A comparison of four methods for simulating the diffusion process

Expressions for Bayesian confidence of drift diffusion observers in fluctuating stimuli tasks

We introduce a new approach to modelling decision confidence, with the aim of enabling computationally cheap predictions while taking into account, and thereby exploiting, trial-by-trial variability in stochastically fluctuating stimuli. Using the framework of the drift diffusion model of decision making, along with time-dependent thresholds and the idea of a Bayesian confidence readout, we derive expressions for the probability distribution over confidence reports. In line with current models of confidence, the derivations allow for the accumulation of "pipeline" evidence that has been received but not processed by the time of response, the effect of drift rate variability, and metacognitive noise. The expressions are valid for stimuli that change over the course of a trial with normally-distributed fluctuations in the evidence they provide. A number of approximations are made to arrive at the final expressions, and we test all approximations via simulation. The derived expressions contain only a small number of standard functions, and require evaluating only once per trial, making trial-by-trial modelling of confidence data in stochastically fluctuating stimuli tasks more feasible. We conclude by using the expressions to gain insight into the confidence of optimal observers, and empirically observed patterns.

An adaptive rejection sampler for sampling from the Wiener diffusion model

The Wiener diffusion model with two absorbing boundaries is one of the most frequently applied models for jointly modeling responses and response latencies in psychological research. We consider four methods for sampling from the model with and without variability in drift rate, starting point, and non-decision time: Inverse transform sampling, rejection sampling, and two new methods based on adaptive rejection sampling (ARS). We implement these four methods in an R package, validate the methods, and compare their sampling speed in different settings. All four implemented methods provide samples that follow the intended distributions. The ARS-based methods, however, outperform the other methods in sampling speed as the requested sample size increases. We provide guidelines for when using ARS is more efficient than using traditional methods and vice versa.

The effect of speed-stress on driving behavior: A diffusion model analysis

In everyday driving on the road, people are often required to make fast decisions that could compromise the accuracy of choices. We present a diffusion model analysis of the adjustments drivers make to the decision process under speed-stress. Participants operated a PC-based driving simulator while performing one of two decision-making tasks that required a driving action as a response to the stimulus. In a one-choice driving task, participants were asked to drive around a lead car when its brake lights were turned on. A two-choice driving task used a brightness-discrimination task in which participants were asked to drive to the left and back behind a lead car if there were more black than white pixels in a display and to the right and back if there were more white than black pixels. Speed-stress was operationalized by instructing drivers to respond as quickly as possible and by manipulating the distance drivers were required to maintain behind the lead car. Results showed the expected speed-accuracy tradeoff; however, the cost on accuracy in the two-choice task was relatively small. The model-based analysis showed that this was achieved by lowering the decision criteria and speeding up nondecision processes without disrupting components that produce evidence for the decision process. In fact, in the one-choice task, evidence accumulation rate in the speed-stress condition was found to be higher than in the accuracy-stress condition. We concluded that drivers were able to comply with speed-stress demands with relatively safe adjustments that imposed minimal costs on the accuracy of choices.

On the Utility of Integrated Speed-Accuracy Measures when Speed-Accuracy Trade-off is Present

In an attempt to simplify data analysis and to avoid confounds due to speed-accuracy trade-off, sometimes integrated measures of speed and accuracy are used. Although it has been claimed that some of these combined measures are insensitive to speed-accuracy trade-off (SAT), a systematic and broad examination of such claims has not been performed thus far. The present article reports the results of four simulation studies in which five established integrated measures were studied in different speed-accuracy trade-off contexts. All four studies used repeated measures designs crossing an experimental factor (variable of interest) with a factor representing SAT settings, with all conditions occurring randomly over the sequence of trials to avoid condition-wise SATs (mixed conditions repeated measures design). The first study used speed modulations that were balanced by accuracy changes in the opposite direction. The other studies were all based on SAT as modeled either by the drift-diffusion model, with pro-active trade-off settings (Study 2) or with reactive trade-off modulations (Study 3) or by a discontinuous two-phase model (Study 4). Only the studies based on balanced trade-offs showed that two of the measures were insensitive to SAT settings, while in all other contexts, all measures were sensitive to SAT. Nevertheless, as the mixed conditions design distributes the SAT effects over the conditions of the variable of interest, all integrated measures reliably detected the effect of this variable in all SAT conditions. Although integrated measures are sensitive to SAT, these effects can be neutralised by using a mixed conditions repeated measures design.

