Stochastic Dynamic Models of Response Time and Accuracy: A Foundational Primer

Place-value and physical size converge in automatic processing of multi-digit numbers

Previous research has shown that multi-digit number processing is modulated by both place-value and physical size of the digits. By pitting place-value against physical size, the present study examined whether one of the attributes had a greater impact on the automatic processing of multi-digit numbers. In three experiments, participants were presented with two-digit number pairs that appeared in frames. They were instructed to select the larger frame while ignoring the numbers within the frames. Importantly, we manipulated the physical size of the digits (i.e., both decade/unit digits were physically larger) within the frames, the unit-decade compatibility (i.e., the relationship between the numerical values of both decade and unit digits was consistent or inconsistent), and the congruity between the numerical values of the decade digits and the frames' physical size (i.e., decade-value-frame-size congruity). In Experiment 1, where all pairs were unit-decade compatible, a decade-value-frame-size congruity effect emerged for pairs with physically larger decade, but not unit, digits. However, when adding unit-decade incompatible pairs (Experiments 2-3), in unit-decade compatible pairs, there was a decade-value-frame-size congruity effect regardless of the digits' physical size. In contrast, in unit-decade incompatible pairs, there was no decade-value-frame-size congruity effect, even when the physically larger digit (i.e., unit) contradicted the place-value information, presumably due to the cancellation of the opposing influences of the digits' physical sizes their place-values. Overall, these findings suggest that place-value and physical size are intertwined in the Hindu-Arabic numerical system and are processed as one.

The neutral condition in conflict tasks: On the violation of the midpoint assumption in reaction time trends

Although the relation between congruent and incongruent conditions in conflict tasks has been the primary focus of cognitive control studies, the neutral condition is often set as a baseline directly between the two conditions. However, empirical evidence suggests that the average neutral reaction time (RT) is not placed evenly between the two opposing conditions. This article set out to establish two things: First, to reinforce the informative nature of the neutral condition and second, to highlight how it can be useful for modelling. We explored how RT in the neutral condition of conflict tasks (Stroop, Flanker, and Simon Tasks) deviated from the predictions of current diffusion models. Current diffusion models of conflict tasks predict a neutral RT that is the average of the congruent and incongruent RT, called the midpoint assumption. To investigate this, we first conducted a cursory limited search that recorded the average RT's of conflict tasks with neutral conditions. Upon finding evidence of a midpoint assumption violation which showed a larger disparity between average neutral and incongruent RT, we tested the previously mentioned conflict tasks with two different sets of stimuli to establish the robustness of the effect. The midpoint assumption violation is sometimes inconsistent with the prediction of diffusion models of conflict processing (e.g., the Diffusion Model of Conflict), suggesting possible elaborations of such models.

PyBEAM: A Bayesian approach to parameter inference for a wide class of binary evidence accumulation models

Many decision-making theories are encoded in a class of processes known as evidence accumulation models (EAM). These assume that noisy evidence stochastically accumulates until a set threshold is reached, triggering a decision. One of the most successful and widely used of this class is the Diffusion Decision Model (DDM). The DDM however is limited in scope and does not account for processes such as evidence leakage, changes of evidence, or time varying caution. More complex EAMs can encode a wider array of hypotheses, but are currently limited by computational challenges. In this work, we develop the Python package PyBEAM (Bayesian Evidence Accumulation Models) to fill this gap. Toward this end, we develop a general probabilistic framework for predicting the choice and response time distributions for a general class of binary decision models. In addition, we have heavily computationally optimized this modeling process and integrated it with PyMC, a widely used Python package for Bayesian parameter estimation. This 1) substantially expands the class of EAM models to which Bayesian methods can be applied, 2) reduces the computational time to do so, and 3) lowers the entry fee for working with these models. Here we demonstrate the concepts behind this methodology, its application to parameter recovery for a variety of models, and apply it to a recently published data set to demonstrate its practical use.

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.

Rat movements reflect internal decision dynamics in an evidence accumulation task

Perceptual decision-making involves multiple cognitive processes, including accumulation of sensory evidence, planning, and executing a motor action. How these processes are intertwined is unclear; some models assume that decision-related processes precede motor execution, whereas others propose that movements reflecting on-going decision processes occur before commitment to a choice. Here we develop and apply two complementary methods to study the relationship between decision processes and the movements leading up to a choice. The first is a free response pulse-based evidence accumulation task, in which stimuli continue until choice is reported. The second is a motion-based drift diffusion model (mDDM), in which movement variables from video pose estimation constrain decision parameters on a trial-by-trial basis. We find the mDDM provides a better model fit to rats' decisions in the free response accumulation task than traditional DDM models. Interestingly, on each trial we observed a period of time, prior to choice, that was characterized by head immobility. The length of this period was positively correlated with the rats' decision bounds and stimuli presented during this period had the greatest impact on choice. Together these results support a model in which internal decision dynamics are reflected in movements and demonstrate that inclusion of movement parameters improves the performance of diffusion-to-bound decision models.

