Parameter variability and distributional assumptions in the diffusion model

Inhibitory dynamics in dual-route evidence accumulation account for response time distributions from conflict tasks

Laboratory data from conflict tasks, e.g. Simon and Eriksen tasks, reveal differences in response time distributions under different experimental conditions. Only recently have evidence accumulation models successfully reproduced these results, in particular the challenging delta plots with negative slopes. They accomplish this with explicit temporal dependencies in their structure or activation functions. In this work, we introduce an alternative approach to the modeling of decision-making in conflict tasks exclusively based on inhibitory dynamics within a dual-route architecture. We consider simultaneous automatic and controlled drift diffusion processes, with the latter inhibiting the former. Our proposed Dual-Route Evidence Accumulation Model (DREAM) achieves equivalent performance to previous works in fitting experimental response time distributions despite having no time-dependent functions. The model can reproduce conditional accuracy functions and delta plots with positive and negative slopes. The implications of these results, including an interpretation of the parameters and potential links to perceptual representations, are discussed. We provide Python code to fit DREAM to experimental data.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-023-09990-8.

An extension of the shifted Wald model of human response times: Capturing the time dynamic properties of human cognition : Trial-varying Wald model

Despite the ubiquitous nature of evidence accumulation models in cognitive and experimental psychology, there has been a comparatively limited uptake of such techniques in the applied literature. While quantifying latent cognitive processing properties has significant potential for applied domains such as adaptive work systems, accumulator models often fall short in practical applications. Two primary reasons for these shortcomings are the complexities and time needed for the application of cognitive models, and the failure of current models to capture systematic trial-to-trial variability in parameters. In this manuscript, we develop a novel, trial-varying extension of the shifted Wald model to address these concerns. By leveraging conjugate properties of the Wald distribution, we derive computationally efficient solutions for threshold and drift parameters which can be updated instantaneously with new data. The resulting model allows the quantification of systematic variation in latent cognitive parameters across trials and we demonstrate the utility of such analyses through simulations and an exemplar application to an existing data set. The analytic nature of our solutions opens the door for real-world applications, significantly extending the reach of computational models of behavioral responses.

EZ-CDM: Fast, simple, robust, and accurate estimation of circular diffusion model parameters

The investigation of cognitive processes that form the basis of decision-making in paradigms involving continuous outcomes has gained the interest of modeling researchers who aim to develop a dynamic decision theory that accounts for both speed and accuracy. One of the most important of these continuous models is the circular diffusion model (CDM, Smith. Psychological Review, 123(4), 425. 2016), which posits a noisy accumulation process mathematically described as a stochastic two-dimensional Wiener process inside a disk. Despite the considerable benefits of this model, its mathematical intricacy has limited its utilization among scholars. Here, we propose a straightforward and user-friendly method for estimating the CDM parameters and fitting the model to continuous-scale data using simple formulas that can be readily computed and do not require theoretical knowledge of model fitting or extensive programming. Notwithstanding its simplicity, we demonstrate that the aforementioned method performs with a level of accuracy that is comparable to that of the maximum likelihood estimation method. Furthermore, a robust version of the method is presented, which maintains its simplicity while exhibiting a high degree of resistance to contaminant responses. Additionally, we show that the approach is capable of reliably measuring the key parameters of the CDM, even when these values are subject to across-trial variability. Finally, we demonstrate the practical application of the method on experimental data. Specifically, an illustrative example is presented wherein the method is employed along with estimating the probability of guessing. It is hoped that the straightforward methodology presented here will, on the one hand, help narrow the divide between theoretical constructs and empirical observations on continuous response tasks and, on the other hand, inspire cognitive psychology researchers to shift their laboratory investigations towards continuous response paradigms.

A revised diffusion model for conflict tasks

The recently developed diffusion model for conflict tasks (DMC) Ulrich et al. (Cognitive Psychology, 78, 148-174, 2015) provides a good account of data from all standard conflict tasks (e.g., Stroop, Simon, and flanker tasks) within a common evidence accumulation framework. A central feature of DMC's processing dynamics is that there is an initial phase of rapid accumulation of distractor evidence that is then selectively withdrawn from the decision mechanism as processing continues. We argue that this assumption is potentially troubling because it could be viewed as implying qualitative changes in the representation of distractor information over the time course of processing. These changes suggest more than simple inhibition or suppression of distractor information, as they involve evidence produced by distractor processing "changing sign" over time. In this article, we (a) develop a revised DMC (RDMC) whose dynamics operate strictly within the limits of inhibition/suppression (i.e., evidence strength can change monotonically, but cannot change sign); (b) demonstrate that RDMC can predict the full range of delta plots observed in the literature (i.e., both positive-going and negative-going); and (c) show that the model provides excellent fits to Simon and flanker data used to benchmark the original DMC at both the individual and group level. Our model provides a novel account of processing differences across Simon and flanker tasks. Specifically, that they differ in how distractor information is processed on congruent trials, rather than incongruent trials: congruent trials in the Simon task show relatively slow attention shifting away from distractor information (i.e., location) while complete and rapid attention shifting occurs in the flanker task. Our new model highlights the importance of considering dynamic interactions between top-down goals and bottom-up stimulus effects in conflict processing.

Differences in Social Expectations About Robot Signals and Human Signals

In our daily lives, we are continually involved in decision-making situations, many of which take place in the context of social interaction. Despite the ubiquity of such situations, there remains a gap in our understanding of how decision-making unfolds in social contexts, and how communicative signals, such as social cues and feedback, impact the choices we make. Interestingly, there is a new social context to which humans are recently increasingly more frequently exposed-social interaction with not only other humans but also artificial agents, such as robots or avatars. Given these new technological developments, it is of great interest to address the question of whether-and in what way-social signals exhibited by non-human agents influence decision-making. The present study aimed to examine whether robot non-verbal communicative behavior has an effect on human decision-making. To this end, we implemented a two-alternative-choice task where participants were to guess which of two presented cups was covering a ball. This game was an adaptation of a "Shell Game." A robot avatar acted as a game partner producing social cues and feedback. We manipulated robot's cues (pointing toward one of the cups) before the participant's decision and the robot's feedback ("thumb up" or no feedback) after the decision. We found that participants were slower (compared to other conditions) when cues were mostly invalid and the robot reacted positively to wins. We argue that this was due to the incongruence of the signals (cue vs. feedback), and thus violation of expectations. In sum, our findings show that incongruence in pre- and post-decision social signals from a robot significantly influences task performance, highlighting the importance of understanding expectations toward social robots for effective human-robot interactions.