How the Working Memory with Distributed Executive Control Model Accounts for Task Switching and Dual-Task Coordination Costs

According to the working memory model with distributed executive control (WMDEC), working memory is not only used for temporary maintenance of information, but it also serves goal-directed action by maintaining task-related information. Such information may include the current action goal, the means selected to attain the goal, situational constraints, and interim processing results. A computational version of the WMDEC model was used to simulate human performance in a series of experiments that examined particular predictions regarding task switching costs, costs due to task and attention switching, to dual-task coordination in working memory tasks, and to experiments that required dual-task coordination of memorisation and task switching demands. The results of these simulations are reported and their implications for accounts of multi- and dual-tasking are discussed.

Comparing Eight Parameter Estimation Methods for the Ratcliff Diffusion Model Using Free Software

The Ratcliff Diffusion Model has become an important and widely used tool for the evaluation of psychological experiments. Concurrently, numerous programs and routines have appeared to estimate the model's parameters. The present study aims at comparing some of the most widely used tools with special focus on freely available routines (i.e., open source). Our simulations show that (1) starting point and non-decision time were recovered better than drift rate, (2) the Bayesian approach outperformed all other approaches when the number of trials was low, (3) the Kolmogorov-Smirnov and χ² approaches revealed more bias than Bayesian or Maximum Likelihood based routines, and (4) EZ produced substantially biased estimates of threshold separation, non-decision time and drift rate when starting point z ≠ a/2. We discuss the implications for the choice of parameter estimation approaches for real data and suggest that if biased starting point cannot be excluded, EZ will produce deviant estimates and should be used with great care.

Bayesian Semiparametric Longitudinal Drift-Diffusion Mixed Models for Tone Learning in Adults

Understanding how adult humans learn nonnative speech categories such as tone information has shed novel insights into the mechanisms underlying experience-dependent brain plasticity. Scientists have traditionally examined these questions using longitudinal learning experiments under a multi-category decision making paradigm. Drift-diffusion processes are popular in such contexts for their ability to mimic underlying neural mechanisms. Motivated by these problems, we develop a novel Bayesian semiparametric inverse Gaussian drift-diffusion mixed model for multi-alternative decision making in longitudinal settings. We design a Markov chain Monte Carlo algorithm for posterior computation. We evaluate the method's empirical performances through synthetic experiments. Applied to our motivating longitudinal tone learning study, the method provides novel insights into how the biologically interpretable model parameters evolve with learning, differ between input-response tone combinations, and differ between well and poorly performing adults. supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

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.

Computational cognitive modeling of the temporal dynamics of fatigue from sleep loss

Computational models have become common tools in psychology. They provide quantitative instantiations of theories that seek to explain the functioning of the human mind. In this paper, we focus on identifying deep theoretical similarities between two very different models. Both models are concerned with how fatigue from sleep loss impacts cognitive processing. The first is based on the diffusion model and posits that fatigue decreases the drift rate of the diffusion process. The second is based on the Adaptive Control of Thought - Rational (ACT-R) cognitive architecture and posits that fatigue decreases the utility of candidate actions leading to microlapses in cognitive processing. A biomathematical model of fatigue is used to control drift rate in the first account and utility in the second. We investigated the predicted response time distributions of these two integrated computational cognitive models for performance on a psychomotor vigilance test under conditions of total sleep deprivation, simulated shift work, and sustained sleep restriction. The models generated equivalent predictions of response time distributions with excellent goodness-of-fit to the human data. More importantly, although the accounts involve different modeling approaches and levels of abstraction, they represent the effects of fatigue in a functionally equivalent way: in both, fatigue decreases the signal-to-noise ratio in decision processes and decreases response inhibition. This convergence suggests that sleep loss impairs psychomotor vigilance performance through degradation of the quality of cognitive processing, which provides a foundation for systematic investigation of the effects of sleep loss on other aspects of cognition. Our findings illustrate the value of treating different modeling formalisms as vehicles for discovery.

