Competing Accumulator Model Research Articles

The Impact of Speed-Accuracy Instructions on Spatial Congruency Effects.

In many tasks humans can trade speed against accuracy. This variation of strategy has different consequences for congruency effects in different conflict tasks. Recently, Mittelstädt et al. (2022) suggested that these differences are related to the dynamics of congruency effects as assessed by delta plots. With increasing delta plots in the Eriksen flanker task congruency effects were larger under accuracy set, and with decreasing delta plots in the Simon task they were smaller. Here we tested the hypothesis for a single task, making use of the observation that for the Simon task delta plots decline when the irrelevant feature is presented first, but increase when the relevant feature leads. The differences between congruency effects under speed and accuracy instructions confirmed the hypothesized relation to the slope of delta plots. In fact, for similar delta plots in the compared speed-accuracy conditions, the relation should be a straightforward consequence of the shorter and longer reaction times with speed and accuracy set, respectively. However, when relevant and irrelevant features were presented simultaneously, congruency effects were stronger under speed set at all reaction times. For this condition, a supplementary model-based analysis with an extended leaky, competing accumulator model suggested a stronger and longer-lasting influence of the irrelevant stimulus feature. The congruency effects for reaction times were accompanied by congruency effects for error rates when delta plots were decreasing, but not when they were increasing.

Staggered Onsets of Processing Relevant and Irrelevant Stimulus Features Produce Different Dynamics of Congruency Effects.

The dynamics of congruency effects in conflict tasks can be analyzed by means of delta plots which depict the reaction-time differences between incongruent and congruent conditions across the quantiles of the reaction-time distributions. Delta plots exhibit a variety of different shapes. Here we test the hypothesis that staggered onsets of processing task-relevant and task-irrelevant features for response selection (together with a declining influence of the irrelevant feature) produce such variety. For this purpose, staggered onsets were implemented in two extensions of the Leaky, Competing Accumulator model. We show the cardinal capability of these models to produce different shapes of delta plots with different assumptions about temporal offsets between processing relevant and irrelevant stimulus features. Applying the models to experimental data, we first show that they can reproduce the delta plots observed with a conflict task with stimulus size as the irrelevant feature. For this task congruency effects are delayed and appear only at longer reaction times. Second, we fit the models to the results of two new Simon-task experiments with an experimentally controlled temporal offset in addition to the internal one. The experimentally induced variations of the shape of delta plots for this task could be reasonably well fitted by one of the two models that assumed an early start of response selection as soon as either the relevant or the irrelevant stimulus feature becomes available. We conclude that delta plots are crucially shaped by staggered onsets of processing relevant and irrelevant features for response selection.

Human-in-the-loop layered architecture for control of a wearable ankle–foot robot

Intelligent wearable robotics is a promising approach for the development of devices that can interact with people and assist them in daily activities. This work presents a novel human-in-the-loop layered architecture to control a wearable robot while interacting with the human body. The proposed control architecture is composed of high-, mid- and low-level computational and control layers, together with wearable sensors, for the control of a wearable ankle–foot robot. The high-level layer uses Bayesian formulation and a competing accumulator model to estimate the human posture during the gait cycle. The mid-level layer implements a Finite State Machine (FSM) to prepare the control parameters for the wearable robot based on the decisions from the high-level layer. The low-level layer is responsible for the precise control of the wearable robot over time using a cascade proportional–integral–derivative (PID) control approach. The human-in-the-loop layered architecture is systematically validated with the control of a 3D printed wearable ankle–foot robot to assist the human foot while walking. The assistance is applied lifting up the human foot when the toe-off event is detected in the walking cycle, and the assistance is removed allowing the human foot to move down and contact the ground when the heel-contact event is detected. Overall, the experiments in offline and real-time modes, undertaken for the validation process, show the potential of the human-in-the-loop layered architecture to develop intelligent wearable robots capable of making decisions and responding fast and accurately based on the interaction with the human body.