A Modeling Framework to Examine Psychological Processes Underlying Ordinal Responses and Response Times of Psychometric Data

This article presents a joint modeling framework of ordinal responses and response times (RTs) for the measurement of latent traits. We integrate cognitive theories of decision-making and confidence judgments with psychometric theories to model individual-level measurement processes. The model development starts with the sequential sampling framework which assumes that when an item is presented, a respondent accumulates noisy evidence over time to respond to the item. Several cognitive and psychometric theories are reviewed and integrated, leading us to three psychometric process models with different representations of the cognitive processes underlying the measurement. We provide simulation studies that examine parameter recovery and show the relationships between latent variables and data distributions. We further test the proposed models with empirical data measuring three traits related to motivation. The results show that all three models provide reasonably good descriptions of observed response proportions and RT distributions. Also, different traits favor different process models, which implies that psychological measurement processes may have heterogeneous structures across traits. Our process of model building and examination illustrates how cognitive theories can be incorporated into psychometric model development to shed light on the measurement process, which has had little attention in traditional psychometric models.

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.

"Reliable organisms from unreliable components" revisited: the linear drift, linear infinitesimal variance model of decision making

Diffusion models of decision making, in which successive samples of noisy evidence are accumulated to decision criteria, provide a theoretical solution to von Neumann's (1956) problem of how to increase the reliability of neural computation in the presence of noise. I introduce and evaluate a new neurally-inspired dual diffusion model, the linear drift, linear infinitesimal variance (LDLIV) model, which embodies three features often thought to characterize neural mechanisms of decision making. The accumulating evidence is intrinsically positively-valued, saturates at high intensities, and is accumulated for each alternative separately. I present explicit integral-equation predictions for the response time distribution and choice probabilities for the LDLIV model and compare its performance on two benchmark sets of data to three other models: the standard diffusion model and two dual diffusion model composed of racing Wiener processes, one between absorbing and reflecting boundaries and one with absorbing boundaries only. The LDLIV model and the standard diffusion model performed similarly to one another, although the standard diffusion model is more parsimonious, and both performed appreciably better than the other two dual diffusion models. I argue that accumulation of noisy evidence by a diffusion process and drift rate variability are both expressions of how the cognitive system solves von Neumann's problem, by aggregating noisy representations over time and over elements of a neural population. I also argue that models that do not solve von Neumann's problem do not address the main theoretical question that historically motivated research in this area.

Decision neuroscience and neuroeconomics: Recent progress and ongoing challenges

In the past decade, decision neuroscience and neuroeconomics have developed many new insights in the study of decision making. This review provides an overarching update on how the field has advanced in this time period. Although our initial review a decade ago outlined several theoretical, conceptual, methodological, empirical, and practical challenges, there has only been limited progress in resolving these challenges. We summarize significant trends in decision neuroscience through the lens of the challenges outlined for the field and review examples where the field has had significant, direct, and applicable impacts across economics and psychology. First, we review progress on topics including reward learning, explore-exploit decisions, risk and ambiguity, intertemporal choice, and valuation. Next, we assess the impacts of emotion, social rewards, and social context on decision making. Then, we follow up with how individual differences impact choices and new exciting developments in the prediction and neuroforecasting of future decisions. Finally, we consider how trends in decision-neuroscience research reflect progress toward resolving past challenges, discuss new and exciting applications of recent research, and identify new challenges for the field. This article is categorized under: Psychology > Reasoning and Decision Making Psychology > Emotion and Motivation.

Vision for the blind: visual psychophysics and blinded inference for decision models

Evidence accumulation models like the diffusion model are increasingly used by researchers to identify the contributions of sensory and decisional factors to the speed and accuracy of decision-making. Drift rates, decision criteria, and nondecision times estimated from such models provide meaningful estimates of the quality of evidence in the stimulus, the bias and caution in the decision process, and the duration of nondecision processes. Recently, Dutilh et al. (Psychonomic Bulletin & Review 26, 1051–1069, 2019) carried out a large-scale, blinded validation study of decision models using the random dot motion (RDM) task. They found that the parameters of the diffusion model were generally well recovered, but there was a pervasive failure of selective influence, such that manipulations of evidence quality, decision bias, and caution also affected estimated nondecision times. This failure casts doubt on the psychometric validity of such estimates. Here we argue that the RDM task has unusual perceptual characteristics that may be better described by a model in which drift and diffusion rates increase over time rather than turn on abruptly. We reanalyze the Dutilh et al. data using models with abrupt and continuous-onset drift and diffusion rates and find that the continuous-onset model provides a better overall fit and more meaningful parameter estimates, which accord with the known psychophysical properties of the RDM task. We argue that further selective influence studies that fail to take into account the visual properties of the evidence entering the decision process are likely to be unproductive.