A spatially continuous diffusion model of visual working memory

We present results from five visual working memory (VWM) experiments in which participants were briefly shown between 2 and 6 colored squares. They were then cued to recall the color of one of the squares and they responded by choosing the color on a continuous color wheel. The experiments provided response proportions and response time (RT) measures as a function of angle for the choices. Current VWM models for this task include discrete models that assume an item is either within working memory or not and resource models that assume that memory strength varies as a function of the number of items. Because these models do not include processes that allow them to account for RT data, we implemented them within the spatially continuous diffusion model (SCDM, Ratcliff, 2018) and use the experimental data to evaluate these combined models. In the SCDM, evidence retrieved from memory is represented as a spatially continuous normal distribution and this drives the decision process until a criterion (represented as a 1-D line) is reached, which produces a decision. Noise in the accumulation process is represented by continuous Gaussian process noise over spatial position. The models that fit best from the discrete and resource-based classes converged on a common model that had a guessing component and that allowed the height of the normal memory-strength distribution to vary with number of items. The guessing component was implemented as a regular decision process driven by a flat evidence distribution, a zero-drift process. The combination of choice and RT data allows models that were not identifiable based on choice data alone to be discriminated.

Stimulus-response congruency effects depend on quality of perceptual evidence: A diffusion model account

Individuals often need to make quick decisions based on incomplete or "noisy" information. This requires the coordination of attentional, perceptual, cognitive, and behavioral mechanisms. This poses a challenge for isolating the unique effects of each subprocess from behavioral data, which reflect the summation of all subprocesses combined. Sequential sampling models offer a more detailed examination of behavioral data, enabling us to separate decisional and non-decisional processes at play in a task. Participants were required to identify briefly presented shapes while perceptual (duration, size, location) and response features (location-congruent/-incongruent/-neutral) of the task were manipulated. The diffusion model (Ratcliff, 1978) was used to dissociate decisional and executive processes in the task. In Experiment 1, stimuli were presented for either 20 or 80 ms to the left or right of a central fixation while response keys were positioned horizontally. In Experiment 2, stimulus size was manipulated rather than duration. In Experiment 3, response keys were positioned vertically. Results showed a duration x response mapping interaction. Participants displayed stimulus-response (S-R) congruency biases only on short-duration trials. This effect was observed for both horizontal and vertical response key mappings. Stimulus size affected participant response speed, but did not elicit S-R congruency biases. The present findings show that when perceptual quality of evidence is poor, individuals rely more heavily on spatial-motor mechanisms when making speeded choice decisions. Furthermore, positioning response keys vertically is insufficient to eliminate S-R congruency effects. Diffusion model parameters are presented and implications of the model are discussed.

Decision times in orthographic processing: a cross-linguistic study

Reading comparisons across transparent and opaque orthographies indicate critical differences that may reveal the mechanisms involved in orthographic decoding across orthographies. Here, we address the role of criterion and speed of processing in accounting for performance differences across languages. We used binary tasks involving orthographic (words-pseudowords), and non-orthographic materials (female-male faces), and analyzed results based on Ratcliff's Diffusion model. In the first study, 29 English and 28 Italian university students were given a lexical decision test. English observers made more errors than Italian observers while showing generally similar reaction times. In terms of the diffusion model, the two groups differed in the decision criterion: English observers used a lower criterion. There was no overall cross-linguistic difference in processing speed, but English observers showed lower values for words (and a smaller lexicality effect) than Italians. In the second study, participants were given a face gender judgment test. Female faces were identified slower than the male ones with no language group differences. In terms of the diffusion model, there was no difference between groups in drift rate and boundary separation. Overall, the new main finding concerns a difference in decision criterion limited to the orthographic task: English individuals showed a more lenient criterion in judging the lexicality of the items, a tendency that may explain why, despite lower accuracy, they were not slower. It is concluded that binary tasks (and the Diffusion model) can reveal cross-linguistic differences in orthographic processing which would otherwise be difficult to detect in standard single-word reading tasks.

Integrated diffusion models for distance effects in number memory

I evaluated three models for the representation of numbers in memory. These were integrated with the diffusion decision model to explain accuracy and response time (RT) data from a recognition memory experiment in which the stimuli were two-digit numbers. The integrated models accounted for distance/confusability effects: when a test number was numerically close to a studied number, accuracy was lower and RTs were longer than when a test number was numerically far from a studied number. For two of the models, the representations of numbers are distributed over number (with Gaussian or exponential distributions) and the overlap between the distributions of a studied number and a test number provides the evidence (drift rate) on which a decision is made. For the third, the exponential gradient model, drift rate is an exponential function of the numerical distance between studied and test numbers. The exponential gradient model fit the data slightly better than the two overlap models. Monte Carlo simulations showed that the variability in the important parameter estimates from fitting data collected over 30-40 min is smaller than the variability among individuals, allowing differences among individuals to be studied. A second experiment compared number memory and number discrimination tasks and results showed different distance effects. Number memory had an exponential-like distance-effect and number discrimination had a linear function which shows radically different representations drive the two tasks.

Rule-based and stimulus-based cues bias auditory decisions via different computational and physiological mechanisms

Expectations, such as those arising from either learned rules or recent stimulus regularities, can bias subsequent auditory perception in diverse ways. However, it is not well understood if and how these diverse effects depend on the source of the expectations. Further, it is unknown whether different sources of bias use the same or different computational and physiological mechanisms. We examined how rule-based and stimulus-based expectations influenced behavior and pupil-linked arousal, a marker of certain forms of expectation-based processing, of human subjects performing an auditory frequency-discrimination task. Rule-based cues consistently biased choices and response times (RTs) toward the more-probable stimulus. In contrast, stimulus-based cues had a complex combination of effects, including choice and RT biases toward and away from the frequency of recently presented stimuli. These different behavioral patterns also had: 1) distinct computational signatures, including different modulations of key components of a novel form of a drift-diffusion decision model and 2) distinct physiological signatures, including substantial bias-dependent modulations of pupil size in response to rule-based but not stimulus-based cues. These results imply that different sources of expectations can modulate auditory processing via distinct mechanisms: one that uses arousal-linked, rule-based information and another that uses arousal-independent, stimulus-based information to bias the speed and accuracy of auditory perceptual decisions.