The Spatial Leaky Competing Accumulator Model

The Leaky Competing Accumulator model (LCA) of Usher and McClelland is able to simulate the time course of perceptual decision making between an arbitrary number of stimuli. Reaction times, such as saccadic latencies, produce a typical distribution that is skewed toward longer latencies and accumulator models have shown excellent fit to these distributions. We propose a new implementation called the Spatial Leaky Competing Accumulator (SLCA), which can be used to predict the timing of subsequent fixation durations during a visual task. SLCA uses a pre-existing saliency map as input and represents accumulation neurons as a two-dimensional grid to generate predictions in visual space. The SLCA builds on several biologically motivated parameters: leakage, recurrent self-excitation, randomness and non-linearity, and we also test two implementations of lateral inhibition. A global lateral inhibition, as implemented in the original model of Usher and McClelland, is applied to all competing neurons, while a local implementation allows only inhibition of immediate neighbors. We trained and compared versions of the SLCA with both global and local lateral inhibition with use of a genetic algorithm, and compared their performance in simulating human fixation latency distribution in a foraging task. Although both implementations were able to produce a positively skewed latency distribution, only the local SLCA was able to match the human data distribution from the foraging task. Our model is discussed for its potential in models of salience and priority, and its benefits as compared to other models like the Leaky integrate and fire network.

A neurocomputational model for the processing of conflicting information in context-dependent decision tasks.

Context-dependent computation is a relevant characteristic of neural systems, endowing them with the capacity of adaptively modifying behavioral responses and flexibly discriminating between relevant and irrelevant information in a stimulus. This ability is particularly highlighted in solving conflicting tasks. A long-standing problem in computational neuroscience, flexible routing of information, is also closely linked with the ability to perform context-dependent associations. Here we present an extension of a context-dependent associative memory model to achieve context-dependent decision-making in the presence of conflicting and noisy multi-attribute stimuli. In these models, the input vectors are multiplied by context vectors via the Kronecker tensor product. To outfit the model with a noisy dynamic, we embedded the context-dependent associative memory in a leaky competing accumulator model, and, finally, we proved the power of the model in the reproduction of a behavioral experiment with monkeys in a context-dependent conflicting decision-making task. At the end, we discuss the neural feasibility of the tensor product and made the suggestive observation that the capacities of tensor context models are surprisingly in alignment with the more recent experimental findings about functional flexibility at different levels of brain organization.

Modified leaky competing accumulator model of decision making with multiple alternatives: the Lie-algebraic approach

In this communication, based upon the stochastic Gompertz law of population growth, we have reformulated the Leaky Competing Accumulator (LCA) model with multiple alternatives such that the positive-definiteness of evidence accumulation is automatically satisfied. By exploiting the Lie symmetry of the backward Kolmogorov equation (or Fokker–Planck equation) assoicated with the modified model and applying the Wei–Norman theorem, we have succeeded in deriving the N-dimensional joint probability density function (p.d.f.) and marginal p.d.f. for each alternative in closed form. With this joint p.d.f., a likelihood function can be constructed and thus model-fitting procedures become feasible. We have also demonstrated that the calibration of model parameters based upon the Monte Carlo simulated time series is indeed both efficient and accurate. Moreover, it should be noted that the proposed Lie-algebraic approach can also be applied to tackle the modified LCA model with time-varying parameters.

Perceptual change-of-mind decisions are sensitive to absolute evidence magnitude

To navigate the world safely, we often need to rapidly ‘change our mind’ about decisions. Current models assume that initial decisions and change-of-mind decisions draw upon common sources of sensory evidence. In two-choice scenarios, this evidence may be ‘relative’ or ‘absolute’. For example, when judging which of two objects is the brightest, the luminance difference and luminance ratio between the two objects are sources of ‘relative’ evidence, which are invariant across additive and multiplicative luminance changes. Conversely, the overall luminance of the two objects combined is a source of ‘absolute’ evidence, which necessarily varies across symmetric luminance manipulations. Previous studies have shown that initial decisions are sensitive to both relative and absolute evidence; however, it is unknown whether change-of-mind decisions are sensitive to absolute evidence. Here, we investigated this question across two experiments. In each experiment participants indicated which of two flickering greyscale squares was brightest. Following an initial decision, the stimuli remained on screen for a brief period and participants could change their response. To investigate the effect of absolute evidence, the overall luminance of the two squares was varied whilst either the luminance difference (Experiment 1) or luminance ratio (Experiment 2) was held constant. In both experiments we found that increases in absolute evidence led to faster, less accurate initial responses and slower changes of mind. Change-of-mind accuracy decreased when the luminance difference was held constant, but remained unchanged when the luminance ratio was fixed. We show that the three existing change-of-mind models cannot account for our findings. We then fit three alternative models, previously used to account for the effect of absolute evidence on one-off decisions, to the data. A leaky competing accumulator model best accounted for the changes in behaviour across absolute evidence conditions – suggesting an important role for input-dependent leak in explaining perceptual changes of mind.

