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

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

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

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

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

A two-layered diffusion model traces the dynamics of information processing in the valuation-and-choice circuit of decision making

A circuit of evaluation and selection of the alternatives is considered a reliable model in neurobiology. The prominent contributions of the literature to this topic are reported. In this study, valuation and choice of a decisional process during Two-Alternative Forced-Choice (TAFC) task are represented as a two-layered network of computational cells, where information accrual and processing progress in nonlinear diffusion dynamics. The evolution of the response-to-stimulus map is thus modeled by two linked diffusive modules (2LDM) representing the neuronal populations involved in the valuation-and-decision circuit of decision making. Diffusion models are naturally appropriate for describing accumulation of evidence over the time. This allows the computation of the response times (RTs) in valuation and choice, under the hypothesis of ex-Wald distribution. A nonlinear transfer function integrates the activities of the two layers. The input-output map based on the infomax principle makes the 2LDM consistent with the reinforcement learning approach. Results from simulated likelihood time series indicate that 2LDM may account for the activity-dependent modulatory component of effective connectivity between the neuronal populations. Rhythmic fluctuations of the estimate gain functions in the delta-beta bands also support the compatibility of 2LDM with the neurobiology of DM.

When learners surpass their models: mathematical modeling of learning from an inconsistent source

It has been reported in the literature that both adults and children can, to a different degree, modify and regularize the often-inconsistent linguistic input they receive. We present a new algorithm to model and investigate the learning process of a learner mastering a set of (grammatical or lexical) forms from an inconsistent source. The algorithm is related to reinforcement learning and drift-diffusion models of decision making, and possesses several psychologically relevant properties such as fidelity, robustness, discounting, and computational simplicity. It demonstrates how a learner can successfully learn from or even surpass its imperfect source. We use the data collected by Singleton and Newport (Cognit Psychol 49(4):370-407, 2004) on the performance of a 7-year-boy Simon, who mastered the American Sign Language (ASL) by learning it from his parents, both of whom were imperfect speakers of ASL. We show that the algorithm possesses a frequency boosting property, whereby the frequency of the most common form of the source is increased by the learner. We also explain several key features of Simon's ASL.

Optimal multisensory decision-making in a reaction-time task

Humans and animals can integrate sensory evidence from various sources to make decisions in a statistically near-optimal manner, provided that the stimulus presentation time is fixed across trials. Little is known about whether optimality is preserved when subjects can choose when to make a decision (reaction-time task), nor when sensory inputs have time-varying reliability. Using a reaction-time version of a visual/vestibular heading discrimination task, we show that behavior is clearly sub-optimal when quantified with traditional optimality metrics that ignore reaction times. We created a computational model that accumulates evidence optimally across both cues and time, and trades off accuracy with decision speed. This model quantitatively explains subjects's choices and reaction times, supporting the hypothesis that subjects do, in fact, accumulate evidence optimally over time and across sensory modalities, even when the reaction time is under the subject's control.

Learning to maximize reward rate: a model based on semi-Markov decision processes

WHEN ANIMALS HAVE TO MAKE A NUMBER OF DECISIONS DURING A LIMITED TIME INTERVAL, THEY FACE A FUNDAMENTAL PROBLEM: how much time they should spend on each decision in order to achieve the maximum possible total outcome. Deliberating more on one decision usually leads to more outcome but less time will remain for other decisions. In the framework of sequential sampling models, the question is how animals learn to set their decision threshold such that the total expected outcome achieved during a limited time is maximized. The aim of this paper is to provide a theoretical framework for answering this question. To this end, we consider an experimental design in which each trial can come from one of the several possible "conditions." A condition specifies the difficulty of the trial, the reward, the penalty and so on. We show that to maximize the expected reward during a limited time, the subject should set a separate value of decision threshold for each condition. We propose a model of learning the optimal value of decision thresholds based on the theory of semi-Markov decision processes (SMDP). In our model, the experimental environment is modeled as an SMDP with each "condition" being a "state" and the value of decision thresholds being the "actions" taken in those states. The problem of finding the optimal decision thresholds then is cast as the stochastic optimal control problem of taking actions in each state in the corresponding SMDP such that the average reward rate is maximized. Our model utilizes a biologically plausible learning algorithm to solve this problem. The simulation results show that at the beginning of learning the model choses high values of decision threshold which lead to sub-optimal performance. With experience, however, the model learns to lower the value of decision thresholds till finally it finds the optimal values.