How do recall requirements affect decision-making in free recall initiation? A linear ballistic accumulator approach

Models of free recall describe free recall initiation as a decision-making process in which items compete to be retrieved. Recently, Osth and Farrell (Psychological Review, 126, 578–609, 2019) applied evidence accumulation models to complete RT distributions and serial positions of participants’ first recalls in free recall, which resulted in some novel conclusions about primacy and recency effects. Specifically, the results of the modeling favored an account in which primacy was due to reinstatement of the start-of-the-list, and recency was found to be exponential in shape. In this work, we examine what happens when participants are given alternative recall instructions. Prior work has demonstrated weaker primacy and greater recency when fewer items are required to report (Ward & Tan, Memory & Cognition, 2019), and a key question is whether this change in instructions qualitatively changes the nature of the recall process, or merely changes the parameters of the recall competition. We conducted an experiment where participants studied six- or 12-item lists and were post-cued as to whether to retrieve a single item, or as many items as possible. Subsequently, we applied LBA models with various assumptions about primacy and recency, implemented using hierarchical Bayesian techniques. While greater recency was observed when only one item was required for output, the model selection did not suggest that there were qualitative differences between the two conditions. Specifically, start-of-list reinstatement and exponential recency functions were favored in both conditions.

Modeling across-trial variability in the Wald drift rate parameter

The shifted-Wald model is a popular analysis tool for one-choice reaction-time tasks. In its simplest version, the shifted-Wald model assumes a constant trial-independent drift rate parameter. However, the presence of endogenous processes—fluctuation in attention and motivation, fatigue and boredom—suggest that drift rate might vary across experimental trials. Here we show how across-trial variability in drift rate can be accounted for by assuming a trial-specific drift rate parameter that is governed by a positive-valued distribution. We consider two candidate distributions: the truncated normal distribution and the gamma distribution. For the resulting distributions of first-arrival times, we derive analytical and sampling-based solutions, and implement the models in a Bayesian framework. Recovery studies and an application to a data set comprised of 1469 participants suggest that (1) both mixture distributions yield similar results; (2) all model parameters can be recovered accurately except for the drift variance parameter; (3) despite poor recovery, the presence of the drift variance parameter facilitates accurate recovery of the remaining parameters; (4) shift, threshold, and drift mean parameters are correlated.

A flexible framework for simulating and fitting generalized drift-diffusion models

The drift-diffusion model (DDM) is an important decision-making model in cognitive neuroscience. However, innovations in model form have been limited by methodological challenges. Here, we introduce the generalized drift-diffusion model (GDDM) framework for building and fitting DDM extensions, and provide a software package which implements the framework. The GDDM framework augments traditional DDM parameters through arbitrary user-defined functions. Models are solved numerically by directly solving the Fokker-Planck equation using efficient numerical methods, yielding a 100-fold or greater speedup over standard methodology. This speed allows GDDMs to be fit to data using maximum likelihood on the full response time (RT) distribution. We demonstrate fitting of GDDMs within our framework to both animal and human datasets from perceptual decision-making tasks, with better accuracy and fewer parameters than several DDMs implemented using the latest methodology, to test hypothesized decision-making mechanisms. Overall, our framework will allow for decision-making model innovation and novel experimental designs.

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.

Audiovisual detection at different intensities and delays

In the redundant signals task, two target stimuli are associated with the same response. If both targets are presented together, redundancy gains are observed, as compared with single-target presentation. Different models explain these redundancy gains, including race and coactivation models (e.g., the Wiener diffusion superposition model, Schwarz, 1994, Journal of Mathematical Psychology, and the Ornstein Uhlenbeck diffusion superposition model, Diederich, 1995, Journal of Mathematical Psychology). In the present study, two monkeys performed a simple detection task with auditory, visual and audiovisual stimuli of different intensities and onset asynchronies. In its basic form, a Wiener diffusion superposition model provided only a poor description of the observed data, especially of the detection rate (i.e., accuracy or hit rate) for low stimulus intensity. We expanded the model in two ways, by (A) adding a temporal deadline, that is, restricting the evidence accumulation process to a stopping time, and (B) adding a second "nogo" barrier representing target absence. We present closed-form solutions for the mean absorption times and absorption probabilities for a Wiener diffusion process with a drift towards a single barrier in the presence of a temporal deadline (A), and numerically improved solutions for the two-barrier model (B). The best description of the data was obtained from the deadline model and substantially outperformed the two-barrier approach.

Validating a dimension of doubt in decision-making: A proposed endophenotype for obsessive-compulsive disorder

Doubt is subjective uncertainty about one's perceptions and recall. It can impair decision-making and is a prominent feature of obsessive-compulsive disorder (OCD). We propose that evaluation of doubt during decision-making provides a useful endophenotype with which to study the underlying pathophysiology of OCD and potentially other psychopathologies. For the current study, we developed a new instrument, the Doubt Questionnaire, to clinically assess doubt. The random dot motion task was used to measure reaction time and subjective certainty, at varying levels of perceptual difficulty, in individuals who scored high and low on doubt, and in individuals with and without OCD. We found that doubt scores were significantly higher in OCD cases than controls. Drift diffusion modeling revealed that high doubt scores predicted slower evidence accumulation than did low doubt scores; and OCD diagnosis lower than controls. At higher levels of dot coherence, OCD participants exhibited significantly slower drift rates than did controls (q<0.05 for 30%, and 45% coherence; q<0.01 for 70% coherence). In addition, at higher levels of coherence, high doubt subjects exhibited even slower drift rates and reaction times than low doubt subjects (q<0.01 for 70% coherence). Moreover, under high coherence conditions, individuals with high doubt scores reported lower certainty in their decisions than did those with low doubt scores. We conclude that the Doubt Questionnaire is a useful instrument for measuring doubt. Compared to those with low doubt, those with high doubt accumulate evidence more slowly and report lower certainty when making decisions under conditions of low uncertainty. High doubt may affect the decision-making process in individuals with OCD. The dimensional doubt measure is a useful endophenotype for OCD research and could enable computationally rigorous and neurally valid understanding of decision-making and its pathological expression in OCD and other disorders.

