Adaptive Spontaneous Transitions between Two Mechanisms of Numerical Averaging

The role of episodic memory sampling in evaluation

Many models of choice assume that people retrieve memories of past experiences and use them to guide evaluation and choice. In this paper, we examine whether samples of recalled past experiences do indeed underpin our evaluations of options. We showed participants sequences of numerical values and asked them to recall as many of those values as possible and also to state how much they would be willing to pay for another draw from the sequence. Using Bayesian mixed effects modeling, we predicted participants' evaluation of the sequences at the group level from either the average of the values they recalled or the average of the values they saw. Contrary to the predictions of recall-based models, people's evaluations appear to be sensitive to information beyond what was actually recalled. Moreover, we did not find consistent evidence that memory for specific items is sufficient to predict evaluation of sequences. We discuss the implications for sampling models of memory and decision-making and alternative explanations.

What's in a sample? Epistemic uncertainty and metacognitive awareness in risk taking

In a fundamentally uncertain world, sound information processing is a prerequisite for effective behavior. Given that information processing is subject to inevitable cognitive imprecision, decision makers should adapt to this imprecision and to the resulting epistemic uncertainty when taking risks. We tested this metacognitive ability in two experiments in which participants estimated the expected value of different number distributions from sequential samples and then bet on their own estimation accuracy. Results show that estimates were imprecise, and this imprecision increased with higher distributional standard deviations. Importantly, participants adapted their risk-taking behavior to this imprecision and hence deviated from the predictions of Bayesian models of uncertainty that assume perfect integration of information. To explain these results, we developed a computational model that combines Bayesian updating with a metacognitive awareness of cognitive imprecision in the integration of information. Modeling results were robust to the inclusion of an empirical measure of participants' perceived variability. In sum, we show that cognitive imprecision is crucial to understanding risk taking in decisions from experience. The results further demonstrate the importance of metacognitive awareness as a cognitive building block for adaptive behavior under (partial) uncertainty.

The Future of Decisions From Experience: Connecting Real-World Decision Problems to Cognitive Processes

In many important real-world decision domains, such as finance, the environment, and health, behavior is strongly influenced by experience. Renewed interest in studying this influence led to important advancements in the understanding of these decisions from experience (DfE) in the last 20 years. Building on this literature, we suggest ways the standard experimental design should be extended to better approach important real-world DfE. These extensions include, for example, introducing more complex choice situations, delaying feedback, and including social interactions. When acting upon experiences in these richer and more complicated environments, extensive cognitive processes go into making a decision. Therefore, we argue for integrating cognitive processes more explicitly into experimental research in DfE. These cognitive processes include attention to and perception of numeric and nonnumeric experiences, the influence of episodic and semantic memory, and the mental models involved in learning processes. Understanding these basic cognitive processes can advance the modeling, understanding and prediction of DfE in the laboratory and in the real world. We highlight the potential of experimental research in DfE for theory integration across the behavioral, decision, and cognitive sciences. Furthermore, this research could lead to new methodology that better informs decision-making and policy interventions.

The role of motion in visual working memory for dynamic stimuli: More lagged but more precise representations of moving objects

While most visual working memory studies use static stimuli with unchanging features, objects in the real world are often dynamic, introducing significant differences in the surface feature information hitting the retina from the same object over time (e.g., changes in orientation, lighting, shadows). Previous research on dynamic stimuli has shown that change detection is improved if objects obey rules of physical motion, but it is unclear how memory for visual features interacts with object motion. In the current study, we investigated whether object motion facilitates greater temporal integration of continuously changing surface feature information. In a series of experiments, participants were asked to report the final color of continuously changing colored dots that were either moving or stationary on the screen. We found that the reported colors "lagged behind" the physical states of the dots when they were in motion. We also observed that the precision of memory responses was significantly higher for stimuli in the moving condition compared to the stationary condition. Together, these findings suggest that memory representation is improved - but lagged - for moving objects, consistent with the idea that object motion may facilitate integration of object information over longer intervals.

A Model of Scientific Data Reasoning

Data reasoning is an essential component of scientific reasoning, as a component of evidence evaluation. In this paper, we outline a model of scientific data reasoning that describes how data sensemaking underlies data reasoning. Data sensemaking, a relatively automatic process rooted in perceptual mechanisms that summarize large quantities of information in the environment, begins early in development, and is refined with experience, knowledge, and improved strategy use. Summarizing data highlights set properties such as central tendency and variability, and these properties are used to draw inferences from data. However, both data sensemaking and data reasoning are subject to cognitive biases or heuristics that can lead to flawed conclusions. The tools of scientific reasoning, including external representations, scientific hypothesis testing, and drawing probabilistic conclusions, can help reduce the likelihood of such flaws and help improve data reasoning. Although data sensemaking and data reasoning are not supplanted by scientific data reasoning, scientific reasoning skills can be leveraged to improve learning about science and reasoning with data.