Decision processes in temporal discrimination

The processing dynamics underlying temporal decisions and the response times they generate have received little attention in the study of interval timing. In contrast, models of other simple forms of decision making have been extensively investigated using response times, leading to a substantial disconnect between temporal and non-temporal decision theories. An overarching decision-theoretic framework that encompasses existing, non-temporal decision models may, however, account both for interval timing itself and for time-based decision-making. We sought evidence for this framework in the temporal discrimination performance of humans tested on the temporal bisection task. In this task, participants retrospectively categorized experienced stimulus durations as short or long based on their perceived similarity to two, remembered reference durations and were rewarded only for correct categorization of these references. Our analysis of choice proportions and response times suggests that a two-stage, sequential diffusion process, parameterized to maximize earned rewards, can account for salient patterns of bisection performance. The first diffusion stage times intervals by accumulating an endogenously noisy clock signal; the second stage makes decisions about the first-stage temporal representation by accumulating first-stage evidence corrupted by endogenous noise. Reward-maximization requires that the second-stage accumulation rate and starting point be based on the state of the first-stage timer at the end of the stimulus duration, and that estimates of non-decision-related delays should decrease as a function of stimulus duration. Results are in accord with these predictions and thus support an extension of the drift-diffusion model of static decision making to the domain of interval timing and temporal decisions.

Relation between belief and performance in perceptual decision making

In an uncertain and ambiguous world, effective decision making requires that subjects form and maintain a belief about the correctness of their choices, a process called meta-cognition. Prediction of future outcomes and self-monitoring are only effective if belief closely matches behavioral performance. Equality between belief and performance is also critical for experimentalists to gain insight into the subjects' belief by simply measuring their performance. Assuming that the decision maker holds the correct model of the world, one might indeed expect that belief and performance should go hand in hand. Unfortunately, we show here that this is rarely the case when performance is defined as the percentage of correct responses for a fixed stimulus, a standard definition in psychophysics. In this case, belief equals performance only for a very narrow family of tasks, whereas in others they will only be very weakly correlated. As we will see it is possible to restore this equality in specific circumstances but this remedy is only effective for a decision-maker, not for an experimenter. We furthermore show that belief and performance do not match when conditioned on task difficulty--as is common practice when plotting the psychometric curve--highlighting common pitfalls in previous neuroscience work. Finally, we demonstrate that miscalibration and the hard-easy effect observed in humans' and other animals' certainty judgments could be explained by a mismatch between the experimenter's and decision maker's expected distribution of task difficulties. These results have important implications for experimental design and are of relevance for theories that aim to unravel the nature of meta-cognition.

Dynamics of multiple-choice decision making

Neuroscientists have carried out comprehensive experiments to reveal the neural mechanisms underlying the perceptual decision making that pervades daily life. These experiments have illuminated salient features of decision making, including probabilistic choice behavior, the ramping activity of decision-related neurons, and the dependence of decision time and accuracy on the difficulty of the task. Spiking network models have reproduced these features, and a two-dimensional mean field model has demonstrated that the saddle node structure underlies two-alternative decision making. Here, we reduced a spiking network model to an analytically tractable, partial integro-differential system and characterized not only multiple-choice decision behaviors but also the time course of neural activities underlying decisions, providing a mechanistic explanation for the observations noted in the experiments. First, we observed that a two-bump unstable steady state of the system is responsible for two-choice decision making, similar to the saddle node structure in the two-dimensional mean field model. However, for four-choice decision making, three types of unstable steady states collectively predominate the time course of the evolution from the initial state to the stable states. Second, the time constant of the unstable steady state can explain the fact that four-choice decision making requires a longer time than two-choice decision making. However, the quicker decision, given a stronger motion strength, cannot be explained by the time constant of the unstable steady state. Rather, the decision time can be attributed to the projection coefficient of the difference between the initial state and the unstable steady state on the eigenvector corresponding to the largest positive eigenvalue.

Unreliable gut feelings can lead to correct decisions: the somatic marker hypothesis in non-linear decision chains