Prior information about upcoming stimuli can bias our perception of those stimuli. Whether different sources of prior information bias perception in similar or distinct ways is not well understood. We compared the influence of two kinds of prior information on tone-frequency discrimination: rule-based cues, in the form of explicit information about the most-likely identity of the upcoming tone; and stimulus-based cues, in the form of sequences of tones presented before the to-be-discriminated tone. Although both types of prior information biased auditory decision-making, they demonstrated distinct behavioral, computational, and physiological signatures. Our results suggest that the brain processes prior information in a form-specific manner rather than utilizing a general-purpose prior. Such form-specific processing has implications for understanding decision biases real-world contexts, in which prior information comes from many different sources and modalities.

Practice-related changes in perceptual evidence accumulation correlate with changes in working memory

It has been proposed that evidence accumulation determines not only the speed and accuracy of simple perceptual decisions but also influences performance on tasks assessing higher-order cognitive abilities, such as working memory (WM). Accordingly, estimates of evidence accumulation based on diffusion decision modeling of perceptual decision-making tasks have been found to correlate with WM performance. Here we use diffusion decision modeling in combination with latent factor modeling to test the stronger prediction that practice-induced changes in evidence accumulation correlate with changes in WM performance. Analyses are based on data from the COGITO Study, in which 101 young adults practiced a battery of cognitive tasks, including three simple two-choice reaction time tasks and three WM tasks, in 100 day-to-day training sessions distributed over 6 months. In initial analyses, drift rates were found to correlate across the three choice tasks, such that latent factors of evidence accumulation could be established. These latent factors of evidence accumulation were positively correlated with latent factors of practiced and unpracticed WM tasks, both before and after practice. As predicted, individual differences in changes of evidence accumulation correlated positively with changes in WM performance. Our findings support the proposition that decision making and WM both rely on the active maintenance of task-relevant internal representations. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Discriminating memory disordered patients from controls using diffusion model parameters from recognition memory

One hundred and five memory disordered (MD) patients and 57 controls were tested on item recognition memory and lexical decision tasks, and diffusion model analyses were conducted on accuracy and response time distributions for correct and error responses. The diffusion model fit the data well for the MD patients and control subjects, the results replicated earlier studies with young and older adults, and individual differences were consistent between the item recognition and lexical decision tasks. In the diffusion model analysis, MD patients had lower drift rates (with mild Alzheimer's [AD] patients lower than mild cognitive impairment [MCI] patients) as well as wider boundaries and longer nondecision times. These data and results were used in a series of studies to examine how well MD patients could be discriminated from controls using machine-learning techniques, linear discriminant analysis, logistic regression, and support vector machines (all of which produced similar results). There was about 83% accuracy in separating MD from controls, and within the MD group, AD patients had about 90% accuracy and MCI patients had about 68% accuracy (controls had about 90% accuracy). These methods might offer an adjunct to traditional clinical diagnosis. Limitations are noted including difficulties in obtaining a matched group of control subjects as well as the possibility of misdiagnosis of MD patients. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

An initial 'snapshot' of sensory information biases the likelihood and speed of subsequent changes of mind

We often need to rapidly change our mind about perceptual decisions in order to account for new information and correct mistakes. One fundamental, unresolved question is whether information processed prior to a decision being made ('pre-decisional information') has any influence on the likelihood and speed with which that decision is reversed. We investigated this using a luminance discrimination task in which participants indicated which of two flickering greyscale squares was brightest. Following an initial decision, the stimuli briefly remained on screen, and participants could change their response. Using psychophysical reverse correlation, we examined how moment-to-moment fluctuations in stimulus luminance affected participants' decisions. This revealed that the strength of even the very earliest (pre-decisional) evidence was associated with the likelihood and speed of later changes of mind. To account for this effect, we propose an extended diffusion model in which an initial 'snapshot' of sensory information biases ongoing evidence accumulation.

Qualitative speed-accuracy tradeoff effects can be explained by a diffusion/fast-guess mixture model

Rafiei and Rahnev (2021) presented an analysis of an experiment in which they manipulated speed-accuracy stress and stimulus contrast in an orientation discrimination task. They argued that the standard diffusion model could not account for the patterns of data their experiment produced. However, their experiment encouraged and produced fast guesses in the higher speed-stress conditions. These fast guesses are responses with chance accuracy and response times (RTs) less than 300 ms. We developed a simple mixture model in which fast guesses were represented by a simple normal distribution with fixed mean and standard deviation and other responses by the standard diffusion process. The model fit the whole pattern of accuracy and RTs as a function of speed/accuracy stress and stimulus contrast, including the sometimes bimodal shapes of RT distributions. In the model, speed-accuracy stress affected some model parameters while stimulus contrast affected a different one showing selective influence. Rafiei and Rahnev's failure to fit the diffusion model was the result of driving subjects to fast guess in their experiment.

Fronto-striatal structures related with model-based control as an endophenotype for obsessive-compulsive disorder

Recent theories suggest a shift from model-based goal-directed to model-free habitual decision-making in obsessive–compulsive disorder (OCD). However, it is yet unclear, whether this shift in the decision process is heritable. We investigated 32 patients with OCD, 27 unaffected siblings (SIBs) and 31 healthy controls (HCs) using the two-step task. We computed behavioral and reaction time analyses and fitted a computational model to assess the balance between model-based and model-free control. 80 subjects also underwent structural imaging. We observed a significant ordered effect for the shift towards model-free control in the direction OCD > SIB > HC in our computational parameter of interest. However less directed analyses revealed no shift towards model-free control in OCDs. Nonetheless, we found evidence for reduced model-based control in OCDs compared to HCs and SIBs via 2nd stage reaction time analyses. In this measure SIBs also showed higher levels of model-based control than HCs. Across all subjects these effects were associated with the surface area of the left medial/right dorsolateral prefrontal cortex. Moreover, correlations between bilateral putamen/right caudate volumes and these effects varied as a function of group: they were negative in SIBs and OCDs, but positive in HCs. Associations between fronto-striatal regions and model-based reaction time effects point to a potential endophenotype for OCD.

Do data from mechanical Turk subjects replicate accuracy, response time, and diffusion modeling results?

