duration that is greater than zero in seconds. Because the result of boxcar convolution can arbitrarily scale a convolved HRF according to how we set duration and resolution, the function normalizes the height of the convolved HRF (i.e., the maximum value of the convolved HRF) to 1.

Line 2 provides an example of the syntax of boxcar convolution within hrf.conv when stimulus duration is nonzero. In this case, we should define duration as a vector representing stimulus duration for each stimulus, and use the R function mapply so that hrf.conv can call the information in the onset and duration vectors for each stimulus simultaneously. Line 4 presents a concrete example when we show four stimuli at *t* = 4,7,12, and 20, and their durations are 2, 10, 10, and 2 s, respectively. The right panel of Fig. 2 shows the resulting convolved neural signal. Here, the HRFs for the second (green) and third (blue) stimuli have extended durations compared to the HRFs for the first (red) and fourth (cyan) stimuli due to longer stimulus durations. As described above, the heights of each individual HRF are scaled to have a maximum value of 1, arbitrarily.

### Summary

In this section, we described how to convolve a canonical HRF with the design matrix for one’s specific experiment. As convolution can be abstruse, we discussed how to perform this operation in the simple case that stimuli are shown briefly (i.e., an impulse function), and then the more general case of stimuli presented for a longer duration. Although the computer code for implementing impulse convolution is relatively simple (see above), performing convolution for the more general case is more difficult. To facilitate this, Appendix A provides a function for the HRF template, Appendix B provides a function that defines a boxcar function, and Appendix C provides a function for general convolution in R. These functions make Bayesian analysis of fMRI time series data quite convenient as they enable the design matrix **X** to be precomputed in R and simply passed to either JAGS or Stan. Within these packages, one can estimate the influence of an individual stimulus (i.e., the *β* parameter) by using the dot product of a vector of coefficients with the design matrix.

## Model Specification

*σ*

^{β}, included in models 2–5, is also not pictured as it is nonessential, but its role will be described in detail in later sections. We now discuss each of the five models in turn.

### Model 1

The first model is the simplest and has no hierarchical component. *N*_{i,t} represents the observed neural data at time *t* for a given region of interest in response to the presentation of a go, no-go, or stop-signal stimulus, referenced as stimulus *j*. The shape of the BOLD response is constructed from convolved hemodynamic response functions discussed in the previous section.

*β*

^{0}) and the convolved hemodynamic responses, with an assumption that the measurement error follows a normal distribution:

*β*

_{i}and

*h*

_{0,i}refer to the neural activation amplitude and nonscaled HRF for the

*i*th stimulus, respectively. R is the number of stimulus presentations, which for one run of our stop-signal task is 240. The notation

*𝜖*(

*t*) denotes residual noise in the neural activity at each point in time that is not accounted for by the model,

*a*and standard deviation

*b*. Given this distributional assumption, we can express the likelihood of the neural data

**N**using matrix notation as

**X**and

*are the design matrix with stimulus-wise regressors and corresponding activation coefficient vector, respectively.*

**β***β*

^{0}(baseline activation), and

*σ*(measurement noise) were freely estimated. For the activation parameters, we imposed diffuse normal priors:

*σ*

^{2}) instead of standard deviation (i.e.,

*σ*) for parameterizing the uncertainty of estimates. However, in this paper, we will report the normal priors in terms of standard deviation.

*r*,

*λ*) is the inverse gamma distribution with shape

*r*and rate

*λ*. As stated previously, JAGS uses precision, but for reporting priors in this paper, we use the more conventional notation of standard deviation. We report the noise parameters in terms of variance, as the inverse gamma distribution is conjugate for the variance parameter in this model, which speeds the sampling procedure in JAGS. We note that applying an inverse gamma prior with shape

*r*and rate

*λ*for variance is equivalent to applying a gamma prior with shape

*r*and rate

*λ*for precision.

### Model 2

The second model adds a hierarchical structure across conditions. For the stop-signal task, there were four conditions: go, no-go, stop-signal presented before a response was made, and a nuisance regressor. Here, the nuisance regressor refers to trials in which a stop-signal trial was presented *after* a response was made. As the response was already made, we considered the late stop-signal to be cognitively unimportant as it is clearly not related to response inhibition. Despite this, we included a parameter to model the effects of this late stop-signal as it may have had some unintended influence on the neural signal on subsequent trials.

*δ*

_{k}where

*k*= {1,…,4} and

*σ*

^{β}to capture the mean and standard deviation of the single-stimulus

*β*s, respectively. Formally, we specified that

*i*corresponds to the individual stimuli, and

*k*corresponds to the four types of stimuli (i.e., go, no-go, stop-signal, and nuisance stop-signal). For

*δ*

_{k}we imposed a diffuse normal prior

*σ*

^{β}such that

We did not assume baseline activation or noise was constrained by condition, so the priors for *β*^{0} and *σ* are equivalent to their specification in model 1.