Learning and forgetting using reinforced Bayesian change detection

Agents living in volatile environments must be able to detect changes in contingencies while refraining to adapt to unexpected events that are caused by noise. In Reinforcement Learning (RL) frameworks, this requires learning rates that adapt to past reliability of the model. The observation that behavioural flexibility in animals tends to decrease following prolonged training in stable environment provides experimental evidence for such adaptive learning rates. However, in classical RL models, learning rate is either fixed or scheduled and can thus not adapt dynamically to environmental changes. Here, we propose a new Bayesian learning model, using variational inference, that achieves adaptive change detection by the use of Stabilized Forgetting, updating its current belief based on a mixture of fixed, initial priors and previous posterior beliefs. The weight given to these two sources is optimized alongside the other parameters, allowing the model to adapt dynamically to changes in environmental volatility and to unexpected observations. This approach is used to implement the "critic" of an actor-critic RL model, while the actor samples the resulting value distributions to choose which action to undertake. We show that our model can emulate different adaptation strategies to contingency changes, depending on its prior assumptions of environmental stability, and that model parameters can be fit to real data with high accuracy. The model also exhibits trade-offs between flexibility and computational costs that mirror those observed in real data. Overall, the proposed method provides a general framework to study learning flexibility and decision making in RL contexts.

Bayesian Modeling of Motion Perception Using Dynamical Stochastic Textures

A common practice to account for psychophysical biases in vision is to frame them as consequences of a dynamic process relying on optimal inference with respect to a generative model. The study presented here details the complete formulation of such a generative model intended to probe visual motion perception with a dynamic texture model. It is derived in a set of axiomatic steps constrained by biological plausibility. We extend previous contributions by detailing three equivalent formulations of this texture model. First, the composite dynamic textures are constructed by the random aggregation of warped patterns, which can be viewed as three-dimensional gaussian fields. Second, these textures are cast as solutions to a stochastic partial differential equation (sPDE). This essential step enables real-time, on-the-fly texture synthesis using time-discretized autoregressive processes. It also allows for the derivation of a local motion-energy model, which corresponds to the log likelihood of the probability density. The log likelihoods are essential for the construction of a Bayesian inference framework. We use the dynamic texture model to psychophysically probe speed perception in humans using zoom-like changes in the spatial frequency content of the stimulus. The human data replicate previous findings showing perceived speed to be positively biased by spatial frequency increments. A Bayesian observer who combines a gaussian likelihood centered at the true speed and a spatial frequency dependent width with a “slow-speed prior” successfully accounts for the perceptual bias. More precisely, the bias arises from a decrease in the observer's likelihood width estimated from the experiments as the spatial frequency increases. Such a trend is compatible with the trend of the dynamic texture likelihood width.

Adaptive Equilibrium Regulation: Modeling Individual Dynamics on Multiple Timescales

Damped Linear Oscillators estimated by 2nd-order Latent Differential Equation have assumed a constant equilibrium and one oscillatory component. Lower-frequency oscillations may come from seasonal background processes, which non-randomly contribute to deviation from equilibrium at each occasion and confound estimation of dynamics over shorter timescales. Boker (2015) proposed a model of individual change on multiple timescales, but implementation, simulation, and applications to data have not been demonstrated. This study implemented a generalization of the proposed model; examined robustness to varied timescale ratios, measurement error, and occasions-per-person in simulated data; and tested for dynamics at multiple timescales in experience sampling affect data. Results show small standard errors and low bias to dynamic estimates at timescale ratios greater than 3:1. Below 3:1, estimate error was sensitive to noise and total occasions; rates of non-convergence increased. For affect data, model comparisons showed statistically significant dynamics at both timescales for both participants.

An entropic barriers diffusion theory of decision-making in multiple alternative tasks

We present a theory of decision-making in the presence of multiple choices that departs from traditional approaches by explicitly incorporating entropic barriers in a stochastic search process. We analyze response time data from an on-line repository of 15 million blitz chess games, and show that our model fits not just the mean and variance, but the entire response time distribution (over several response-time orders of magnitude) at every stage of the game. We apply the model to show that (a) higher cognitive expertise corresponds to the exploration of more complex solution spaces, and (b) reaction times of users at an on-line buying website can be similarly explained. Our model can be seen as a synergy between diffusion models used to model simple two-choice decision-making and planning agents in complex problem solving.

Decision-making has been studied in great detail relying on binary choices, modeled as the noisy accumulation of a decision variable to a threshold. We show that it breaks down when used to describe real-life human decision involving multiple options. We show instead that including obstacles in the diffusion model (a natural conceptual extension) can describe the data with great degree of accuracy. We evaluate this new model by capitalizing on the advent of big data, analyzing a vast corpus of decision making obtained from on-line chess servers. The present manuscript resolves a conflict between current theories of decision-making and concrete data, it solves this data with a concrete theoretical proposal and analyzes specific predictions of the model.