Online data collection is being used more and more, especially in the face of the COVID crisis. To examine the quality of such data, we chose to replicate lexical decision and item recognition paradigms from Ratcliff et al. (Cognitive Psychology, 60, 127-157, 2010) and numerosity discrimination paradigms from Ratcliff and McKoon (Psychological Review, 125, 183-217, 2018) with subjects recruited from Amazon Mechanical Turk (AMT). Along with these tasks, we collected data from either an IQ test or a math computation test. Subjects in the lexical decision and item recognition tasks were relatively well-behaved, with only a few giving a significant number of responses with response times (RTs) under 300 ms at chance accuracy, i.e., fast guesses, and a few with unstable RTs across a session. But in the numerosity discrimination tasks, almost half of the subjects gave a significant number of fast guesses and/or unstable RTs across the session. Diffusion model parameters were largely consistent with the earlier studies as were correlations across tasks and correlations with IQ and age. One surprising result was that eliminating fast outliers from subjects with highly variable RTs (those eliminated from the main analyses) produced diffusion model analyses that showed patterns of correlations similar to the subjects with stable performance. Methods for displaying data to examine stability, eliminating subjects, and implementing RT data collection on AMT including checks on timing are also discussed.

Sequential sampling models without random between-trial variability: the racing diffusion model of speeded decision making

Most current sequential sampling models have random between-trial variability in their parameters. These sources of variability make the models more complex in order to fit response time data, do not provide any further explanation to how the data were generated, and have recently been criticised for allowing infinite flexibility in the models. To explore and test the need of between-trial variability parameters we develop a simple sequential sampling model of N-choice speeded decision making: the racing diffusion model. The model makes speeded decisions from a race of evidence accumulators that integrate information in a noisy fashion within a trial. The racing diffusion does not assume that any evidence accumulation process varies between trial, and so, the model provides alternative explanations of key response time phenomena, such as fast and slow error response times relative to correct response times. Overall, our paper gives good reason to rethink including between-trial variability parameters in sequential sampling models.

Nonlinear neural network dynamics accounts for human confidence in a sequence of perceptual decisions

Electrophysiological recordings during perceptual decision tasks in monkeys suggest that the degree of confidence in a decision is based on a simple neural signal produced by the neural decision process. Attractor neural networks provide an appropriate biophysical modeling framework, and account for the experimental results very well. However, it remains unclear whether attractor neural networks can account for confidence reports in humans. We present the results from an experiment in which participants are asked to perform an orientation discrimination task, followed by a confidence judgment. Here we show that an attractor neural network model quantitatively reproduces, for each participant, the relations between accuracy, response times and confidence. We show that the attractor neural network also accounts for confidence-specific sequential effects observed in the experiment (participants are faster on trials following high confidence trials). Remarkably, this is obtained as an inevitable outcome of the network dynamics, without any feedback specific to the previous decision (that would result in, e.g., a change in the model parameters before the onset of the next trial). Our results thus suggest that a metacognitive process such as confidence in one's decision is linked to the intrinsically nonlinear dynamics of the decision-making neural network.

Decision making in numeracy tasks with spatially continuous scales

A diffusion model of decision making on continuous response scales is applied to three numeracy tasks. The goal is to explain the distributions of responses on the continuous response scale and the time taken to make decisions. In the model, information from a stimulus is spatially continuously distributed, the response is made by accumulating information to a criterion, which is a 1D line, and the noise in the accumulation process is continuous Gaussian process noise over spatial position. The model is fit to the data from three experiments. In one experiment, a one or two digit number is displayed and the task is to point to its location on a number line ranging from 1 to 100. This task is used extensively in research in education but there has been no model for it that accounts for both decision times and decision choices. In the second task, an array of dots is displayed and the task is to point to the position of the number of dots on an arc ranging from 11 to 90. In a third task, an array of dots is displayed and the task is to speak aloud the number of dots. The model we propose accounts for both accuracy and response time variables, including the full distributions of response times. It also provides estimates of the acuity of decisions (standard deviations in the evidence distributions) and it shows how representations of numeracy information are task-dependent. We discuss how our model relates to research on numeracy and the neuroscience of numeracy, and how it can produce more comprehensive measures of individual differences in numeracy skills in tasks with continuous response scales than have hitherto been available.

ChaRTr: An R toolbox for modeling choices and response times in decision-making tasks

BACKGROUND

Decision-making is the process of choosing and performing actions in response to sensory cues to achieve behavioral goals. Many mathematical models have been developed to describe the choice behavior and response time (RT) distributions of observers performing decision-making tasks. However, relatively few researchers use these models because it demands expertise in various numerical, statistical, and software techniques.

NEW METHOD

We present a toolbox - Choices and Response Times in R, or ChaRTr - that provides the user the ability to implement and test a wide variety of decision-making models ranging from classic through to modern versions of the diffusion decision model, to models with urgency signals, or collapsing boundaries.

RESULTS

In three different case studies, we demonstrate how ChaRTr can be used to effortlessly discriminate between multiple models of decision-making behavior. We also provide guidance on how to extend the toolbox to incorporate future developments in decision-making models.

COMPARISON WITH EXISTING METHOD(S)

Existing software packages surmounted some of the numerical issues but have often focused on the classical decision-making model, the diffusion decision model. Recent models that posit roles for urgency, time-varying decision thresholds, noise in various aspects of the decision-formation process or low pass filtering of sensory evidence have proven to be challenging to incorporate in a coherent software framework that permits quantitative evaluation among these competing classes of decision-making models.

CONCLUSION

ChaRTr can be used to make insightful statements about the cognitive processes underlying observed decision-making behavior and ultimately for deeper insights into decision mechanisms.

Combining error-driven models of associative learning with evidence accumulation models of decision-making

As people learn a new skill, performance changes along two fundamental dimensions: Responses become progressively faster and more accurate. In cognitive psychology, these facets of improvement have typically been addressed by separate classes of theories. Reductions in response time (RT) have usually been addressed by theories of skill acquisition, whereas increases in accuracy have been explained by associative learning theories. To date, relatively little work has examined how changes in RT relate to changes in response accuracy, and whether these changes can be accounted for quantitatively within a single theoretical framework. The current work examines joint changes in accuracy and RT in a probabilistic category learning task. We report a model-based analysis of changes in the shapes of RT distributions for different category responses at the level of individual stimuli over the course of learning. We show that changes in performance are determined solely by changes in the quality of information entering the decision process. We then develop a new model that combines an associative learning front end with a sequential sampling model of the decision process, showing that the model provides a good account of all aspects of the learning data. We conclude by discussing potential extensions of the model and future directions for theoretical development that are opened up by our findings.