Out of the five models, model 2 most closely resembles the standard GLM analysis that would only consider condition-level effects. Compared to the standard GLM, one key difference in our analysis is that we obtain parameter estimates for every stimulus presented in the experiment, instead of averaging this effect out across trials (i.e., pooling trials within the same condition). Because the *δ*_{k} parameter captures the central tendency of the single-stimulus *β*s within a given condition *k*, the *δ*_{k} parameter conveys the same information as a *β* estimate would in the standard GLM. However, one key difference is that the variance of the posterior of *δ*_{k} will likely be larger than the standard GLM *β*.

### Model 3

The third model adds an additional hierarchical structure across subjects. The primary difference between models 2 and 3 is that model 3 assumes that each subject may have a different generative model to describe their brain activation. Formally, this assumption relates to adding subject-specific parameters for the single-stimulus *β*s and the baseline activation levels *β*^{0}.

*μ*

^{0}to describe the mean baseline for each of the

*j*terms of

*β*

^{0}. We specified that each subject’s baseline activation is sampled from a common normal distribution such that

*j*indexes the subject number. For the hypermean

*μ*

^{0}, we imposed a diffuse normal prior such that

*μ*

_{j}to center each activation parameter

*δ*

_{j,k}. Similarly with

*μ*

^{0}, we assume that the condition-level mean activation

*δ*

_{j,k}s differ among subjects and are sampled from a normal distribution, which incorporates individual differences in stimulus-wise brain activation in terms of

*β*

_{i,j,k}:

*j*to refer to the

*j*th subject. We specified a similarly diffuse prior for

*μ*

_{j}, such that

*β*s (i.e.,

*σ*

^{β}), we do not assume individual differences and therefore the prior for this parameter is defined in the same way as for model 2.

*σ*should also vary freely across subjects. However, we did not build a hierarchy on the resulting

*σ*

_{j}s as it was not clear what distribution they should take. For each

*σ*

_{j}, we specified a similar prior as was declared in model 2:

### Model 4

The fourth model investigates patterns of coactivation in the ROIs across the brain. The rationale for including patterns of coactivity was to build in *functional* constraint such that ROIs that had similar profiles of activity through time could constrain one another. In modeling perceptual decision-making in a random dot motion task, Turner et al. (2015) found that models including patterns of single-trial coactivation performed better than models that did not, suggesting that information in different ROIs can be used effectively to constrain estimates of single-trial activation. This result naturally follows from the conditional distribution of a multivariate normal distribution when at least two ROIs have nonzero functional correlation (Turner 2015).

*to control the distribution of the activation parameters*

**μ***δ*. Specifically, letting the subscripts

*j*,

*k*, and

*r*represent subject, condition, and ROI, respectively, we assumed

*p*-dimensional multivariate normal distribution with mean vector

*a*and variance-covariance matrix

*b*. We introduce the notation

**μ**_{k,1:R}to refer to the

*k*th row of

*such that*

**μ***.*

**δ***β*was changed to index each ROI:

**μ**_{k,1:R}is a 24-dimensional multivariate normal distribution

**ϕ**_{0}is a 24-dimensional vector of zeros and

*s*

_{0}is a (24 × 24) identity matrix (i.e., a diagonal matrix with ones on the diagonal). We specified an inverse Wishart distribution as the prior on Σ such that

*I*

_{0}is a (24 × 24) identity matrix representing the scale of the distribution, and

*n*

_{0}= 24 is the degrees of freedom of the inverse Wishart. It should be noted that many different priors for the variance-covariance matrix are possible. We chose the inverse-Wishart distribution because of its mathematical convenience: it is a conjugate prior for the variance-covariance matrix Σ. However, the inverse-Wishart distribution is quite inflexible in that it only contains a precision matrix

*I*

_{0}and a degrees of freedom parameter

*n*

_{0}. Other, more flexible, priors decompose the Σ matrix and place priors on the resulting components (see Gelman et al. 2004 for more details). Fortunately, software packages, such as JAGS and Stan, make modification of the prior on Σ straightforward, and so we recommend a sensitivity analysis for this particular set of priors if one is interested in their influence on the estimates of Σ.