When uncertain, does human self-motion decision-making fully utilize complete information?

When forced to choose humans often feel uncertain. Investigations of human perceptual decision-making often employ signal detection theory, which assumes that even when uncertain all available information is fully utilized. However, other studies have suggested or assumed that, when uncertain, human subjects guess totally at random, ignoring available information. When uncertain, do humans simply guess totally at random? Or do humans fully utilize complete information? Or does behavior fall between these two extremes yielding "above chance" performance without fully utilizing complete information? While it is often assumed complete information is fully utilized, even when uncertain, to our knowledge this has never been experimentally confirmed. To answer this question, we combined numerical simulations, theoretical analyses, and human studies performed using a self-motion direction-recognition perceptual decision-making task (did I rotate left or right?). Subjects were instructed to make forced-choice binary (left/right) and trinary (left/right/uncertain) decisions when cued following each stimulus. Our results show that humans 1) do not guess at random when uncertain and 2) make binary and trinary decisions equally well. These findings show that humans fully utilize complete information when uncertain for our perceptual decision-making task. This helps unify signal detection theory and other models of forced-choice decision-making which allow for uncertain responses. NEW & NOTEWORTHY Humans make many perceptual decisions every day. But what if we are uncertain? While many studies assume that humans fully utilize complete information, other studies have suggested and/or assumed that when we're uncertain and forced to decide, information is not fully utilized. While humans tend to perform above chance when uncertain, no earlier study has tested whether available information is fully utilized. Our results show that humans make fully informed decisions even when uncertain.

The determinants of response time in a repeated constant-sum game: A robust Bayesian hierarchical dual-process model

The investigation of response time and behavior has a long tradition in cognitive psychology, particularly for non-strategic decision-making. Recently, experimental economists have also studied response time in strategic interactions, but with an emphasis on either one-shot games or repeated social-dilemmas. I investigate the determinants of response time in a repeated (pure-conflict) game, admitting a unique mixed strategy Nash equilibrium, with fixed partner matching. Response times depend upon the interaction of two decision models embedded in a dual-process framework (Achtziger and Alós-Ferrer, 2014; Alós-Ferrer, 2016). The first decision model is the commonly used win-stay/lose-shift heuristic and the second the pattern-detecting reinforcement learning model in Spiliopoulos (2013b). The former is less complex and can be executed more quickly than the latter. As predicted, conflict between these two models (i.e., each one recommending a different course of action) led to longer response times than cases without conflict. The dual-process framework makes other qualitative response time predictions arising from the interaction between the existence (or not) of conflict and which one of the two decision models the chosen action is consistent with-these were broadly verified by the data. Other determinants of RT were hypothesized on the basis of existing theory and tested empirically. Response times were strongly dependent on the actions chosen by both players in the previous rounds and the resulting outcomes. Specifically, response time was shortest after a win in the previous round where the maximum possible payoff was obtained; response time after losses was significantly longer. Strongly auto-correlated behavior (regardless of its sign) was also associated with longer response times. I conclude that, similar to other tasks, there is a strong coupling in repeated games between behavior and RT, which can be exploited to further our understanding of decision making.

The elimination of positive priming with increasing prime duration reflects a transition from perceptual fluency to disfluency rather than bias against primed words

With immediate repetition priming of forced choice perceptual identification, short prime durations produce positive priming (i.e., priming the target leads to higher accuracy, while priming the foil leads to lower accuracy). Many theories explain positive priming following short duration primes as reflecting increased perceptual fluency for the primed target (i.e., decreased identification latency). However, most studies only examine either accuracy or response times, rather than considering the joint constraints of response times and accuracy to properly address the role of decision biases and response caution. This is a critical oversight because several theories propose that the transition to negative priming following a long duration prime reflects a decision strategy to compensate for the effect of increased perceptual fluency. In contrast, the nROUSE model of Huber and O'Reilly (2003) explains this transition as reflecting perceptual habituation, and thus a change to perceptual disfluency. We confirmed this prediction by applying a sequential sampling model (the diffusion race model) to accuracy and response time distributions from a new single item same-different version of the priming task. In this way, we measured strategic biases and perceptual fluency in each condition for each subject. The nROUSE model was only applied to accuracy from the original forced-choice version of the priming task. This application of nROUSE produced separate predictions for each subject regarding the degree of fluency and disfluency in each condition, and these predictions were confirmed by the drift rate parameters (i.e., fluency) from the response time model in contrast to the threshold parameters (i.e., bias).