Dual-process approaches of decision-making examine the interaction between affective/intuitive and deliberative processes underlying value judgment. From this perspective, decisions are supported by a combination of relatively explicit capabilities for abstract reasoning and relatively implicit evolved domain-general as well as learned domain-specific affective responses. One such approach, the somatic markers hypothesis (SMH), expresses these implicit processes as a system of evolved primary emotions supplemented by associations between affect and experience that accrue over lifetime, or somatic markers. In this view, somatic markers are useful only if their local capability to predict the value of an action is above a baseline equal to the predictive capability of the combined rational and primary emotional subsystems. We argue that decision-making has often been conceived of as a linear process: the effect of decision sequences is additive, local utility is cumulative, and there is no strong environmental feedback. This widespread assumption can have consequences for answering questions regarding the relative weight between the systems and their interaction within a cognitive architecture. We introduce a mathematical formalization of the SMH and study it in situations of dynamic, non-linear decision chains using a discrete-time stochastic model. We find, contrary to expectations, that decision-making events can interact non-additively with the environment in apparently paradoxical ways. We find that in non-lethal situations, primary emotions are represented globally over and above their local weight, showing a tendency for overcautiousness in situated decision chains. We also show that because they tend to counteract this trend, poorly attuned somatic markers that by themselves do not locally enhance decision-making, can still produce an overall positive effect. This result has developmental and evolutionary implications since, by promoting exploratory behavior, somatic markers would seem to be beneficial even at early stages when experiential attunement is poor. Although the model is formulated in terms of the SMH, the implications apply to dual systems theories in general since it makes minimal assumptions about the nature of the processes involved.

Decision making and confidence given uncertain advice

We study human decision making in a simple forced-choice task that manipulates the frequency and accuracy of available information. Empirically, we find that people make decisions consistent with the advice provided, but that their subjective confidence in their decisions shows 2 interesting properties. First, people's confidence does not depend solely on the accuracy of the advice. Rather, confidence seems to be influenced by both the frequency and accuracy of the advice. Second, people are less confident in their guessed decisions when they have to make relatively more of them. Theoretically, we develop and evaluate a type of sequential sampling process model-known as a self-regulating accumulator-that accounts for both decision making and confidence. The model captures the regularities in people's behavior with interpretable parameter values, and we show its ability to fit the data is not due to excessive model complexity. Using the model, we draw conclusions about some properties of human reasoning under uncertainty.

Is saccade averaging determined by visual processing or movement planning?

Saccadic averaging that causes subjects' gaze to land between the location of two targets when faced with simultaneously or sequentially presented stimuli has been often used as a probe to investigate the nature of computations that transform sensory representations into an oculomotor plan. Since saccadic movements involve at least two processing stages-a visual stage that selects a target and a movement stage that prepares the response-saccade averaging can either occur due to interference in visual processing or movement planning. By having human subjects perform two versions of a saccadic double-step task, in which the stimuli remained the same, but different instructions were provided (REDIRECT gaze to the later-appearing target vs. FOLLOW the sequence of targets in their order of appearance), we tested two alternative hypotheses. If saccade averaging were due to visual processing alone, the pattern of saccade averaging is expected to remain the same across task conditions. However, whereas subjects produced averaged saccades between two targets in the FOLLOW condition, they produced hypometric saccades in the direction of the initial target in the REDIRECT condition, suggesting that the interaction between competing movement plans produces saccade averaging.

Accounting for speed-accuracy tradeoff in perceptual learning

In the perceptual learning (PL) literature, researchers typically focus on improvements in accuracy, such as d'. In contrast, researchers who investigate the practice of cognitive skills focus on improvements in response times (RT). Here, we argue for the importance of accounting for both accuracy and RT in PL experiments, due to the phenomenon of speed-accuracy tradeoff (SAT): at a given level of discriminability, faster responses tend to produce more errors. A formal model of the decision process, such as the diffusion model, can explain the SAT. In this model, a parameter known as the drift rate represents the perceptual strength of the stimulus, where higher drift rates lead to more accurate and faster responses. We applied the diffusion model to analyze responses from a yes-no coherent motion detection task. The results indicate that observers do not use a fixed threshold for evidence accumulation, so changes in the observed accuracy may not provide the most appropriate estimate of learning. Instead, our results suggest that SAT can be accounted for by a modeling approach, and that drift rates offer a promising index of PL.

The cost of accumulating evidence in perceptual decision making

Decision making often involves the accumulation of information over time, but acquiring information typically comes at a cost. Little is known about the cost incurred by animals and humans for acquiring additional information from sensory variables due, for instance, to attentional efforts. Through a novel integration of diffusion models and dynamic programming, we were able to estimate the cost of making additional observations per unit of time from two monkeys and six humans in a reaction time (RT) random-dot motion discrimination task. Surprisingly, we find that the cost is neither zero nor constant over time, but for the animals and humans features a brief period in which it is constant but increases thereafter. In addition, we show that our theory accurately matches the observed reaction time distributions for each stimulus condition, the time-dependent choice accuracy both conditional on stimulus strength and independent of it, and choice accuracy and mean reaction times as a function of stimulus strength. The theory also correctly predicts that urgency signals in the brain should be independent of the difficulty, or stimulus strength, at each trial.