Parameter Inference for Computational Cognitive Models with Approximate Bayesian Computation

This paper addresses a common challenge with computational cognitive models: identifying parameter values that are both theoretically plausible and generate predictions that match well with empirical data. While computational models can offer deep explanations of cognition, they are computationally complex and often out of reach of traditional parameter fitting methods. Weak methodology may lead to premature rejection of valid models or to acceptance of models that might otherwise be falsified. Mathematically robust fitting methods are, therefore, essential to the progress of computational modeling in cognitive science. In this article, we investigate the capability and role of modern fitting methods—including Bayesian optimization and approximate Bayesian computation—and contrast them to some more commonly used methods: grid search and Nelder–Mead optimization. Our investigation consists of a reanalysis of the fitting of two previous computational models: an Adaptive Control of Thought—Rational model of skill acquisition and a computational rationality model of visual search. The results contrast the efficiency and informativeness of the methods. A key advantage of the Bayesian methods is the ability to estimate the uncertainty of fitted parameter values. We conclude that approximate Bayesian computation is (a) efficient, (b) informative, and (c) offers a path to reproducible results.

DstarM: an R package for analyzing two-choice reaction time data with the D∗M method

The decision process in choice reaction time data is traditionally described in detail with diffusion models. However, the total reaction time is assumed to consist of the sum of a decision time (as modeled by the diffusion process) and the time devoted to nondecision processes (e.g., perceptual and motor processes). It has become standard practice to assume that the nondecision time is uniformly distributed. However, a misspecification of the nondecision time distribution introduces bias in the parameter estimates for the decision model. Recently, a new method has been proposed (called the D∗M method) that allows the estimation of the decision model parameters, while leaving the nondecision time distribution unspecified. In a second step, a nonparametric estimate of the nondecision time distribution may be retrieved. In this paper, we present an R package that estimates parameters of several diffusion models via the D∗M method. Moreover, it is shown in a series of extensive simulation studies that the parameters of the decision model and the nondecision distributions are correctly retrieved.

Improving the reliability of model-based decision-making estimates in the two-stage decision task with reaction-times and drift-diffusion modeling

A well-established notion in cognitive neuroscience proposes that multiple brain systems contribute to choice behaviour. These include: (1) a model-free system that uses values cached from the outcome history of alternative actions, and (2) a model-based system that considers action outcomes and the transition structure of the environment. The widespread use of this distinction, across a range of applications, renders it important to index their distinct influences with high reliability. Here we consider the two-stage task, widely considered as a gold standard measure for the contribution of model-based and model-free systems to human choice. We tested the internal/temporal stability of measures from this task, including those estimated via an established computational model, as well as an extended model using drift-diffusion. Drift-diffusion modeling suggested that both choice in the first stage, and RTs in the second stage, are directly affected by a model-based/free trade-off parameter. Both parameter recovery and the stability of model-based estimates were poor but improved substantially when both choice and RT were used (compared to choice only), and when more trials (than conventionally used in research practice) were included in our analysis. The findings have implications for interpretation of past and future studies based on the use of the two-stage task, as well as for characterising the contribution of model-based processes to choice behaviour.

Decision making on spatially continuous scales

A new diffusion model of decision making in continuous space is presented and tested. The model is a sequential sampling model in which both spatially continuously distributed evidence and noise are accumulated up to a decision criterion (a 1 dimensional [1D] line or a 2 dimensional [2D] plane). There are two major advances represented in this research. The first is to use spatially continuously distributed Gaussian noise in the decision process (Gaussian process or Gaussian random field noise) which allows the model to represent truly spatially continuous processes. The second is a series of experiments that collect data from a variety of tasks and response modes to provide the basis for testing the model. The model accounts for the distributions of responses over position and response time distributions for the choices. The model applies to tasks in which the stimulus and the response coincide (moving eyes or fingers to brightened areas in a field of pixels) and ones in which they do not (color, motion, and direction identification). The model also applies to tasks in which the response is made with eye movements, finger movements, or mouse movements. This modeling offers a wide potential scope of applications including application to any device or scale in which responses are made on a 1D continuous scale or in a 2D spatial field. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

The Psychophysiology of Action: A Multidisciplinary Endeavor for Integrating Action and Cognition

There is a vast amount of literature concerning the integration of action and cognition. Although this broad research area is of great interest for many disciplines like sports, psychology and cognitive neuroscience, only a few attempts tried to bring together different perspectives so far. Our goal is to provide a perspective to spark a debate across theoretical borders and integration of different disciplines via psychophysiology. In order to boost advances in this research field it is not only necessary to become aware of the different areas that are relevant but also to consider methodological aspects and challenges. We briefly describe the most relevant theoretical accounts to the question of how internal and external information processes or factors interact and, based on this, argue that research programs should consider the three dimensions: (a) dynamics of movements; (b) multivariate measures and; (c) dynamic statistical parameters. Only with an extended perspective on theoretical and methodological accounts, one would be able to integrate the dynamics of actions into theoretical advances.

Modeling 2-alternative forced-choice tasks: Accounting for both magnitude and difference effects

We present a model-based analysis of two-alternative forced-choice tasks in which two stimuli are presented side by side and subjects must make a comparative judgment (e.g., which stimulus is brighter). Stimuli can vary on two dimensions, the difference in strength of the two stimuli and the magnitude of each stimulus. Differences between the two stimuli produce typical RT and accuracy effects (i.e., subjects respond more quickly and more accurately when there is a larger difference between the two). However, the overall magnitude of the pair of stimuli also affects RT and accuracy. In the more common two-choice task, a single stimulus is presented and the stimulus varies on only one dimension. In this two-stimulus task, if the standard diffusion decision model is fit to the data with only drift rate (evidence accumulation rate) differing among conditions, the model cannot fit the data. However, if either of one of two variability parameters is allowed to change with stimulus magnitude, the model can fit the data. This results in two models that are extremely constrained with about one tenth of the number of parameters than there are data points while at the same time the models account for accuracy and correct and error RT distributions. While both of these versions of the diffusion model can account for the observed data, the model that allows across-trial variability in drift to vary might be preferred for theoretical reasons. The diffusion model fits are compared to the leaky competing accumulator model which did not perform as well.