Perceptual Decisions in the Presence of Relevant and Irrelevant Sensory Evidence

Perceptual decisions in the presence of decision-irrelevant sensory information require a selection of decision-relevant sensory evidence. To characterize the mechanism that is responsible for separating decision-relevant from irrelevant sensory information we asked human subjects to make judgments about one of two simultaneously present motion components in a random dot stimulus. Subjects were able to ignore the decision-irrelevant component to a large degree, but their decisions were still influenced by the irrelevant sensory information. Computational modeling revealed that this influence was not simply the consequence of subjects forgetting at times which stimulus component they had been instructed to base their decision on. Instead, residual irrelevant information always seems to be leaking through, and the decision process is captured by a net sensory evidence signal being accumulated to a decision threshold. This net sensory evidence is a linear combination of decision-relevant and irrelevant sensory information. The selection process is therefore well-described by a strong linear gain modulation, which, in our experiment, resulted in the relevant sensory evidence having at least 10 times more impact on the decision than the irrelevant evidence.

The computations that support simple decision-making: A comparison between the diffusion and urgency-gating models

We investigate a question relevant to the psychology and neuroscience of perceptual decision-making: whether decisions are based on steadily accumulating evidence, or only on the most recent evidence. We report an empirical comparison between two of the most prominent examples of these theoretical positions, the diffusion model and the urgency-gating model, via model-based qualitative and quantitative comparisons. Our findings support the predictions of the diffusion model over the urgency-gating model, and therefore, the notion that evidence accumulates without much decay. Gross qualitative patterns and fine structural details of the data are inconsistent with the notion that decisions are based only on the most recent evidence. More generally, we discuss some strengths and weaknesses of scientific methods that investigate quantitative models by distilling the formal models to qualitative predictions.

The BCD of response time analysis in experimental economics

For decisions in the wild, time is of the essence. Available decision time is often cut short through natural or artificial constraints, or is impinged upon by the opportunity cost of time. Experimental economists have only recently begun to conduct experiments with time constraints and to analyze response time (RT) data, in contrast to experimental psychologists. RT analysis has proven valuable for the identification of individual and strategic decision processes including identification of social preferences in the latter case, model comparison/selection, and the investigation of heuristics that combine speed and performance by exploiting environmental regularities. Here we focus on the benefits, challenges, and desiderata of RT analysis in strategic decision making. We argue that unlocking the potential of RT analysis requires the adoption of process-based models instead of outcome-based models, and discuss how RT in the wild can be captured by time-constrained experiments in the lab. We conclude that RT analysis holds considerable potential for experimental economics, deserves greater attention as a methodological tool, and promises important insights on strategic decision making in naturally occurring environments.

Learning to allocate limited time to decisions with different expected outcomes

The goal of this article is to investigate how human participants allocate their limited time to decisions with different properties. We report the results of two behavioral experiments. In each trial of the experiments, the participant must accumulate noisy information to make a decision. The participants received positive and negative rewards for their correct and incorrect decisions, respectively. The stimulus was designed such that decisions based on more accumulated information were more accurate but took longer. Therefore, the total outcome that a participant could achieve during the limited experiments' time depended on her "decision threshold", the amount of information she needed to make a decision. In the first experiment, two types of trials were intermixed randomly: hard and easy. Crucially, the hard trials were associated with smaller positive and negative rewards than the easy trials. A cue presented at the beginning of each trial would indicate the type of the upcoming trial. The optimal strategy was to adopt a small decision threshold for hard trials. The results showed that several of the participants did not learn this simple strategy. We then investigated how the participants adjusted their decision threshold based on the feedback they received in each trial. To this end, we developed and compared 10 computational models for adjusting the decision threshold. The models differ in their assumptions on the shape of the decision thresholds and the way the feedback is used to adjust the decision thresholds. The results of Bayesian model comparison showed that a model with time-varying thresholds whose parameters are updated by a reinforcement learning algorithm is the most likely model. In the second experiment, the cues were not presented. We showed that the optimal strategy is to use a single time-decreasing decision threshold for all trials. The results of the computational modeling showed that the participants did not use this optimal strategy. Instead, they attempted to detect the difficulty of the trial first and then set their decision threshold accordingly.

A martingale analysis of first passage times of time-dependent Wiener diffusion models

Research in psychology and neuroscience has successfully modeled decision making as a process of noisy evidence accumulation to a decision bound. While there are several variants and implementations of this idea, the majority of these models make use of a noisy accumulation between two absorbing boundaries. A common assumption of these models is that decision parameters, e.g., the rate of accumulation (drift rate), remain fixed over the course of a decision, allowing the derivation of analytic formulas for the probabilities of hitting the upper or lower decision threshold, and the mean decision time. There is reason to believe, however, that many types of behavior would be better described by a model in which the parameters were allowed to vary over the course of the decision process. In this paper, we use martingale theory to derive formulas for the mean decision time, hitting probabilities, and first passage time (FPT) densities of a Wiener process with time-varying drift between two time-varying absorbing boundaries. This model was first studied by Ratcliff (1980) in the two-stage form, and here we consider the same model for an arbitrary number of stages (i.e. intervals of time during which parameters are constant). Our calculations enable direct computation of mean decision times and hitting probabilities for the associated multistage process. We also provide a review of how martingale theory may be used to analyze similar models employing Wiener processes by re-deriving some classical results. In concert with a variety of numerical tools already available, the current derivations should encourage mathematical analysis of more complex models of decision making with time-varying evidence.