*β*

^{0}, we added an index for the

*r*th ROI, so that

*σ*to index each subject and ROI, but did not constrain these hierarchically for similar reasons as in model 3:

### Model 5

*δ*in model 5 is

*j*th subject and

*k*th condition. As in model 4,

**μ**_{j,k,1:R}and

**δ**_{j,k,1:R}are 24-dimensional vectors where each element represents a different ROI. The hyperprior for Σ

_{j}is equivalent to the specification in model 4 (i.e., Eq. 7), except for the new index on Σ

_{j}:

### Model Summary

*β*s by the type of stimulus that was presented (i.e., go, no-go, stop-signal, or nuisance stop-signal). The third model constructs a hierarchy on the subject-level, assuming individual differences in measurement noise, baseline neural activity, and the condition-level constraint on single-stimulus

*β*s. The fourth and fifth models construct a hierarchy on the ROI-level, building off of all previous assumptions, but also add the additional feature of ROI coactivation. Model 5 differs from model 4 in that it has a covariance matrix for each subject, whereas model 4 has one group-level covariance matrix. As the five models have somewhat similar JAGS code, Appendix A provides JAGS code for model 3 as an example.

Conceptual summary of the different types of hierarchical constraints used in each of the five models

Model | Hierarchy | ROI covariance matrix | ||
---|---|---|---|---|

Condition-level | Subject-level | ROI-level | ||

Model 1 | ||||

Model 2 | X | |||

Model 3 | X | X | ||

Model 4 | X | X | X | Collapsed across subjects |

Model 5 | X | X | X | One per individual |

Assessing which hierarchical constraints are appropriate is a tricky problem: we want the models to be complex enough to fit data well, but we also want the models to be able to generalize to new data accurately (Pitt and Myung 2002). In order to explore these considerations, we fit the models to real experimental data. We chose to use the stop-signal task because of its interesting cognitive applications and relevance to missing data problems present in a variety of tasks. Before discussing model fit and generalizability, we review our experimental design and MRI procedures.

## The Stop-signal Task

As discussed in the introduction, the stop-signal task is a widely used paradigm in studying response inhibition (Logan and Cowan 1984). Our task design differs from the standard stop-signal task in that it also incorporates aspects of the go/no-go task. The purpose of this incorporation is to be able to distinguish between different types of response inhibition: not going and stopping. In this section, we describe the experimental methods of the task as well as the initial fMRI processing.

### Participants

The eleven participants analyzed in this study completed the stop-signal task in the MRI scanner. All participants were recruited from The Ohio State University and its surrounding community and provided informed consent. The study was approved by the Institutional Review Board of the university. Among the eleven participants (mean age = 24.6 years; range from 18 to 48) included in the analysis, there were five females and six males.

### Stimuli

### Behavioral Results

*p*(

*i*

*n*

*h*

*i*

*b*

*i*

*t*|

*s*

*i*

*g*

*n*

*a*

*l*). The black “+” signs refer to the group mean

*p*(

*i*

*n*

*h*

*i*

*b*

*i*

*t*|

*s*

*i*

*g*

*n*

*a*

*l*) at that given SSD. The squares indicate an individual subjects’

*p*(

*i*

*n*

*h*

*i*

*b*

*i*

*t*|

*s*

*i*

*g*

*n*

*a*

*l*) for a given SSD, where every subject has a distinct color. Each subject had three different SSD conditions and thus three points on the plot. For the majority of subjects, as SSD increases,

*p*(

*i*

*n*

*h*

*i*

*b*

*i*

*t*|

*s*

*i*

*g*

*n*

*a*

*l*) decreases; in other words, the longer the delay, the less likely the subject was to successfully inhibit the response. It is important to emphasize that while we show accuracy in these figures, incorrect trials were modeled the same way as correct trials in our analyses—we neither treated errors differently nor removed error trials.

### MRI Data Acquisition

MRI recording was performed using a 12-channel head coil in a Siemens 3T Trio Magnetic Resonance Imaging System with TIM, housed in the Center for Cognitive and Behavioral Brain Imaging at The Ohio State University. BOLD functional activations were measured with a T2*-weighted EPI sequence (repetition time = 2000 m/s, echo time = 28 m/s, flip angle = 72^{∘}, field of view = 222×222 mm, in-plane resolution = 74×74 pixels or 3×3 mm, and 38 axial slices with 3-mm thickness to cover the entire cerebral cortex and most of the cerebellum). In addition, the anatomical structure of the brain was acquired with the three-dimensional MPRAGE sequence (1× 1 ×1 mm^{3} resolution, inversion time = 950 m/s, repetition time = 1950 m/s, echo time = 4.44 m/s, flip angle = 12^{∘}, matrix size = 256×224, 176 sagittal slices per slab; scan time 7.5 min) for each participant.