Cortical topography of intracortical inhibition influences the speed of decision making

The neocortex contains orderly topographic maps; however, their functional role remains controversial. Theoretical studies have suggested a role in minimizing computational costs, whereas empirical studies have focused on spatial localization. Using a tactile multiple-choice reaction time (RT) task before and after the induction of perceptual learning through repetitive sensory stimulation, we extend the framework of cortical topographies by demonstrating that the topographic arrangement of intracortical inhibition contributes to the speed of human perceptual decision-making processes. RTs differ among fingers, displaying an inverted U-shaped function. Simulations using neural fields show the inverted U-shaped RT distribution as an emergent consequence of lateral inhibition. Weakening inhibition through learning shortens RTs, which is modeled through topographically reorganized inhibition. Whereas changes in decision making are often regarded as an outcome of higher cortical areas, our data show that the spatial layout of interaction processes within representational maps contributes to selection and decision-making processes.

Evaluating the unequal-variance and dual-process explanations of zROC slopes with response time data and the diffusion model

We tested two explanations for why the slope of the z-transformed receiver operating characteristic (zROC) is less than 1 in recognition memory: the unequal-variance account (target evidence is more variable than lure evidence) and the dual-process account (responding reflects both a continuous familiarity process and a threshold recollection process). These accounts are typically implemented in signal detection models that do not make predictions for response time (RT) data. We tested them using RT data and the diffusion model. Participants completed multiple study/test blocks of an "old"/"new" recognition task with the proportion of targets and the test varying from block to block (.21, .32, .50, .68, or .79 targets). The same participants completed sessions with both speed-emphasis and accuracy-emphasis instructions. zROC slopes were below one for both speed and accuracy sessions, and they were slightly lower for speed. The extremely fast pace of the speed sessions (mean RT=526) should have severely limited the role of the slower recollection process relative to the fast familiarity process. Thus, the slope results are not consistent with the idea that recollection is responsible for slopes below 1. The diffusion model was able to match the empirical zROC slopes and RT distributions when between-trial variability in memory evidence was greater for targets than for lures, but missed the zROC slopes when target and lure variability were constrained to be equal. Therefore, unequal variability in continuous evidence is supported by RT modeling in addition to signal detection modeling. Finally, we found that a two-choice version of the RTCON model could not accommodate the RT distributions as successfully as the diffusion model.

What can the diffusion model tell us about prospective memory?

Cognitive process models, such as Ratcliff's (1978) diffusion model, are useful tools for examining cost or interference effects in event-based prospective memory (PM). The diffusion model includes several parameters that provide insight into how and why ongoing-task performance may be affected by a PM task and is ideally suited to analyse performance because both reaction time and accuracy are taken into account. Separate analyses of these measures can easily yield misleading interpretations in cases of speed-accuracy trade-offs. The diffusion model allows us to measure possible criterion shifts and is thus an important methodological improvement over standard analyses. Performance in an ongoing lexical-decision task was analysed with the diffusion model. The results suggest that criterion shifts play an important role when a PM task is added, but do not fully explain the cost effect on reaction time.

Diffusive information accumulation by minimal recurrent neural models of decision making

An important class of psychological models of decision making assumes that evidence is accumulated by a diffusion process to a response criterion. These models have successfully accounted for reaction time (RT) distributions and choice probabilities from a wide variety of experimental tasks. An outstanding theoretical problem is how the integration process that underlies diffusive evidence accumulation can be realized neurally. Wang ( 2001 , 2002 ) has suggested that long timescale neural integration may be implemented by persistent activity in reverberation loops. We analyze a simple recurrent decision making architecture and show that it leads to a diffusive accumulation process. The process has the form of a time-inhomogeneous Ornstein-Uhlenbeck velocity process with linearly increasing drift and diffusion coefficients. The resulting model predicts RT distributions and choice probabilities that closely approximate those found in behavioral data.

Inhibition in superior colliculus neurons in a brightness discrimination task?

Simultaneous recordings were collected from between two and four buildup neurons from the left and right superior colliculi in rhesus monkeys in a simple two-choice brightness discrimination task. The monkeys were required to move their eyes to one of two response targets to indicate their decision. Neurons were identified whose receptive fields were centered on the response targets. The functional role of inhibition was examined by conditionalizing firing rate on a high versus low rate in target neurons 90 ms to 30 ms before the saccade and examining the firing rate in both contralateral and ipsilateral neurons. Two models with racing diffusion processes were fit to the behavioral data, and the same analysis was performed on simulated paths in the diffusion processes that have been found to represent firing rate. The results produce converging evidence for the lack of a functional role for inhibition between neural populations corresponding to the two decisions.