Modeling numerosity representation with an integrated diffusion model

Models of the representation of numerosity information used in discrimination tasks are integrated with a diffusion decision model. The representation models assume distributions of numerosity either with means and SD that increase linearly with numerosity or with means that increase logarithmically with constant SD. The models produce coefficients that are applied to differences between two numerosities to produce drift rates and these drive the decision process. The linear and log models make differential predictions about how response time (RT) distributions and accuracy change with numerosity and which model is successful depends on the task. When the task is to decide which of two side-by-side arrays of dots has more dots, the log model fits decreasing accuracy and increasing RT as numerosity increases. When the task is to decide, for dots of two colors mixed in a single array, which color has more dots, the linear model fits decreasing accuracy and decreasing RT as numerosity increases. For both tasks, variables such as the areas covered by the dots affect performance, but if the task is changed to one in which the subject has to decide whether the number of dots in a single array is more or less than a standard, the variables have little effect on performance. Model parameters correlate across tasks suggesting commonalities in the abilities to perform them. Overall, results show that the representation used depends on the task and no single representation can account for the data from all the paradigms. (PsycINFO Database Record

Internal and external sources of variability in perceptual decision-making

It is important to identify sources of variability in processing to understand decision-making in perception and cognition. There is a distinction between internal and external variability in processing, and double-pass experiments have been used to estimate their relative contributions. In these and our experiments, exact perceptual stimuli are repeated later in testing, and agreement on the 2 trials is examined to see if it is greater than chance. In recent research in modeling decision processes, some models implement only (internal) variability in the decision process whereas others explicitly represent multiple sources of variability. We describe 5 perceptual double-pass experiments that show greater than chance agreement, which is inconsistent with models that assume internal variability alone. Estimates of total trial-to-trial variability in the evidence accumulation (drift) rate (the decision-relevant stimulus information) were estimated from fits of the standard diffusion decision-making model to the data. The double-pass procedure provided estimates of how much of this total variability was systematic and dependent on the stimulus. These results provide the first behavioral evidence independent of model fits for trial-to-trial variability in drift rate in tasks used in examining perceptual decision-making. (PsycINFO Database Record

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.

PhotoGate microscopy to track single molecules in crowded environments

Tracking single molecules inside cells reveals the dynamics of biological processes, including receptor trafficking, signalling and cargo transport. However, individual molecules often cannot be resolved inside cells due to their high density. Here we develop the PhotoGate technique that controls the number of fluorescent particles in a region of interest by repeatedly photobleaching its boundary. PhotoGate bypasses the requirement of photoactivation to track single particles at surface densities two orders of magnitude greater than the single-molecule detection limit. Using this method, we observe ligand-induced dimerization of a receptor tyrosine kinase at the cell surface and directly measure binding and dissociation of signalling molecules from early endosomes in a dense cytoplasm with single-molecule resolution. We additionally develop a numerical simulation suite for rapid quantitative optimization of Photogate experimental conditions. PhotoGate yields longer tracking times and more accurate measurements of complex stoichiometry than existing single-molecule imaging methods.

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.

Modeling Individual Differences in the Go/No-go Task with a Diffusion Model

The go/no-go task is one in which there are two choices, but the subject responds only to one of them, waiting out a time-out for the other choice. The task has a long history in psychology and modern applications in the clinical/neuropsychological domain. In this article we fit a diffusion model to both experimental and simulated data. The model is the same as the two-choice model and assumes that there are two decision boundaries and termination at one of them produces a response and at the other, the subject waits out the trial. In prior modeling, both two-choice and go/no-go data were fit simultaneously and only group data were fit. Here the model is fit to just go/no-go data for individual subjects. This allows analyses of individual differences which is important for clinical applications. First, we fit the standard two-choice model to two-choice data and fit the go/no-go model to RTs from one of the choices and accuracy from the two-choice data. Parameter values were similar between the models and had high correlations. The go/no-go model was also fit to data from a go/no-go version of the task with the same subjects as the two-choice task. A simulation study with ranges of parameter values that are obtained in practice showed similar parameter recovery between the two-choice and go/no-go models. Results show that a diffusion model with an implicit (no response) boundary can be fit to data with almost the same accuracy as fitting the two-choice model to two-choice data.