### Image Preprocessing and Analysis

The fMRI preprocessing was carried out using FEAT (FMRI Expert Analysis Tool) in FSL (FMRIB software library, version 5.0.8; Smith et al. 2004). The first six volumes were discarded to allow for T1 equilibrium. The remaining images were then realigned to correct for head motion. Data were spatially smoothed using a 6-mm full-width-half maximum Gaussian kernel. The data were filtered in the temporal domain using a nonlinear high-pass filter with a 90-s cutoff. A two-step registration procedure was used whereby EPI images were first registered to the MPRAGE structural image, and then into the standard (MNI) space, using affine transformations. Registration from the MPRAGE structural image to the standard space was further refined using FNIRT nonlinear registration.

Regions of interest. The table shows the number index, name of each region of interest, MNI coordinates, and the number of voxels. *MNI*: Montreal Neurological Institute

Number | Name | MNI [ | nVox (< 40) |
---|---|---|---|

1. | Callosum | [3 −23 29] | 208 |

2. | PCC (posterior cingulate cortex) | [−2 −56 22] | 957 |

3. | preSMA (presupplementary motor area) | [4 21 47] | 1952 |

4. | Left angular gyrus | [−44 −72 30] | 328 |

5. | Left fusiform gyrus | [−43 −60 −17] | 84 |

6. | Left IFG-1 (inferior frontal gyrus 1) | [−37 18 −4] | 912 |

7. | Left IFG-2 (inferior frontal gyrus 2) | [−44 9 29] | 426 |

8. | Left IPL (left inferior parietal lobe) | [−34 −52 46] | 459 |

9. | Left ITG (left inferior temporal gyrus) | [−56 −10 −20] | 44 |

10. | Left insula | [−39 −3 7] | 41 |

11. | Left MFG (left middle frontal gyrus) | [−3 50 −9] | 477 |

12. | Left putamen | [−27 −13 7] | 48 |

13. | Left SFG (left superior frontal gyrus) | [−9 57 35] | 128 |

14. | Left thalamus | [−6 −16 −2] | 72 |

15. | Left ventral striatum | [−1 16 −9] | 100 |

16. | Right caudate | [13 10 6] | 55 |

17. | Right IFG (right inferior frontal gyrus) | [43 20 12] | 2830 |

18. | Right IPL (right inferior parietal lobe) | [48 −44 43] | 1400 |

19. | Right MFG (right middle frontal gyrus) | [38 48 −10] | 83 |

20. | Right MTG (right middle temporal gyrus) | [49 −66 26] | 60 |

21. | Right precuneus | [12 −67 42] | 83 |

22. | Right putamen | [31 −11 4] | 44 |

23. | Right SFG (right superior frontal gyrus) | [21 49 31] | 45 |

24. | Right thalamus | [9 −16 3] | 154 |

As stated previously, the ideal model should be able to fit data well, and be generalizable. To address both concerns, we present our modeling analyses in two stages. First, we fit the five models to one run of data from the stop-signal task, assess the models’ ability to fit data from the first run, and examine how the model structure constrains parameter estimates, in particular the neural activation parameter *β*. Second, we use the fitted models to generate predictions for two additional runs of the same task, using the parameters estimated from the first run, and the experimental design variables from the remaining two runs. Together, the results from these two analyses should help to identify models that not only fit data well, but are also not too complex relative to the data.

## Run 1 Model Fitting

The first step of our analysis is to see how well the five models can fit the data. The models increase in complexity, and so we might expect from the outset that the more complex model will provide the best fits to data. However, this is not always the case, depending on how the model is structured, and which aspects of the data inform the parameter estimation procedure. In this section, we fit all five models to the first run of the stop-signal experiment (runs 2 and 3 will be used in the validation analysis in the following section). We compare the neural predictions to the actual observed data, the single-stimulus *β* estimates, and, for models 2 through 5, the constraint on the hyperparameters. Within this analysis, an “ideal” model would have neural predictions that closely resemble the observed data and provide constrained and reasonable estimates for *β*.

### Methods

#### Fitting Details

We used JAGS to fit all five models. All of the models had three chains, but took on one of two combinations of adaptation, burn-in, and sampling iterations. The first, longer procedure, was used for just model 1. In this procedure, model initialization ran for 2000 adaptations. After initialization, 4000 samples were discarded as burn-in. Then, the posterior sampling ran for 6000 iterations. Thus, with three chains, there was a total of 18,000 samples for each parameter. Because of computational complications, the sampling lengths were shortened for models 2 through 5. In this procedure, model initialization ran for 1000 adaptations, followed by a burn-in period of 2000 iterations. The posterior sampling then ran for 3000 iterations. With the three chains again, there were a total of 9000 samples for each parameter. For all models, the chains were plotted and visually checked for convergence.