How numbers mean: Comparing random walk models of numerical cognition varying both encoding processes and underlying quantity representations

How do people derive meaning from numbers? Here, we instantiate the primary theories of numerical representation in computational models and compare simulated performance to human data. Specifically, we fit simulated data to the distributions for correct and incorrect responses, as well as the pattern of errors made, in a traditional "relative quantity" task. The results reveal that no current theory of numerical representation can adequately account for the data without additional assumptions. However, when we introduce repeated, error-prone sampling of the stimulus (e.g., Cohen, 2009) superior fits are achieved when the underlying representation of integers reflects linear spacing with constant variance. These results provide new insights into (i) the detailed nature of mental numerical representation, and, (ii) general perceptual processes implemented by the human visual system.

Global gain modulation generates time-dependent urgency during perceptual choice in humans

Decision-makers must often balance the desire to accumulate information with the costs of protracted deliberation. Optimal, reward-maximizing decision-making can require dynamic adjustment of this speed/accuracy trade-off over the course of a single decision. However, it is unclear whether humans are capable of such time-dependent adjustments. Here, we identify several signatures of time-dependency in human perceptual decision-making and highlight their possible neural source. Behavioural and model-based analyses reveal that subjects respond to deadline-induced speed pressure by lowering their criterion on accumulated perceptual evidence as the deadline approaches. In the brain, this effect is reflected in evidence-independent urgency that pushes decision-related motor preparation signals closer to a fixed threshold. Moreover, we show that global modulation of neural gain, as indexed by task-related fluctuations in pupil diameter, is a plausible biophysical mechanism for the generation of this urgency. These findings establish context-sensitive time-dependency as a critical feature of human decision-making.

Decision-making balances the benefits of additional information with the cost of time, but it is unclear whether humans adjust this balance within individual decisions. Here, authors show that we do make such adjustments to suit contextual demands and suggest that these are driven by modulation of neural gain.

The attention-weighted sample-size model of visual short-term memory: Attention capture predicts resource allocation and memory load

We investigated the capacity of visual short-term memory (VSTM) in a phase discrimination task that required judgments about the configural relations between pairs of black and white features. Sewell et al. (2014) previously showed that VSTM capacity in an orientation discrimination task was well described by a sample-size model, which views VSTM as a resource comprised of a finite number of noisy stimulus samples. The model predicts the invariance of [Formula: see text] , the sum of squared sensitivities across items, for displays of different sizes. For phase discrimination, the set-size effect significantly exceeded that predicted by the sample-size model for both simultaneously and sequentially presented stimuli. Instead, the set-size effect and the serial position curves with sequential presentation were predicted by an attention-weighted version of the sample-size model, which assumes that one of the items in the display captures attention and receives a disproportionate share of resources. The choice probabilities and response time distributions from the task were well described by a diffusion decision model in which the drift rates embodied the assumptions of the attention-weighted sample-size model.

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.

Collective Activity of Many Bistable Assemblies Reproduces Characteristic Dynamics of Multistable Perception

The timing of perceptual decisions depends on both deterministic and stochastic factors, as the gradual accumulation of sensory evidence (deterministic) is contaminated by sensory and/or internal noise (stochastic). When human observers view multistable visual displays, successive episodes of stochastic accumulation culminate in repeated reversals of visual appearance. Treating reversal timing as a "first-passage time" problem, we ask how the observed timing densities constrain the underlying stochastic accumulation. Importantly, mean reversal times (i.e., deterministic factors) differ enormously between displays/observers/stimulation levels, whereas the variance and skewness of reversal times (i.e., stochastic factors) keep characteristic proportions of the mean. What sort of stochastic process could reproduce this highly consistent "scaling property?" Here we show that the collective activity of a finite population of bistable units (i.e., a generalized Ehrenfest process) quantitatively reproduces all aspects of the scaling property of multistable phenomena, in contrast to other processes under consideration (Poisson, Wiener, or Ornstein-Uhlenbeck process). The postulated units express the spontaneous dynamics of attractor assemblies transitioning between distinct activity states. Plausible candidates are cortical columns, or clusters of columns, as they are preferentially connected and spontaneously explore a restricted repertoire of activity states. Our findings suggests that perceptual representations are granular, probabilistic, and operate far from equilibrium, thereby offering a suitable substrate for statistical inference.

SIGNIFICANCE STATEMENT

Spontaneous reversals of high-level perception, so-called multistable perception, conform to highly consistent and characteristic statistics, constraining plausible neural representations. We show that the observed perceptual dynamics would be reproduced quantitatively by a finite population of distinct neural assemblies, each with locally bistable activity, operating far from the collective equilibrium (generalized Ehrenfest process). Such a representation would be consistent with the intrinsic stochastic dynamics of neocortical activity, which is dominated by preferentially connected assemblies, such as cortical columns or clusters of columns. We predict that local neuron assemblies will express bistable dynamics, with spontaneous active-inactive transitions, whenever they contribute to high-level perception.