Frequency-Sensitivity and Magnitude-Sensitivity in Decision-Making: Predictions of a Theoretical Model-Based Study

We theoretically study decision-making behaviour in a model-based analysis related to binary choices with pulsed stimuli. Assuming a strong coupling between external stimulus and its internal representation, we argue that the frequency of external periodic stimuli represents an important degree of freedom in decision-making which may modulate behavioural responses. We consider various different stimulus conditions, including varying overall magnitudes and magnitude ratios as well as varying overall frequencies and frequency ratios, and different duty cycles of the pulsed stimuli. Decision time distributions, mean decision times and choice probabilities are simulated and compared for two different models—a leaky competing accumulator model and a diffusion-type model with multiplicative noise. Our results reveal an interplay between the sensitivity of the model systems to both frequency and magnitude of the stimuli. In particular, we find that periodic stimuli may shape the decision time distributions resulting from both models by resembling the frequencies of the pulsed stimuli. We obtain significant frequency-sensitive effects on mean decision time and choice probability for a range of overall frequencies and frequency ratios. Our simulation analysis makes testable predictions that frequencies comparable with typical sensory processing and decision-making timescales may influence choice and response times in perceptual decisions. A possible experimental implementation is proposed.

The Neurodynamic Decision Variable in Human Multi-alternative Perceptual Choice.

The neural dynamics underpinning binary perceptual decisions and their transformation into actions are well studied, but real-world decisions typically offer more than two response alternatives. How does decision-related evidence accumulation dynamically influence multiple action representations in humans? The heightened conservatism required in multiple compared with binary choice scenarios suggests a mechanism that compensates for increased uncertainty when multiple choices are present by suppressing baseline activity. Here, we tracked action representations using corticospinal excitability during four- and two-choice perceptual decisions and modeled them using a sequential sampling framework. We found that the predictions made by leaky competing accumulator models to accommodate multiple choices (i.e., reduced baseline activity to compensate increased uncertainty) were borne out by dynamic changes in human action representations. This suggests a direct and continuous influence of interacting evidence accumulators, each favoring a different decision alternative, on downstream corticospinal excitability during complex choice.

Correction: Value-based decision making via sequential sampling with hierarchical competition and attentional modulation.

[This corrects the article DOI: 10.1371/journal.pone.0186822.].

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

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

Value-based decision making via sequential sampling with hierarchical competition and attentional modulation.

In principle, formal dynamical models of decision making hold the potential to represent fundamental computations underpinning value-based (i.e., preferential) decisions in addition to perceptual decisions. Sequential-sampling models such as the race model and the drift-diffusion model that are grounded in simplicity, analytical tractability, and optimality remain popular, but some of their more recent counterparts have instead been designed with an aim for more feasibility as architectures to be implemented by actual neural systems. Connectionist models are proposed herein at an intermediate level of analysis that bridges mental phenomena and underlying neurophysiological mechanisms. Several such models drawing elements from the established race, drift-diffusion, feedforward-inhibition, divisive-normalization, and competing-accumulator models were tested with respect to fitting empirical data from human participants making choices between foods on the basis of hedonic value rather than a traditional perceptual attribute. Even when considering performance at emulating behavior alone, more neurally plausible models were set apart from more normative race or drift-diffusion models both quantitatively and qualitatively despite remaining parsimonious. To best capture the paradigm, a novel six-parameter computational model was formulated with features including hierarchical levels of competition via mutual inhibition as well as a static approximation of attentional modulation, which promotes “winner-take-all” processing. Moreover, a meta-analysis encompassing several related experiments validated the robustness of model-predicted trends in humans’ value-based choices and concomitant reaction times. These findings have yet further implications for analysis of neurophysiological data in accordance with computational modeling, which is also discussed in this new light.