### Results

To assess the models’ ability to fit data, we provide three types of analyses. First, we examine the models’ predictions of neural responses across time. Because all models were fitted to the same time series, we can generate predictions from the model and assess how similar they are to the data to which they were applied. Second, we compare estimates of the trial-level activation parameter *β* across the five models. Unlike in the first comparison, *β* is latent and thus cannot be examined in relation to any true metric. Third, we compare estimates of the condition-level activation parameter *δ*. Because *δ* is the hyperparameter of *β*, it is only present in the models with a hierarchical component (i.e., only models 2–5 can be compared). We conclude by showing how these *δ* estimates can be used to understand neural activation in the stop-signal task.

#### Time Series Predictions

As seen in Fig. 7, and more generally in Fig. 8, model 1 makes predictions for the neural time series that clearly fluctuate in ways that nearly match the data, whereas models 2 and 3 make predictions that are relatively stationary across time. This difference across model predictions can be explained by considering the model structure. Model 1 considers each neural time series as a separate entity, where the estimates for each *β* parameter are purely influenced by the neural time series shown in Fig. 7. Having only one source of influence allows the estimates to be very sensitive to the particularities of each time series. By contrast, the *β* parameters for models 2 and 3 are influenced by multiple time series, either across conditions (i.e., model 2), or both condition and subject (i.e., model 3). In this way, models 2 and 3 are more constrained, and this constraint influences the quality of the fits to data. For some aspects of the data, such as predicting the neural time series, this added constraint manifests in negative ways (e.g., as shown by the quality of the fits). However, as the results in this article will make clear, these constraints can manifest in positive ways, too.

#### Constraint on Beta Estimates

*β*. Figure 9 compares estimates of

*β*from the first three models, using the left fusiform gyrus as an example. Similar to Fig. 7, the rows correspond to the first three subjects and the columns correspond to the first three models. Unlike in Fig. 7, however, there is no evaluative metric for comparing the accuracy of the estimated

*β*values, as the parameter is latent, or unobserved. The dotted lines refer to the range of the posterior estimates and the red boxes denote the interquartile range.

Figure 9 shows the effects of shrinkage on the estimates of the parameter *β*, especially when going from a nonhierarchical model (model 1) to a hierarchical model (model 2). For example, the differences in the variance of the estimated posteriors are so extreme that they distort the *y*-axis (which changes across columns for visual clarity). However, the improvement in the reduction of parameter uncertainty has its limitations, as the estimates for models 4 and 5 did not drastically improve upon the estimates shown for model 3, so they are not pictured. Importantly, models 4 and 5 provide more information regarding correlations between ROIs. Additionally, models 4 and 5 provide more constraint on the posteriors of other parameters, such as the *β* hyperparameters (*δ*), which we now discuss.

#### Constraint on Delta Estimates

*δ*. As additional details of the experiment are incorporated into the model, we should expect the estimates for

*δ*to improve because the amount of data it is directly affected by continue to increase. Figure 10 shows the joint distribution of

*δ*

_{Go}and

*δ*

_{Stop}for the left fusiform gyrus in model 2 (left panel) and model 5 (right panel). In this figure, each red point corresponds to a different subject. The

*x*-coordinate of the point is the mean of the posterior estimate for

*δ*in the go condition and the

*y*-coordinate is the mean of the posterior estimate for

*δ*in the stop condition. The error bars represent two standard deviations away from either the

*δ*

_{Go}mean if oriented horizontally, or the

*δ*

_{Stop}mean if oriented vertically. By plotting the mean activation of

*δ*by trial type, we are better able to visually assess which subjects had activation that was differentially activated across the two conditions. For example, if a point appears in the bottom right triangular area, it suggests that the mean activation in go trials is higher than during stop trials.

In Model 5, all eleven subjects show more activation in the go condition than in the stop condition. In model 2, however, only nine subjects show this pattern of activation. Model 2 also inferred that subject 3 exhibited greater activation for *δ*_{Stop} than *δ*_{Go}, with a relatively larger difference (or distance from the line of indifference) than observed in other subjects. However, in the model 5 subplot, subject 3 shows the opposite pattern, with more activation in *δ*_{Go} than in *δ*_{Stop}, and is closer to the group mean. Furthermore, in model 5, the posteriors for the *δ* parameters for each subject are more constrained than the posteriors from fitting model 2.

### Delta and Response Inhibition

The additional constraint of the hierarchy on *δ* has important implications in understanding the task that is being modeled, because we are often interested in condition-level differences. For example, in the stop-signal task, researchers are interested in understanding the differences between go and stop trials. Because *δ* is a conditional-level hierarchy built on single-stimulus *β* estimates, we can use *δ* estimates to compare activation levels in ROIs and subjects in different conditions. Further constraining *δ* with other hierarchical levels, such as in models 3, 4, and 5, help us to make more informed conclusions on condition-level differences.