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 one-choice and two-choice driving tasks

An experiment is presented in which subjects were tested on both one-choice and two-choice driving tasks and on non-driving versions of them. Diffusion models for one- and two-choice tasks were successful in extracting model-based measures from the response time and accuracy data. These include measures of the quality of the information from the stimuli that drove the decision process (drift rate in the model), the time taken up by processes outside the decision process and, for the two-choice model, the speed/accuracy decision criteria that subjects set. Drift rates were only marginally different between the driving and non-driving tasks, indicating that nearly the same information was used in the two kinds of tasks. The tasks differed in the time taken up by other processes, reflecting the difference between them in response processing demands. Drift rates were significantly correlated across the two two-choice tasks showing that subjects that performed well on one task also performed well on the other task. Nondecision times were correlated across the two driving tasks, showing common abilities on motor processes across the two tasks. These results show the feasibility of using diffusion modeling to examine decision making in driving and so provide for a theoretical examination of factors that might impair driving, such as extreme aging, distraction, sleep deprivation, and so on.

Fast and accurate Monte Carlo sampling of first-passage times from Wiener diffusion models

We present a new, fast approach for drawing boundary crossing samples from Wiener diffusion models. Diffusion models are widely applied to model choices and reaction times in two-choice decisions. Samples from these models can be used to simulate the choices and reaction times they predict. These samples, in turn, can be utilized to adjust the models' parameters to match observed behavior from humans and other animals. Usually, such samples are drawn by simulating a stochastic differential equation in discrete time steps, which is slow and leads to biases in the reaction time estimates. Our method, instead, facilitates known expressions for first-passage time densities, which results in unbiased, exact samples and a hundred to thousand-fold speed increase in typical situations. In its most basic form it is restricted to diffusion models with symmetric boundaries and non-leaky accumulation, but our approach can be extended to also handle asymmetric boundaries or to approximate leaky accumulation.

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.

Modeling simple driving tasks with a one-boundary diffusion model

A one-boundary diffusion model was applied to the data from two experiments in which subjects were performing a simple simulated driving task. In the first experiment, the same subjects were tested on two driving tasks using a PC-based driving simulator and the psychomotor vigilance test. The diffusion model fit the response time distributions for each task and individual subject well. Model parameters were found to correlate across tasks, which suggests that common component processes were being tapped in the three tasks. The model was also fit to a distracted driving experiment of Cooper and Strayer (Human Factors, 50, 893-902, 2008). Results showed that distraction altered performance by affecting the rate of evidence accumulation (drift rate) and/or increasing the boundary settings. This provides an interpretation of cognitive distraction whereby conversing on a cell phone diverts attention from the normal accumulation of information in the driving environment.

Individual differences in attention influence perceptual decision making

Sequential sampling decision-making models have been successful in accounting for reaction time (RT) and accuracy data in two-alternative forced choice tasks. These models have been used to describe the behavior of populations of participants, and explanatory structures have been proposed to account for between individual variability in model parameters. In this study we show that individual differences in behavior from a novel perceptual decision making task can be attributed to (1) differences in evidence accumulation rates, (2) differences in variability of evidence accumulation within trials, and (3) differences in non-decision times across individuals. Using electroencephalography (EEG), we demonstrate that these differences in cognitive variables, in turn, can be explained by attentional differences as measured by phase-locking of steady-state visual evoked potential (SSVEP) responses to the signal and noise components of the visual stimulus. Parameters of a cognitive model (a diffusion model) were obtained from accuracy and RT distributions and related to phase-locking indices (PLIs) of SSVEPs with a single step in a hierarchical Bayesian framework. Participants who were able to suppress the SSVEP response to visual noise in high frequency bands were able to accumulate correct evidence faster and had shorter non-decision times (preprocessing or motor response times), leading to more accurate responses and faster response times. We show that the combination of cognitive modeling and neural data in a hierarchical Bayesian framework relates physiological processes to the cognitive processes of participants, and that a model with a new (out-of-sample) participant's neural data can predict that participant's behavior more accurately than models without physiological data.