Lie-Algebraic Approach for the Leaky Competing Accumulator Model of Decision Making

In this communication, based upon the stochastic Gompertz law of population growth, we have reformulated the Leaky Competing Accumulator (LCA) model with multiple alternatives such that the positive-definiteness of evidence accumulation is automatically satisfied. By exploiting the Lie symmetry of the backward Kolmogorov equation (or Fokker-Planck equation) assoicated with the model and applying the Wei-Norman theorem, we have succeeded to derive the N-dimensional joint probability density function (p.d.f.) and marginal p.d.f. for each alternative in closed form. With this p.d.f., a likelihood function can be constructed and thus model-fitting procedures become feasible. Moreover, by the method of images the closed-form first-passage-time distribution function across a parametric time-varying evidence boundary for each alternative has been obtained. The first-passage-time distribution function is useful for the study of response times in decision making as well as the investigation on how the evidence boundary can influence decision performance. Lastly, it should be noted that the proposed Lie-algebraic approach can also be applied to tackle the LCA model with time-varying parameters.

Parameter recovery for the Leaky Competing Accumulator model

The Leaky Competitive Accumulator (LCA) model for perceptual discrimination is rapidly growing in popularity due to its neural plausibility. The model assumes that perceptual choices and associated response times are the consequence of the accrual of evidence for the various response alternatives up to a certain predetermined threshold. In addition, accrual of evidence is influenced by temporal leakage of information and mutual inhibition between the accumulators. In this paper we provide an overview of fitting routines that may be used to identify the parameter values used for generating data under the LCA assumptions. We find that because (a) there is no closed-form solution to the likelihood function of the LCA model, and (b) there are strong trade-offs between accumulation rate, leakage, and inhibition, it is extremely difficult to faithfully recover the parameters of the LCA model. To minimize these problems, we recommend to use DE-MCMC sampling, collect very large amounts of data, and constrain the parameter space where possible.

Copy Alert: A Method and Metric to Detect Visual Copycat Brands

The authors propose a method and metric to quantify the consumer confusion between leading brands and copycat brands that results from the visual similarity of their packaging designs. The method has three components. First, image processing techniques establish the objective similarity of the packages of leading and copycat brands on the basis of their colors and textures. Second, a perceptual decision task (triangle test) assesses the accuracy and speed with which consumers can identify differences between brands from rapidly (300 milliseconds) flashed images of their packages. Third, a competing accumulator model describes the buildup of evidence on each of the alternative brands during consumers’ perceptual decisions and predicts the accuracy and response time of brand identification. Jointly, these components establish the impact of copycat packaging's visual features on consumer confusion. The method is applied in a test of experimentally designed copycats and market copycats in 15 product categories. A three-tiered metric (“copy alert,” “copy watch,” and “copy safe”) establishes the extent to which copycat brands imitate the package designs of target brands and identifies which visual features are responsible.

A generalized, likelihood-free method for posterior estimation.

Recent advancements in Bayesian modeling have allowed for likelihood-free posterior estimation. Such estimation techniques are crucial to the understanding of simulation-based models, whose likelihood functions may be difficult or even impossible to derive. However, current approaches are limited by their dependence on sufficient statistics and/or tolerance thresholds. In this article, we provide a new approach that requires no summary statistics, error terms, or thresholds and is generalizable to all models in psychology that can be simulated. We use our algorithm to fit a variety of cognitive models with known likelihood functions to ensure the accuracy of our approach. We then apply our method to two real-world examples to illustrate the types of complex problems our method solves. In the first example, we fit an error-correcting criterion model of signal detection, whose criterion dynamically adjusts after every trial. We then fit two models of choice response time to experimental data: the linear ballistic accumulator model, which has a known likelihood, and the leaky competing accumulator model, whose likelihood is intractable. The estimated posterior distributions of the two models allow for direct parameter interpretation and model comparison by means of conventional Bayesian statistics-a feat that was not previously possible.

How to say “no” to a nonword: A leaky competing accumulator model of lexical decision.