*δ*(estimated, in this case, using model 3) can be used to understand conditional differences in four key ROIs: presupplementary motor area (preSMA; ROI 3), left fusiform gyrus (ROI 5), left ventral striatum (ROI 15), and right inferior frontal gyrus (rIFG; ROI 17). The left panel of Fig. 11 shows the mean of

*δ*for go stimuli (denoted by green), no-go stimuli (denoted by blue), and stop-signal stimuli (denoted by red). The rectangles show the mean for each subject, and the cross shows the mean for the group. For the preSMA and left fusiform, there is less activation in the stop condition than in either the go or no-go conditions. Additionally, the subject-to-subject variability within an ROI differs. For example, the variability in the left fusiform is larger, whereas the variability is smallest for the right IFG. The right panel of Fig. 11 visualizes the locations of each ROI in the brain. ROI 5 (left fusiform gyrus) and ROI 15 (left ventral striatum) were chosen because they were used earlier in this paper as illustrative examples to show how the subject-level hierarchy constrains neural predictions and single-stimulus

*β*estimates. ROI 3 (presupplementary motor area) and ROI 17 (right inferior frontal cortex) were included because of their relevance to the go/no-go and stop-signal tasks in the cognitive neuroscience literature (Simmonds et al. 2008; Aron et al. 2014).

## Validation Analysis on Runs 2 and 3

The model-fit analysis in the last section evaluated how well a particular model fit data from one run of the experiment. However, it is also important that our models generally characterize how the stimuli interact with mental operations. To evaluate which models generalize to new data well, we can use out-of-sample prediction on the other runs of the stop-signal task. The validation analysis could help explain why model 1 closely captured the trend of the neural data, yet did not provide much constraint on the estimates of the single-stimulus activation parameters *β*. In this section, we use out-of-sample prediction to evaluate each model’s ability to generalize to new data.

### Methods

As each subject in our task provided data from three different experimental runs, we used the estimated posterior distributions from the first run (i.e., the previous section) to generate predictions from the second and third runs. Hence, the model predictions for the neural time series (i.e., the BOLD response) for both runs are not only based on the parameter estimates from the first run, but the design matrices **X** from runs 2 and 3. If a model variant can generate out-of-sample predictions that more closely match the data from runs 2 and 3 in terms of both central tendency and variance, it can be said that the variant generalizes well.

#### Generating Out-of-sample Predictions

*y*

^{∗}based on the posterior distributions of model parameters

*𝜃*estimated on data

**y**. Mathematically, the PPD is expressed as

*π*(

*𝜃*|

**y**) denotes the posterior distribution—the estimates derived from fitting the model to run 1—and the term

*f*(

*y*

^{∗}|

*𝜃*) denotes a prediction from the model for new data

*y*

^{∗}, using the parameters

*𝜃*and the design matrix corresponding to each run. By integrating over all possible values of

*𝜃*in the parameter space, we can incorporate the relative probabilities of

*𝜃*in our predictions of

*y*

^{∗}. In other words, values of

*𝜃*that are more likely (i.e., values of

*𝜃*with more posterior mass) will place higher predictive mass in the data space corresponding to a prediction from the model at the location of

*𝜃*.

*β*

_{i}. To generate predictions about

*β*

_{i}, we must refer to the hyper structure that governs the shape of each

*β*

_{i}across trials. For models 2–5, the hyper structure is defined by the condition-level hyper mean

*δ*

_{j,k,r}and the hyper standard deviation

*σ*

^{β}. Using the design matrix for runs 2 and 3, we can generate samples from the distribution of predicted single-stimulus

*β*

_{i}s by sampling

*δ*

_{j,k,r}to use as the hyper mean, namely the

*j*th subject,

*k*th condition, and

*r*th ROI.

Unlike the other models, model 1 does not have a condition-level hierarchy, and so we have no guide in selecting single-stimulus *β*s when generating out-of-sample predictions. As a remedy, we “pooled” all of the estimated *β*_{i}s from run 1, and selected one at random when generating each predictive distribution. Because the *β*_{i}s are pooled and the experimental design is not used when generating predictions for model 1, its predictions should remain fixed regardless of stimulus type, subject, and ROI.

Once single-stimulus *β* predictions were sampled, they were convolved with a canonical HRF to generate a mean neural time series (i.e., **X*** β* in Eq. 5) to match the BOLD response from the experiment. Because the residual noise term

*σ*in Eq. 5 was also estimated when fitting the models to data, we could calculate the probability density of observing the withheld data by evaluating the density of the PPD at the location of each data point at each point in time analytically. For every sampled posterior value, we computed the density of the withheld data, and repeated this process for every posterior sample acquired after burn-in: 18,000 samples per time point in model 1, and 9000 samples per time point in models 2–5.