Extracting Summary Statistics of Rapid Numerical Sequences

We examine the ability of observers to extract summary statistics (such as the mean and the relative-variance) from rapid numerical sequences of two digit numbers presented at a rate of 4/s. In four experiments (total N = 100), we find that the participants show a remarkable ability to extract such summary statistics and that their precision in the estimation of the sequence-mean improves with the sequence-length (subject to individual differences). Using model selection for individual participants we find that, when only the sequence-average is estimated, most participants rely on a holistic process of frequency based estimation with a minority who rely on a (rule-based and capacity limited) mid-range strategy. When both the sequence-average and the relative variance are estimated, about half of the participants rely on these two strategies. Importantly, the holistic strategy appears more efficient in terms of its precision. We discuss implications for the domains of two pathways numerical processing and decision-making.

Perceiving ensemble statistics of novel image sets

Perception, representation, and memory of ensemble statistics has attracted growing interest. Studies found that, at different abstraction levels, the brain represents similar items as unified percepts. We found that global ensemble perception is automatic and unconscious, affecting later perceptual judgments regarding individual member items. Implicit effects of set mean and range for low-level feature ensembles (size, orientation, brightness) were replicated for high-level category objects. This similarity suggests that analogous mechanisms underlie these extreme levels of abstraction. Here, we bridge the span between visual features and semantic object categories using the identical implicit perception experimental paradigm for intermediate novel visual-shape categories, constructing ensemble exemplars by introducing systematic variations of a central category base or ancestor. In five experiments, with different item variability, we test automatic representation of ensemble category characteristics and its effect on a subsequent memory task. Results show that observer representation of ensembles includes the group's central shape, category ancestor (progenitor), or group mean. Observers also easily reject memory of shapes belonging to different categories, i.e. originating from different ancestors. We conclude that complex categories, like simple visual form ensembles, are represented in terms of statistics including a central object, as well as category boundaries. We refer to the model proposed by Benna and Fusi (bioRxiv 624239, 2019) that memory representation is compressed when related elements are represented by identifying their ancestor and each one's difference from it. We suggest that ensemble mean perception, like category prototype extraction, might reflect employment at different representation levels of an essential, general representation mechanism.

Ensemble perception: Extracting the average of perceptual versus numerical stimuli

Recent research has established that humans can extract the average perceptual feature over briefly presented arrays of visual elements or the average of a rapid temporal sequence of numbers. Here we compared the extraction of the average over briefly presented arrays, for a perceptual feature (orientations) and for numerical values (1-9 digits), using an identical experimental design for the two tasks. We hypothesized that the averaging of numbers, more than of orientations, would be constrained by capacity limitations. Arrays of Gabor elements or digits were simultaneously presented for 300 ms and observers were required to estimate the average on a continuous response scale. In each trial the elements were sampled from normal distributions (of various means) and we varied the set size (4-12). We found that while for orientation the averaging precision remained constant with set size, for numbers it decreased with set size. Using computational modeling we also extracted capacity parameters (the number of elements that are pooled in the average extraction). Despite marked heterogeneity between observers, the capacity for orientations (around eight items) was much larger than for numbers (around four items). The orientation task also had a larger fraction of participants relying on distributed attention to all elements. Our study thus supports the idea that numbers more than perceptual features are subject to capacity or attentional limitations when observers need to evaluate the average over an ensemble of stimuli.

The averaging of numerosities: A psychometric investigation of the mental line

Humans and animals are capable of estimating and discriminating nonsymbolic numerosities via mental representation of magnitudes—the approximate number system (ANS). There are two models of the ANS system, which are similar in their prediction in numerosity discrimination tasks. The log-Gaussian model, which assumes numerosities are represented on a compressed logarithmic scale, and the scalar variability model, which assumes numerosities are represented on a linear scale. In the first experiment of this paper, we contrasted these models using averaging of numerosities. We examined whether participants generate a compressed mean (i.e., geometric mean) or a linear mean when averaging two numerosities. Our results demonstrated that half of the participants are linear and half are compressed; however, in general, the compression is milder than a logarithmic compression. In Experiments 2 and 3, we examined averaging of numerosities in sequences larger than two. We found that averaging precision increases with sequence length. These results are in line with previous findings, suggesting a mechanism in which the estimate is generated by population averaging of the responses each stimulus generates on the numerosity representation.