We describe a leaky competing accumulator (LCA) model of the lexical decision task that can be used as a response/decision module for any computational model of word recognition. The LCA model uses evidence for a word, operationalized as some measure of lexical activity, as input to the YES decision node. Input to the NO decision node is simply a constant value minus evidence for a word. In this way, evidence for a nonword is a function of time from stimulus onset (as in standard deadline models) modulated by lexical activity via the competitive dynamics of the LCA. We propose a simple mechanism for determining the value of this constant online during the first trials of a lexical decision experiment, such that the model can rapidly optimize speed and accuracy in discriminating words from nonwords. Further optimization is achieved via trial-by-trial adjustments in response criteria as a function of task demands and list context. We show that the LCA model can simulate mean response times and response distributions for correct and incorrect YES and NO decisions for a number of benchmark experiments that have been shown to be fatal for deadline models of lexical decision. Finally, using lexical activity calculated by a computational model of word recognition as input to the LCA decision module, we provide the first item-level simulation of both word and nonword responses in a large-scale database.

The 2N-ary Choice Tree Model for N-Alternative Preferential Choice

The 2N-ary choice tree model accounts for response times and choice probabilities in multi-alternative preferential choice. It implements pairwise comparison of alternatives on weighted attributes into an information sampling process which, in turn, results in a preference process. The model provides expected choice probabilities and response time distributions in closed form for optional and fixed stopping times. The theoretical background of the 2N-ary choice tree model is explained in detail with focus on the transition probabilities that take into account constituents of human preferences such as expectations, emotions, or socially influenced attention. Then it is shown how the model accounts for several context-effects observed in human preferential choice like similarity, attraction, and compromise effects and how long it takes, on average, for the decision. The model is extended to deal with more than three choice alternatives. A short discussion on how the 2N-ary choice tree model differs from the multi-alternative decision field theory and the leaky competing accumulator model is provided.

Using Time-Varying Evidence to Test Models of Decision Dynamics: Bounded Diffusion vs. the Leaky Competing Accumulator Model.

When people make decisions, do they give equal weight to evidence arriving at different times? A recent study (Kiani et al., 2008) using brief motion pulses (superimposed on a random moving dot display) reported a primacy effect: pulses presented early in a motion observation period had a stronger impact than pulses presented later. This observation was interpreted as supporting the bounded diffusion (BD) model and ruling out models in which evidence accumulation is subject to leakage or decay of early-arriving information. We use motion pulses and other manipulations of the timing of the perceptual evidence in new experiments and simulations that support the leaky competing accumulator (LCA) model as an alternative to the BD model. While the LCA does include leakage, we show that it can exhibit primacy as a result of competition between alternatives (implemented via mutual inhibition), when the inhibition is strong relative to the leak. Our experiments replicate the primacy effect when participants must be prepared to respond quickly at the end of a motion observation period. With less time pressure, however, the primacy effect is much weaker. For 2 (out of 10) participants, a primacy bias observed in trials where the motion observation period is short becomes weaker or reverses (becoming a recency effect) as the observation period lengthens. Our simulation studies show that primacy is equally consistent with the LCA or with BD. The transition from primacy-to-recency can also be captured by the LCA but not by BD. Individual differences and relations between the LCA and other models are discussed.

Optimal decision making in neural inhibition models.

In their influential Psychological Review article, Bogacz, Brown, Moehlis, Holmes, and Cohen (2006) discussed optimal decision making as accomplished by the drift diffusion model (DDM). The authors showed that neural inhibition models, such as the leaky competing accumulator model (LCA) and the feedforward inhibition model (FFI), can mimic the DDM and accomplish optimal decision making. Here we show that these conclusions depend on how the models handle negative activation values and (for the LCA) across-trial variability in response conservativeness. Negative neural activations are undesirable for both neurophysiological and mathematical reasons. However, when negative activations are truncated to 0, the equivalence to the DDM is lost. Simulations show that this concern has practical ramifications: the DDM generally outperforms truncated versions of the LCA and the FFI, and the parameter estimates from the neural models can no longer be mapped onto those of the DDM in a simple fashion. We show that for both models, truncation may be avoided by assuming a baseline activity for each accumulator. This solution allows the LCA to approximate the DDM and the FFI to be identical to the DDM.