### Results

Across almost every subject, it is difficult to differentiate the prediction performance between models 3, 4, and 5 by the log PPD. Model 2 seems to suffer from a similar problem as model 1, where its predictions are often too under constrained to make consistently accurate predictions for the entire time series (e.g., see the green lines in Fig. 13). Together, these results suggest that models 3–5 have more flexibility when explaining unobserved data by embedding individual- and ROI-level covariance structure.

Unfortunately, it does not appear as though our validation analysis provides any justification for the additional ROI-based constraint used in models 4 and 5. This suggests that while ROIs are clearly correlated within the task^{1}, the information does not seem to provide additional constraints that are generalizable to new data.

## Discussion

We used hierarchical Bayesian modeling to improve the constraint and generalizability of time series models of fMRI data collected in a stop-signal task. First, we found evidence that hierarchical Bayesian modeling improves single-stimulus *β* estimates by reducing the variability of their posterior distributions. Increasing the levels of the hierarchy improved estimation of these parameters, both at the hyper parameter level and the single-stimulus level. However, we also found some contradictory results from model 1. Of all the models, model 1 best fit the observed neural time series data, but had the least constrained single-stimulus *β* estimates by far. This is contradictory, because the *β* estimates inform the neural predictions, so we hypothesized that if the neural predictions are similar to the real data (i.e., they are accurate), then *β* estimates would be constrained appropriately. These two aspects of the parameter estimates for model 1 signify the hallmark of overfitting: the parameters are weakly constrained by the data, and so the model is too flexible.

To elucidate this property of the model, we performed a validation analysis. We took the parameter estimates from the first set of data (i.e., the first run) and used them to generate predictions for two new time series of the same task. Here, the models were only guided by the information in the design matrix, i.e., the timing of the stimulus presentation and the type of stimulus presented (e.g., a go or stop signal). The results of the validation analysis clarify the apparent contradiction found in the first set of analyses. Model 1 focused on providing the best fit to the neural data as no formal constraint was imposed to single-stimulus *β*s. However, in the validation analyses, it allowed an overly broad range of predictions for the new time series data, making its predictions less accurate overall. On the contrary, model 2 did not allow any variability in the single-stimulus *β* estimates for different individuals or ROIs, solely focusing on experimental conditions. As a consequence, it was also unable to generalize well, although its performance in the validation analysis was consistently better than that of model 1.

This pattern of results can be viewed as symptomatic of overfitting. Model 1 overfits the neural data (time-series predictions) because the single-stimulus *β*s are freely estimated for every stimulus presentation. This gives the model practically unlimited ability to match the precise shape of the neural time series. Meanwhile, model 2 tends to underfit the neural data because the condition-level hierarchy, collapsing across individual or ROI-level differences, does not allow for flexible predictions. While both of these models fit the time series data well, they were unable to generalize to new data well because they did not learn features of the neural data that were consistent from one scanning run to the next. By contrast, models 3–5 imposed several different constraints that made their particular fit to neural data appear visually worse (compared to models 1 and 2), yet they were able to generalize well because of the neural features they learned in the fitting process. To summarize, by applying the right type of constraints (condition, subject, and ROI), better models were developed that both provided adequate fits to data and generalized well to new data.

### Establishing Brain-Behavior Relations

While our analyses provide insight into neural dynamics, they do not connect patterns of brain activation to behavioral responses. To do this, one would have to sort estimates for the neural activation parameter *β* not only by the type of condition (e.g., go trials), but also by the type of behavior that was observed. For example, with the stop trials, one could manually divide the estimates for *β* into trials in which the subject correctly inhibited their response as instructed, and into another group where the subject failed to do so. The distribution of *β*s across these two groups could then reveal how the pattern of ROI activations was different across the two response contingencies, and one could subsequently speculate about the importance of each ROI in successfully implementing executive control.

As mentioned in the introduction, there are several computational models of executive control, designed specifically for the stop-signal task (Logan and Cowan 1984; Logan et al. 2014; Matzke et al. 2013). These models make strong commitments to particular theories about control by instantiating theoretical assumptions within a complete computational model. The advantage of using a computational model is that the patterns of behavioral data—both choice and response time—can be understood through theoretically motivated mechanisms in the model. As many authors have argued, including both choice and response time can provide stringent tests of the suitability of various mechanisms of such models (Ratcliff 1978; Luce 1986; Ratcliff and Rouder 1998; Ratcliff et al. 1999; Van Zandt 2000; Molloy et al. in press). Furthermore, new tech- niques have been established for linking the mechanisms in these computational models to patterns of neural data, providing even greater constraints on the model (Turner et al. 2013, 2015, 2016, 2017, 2018). An important next step will be to investigate the role that neural data play in constraining computational models of the stop-signal task.