Limited information estimation of the diffusion-based item response theory model for responses and response times

Psychological tests are usually analysed with item response models. Recently, some alternative measurement models have been proposed that were derived from cognitive process models developed in experimental psychology. These models consider the responses but also the response times of the test takers. Two such models are the Q-diffusion model and the D-diffusion model. Both models can be calibrated with the diffIRT package of the R statistical environment via marginal maximum likelihood (MML) estimation. In this manuscript, an alternative approach to model calibration is proposed. The approach is based on weighted least squares estimation and parallels the standard estimation approach in structural equation modelling. Estimates are determined by minimizing the discrepancy between the observed and the implied covariance matrix. The estimator is simple to implement, consistent, and asymptotically normally distributed. Least squares estimation also provides a test of model fit by comparing the observed and implied covariance matrix. The estimator and the test of model fit are evaluated in a simulation study. Although parameter recovery is good, the estimator is less efficient than the MML estimator.

Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

Consider a one-dimensional diffusion process on the diffusion interval I originated in x0 ∈ I. Let a(t) and b(t) be two continuous functions of t, t > t0, with bounded derivatives, a(t) < b(t), and a(t), b(t) ∈ I, for all t > t0. We study the joint distribution of the two random variables Ta and Tb, the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of Ta and Tb in terms of ℙ(Ta < t, Ta < Tb) and ℙ(Tb < t, Ta > Tb), and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.

Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

Consider a one-dimensional diffusion process on the diffusion interval I originated in x 0 ∈ I. Let a(t) and b(t) be two continuous functions of t, t &gt; t 0, with bounded derivatives, a(t) &lt; b(t), and a(t), b(t) ∈ I, for all t &gt; t 0. We study the joint distribution of the two random variables T a and T b , the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of T a and T b in terms of ℙ(T a &lt; t, T a &lt; T b ) and ℙ(T b &lt; t, T a &gt; T b ), and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.

Dynamics of individual perceptual decisions

Perceptual decision making is fundamental to a broad range of fields including neurophysiology, economics, medicine, advertising, law, etc. Although recent findings have yielded major advances in our understanding of perceptual decision making, decision making as a function of time and frequency (i.e., decision-making dynamics) is not well understood. To limit the review length, we focus most of this review on human findings. Animal findings, which are extensively reviewed elsewhere, are included when beneficial or necessary. We attempt to put these various findings and data sets, which can appear to be unrelated in the absence of a formal dynamic analysis, into context using published models. Specifically, by adding appropriate dynamic mechanisms (e.g., high-pass filters) to existing models, it appears that a number of otherwise seemingly disparate findings from the literature might be explained. One hypothesis that arises through this dynamic analysis is that decision making includes phasic (high pass) neural mechanisms, an evidence accumulator and/or some sort of midtrial decision-making mechanism (e.g., peak detector and/or decision boundary).

Sequential Sampling Models in Cognitive Neuroscience: Advantages, Applications, and Extensions

Sequential sampling models assume that people make speeded decisions by gradually accumulating noisy information until a threshold of evidence is reached. In cognitive science, one such model--the diffusion decision model--is now regularly used to decompose task performance into underlying processes such as the quality of information processing, response caution, and a priori bias. In the cognitive neurosciences, the diffusion decision model has recently been adopted as a quantitative tool to study the neural basis of decision making under time pressure. We present a selective overview of several recent applications and extensions of the diffusion decision model in the cognitive neurosciences.

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.

Discriminating evidence accumulation from urgency signals in speeded decision making

The dominant theoretical paradigm in explaining decision making throughout both neuroscience and cognitive science is known as “evidence accumulation”--The core idea being that decisions are reached by a gradual accumulation of noisy information. Although this notion has been supported by hundreds of experiments over decades of study, a recent theory proposes that the fundamental assumption of evidence accumulation requires revision. The "urgency gating" model assumes decisions are made without accumulating evidence, using only moment-by-moment information. Under this assumption, the successful history of evidence accumulation models is explained by asserting that the two models are mathematically identical in standard experimental procedures. We demonstrate that this proof of equivalence is incorrect, and that the models are not identical, even when both models are augmented with realistic extra assumptions. We also demonstrate that the two models can be perfectly distinguished in realistic simulated experimental designs, and in two real data sets; the evidence accumulation model provided the best account for one data set, and the urgency gating model for the other. A positive outcome is that the opposing modeling approaches can be fruitfully investigated without wholesale change to the standard experimental paradigms. We conclude that future research must establish whether the urgency gating model enjoys the same empirical support in the standard experimental paradigms that evidence accumulation models have gathered over decades of study.

The Lognormal Race: A Cognitive-Process Model of Choice and Latency with Desirable Psychometric Properties

We present a cognitive process model of response choice and response time performance data that has excellent psychometric properties and may be used in a wide variety of contexts. In the model there is an accumulator associated with each response option. These accumulators have bounds, and the first accumulator to reach its bound determines the response time and response choice. The times at which accumulator reaches its bound is assumed to be lognormally distributed, hence the model is race or minima process among lognormal variables. A key property of the model is that it is relatively straightforward to place a wide variety of models on the logarithm of these finishing times including linear models, structural equation models, autoregressive models, growth-curve models, etc. Consequently, the model has excellent statistical and psychometric properties and can be used in a wide range of contexts, from laboratory experiments to high-stakes testing, to assess performance. We provide a Bayesian hierarchical analysis of the model, and illustrate its flexibility with an application in testing and one in lexical decision making, a reading skill.