Predicting the timing of wrong decisions with LATER

Response time, or latency, is increasingly being used to provide information about neural decision processes. LATER (Linear Approach to Threshold with Ergodic Rate) is a quasi-Bayesian model of decision-making, with the additional feature that it introduces a degree of gratuitous randomisation into the decision process. It has had some success in predicting latencies under various conditions, but has not specifically been applied to an equally important aspect of decision-making, namely errors: a complete model of decision-making should not only account for latency distributions of correct decisions but also of wrong ones. We therefore used a decision task that generates large numbers of errors: subjects are told to look at suddenly appearing targets of one colour, but not another. We found that subjects' faster responses are as likely to be correct as wrong, but eventually the latency distributions diverge, with errors becoming infrequent. It seems that colour information, arriving after a delay, results both in cancellation of the developing response to the mere existence of the target and in delayed initiation of the correct response. A simple model, using LATER units in a similar way to one that has previously successfully modelled countermanding, accurately predicts latency distributions and proportions of all responses, whether correct or incorrect, demonstrating that the LATER model can indeed account for errors as well as correct responses.

Neurally constrained modeling of perceptual decision making

Stochastic accumulator models account for response time in perceptual decision-making tasks by assuming that perceptual evidence accumulates to a threshold. The present investigation mapped the firing rate of frontal eye field (FEF) visual neurons onto perceptual evidence and the firing rate of FEF movement neurons onto evidence accumulation to test alternative models of how evidence is combined in the accumulation process. The models were evaluated on their ability to predict both response time distributions and movement neuron activity observed in monkeys performing a visual search task. Models that assume gating of perceptual evidence to the accumulating units provide the best account of both behavioral and neural data. These results identify discrete stages of processing with anatomically distinct neural populations and rule out several alternative architectures. The results also illustrate the use of neurophysiological data as a model selection tool and establish a novel framework to bridge computational and neural levels of explanation.

Exit times in non-Markovian drifting continuous-time random-walk processes

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

Individual differences, aging, and IQ in two-choice tasks

The effects of aging and IQ on performance were examined in three two-choice tasks: numerosity discrimination, recognition memory, and lexical decision. The experimental data, accuracy, correct and error response times, and response time distributions, were well explained by Ratcliff's (1978) diffusion model. The components of processing identified by the model were compared across levels of IQ (ranging from 83 to 146) and age (college students, 60-74, and 75-90 year olds). Declines in performance with age were not significantly different for low compared to high IQ subjects. IQ but not age had large effects on the quality of the evidence that was obtained from a stimulus or memory, that is, the evidence upon which decisions were based. Applying the model to individual subjects, the components of processing identified by the model for individuals correlated across tasks. In addition, the model's predictions and the data were examined for the "worst performance rule", the finding that age and IQ have larger effects on slower responses than faster responses.

Perceptual discrimination in static and dynamic noise: the temporal relation between perceptual encoding and decision making

The authors report 9 new experiments and reanalyze 3 published experiments that investigate factors affecting the time course of perceptual processing and its effects on subsequent decision making. Stimuli in letter-discrimination and brightness-discrimination tasks were degraded with static and dynamic noise. The onset and the time course of decision making were quantified by fitting the data with the diffusion model. Dynamic noise and, to a lesser extent, static noise, produced large shifts in the leading edge of the response-time distribution in letter discrimination but had little effect in brightness discrimination. The authors interpret these shifts as changes in the onset of decision making. The different pattern of shifts in letter discrimination and brightness discrimination implies that decision making in the 2 tasks was affected differently by noise. The changes in response-time distributions found with letter stimuli are inconsistent with the hypothesis that noise increases response times to letter stimuli simply by reducing the rate at which evidence accumulates in the decision process. Instead, they imply that noise also delays the time at which evidence accumulation begins. The delay is shown not to be the result of strategic processes or the result of using different stimuli in different tasks. The results imply, rather, that the onset of evidence accumulation in the decision process is time-locked to the perceptual encoding of the stimulus features needed to do the task. Two mechanisms that could produce this time-locking are described.