A model-based quantification of action control deficits in Parkinson's disease

Basal ganglia dysfunction in Parkinson's disease (PD) is thought to generate deficits in action control, but the characterization of these deficits have been qualitative rather than quantitative. Patients with PD typically show prolonged response times on tasks that instantiate a conflict between goal-directed processing and automatic response tendencies. In the Simon task, for example, the irrelevant location of the stimulus automatically activates a corresponding lateralized response, generating a potential conflict with goal-directed choices. We applied a new computational model of conflict processing to two sets of behavioral data from the Simon task to quantify the effects of PD and dopaminergic (DA) medication on action control mechanisms. Compared to healthy controls (HC) matched in age gender and education, patients with PD showed a deficit in goal-directed processing, and the magnitude of this deficit positively correlated with cognitive symptoms. Analyses of the time-course of the location-based automatic activation yielded mixed findings. In both datasets, we found that the peak amplitude of the automatic activation was similar between PD and HC, demonstrating a similar degree of response capture. However, PD patients showed a prolonged automatic activation in only one dataset. This discrepancy was resolved by theoretical analyses of conflict resolution in the Simon task. The reduction of interference generated by the automatic activation appears to be driven by a mixture of passive decay and top-down inhibitory control, the contribution of each component being modulated by task demands. Our results suggest that PD selectively impairs the inhibitory control component, a deficit likely remediated by DA medication. This work advances our understanding of action control deficits in PD, and illustrates the benefit of using computational models to quantitatively measure cognitive processes in clinical populations.

A Bayesian Reformulation of the Extended Drift-Diffusion Model in Perceptual Decision Making

Perceptual decision making can be described as a process of accumulating evidence to a bound which has been formalized within drift-diffusion models (DDMs). Recently, an equivalent Bayesian model has been proposed. In contrast to standard DDMs, this Bayesian model directly links information in the stimulus to the decision process. Here, we extend this Bayesian model further and allow inter-trial variability of two parameters following the extended version of the DDM. We derive parameter distributions for the Bayesian model and show that they lead to predictions that are qualitatively equivalent to those made by the extended drift-diffusion model (eDDM). Further, we demonstrate the usefulness of the extended Bayesian model (eBM) for the analysis of concrete behavioral data. Specifically, using Bayesian model selection, we find evidence that including additional inter-trial parameter variability provides for a better model, when the model is constrained by trial-wise stimulus features. This result is remarkable because it was derived using just 200 trials per condition, which is typically thought to be insufficient for identifying variability parameters in DDMs. In sum, we present a Bayesian analysis, which provides for a novel and promising analysis of perceptual decision making experiments.

RELATING ACCUMULATOR MODEL PARAMETERS AND NEURAL DYNAMICS

Accumulator models explain decision-making as an accumulation of evidence to a response threshold. Specific model parameters are associated with specific model mechanisms, such as the time when accumulation begins, the average rate of evidence accumulation, and the threshold. These mechanisms determine both the within-trial dynamics of evidence accumulation and the predicted behavior. Cognitive modelers usually infer what mechanisms vary during decision-making by seeing what parameters vary when a model is fitted to observed behavior. The recent identification of neural activity with evidence accumulation suggests that it may be possible to directly infer what mechanisms vary from an analysis of how neural dynamics vary. However, evidence accumulation is often noisy, and noise complicates the relationship between accumulator dynamics and the underlying mechanisms leading to those dynamics. To understand what kinds of inferences can be made about decision-making mechanisms based on measures of neural dynamics, we measured simulated accumulator model dynamics while systematically varying model parameters. In some cases, decision- making mechanisms can be directly inferred from dynamics, allowing us to distinguish between models that make identical behavioral predictions. In other cases, however, different parameterized mechanisms produce surprisingly similar dynamics, limiting the inferences that can be made based on measuring dynamics alone. Analyzing neural dynamics can provide a powerful tool to resolve model mimicry at the behavioral level, but we caution against drawing inferences based solely on neural analyses. Instead, simultaneous modeling of behavior and neural dynamics provides the most powerful approach to understand decision-making and likely other aspects of cognition and perception.

Linking Theoretical Decision-making Mechanisms in the Simon Task with Electrophysiological Data: A Model-based Neuroscience Study in Humans

A current challenge for decision-making research is in extending models of simple decisions to more complex and ecological choice situations. Conflict tasks (e.g., Simon, Stroop, Eriksen flanker) have been the focus of much interest, because they provide a decision-making context representative of everyday life experiences. Modeling efforts have led to an elaborated drift diffusion model for conflict tasks (DMC), which implements a superimposition of automatic and controlled decision activations. The DMC has proven to capture the diversity of behavioral conflict effects across various task contexts. This study combined DMC predictions with EEG and EMG measurements to test a set of linking propositions that specify the relationship between theoretical decision-making mechanisms involved in the Simon task and brain activity. Our results are consistent with a representation of the superimposed decision variable in the primary motor cortices. The decision variable was also observed in the EMG activity of response agonist muscles. These findings provide new insight into the neurophysiology of human decision-making. In return, they provide support for the DMC model framework.

A single trial analysis of EEG in recognition memory: Tracking the neural correlates of memory strength

Recent work in perceptual decision-making has shown that although two distinct neural components differentiate experimental conditions (e.g., did you see a face or a car), only one tracked the evidence guiding the decision process. In the memory literature, there is a distinction between a fronto-central evoked potential measured with EEG beginning at 350ms that seems to track familiarity and a late parietal evoked potential that peaks at 600ms that tracks recollection. Here, we applied single-trial regressor analysis (similar to multivariate pattern analysis, MVPA) and diffusion decision modeling to EEG and behavioral data from two recognition memory experiments to test whether these two components contribute to the recognition decision process. The regressor analysis only involved whether an item was studied or not and did not involve any use of the behavioral data. Only late EEG activity distinguishes studied from not studied items that peaks at about 600ms following each test item onset predicted the diffusion model drift rate derived from the behavioral choice and reaction times (but only for studied items). When drift rate was made a linear function of the trial-level regressor values, the estimate for studied items was different than zero. This showed that the later EEG activity indexed the trial-to-trial variability in drift rate for studied items. Our results provide strong evidence that only a single EEG component reflects evidence being used in the recegnition decision process.