Modeling Eye Movements and Response Times in Consumer Choice

Peoples’ choices are not instantaneous, nor are they perfectly self consistent. While these two facts may at first seem unrelated, they are in fact inextricably linked. Decision scientists are accustomed to using logit and probit models to account for “noise” in their choice data. But what is the driving force behind these behavioral inconsistencies? Random utility theory (RUT) provides little guidance in this respect. While providing a mathematical basis for dealing with stochastic choice, RUT is agnostic about whether the noise is due to unobserved characteristics of the decision maker and/or the choice environment, or due to actual “mistakes.” The distinction is important because the former implies that from the point of view of the decision maker, her choices are perfectly consistent, while the latter implies that the decision maker herself may be surprised by her set of choices. Here we argue that non-choice (“process”) data strongly favors the latter explanation. Rather than thinking of choice as an instantaneous realization of stored preferences, we instead conceptualize choice as a dynamical process of information accumulation and comparison. Adapting “sequential sampling models” from cognitive psychology to economic choice, we illustrate the surprisingly complex relationship between choice and response-time data. Finally, we review recent data demonstrating how other process measures such as eye-tracking and neural recordings can be incorporated into this modeling approach, yielding further insights into the choice process.

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.

Reinforcement-Based Decision Making in Corticostriatal Circuits: Mutual Constraints by Neurocomputational and Diffusion Models

In this letter, we examine the computational mechanisms of reinforce-ment-based decision making. We bridge the gap across multiple levels of analysis, from neural models of corticostriatal circuits—the basal ganglia (BG) model (Frank, 2005 , 2006 ) to simpler but mathematically tractable diffusion models of two-choice decision making. Specifically, we generated simulated data from the BG model and fit the diffusion model (Ratcliff, 1978 ) to it. The standard diffusion model fits underestimated response times under conditions of high response and reinforcement conflict. Follow-up fits showed good fits to the data both by increasing nondecision time and by raising decision thresholds as a function of conflict and by allowing this threshold to collapse with time. This profile captures the role and dynamics of the subthalamic nucleus in BG circuitry, and as such, parametric modulations of projection strengths from this nucleus were associated with parametric increases in decision boundary and its modulation by conflict. We then present data from a human reinforcement learning experiment involving decisions with low- and high-reinforcement conflict. Again, the standard model failed to fit the data, but we found that two variants similar to those that fit the BG model data fit the experimental data, thereby providing a convergence of theoretical accounts of complex interactive decision-making mechanisms consistent with available data. This work also demonstrates how to make modest modifications to diffusion models to summarize core computations of the BG model. The result is a better fit and understanding of reinforcement-based choice data than that which would have occurred with either model alone.

Diffusion model for one-choice reaction-time tasks and the cognitive effects of sleep deprivation

One-choice reaction-time (RT) tasks are used in many domains, including assessments of motor vehicle driving and assessments of the cognitive/behavioral consequences of sleep deprivation. In such tasks, subjects are asked to respond when they detect the onset of a stimulus; the dependent variable is RT. We present a cognitive model for one-choice RT tasks that uses a one-boundary diffusion process to represent the accumulation of stimulus information. When the accumulated evidence reaches a decision criterion, a response is initiated. This model is distinct in accounting for the RT distributions observed for one-choice RT tasks, which can have long tails that have not been accurately captured by earlier cognitive modeling approaches. We show that the model explains performance on a brightness-detection task (a "simple RT task") and on a psychomotor vigilance test. The latter is used extensively to examine the clinical and behavioral effects of sleep deprivation. For the brightness-detection task, the model explains the behavior of RT distributions as a function of brightness. For the psychomotor vigilance test, it accounts for lapses in performance under conditions of sleep deprivation and for changes in the shapes of RT distributions over the course of sleep deprivation. The model also successfully maps the rate of accumulation of stimulus information onto independently derived predictions of alertness. The model is a unified, mechanistic account of one-choice RT under conditions of sleep deprivation.

The Drift Diffusion Model can account for the accuracy and reaction time of value-based choices under high and low time pressure

An important open problem is how values are compared to make simple choices. A natural hypothesis is that the brain carries out the computations associated with the value comparisons in a manner consistent with the Drift Diffusion Model (DDM), since this model has been able to account for a large amount of data in other domains. We investigated the ability of four different versions of the DDM to explain the data in a real binary food choice task under conditions of high and low time pressure. We found that a seven-parameter version of the DDM can account for the choice and reaction time data with high-accuracy, in both the high and low time pressure conditions. The changes associated with the introduction of time pressure could be traced to changes in two key model parameters: the barrier height and the noise in the slope of the drift process.