Modeling reaction time and accuracy of multiple-alternative decisions

Several sequential-sampling models using racing diffusion processes for multiple-alternative decisions were evaluated, using data from two perceptual discrimination experiments. The structures of the models differed on a number of dimensions, including whether there was lateral inhibition between accumulators, whether there was decay in evidence, whether evidence could be negative, and whether there was variability in starting points. Data were collected from a letter discrimination task in which stimulus difficulty and probability of the response alternatives were varied along with number of response alternatives. Model-fitting results ruled out a large number of model classes in favor of a smaller number of specific models, most of which showed a moderate to high degree of mimicking. The best-fitting models had zero to moderate values of decay, had no inhibition, and assumed that the addition of alternatives affected the subprocesses contributing to the nondecisional time, the degree of caution, or the quality of evidence extracted from stimuli.

On the effects of signal acuity in a multi-alternative model of decision making

We consider the effects of signal sharpness or acuity on the performance of neural models of decision making. In these models, a vector of signals is presented, and the subject must decide which of the elements of the vector is the largest. McMillen and Holmes ( 2006 ) derived asymptotically optimal tests under the assumption that the elements of the signal vector were all equal except one. In this letter, we consider the case of signals spread around a peak. The acuity is a measure of how strongly peaked the signal is. We find that the optimal test is one in which the detectors are passed through an output layer that encodes knowledge of the possible shapes of the incoming signals. The incorporation of such an output layer can lead to significant improvements in decision-making tasks.

Sleep deprivation affects multiple distinct cognitive processes

Sleep deprivation adversely affects the ability to perform cognitive tasks, but theories range from predicting an overall decline in cognitive functioning (because of reduced stability in attentional networks) to claiming specific deficits in executive functions. In the present study, we measured the effects of sleep deprivation on a two-choice numerosity discrimination task. A diffusion model was used to decompose accuracy and response time distributions in order to produce estimates of distinct components of cognitive processing. The model assumes that, over time, noisy evidence from the task stimulus is accumulated to one of two decision criteria and that parameters governing this process can be extracted and interpreted in terms of distinct cognitive processes. The results showed that sleep deprivation affects multiple components of cognitive processing, ranging from stimulus processing to peripheral nondecision processes. Thus, sleep deprivation appears to have wide-ranging effects: Reduced attentional arousal and impaired central processing combine to produce an overall decline in cognitive functioning.

A comparison of bounded diffusion models for choice in time controlled tasks

The Wiener diffusion model (WDM) for 2-alternative tasks assumes that sensory information is integrated over time. Recent neurophysiological studies have found neural correlates of this integration process in certain neuronal populations. This paper analyses the properties of the WDM with two different boundary conditions in decision making tasks in which the time of response is indicated by a cue. A dual reflecting boundary mechanism is proposed and its performance is compared with a well-established absorbing boundary in the cases of the WDM, the WDM with extensions, and the WDM with prior probability. The two types of boundary influence the dynamics of the model and introduce differential weighting of evidence. Comparisons with Ornstein-Uhlenbeck models are also done, and it is shown that the WDM with both types of boundaries achieves similar performance and produce similar fits to existing behavioural data. Further studies are proposed to distinguish which boundary mechanism is more consistent with experimental data.

Larger variability of saccadic reaction times in schizophrenia patients

Slower mean reaction time (RT), known as psychomotor slowing, is well documented in patients with schizophrenia. Fewer studies have shown increased variability of RT in these patients suggesting a basic difference in the distribution of RT. In this study median RT and its variability were measured for visually guided saccades performed by 53 patients and 1089 control subjects. Then average cumulative RT distributions were derived for each group and the RT distribution for each group was modeled using a decision signal rising linearly to a threshold signaling the beginning of the visually guided saccade. There was a small increase in the median RT for patients while their RTs were much more variable from trial to trial leading to a difference in the average RT distribution of the patient group. The model application led to the conclusion that this difference in the distribution of RT for patients could be attributed to a basic difference in information processing leading to the decision to move the eyes to the visually presented target. This information-processing difference could be the result of a difference in the build-up of neuronal activity involved in the generation of visually guided saccades in the frontal cortex.

Closed-Form Approximations of First-Passage Distributions for a Stochastic Decision-Making Model

In free response choice tasks, decision making is often modeled as a first-passage problem for a stochastic differential equation. In particular, drift-diffusion processes with constant or time-varying drift rates and noise can reproduce behavioral data (accuracy and response-time distributions) and neuronal firing rates. However, no exact solutions are known for the first-passage problem with time-varying data. Recognizing the importance of simple closed-form expressions for modeling and inference, we show that an interrogation or cued-response protocol, appropriately interpreted, can yield approximate first-passage (response time) distributions for a specific class of time-varying processes used to model evidence accumulation. We test these against exact expressions for the constant drift case and compare them with data from a class of sigmoidal functions. We find that both the direct interrogation approximation and an error-minimizing interrogation approximation can capture a variety of distribution shapes and mode numbers but that the direct approximation, in particular, is systematically biased away from the correct free response distribution.