Comparing fixed and collapsing boundary versions of the diffusion model

Optimality studies and studies of decision-making in monkeys have been used to support a model in which the decision boundaries used to evaluate evidence collapse over time. This article investigates whether a diffusion model with collapsing boundaries provides a better account of human data than a model with fixed boundaries. We compared the models using data from four new numerosity discrimination experiments and two previously published motion discrimination experiments. When model selection was based on BIC values, the fixed boundary model was preferred over the collapsing boundary model for all of the experiments. When model selection was carried out using a parametric bootstrap cross-fitting method (PBCM), which takes into account the flexibility of the alternative models and the ability of one model to account for data from another model, data from 5 of 6 experiments favored either fixed boundaries or boundaries with only negligible collapse. We found that the collapsing boundary model produces response times distributions with the same shape as those produced by the fixed boundary model and that its parameters were not well-identified and were difficult to recover from data. Furthermore, the estimated boundaries of the best-fitting collapsing boundary model were relatively flat and very similar to those of the fixed-boundary model. Overall, a diffusion model with decision boundaries that converge over time does not provide an improvement over the standard diffusion model for our tasks with human data.

Individual differences in the components of children's and adults' information processing for simple symbolic and non-symbolic numeric decisions

How do speed and accuracy trade off, and what components of information processing develop as children and adults make simple numeric comparisons? Data from symbolic and non-symbolic number tasks were collected from 19 first graders (Mage=7.12 years), 26 second/third graders (Mage=8.20 years), 27 fourth/fifth graders (Mage=10.46 years), and 19 seventh/eighth graders (Mage=13.22 years). The non-symbolic task asked children to decide whether an array of asterisks had a larger or smaller number than 50, and the symbolic task asked whether a two-digit number was greater than or less than 50. We used a diffusion model analysis to estimate components of processing in tasks from accuracy, correct and error response times, and response time (RT) distributions. Participants who were accurate on one task were accurate on the other task, and participants who made fast decisions on one task made fast decisions on the other task. Older participants extracted a higher quality of information from the stimulus arrays, were more willing to make a decision, and were faster at encoding, transforming the stimulus representation, and executing their responses. Individual participants' accuracy and RTs were uncorrelated. Drift rate and boundary settings were significantly related across tasks, but they were unrelated to each other. Accuracy was mainly determined by drift rate, and RT was mainly determined by boundary separation. We concluded that RT and accuracy operate largely independently.

Diffusion Decision Model: Current Issues and History

There is growing interest in diffusion models to represent the cognitive and neural processes of speeded decision making. Sequential-sampling models like the diffusion model have a long history in psychology. They view decision making as a process of noisy accumulation of evidence from a stimulus. The standard model assumes that evidence accumulates at a constant rate during the second or two it takes to make a decision. This process can be linked to the behaviors of populations of neurons and to theories of optimality. Diffusion models have been used successfully in a range of cognitive tasks and as psychometric tools in clinical research to examine individual differences. In this review, we relate the models to both earlier and more recent research in psychology.

Modeling Regularities in Response Time and Accuracy Data with the Diffusion Model

Diffusion models for simple two-choice decision-making have achieved prominence in psychology and neuroscience. The standard model views decision-making as a process in which noisy evidence is accumulated until one of the two response criteria is reached, at which point the associated response is made. The criteria represent the amount of evidence needed to make a decision and they reflect the decision maker's response biases and speed-accuracy trade-off settings. In this article, we review the regularities in experimental data that a model must explain. These include the relation between accuracy and mean response times, the shapes of response time distributions for correct and error responses and how they change with experimental variables, and individual differences in response time and accuracy. These relations are sometimes overlooked by researchers, but, taken together, they provide extremely strong tests of models.

Individual Differences and Fitting Methods for the Two-Choice Diffusion Model of Decision Making

Methods of fitting the diffusion model were examined with a focus on what the model can tell us about individual differences. Diffusion model parameters were obtained from the fits to data from two experiments and consistency of parameter values, individual differences, and practice effects were examined using different numbers of observations from each subject. Two issues were examined, first, what sizes of differences between groups can be obtained to distinguish between groups and second, what sizes of differences would be needed to find individual subjects that had a deficit relative to a control group. The parameter values from the experiments provided ranges that were used in a simulation study to examine recovery of individual differences. This study used several diffusion model fitting programs, fitting methods, and published packages. In a second simulation study, 64 sets of simulated data from each of 48 sets of parameter values (spanning the range of typical values obtained from fits to data) were fit with the different methods and biases and standard deviations in recovered model parameters were compared across methods. Finally, in a third simulation study, a comparison between a standard chi-square method and a hierarchical Bayesian method was performed. The results from these studies can be used as a starting point for selecting fitting methods and as a basis for understanding the strengths and weaknesses of using diffusion model analyses to examine individual differences in clinical, neuropsychological, and educational testing.

Automatic and controlled stimulus processing in conflict tasks: Superimposed diffusion processes and delta functions

An elaborated diffusion process model (a Diffusion Model for Conflict Tasks, DMC) is introduced that combines conceptual features of standard diffusion models with the notion of controlled and automatic processes. DMC can account for a variety of distributional properties of reaction time (RT) in conflict tasks (e.g., Eriksen flanker, Simon, Stroop). Specifically, DMC is compatible with all observed shapes of delta functions, including negative-going delta functions that are particularly challenging for the class of standard diffusion models. Basically, DMC assumes that the activations of controlled and automatic processes superimpose to trigger a response. Monte Carlo simulations demonstrate that the unfolding of automatic activation in time largely determines the shape of delta functions. Furthermore, the predictions of DMC are consistent with other phenomena observed in conflict tasks such as error rate patterns. In addition, DMC was successfully fitted to experimental data of the standard Eriksen flanker and the Simon task. Thus, the present paper reconciles the prominent and successful class of diffusion models with the empirical finding of negative-going delta functions.