Reward rate optimization in two-alternative decision making: empirical tests of theoretical predictions

The drift-diffusion model (DDM) implements an optimal decision procedure for stationary, 2-alternative forced-choice tasks. The height of a decision threshold applied to accumulating information on each trial determines a speed-accuracy tradeoff (SAT) for the DDM, thereby accounting for a ubiquitous feature of human performance in speeded response tasks. However, little is known about how participants settle on particular tradeoffs. One possibility is that they select SATs that maximize a subjective rate of reward earned for performance. For the DDM, there exist unique, reward-rate-maximizing values for its threshold and starting point parameters in free-response tasks that reward correct responses (R. Bogacz, E. Brown, J. Moehlis, P. Holmes, & J. D. Cohen, 2006). These optimal values vary as a function of response-stimulus interval, prior stimulus probability, and relative reward magnitude for correct responses. We tested the resulting quantitative predictions regarding response time, accuracy, and response bias under these task manipulations and found that grouped data conformed well to the predictions of an optimally parameterized DDM.

Diffusion model analysis with MATLAB: a DMAT primer

The Ratcliff diffusion model has proved to be a useful tool in reaction time analysis. However, its use has been limited by the practical difficulty of estimating the parameters. We present a software tool, the Diffusion Model Analysis Toolbox (DMAT), intended to make the Ratcliff diffusion model for reaction time and accuracy data more accessible to experimental psychologists. The tool takes the form of a MATLAB toolbox and can be freely downloaded from ppw.kuleuven.be/okp/dmatoolbox. Using the program does not require a background in mathematics, nor any advanced programming experience (but familiarity with MATLAB is useful). We demonstrate the basic use of DMAT with two examples.

Evaluating methods for approximating stochastic differential equations

Models of decision making and response time (RT) are often formulated using stochastic differential equations (SDEs). Researchers often investigate these models using a simple Monte Carlo method based on Euler's method for solving ordinary differential equations. The accuracy of Euler's method is investigated and compared to the performance of more complex simulation methods. The more complex methods for solving SDEs yielded no improvement in accuracy over the Euler method. However, the matrix method proposed by Diederich and Busemeyer (2003) yielded significant improvements. The accuracy of all methods depended critically on the size of the approximating time step. The large (∼10 ms) step sizes often used by psychological researchers resulted in large and systematic errors in evaluating RT distributions.

An efficient stochastic diffusion algorithm for modeling second messengers in dendrites and spines

Intracellular signaling pathways, which encompass both biochemical reactions and second messenger diffusion, interact non-linearly with neuronal membrane properties in their role as essential intermediaries for synaptic plasticity and neuromodulation. Computational modeling is a productive approach for investigating these phenomena; however, most current strategies for modeling neurons exclude signaling pathways. To overcome this deficiency, a new algorithm is presented to simulate stochastic diffusion in a highly efficient manner. The gain in speed is obtained by considering collections of molecules, instead of tracking the movement of individual molecules. The probability of a molecule leaving a spatially discrete compartment is used to create a lookup table that stores the probability of k(m) molecules leaving the compartment as a function of the total number of molecules in the compartment. During the simulation, the number of molecules leaving the compartment is determined using a uniform random number as an index into the lookup table. Simulations illustrate the accuracy of this algorithm by comparing it with the theoretical solution for deterministic diffusion. Additional simulations show how spines on a dendritic branch compartmentalize diffusible molecules. The efficiency of the algorithm is sufficient to allow simulation of second messenger pathways in a multitude of spines on an entire neuron.

Estimating parameters of the diffusion model: approaches to dealing with contaminant reaction times and parameter variability

Three methods for fitting the diffusion model (Ratcliff, 1978) to experimental data are examined. Sets of simulated data were generated with known parameter values, and from fits of the model, we found that the maximum likelihood method was better than the chi-square and weighted least squares methods by criteria of bias in the parameters relative to the parameter values used to generate the data and standard deviations in the parameter estimates. The standard deviations in the parameter values can be used as measures of the variability in parameter estimates from fits to experimental data. We introduced contaminant reaction times and variability into the other components of processing besides the decision process and found that the maximum likelihood and chi-square methods failed, sometimes dramatically. But the weighted least squares method was robust to these two factors. We then present results from modifications of the maximum likelihood and chi-square methods, in which these factors are explicitly modeled, and show that the parameter values of the diffusion model are recovered well. We argue that explicit modeling is an important method for addressing contaminants and variability in nondecision processes and that it can be applied in any theoretical approach to modeling reaction time.