Modeling aging effects on two-choice tasks: response signal and response time data

In the response signal paradigm, a test stimulus is presented, and then at one of a number of experimenter-determined times, a signal to respond is presented. Response signal, standard response time (RT), and accuracy data were collected from 19 college-age and 19 60- to 75-year-old participants in a numerosity discrimination task. The data were fit with 2 versions of the diffusion model. Response signal data were modeled by assuming a mixture of processes, those that have terminated before the signal and those that have not terminated; in the latter case, decisions are based on either partial information or guessing. The effects of aging on performance in the regular RT task were explained the same way in the models, with a 70- to 100-ms increase in the nondecision component of processing, more conservative decision criteria, and more variability across trials in drift and the nondecision component of processing, but little difference in drift rate (evidence). In the response signal task, the primary reason for a slower rise in the response signal functions for older participants was variability in the nondecision component of processing. Overall, the results were consistent with earlier fits of the diffusion model to the standard RT task for college-age participants and to the data from aging studies using this task in the standard RT procedure.

Comparing time-accuracy curves: beyond goodness-of-fit measures

The speed-accuracy trade-off (SAT) is a ubiquitous phenomenon in experimental psychology. One popular strategy for controlling SAT is to use the response signal paradigm. This paradigm produces time-accuracy curves (or SAT functions), which can be compared across different experimental conditions. The typical approach to analyzing time-accuracy curves involves the comparison of goodness-of-fit measures (e.g., adjusted-R2), as well as interpretation of point estimates. In this article, we examine the implications of this approach and discuss a number of alternative methods that have been successfully applied in the cognitive modeling literature. These methods include model selection criteria (the Akaike information criterion and the Bayesian information criterion) and interval estimation procedures (bootstrap and Bayesian). We demonstrate the utility of these methods with a hypothetical data set.

Modeling confidence and response time in recognition memory

A new model for confidence judgments in recognition memory is presented. In the model, the match between a single test item and memory produces a distribution of evidence, with better matches corresponding to distributions with higher means. On this match dimension, confidence criteria are placed, and the areas between the criteria under the distribution are used as drift rates to drive racing Ornstein-Uhlenbeck diffusion processes. The model is fit to confidence judgments and quantile response times from two recognition memory experiments that manipulated word frequency and speed versus accuracy emphasis. The model and data show that the standard signal detection interpretation of z-transformed receiver operating characteristic (z-ROC) functions is wrong. The model also explains sequential effects in which the slope of the z-ROC function changes by about 10% as a function of the prior response in the test list.

Temporal dynamics of decision-making during motion perception in the visual cortex

How does the brain make decisions? Speed and accuracy of perceptual decisions covary with certainty in the input, and correlate with the rate of evidence accumulation in parietal and frontal cortical "decision neurons". A biophysically realistic model of interactions within and between Retina/LGN and cortical areas V1, MT, MST, and LIP, gated by basal ganglia, simulates dynamic properties of decision-making in response to ambiguous visual motion stimuli used by Newsome, Shadlen, and colleagues in their neurophysiological experiments. The model clarifies how brain circuits that solve the aperture problem interact with a recurrent competitive network with self-normalizing choice properties to carry out probabilistic decisions in real time. Some scientists claim that perception and decision-making can be described using Bayesian inference or related general statistical ideas, that estimate the optimal interpretation of the stimulus given priors and likelihoods. However, such concepts do not propose the neocortical mechanisms that enable perception, and make decisions. The present model explains behavioral and neurophysiological decision-making data without an appeal to Bayesian concepts and, unlike other existing models of these data, generates perceptual representations and choice dynamics in response to the experimental visual stimuli. Quantitative model simulations include the time course of LIP neuronal dynamics, as well as behavioral accuracy and reaction time properties, during both correct and error trials at different levels of input ambiguity in both fixed duration and reaction time tasks. Model MT/MST interactions compute the global direction of random dot motion stimuli, while model LIP computes the stochastic perceptual decision that leads to a saccadic eye movement.