A perspective on neural and cognitive mechanisms of error commission

Behavioral adaptation and cognitive control are crucial for goal-reaching behaviors. Every creature is ubiquitously faced with choices between behavioral alternatives. Common sense suggests that errors are an important source of information in the regulation of such processes. Several theories exist regarding cognitive control and the processing of undesired outcomes. However, most of these models focus on the consequences of an error, and less attention has been paid to the mechanisms that underlie the commissioning of an error. In this article, we present an integrative review of neuro-cognitive models that detail the determinants of the occurrence of response errors. The factors that may determine the likelihood of committing errors are likely related to the stability of task-representations in prefrontal networks, attentional selection mechanisms and mechanisms of action selection in basal ganglia circuits. An important conclusion is that the likelihood of committing an error is not stable over time but rather changes depending on the interplay of different functional neuro-anatomical and neuro-biological systems. We describe factors that might determine the time-course of cognitive control and the need to adapt behavior following response errors. Finally, we outline the mechanisms that may proof useful for predicting the outcomes of cognitive control and the emergence of response errors in future research.

Modeling individual differences in response time and accuracy in numeracy

In the study of numeracy, some hypotheses have been based on response time (RT) as a dependent variable and some on accuracy, and considerable controversy has arisen about the presence or absence of correlations between RT and accuracy, between RT or accuracy and individual differences like IQ and math ability, and between various numeracy tasks. In this article, we show that an integration of the two dependent variables is required, which we accomplish with a theory-based model of decision making. We report data from four tasks: numerosity discrimination, number discrimination, memory for two-digit numbers, and memory for three-digit numbers. Accuracy correlated across tasks, as did RTs. However, the negative correlations that might be expected between RT and accuracy were not obtained; if a subject was accurate, it did not mean that they were fast (and vice versa). When the diffusion decision-making model was applied to the data (Ratcliff, 1978), we found significant correlations across the tasks between the quality of the numeracy information (drift rate) driving the decision process and between the speed/accuracy criterion settings, suggesting that similar numeracy skills and similar speed-accuracy settings are involved in the four tasks. In the model, accuracy is related to drift rate and RT is related to speed-accuracy criteria, but drift rate and criteria are not related to each other across subjects. This provides a theoretical basis for understanding why negative correlations were not obtained between accuracy and RT. We also manipulated criteria by instructing subjects to maximize either speed or accuracy, but still found correlations between the criteria settings between and within tasks, suggesting that the settings may represent an individual trait that can be modulated but not equated across subjects. Our results demonstrate that a decision-making model may provide a way to reconcile inconsistent and sometimes contradictory results in numeracy research.

Time-varying boundaries for diffusion models of decision making and response time

Diffusion models are widely-used and successful accounts of the time course of two-choice decision making. Most diffusion models assume constant boundaries, which are the threshold levels of evidence that must be sampled from a stimulus to reach a decision. We summarize theoretical results from statistics that relate distributions of decisions and response times to diffusion models with time-varying boundaries. We then develop a computational method for finding time-varying boundaries from empirical data, and apply our new method to two problems. The first problem involves finding the time-varying boundaries that make diffusion models equivalent to the alternative sequential sampling class of accumulator models. The second problem involves finding the time-varying boundaries, at the individual level, that best fit empirical data for perceptual stimuli that provide equal evidence for both decision alternatives. We discuss the theoretical and modeling implications of using time-varying boundaries in diffusion models, as well as the limitations and potential of our approach to their inference.

Empirical validation of the diffusion model for recognition memory and a comparison of parameter-estimation methods

The diffusion model introduced by Ratcliff (Psychol Rev 85:59–108, 1978) has been applied to many binary decision tasks including recognition memory. It describes dynamic evidence accumulation unfolding over time and models choice accuracy as well as response-time distributions. Various parameters describe aspects of decision quality and response bias. In three recognition-memory experiments, the validity of the model was tested experimentally and analyzed with three different programs: fast-dm, EZ, and DMAT. Each of three central model parameters was targeted via specific experimental manipulations. All manipulations affected mainly the corresponding parameters, thus supporting the convergent validity of the measures. There were, however, smaller effects on other parameters, showing some limitations in discriminant validity.

The diffusion model is not a deterministic growth model: comment on Jones and Dzhafarov (2014)

Jones and Dzhafarov (2014) claim that several current models of speeded decision making in cognitive tasks, including the diffusion model, can be viewed as special cases of other general models or model classes. The general models can be made to match any set of response time (RT) distribution and accuracy data exactly by a suitable choice of parameters and so are unfalsifiable. The implication of their claim is that models like the diffusion model are empirically testable only by artificially restricting them to exclude unfalsifiable instances of the general model. We show that Jones and Dzhafarov's argument depends on enlarging the class of "diffusion" models to include models in which there is little or no diffusion. The unfalsifiable models are deterministic or near-deterministic growth models, from which the effects of within-trial variability have been removed or in which they are constrained to be negligible. These models attribute most or all of the variability in RT and accuracy to across-trial variability in the rate of evidence growth, which is permitted to be distributed arbitrarily and to vary freely across experimental conditions. In contrast, in the standard diffusion model, within-trial variability in evidence is the primary determinant of variability in RT. Across-trial variability, which determines the relative speed of correct responses and errors, is theoretically and empirically constrained. Jones and Dzhafarov's attempt to include the diffusion model in a class of models that also includes deterministic growth models misrepresents and trivializes it and conveys a misleading picture of cognitive decision-making research. (PsycINFO Database Record (c) 2014 APA, all rights reserved).

Using response time modeling to distinguish memory and decision processes in recognition and source tasks

Receiver operating characteristic (ROC) functions are often used to make inferences about memory processes, such as claiming that memory strength is more variable for studied versus nonstudied items. However, decision processes can produce the ROC patterns that are usually attributed to memory, so independent forms of data are needed to support strong conclusions. The present experiments tested ROC-based claims about the variability of memory evidence by modeling response time (RT) data with the diffusion model. To ensure that the model can correctly discriminate equal- and unequal-variance distributions, Experiment 1 used a numerousity discrimination task that had a direct manipulation of evidence variability. Fits of the model produced correct conclusions about evidence variability in all cases. Experiments 2 and 3 explored the effect of repeated learning trials on evidence variability in recognition and source memory tasks, respectively. Fits of the diffusion model supported the same conclusions about variability as the ROC literature. For recognition, evidence variability was higher for targets than for lures, but it did not differ on the basis of the number of learning trials for target items. For source memory, evidence variability was roughly equal for source 1 and source 2 items, and variability increased for items with additional learning attempts. These results demonstrate that RT modeling can help resolve ambiguities regarding the processes that produce different patterns in ROC data. The results strengthen the evidence that memory strength distributions have unequal variability across item types in recognition and source memory tasks.