### Limitations

Although the analyses presented here are suitable for understanding complex patterns of brain activation, they are not without their limitations. In our view, there are four limitations that merit further consideration: handling autocorrelation of the BOLD response, implementing “boxcar” convolution, using JAGS as a sampler, and performing voxel-based analyses. Below, we provide a discussion of these limitations, as well as some strategies for improving upon the models we have presented here.

#### Autocorrelation of BOLD Response

One potentially problematic assumption of the five models we have discussed here is the assumption of independent, identically distributed Gaussian noise surrounding the time course of each time series. We made this assumption as it is conventional in both frequentist and Bayesian applications of the general linear model (GLM) to neural time series data (Friston et al. 2002; Penny and Friston 2004). However, BOLD responses measured in fMRI experiments are known to have temporal autocorrelation due to task-irrelevant factors such as thermal noises from the scanner, and physiological noises from participants (Lindquist 2008). Ignoring the temporal autocorrelation has been shown to cause higher false-positive rates (Zarahn et al. 1997; Purdon and Weisskoff 1998) and overestimation of power (Mumford and Nichols 2008).

Fortunately, the autocorrelation of BOLD responses can be accommodated by incorporating autoregressive error models into the typical GLM structure (Purdon et al. 2001; Lindquist 2008; Leonski et al. 2008). For example, a simple autoregressive error model can be constructed such that measurement noise at a given time point *t* is based on (i.e., correlated with) measurement noise at time *t* − *p*. These models are denoted AR(*p*), where *p* denotes the time dependence in the noise process. More advanced models use a combination of autoregressive structures and moving averages to allow for known physical properties of fMRI scanners, such as the so-called “scanner drift,” where the average BOLD response gradually changes throughout the duration of the experiment (Poldrack et al. 2011).

#### JAGS as a Sampler

One of the major goals of this paper is to present hierarchical Bayesian modeling on time series data in an accessible way. To achieve this, we used JAGS to perform the difficult process of collecting posterior samples. While JAGS is user-friendly and widely used, it is not without limitations. One of the major limitations is that JAGS uses Gibbs sampling which can sometimes produce large autocorrelations. JAGS has built-in functions to plot or obtain a numerical estimate of autocorrelation, as a check on the quality of the samples. In cases where autocorrelation is too large, we recommend using advanced techniques such as differential evolution MCMC (DE-MCMC; ter Braak 2006; Turner et al. 2013), as it is especially powerful in reducing autocorrelation among posterior samples.

Another limitation in the results presented here was the computational burden for the more complicated models. Because of the size of the data and the variables being estimated, we experienced limitations of memory on our system, especially when we saved the mean activation and posterior predictive samples for the raw BOLD responses. Here, custom-built samplers have another advantage over programs like JAGS because the user can define when samples are stored, such as allocating them to memory or writing them out to a file during the sampling procedure. Finally, while JAGS is not directly parallelizable, a custom-build function can dramatically increase the efficiency of collecting posterior samples such as with the R package snow or snowfall.

#### Voxel-based Versus ROI-based

A final limitation of the current analyses presented here is the focus on ROIs rather than individual voxels. We decided to focus on ROI-based analyses out of practical concerns. While it is not impossible to perform voxel-based analyses, it dramatically increases the computational burden. To implement a voxel-based analysis, one simply needs to replace the time series from a given ROI with the time series data from each individual voxel. Unfortunately, JAGS has memory limitations that make it difficult to store all of the results from extensive estimation procedures. One approach may be to model each voxel independently, although we discourage this approach as it neglects important spatial autocorrelations that exist between nearby voxels. An ideal solution is to apply priors to the parameters governing the time series of each voxel, where the priors reflect information about the distance between voxels (Bowman et al. 2008). To circumvent the memory capacity limitations of JAGS, one could program up the sampler as we suggested above. Although this approach is more difficult to program, it allows the user to print out the samples from each iteration, and so memory would only need to be allocated to store the current position (i.e., within one iteration) of each chain.

### Conclusions

This article has illustrated the importance of including appropriate constraints on models of BOLD time series data. We found that applying no constraints allowed models to be too flexible where they fit data well, but performed poorly in validation analyses. On the other hand, imposing constraints on the bases of only condition-level variables masked important features of the data that prohibited the model’s ability to capture data well and perform well in validation analyses. The best constraints for our data imposed structure that modulated parameters across both condition and subject variables. While it is certainly true that BOLD measurements from fMRI experiments are often noisy, the results presented here reveal that the data can be purified by applying appropriately constrained hierarchical (Bayesian) models.

## Footnotes

- 1.
We began our analysis by first examining pairwise functional correlations of each time series across all ROIs. These analyses revealed a potential need for ROI-based constraints, motivating the development of models 4 and 5.

## Notes

### Compliance with Ethical Standards

The study was approved by the Institutional Review Board of the university.

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