A dynamic extraversion model. The brain's response to a single dose of a stimulant drug

The aim of this paper is to present a mathematical dynamic modelling of the effect a stimulant drug has on different people which, at the same time, can be a useful tool for future brain studies. To this end, a dynamic model of the evolution of extraversion (considering its tonic and phasic aspects) has been constructed taking into account the unique personality trait theory and the general modelling methodology. This model consists of a delayed differential equation which, on one hand, considers that the active stimulus, a consequence of a single intake, is not constant; on the other hand, it contemplates that the state variable representing the phasic extraversion also represents the brain activation. The derivative of this state variable is calculated as the sum of the homeostatic control flow, the excitatory effect flow and the inhibitor effect flow. The solutions of this equation relate the tonic activation of an individual (that characterizes his or her personality) with his or her phasic activation level, whose evolution over time describes the organism's response to a single drug intake. These solutions quantitatively reproduce the predictions of current personality theories and anticipate vulnerability to drug misuse and addiction development.

Optimal temporal decoding of neural population responses in a reaction-time visual detection task

Behavioral performance in detection and discrimination tasks is likely to be limited by the quality and nature of the signals carried by populations of neurons in early sensory cortical areas. Here we used voltage-sensitive dye imaging (VSDI) to directly measure neural population responses in the primary visual cortex (V1) of monkeys performing a reaction-time detection task. Focusing on the temporal properties of the population responses, we found that V1 responses are consistent with a stimulus-evoked response with amplitude and latency that depend on target contrast and a stimulus-independent additive noise with long-lasting temporal correlations. The noise had much lower amplitude than the ongoing activity reported previously in anesthetized animals. To understand the implications of these properties for subsequent processing stages that mediate behavior, we derived the Bayesian ideal observer that specifies how to optimally use neural responses in reaction time tasks. Using the ideal observer analysis, we show that 1) the observed temporal correlations limit the performance benefit that can be attained by accumulating V1 responses over time, 2) a simple temporal decorrelation operation with time-lagged excitation and inhibition minimizes the detrimental effect of these correlations, 3) the neural information relevant for target detection is concentrated in the initial response following stimulus onset, and 4) a decoder that optimally uses V1 responses far outperforms the monkey in both speed and accuracy. Finally, we demonstrate that for our particular detection task, temporal decorrelation followed by an appropriate running integrator can approach the speed and accuracy of the optimal decoder.

A model of the go/no-go task

In this article, the first explicit, theory-based comparison of 2-choice and go/no-go variants of 3 experimental tasks is presented. Prior research has questioned whether the underlying core-information processing is different for the 2 variants of a task or whether they differ mostly in response demands. The authors examined 4 different diffusion models for the go/no-go variant of each task along with a standard diffusion model for the 2-choice variant (R. Ratcliff, 1978). The 2-choice and the go/no-go models were fit to data from 4 lexical decision experiments, 1 numerosity discrimination experiment, and 1 recognition memory experiment, each with 2-choice and go/no-go variants. The models that assumed an implicit decision criterion for no-go responses produced better fits than models that did not. The best model was one in which only response criteria and the nondecisional components of processing changed between the 2 variants, supporting the view that the core information on which decisions are based is not different between them.

A model of the go/no-go task

In this article, the first explicit, theory-based comparison of 2-choice and go/no-go variants of 3 experimental tasks is presented. Prior research has questioned whether the underlying core-information processing is different for the 2 variants of a task or whether they differ mostly in response demands. The authors examined 4 different diffusion models for the go/no-go variant of each task along with a standard diffusion model for the 2-choice variant (R. Ratcliff, 1978). The 2-choice and the go/no-go models were fit to data from 4 lexical decision experiments, 1 numerosity discrimination experiment, and 1 recognition memory experiment, each with 2-choice and go/no-go variants. The models that assumed an implicit decision criterion for no-go responses produced better fits than models that did not. The best model was one in which only response criteria and the nondecisional components of processing changed between the 2 variants, supporting the view that the core information on which decisions are based is not different between them.

An EZ-diffusion model for response time and accuracy

The EZ-diffusion model for two-choice response time tasks takes mean response time, the variance of response time, and response accuracy as inputs. The model transforms these data via three simple equations to produce unique values for the quality of information, response conservativeness, and nondecision time. This transformation of observed data in terms of unobserved variables addresses the speed-accuracy trade-off and allows an unambiguous quantification of performance differences in two-choice response time tasks. The EZ-diffusion model can be applied to data-sparse situations to facilitate individual subject analysis. We studied the performance of the EZ-diffusion model in terms of parameter recovery and robustness against misspecification by using Monte Carlo simulations. The EZ model was also applied to a real-world data set.