Perceptual learning of ensemble and outlier perception

Previous studies have demonstrated a complex relationship between ensemble perception and outlier detection. We presented two array of heterogeneously oriented stimulus bars and different mean orientations and/or a bar with an outlier orientation, asking participants to discriminate the mean orientations or detect the outlier. Perceptual learning was found in every case, with improved performance accuracy and speeded responses. Testing for improved accuracy through cross-task transfer, we found considerable transfer from training outlier detection to mean discrimination performance, and none in the opposite direction. Implicit learning in terms of increased accuracy was not found in either direction when participants performed one task, and the second task's stimulus features were present. Reaction time improvement was found to transfer in all cases. This study adds to the already broad knowledge concerning perceptual learning and cross-task transfer of training effects.

Distinct strategies for estimating the temporal average of numerical and perceptual information

Humans can estimate global trends in dynamic information presented either as perceptual features or as symbolic codes such as numbers. Previous studies on temporal statistics estimation have shown that observers judge the temporal average of visual attributes according to information from the last few frames of the presentation sequence (in what is referred to as the recency effect). Here, we investigated how humans estimate the temporal average of number vs. orientation using identical stimuli for the two tasks. In Experiment 1, a randomly-selected single-digit number was serially presented at orientations randomly varying over time. In Experiment 2, a texture comprising a random number of Gabor elements was shown at orientations randomly varying over time. In both experiments, observers judged the temporal averages of the numerical values and orientations in separate blocks. Results showed that observers judging the temporal average of orientation relied upon information from later frames as predicted by a typical model of perceptual decision making. By contrast, for the judgement of numerical values, we found that the impacts of each temporal frame were constant or varied little across temporal frames regardless of whether the numerical information was given as digits or by the number of texture elements. The results are interpreted as evidence that distinct computational strategies may be involved in estimating the temporal averages of perceptual features and numerical information.

Relating categorization to set summary statistics perception

Two cognitive processes have been explored that compensate for the limited information that can be perceived and remembered at any given moment. The first parsimonious cognitive process is object categorization. We naturally relate objects to their category, assume they share relevant category properties, often disregarding irrelevant characteristics. Another scene organizing mechanism is representing aspects of the visual world in terms of summary statistics. Spreading attention over a group of objects with some similarity, one perceives an ensemble representation of the group. Without encoding detailed information of individuals, observers process summary data concerning the group, including set mean for various features (from circle size to face expression). Just as categorization may include/depend on prototype and intercategory boundaries, so set perception includes property mean and range. We now explore common features of these processes. We previously investigated summary perception of low-level features with a rapid serial visual presentation (RSVP) paradigm and found that participants perceive both the mean and range extremes of stimulus sets, automatically, implicitly, and on-the-fly, for each RSVP sequence, independently. We now use the same experimental paradigm to test category representation of high-level objects. We find participants perceive categorical characteristics better than they code individual elements. We relate category prototype to set mean and same/different category to in/out-of-range elements, defining a direct parallel between low-level set perception and high-level categorization. The implicit effects of mean or prototype and set or category boundaries are very similar. We suggest that object categorization may share perceptual-computational mechanisms with set summary statistics perception.

Goal-dependent flexibility in preferences formation from rapid payoff sequences

The formation of attitudes or preferences for alternatives that consist of rapid numerical sequences has been suggested to reflect either a summation or an averaging principle. Previous studies indicate the presence of two mechanisms, accumulators (that mediate summation) and population-coding (that mediate averaging), which operate in preference formation tasks of rapid numerical sequences, and are subject to task-demands and individual differences. Here, we test whether participants can flexibly control the preference mechanism they deploy as a function of the reward contingency. Towards this aim, participants in two studies ( N1 = 21, N2 = 23) made choices between the same sequence alternatives in two task-framing sessions, which made the reward dependent on the sequence-sum or sequence-average, respectively. The results demonstrate that although participants show an overall bias in favour of averaging, they are also remarkably flexible in deploying an averaging or a summation type mechanism that matches the reward contingency

Integration to boundary in decisions between numerical sequences

Integration-to-boundary is a prominent normative principle used in evidence-based decisions to explain the speed-accuracy trade-off and determine the decision-time. Despite its prominence, however, the decision boundary is not directly observed, but rather is theoretically assumed, and there is still an ongoing debate regarding its form: fixed vs. collapsing. The aim of this study is to show that the integration-to-boundary process extends to decisions between rapid pairs of numerical sequences (2 Hz rate), and to determine the boundary type by directly monitoring the noisy accumulated evidence. In a set of two experiments (supplemented by computational modelling), we demonstrate that integration to a collapsing-boundary takes place in such tasks, ruling out non-integration heuristic strategies. Moreover, we show that participants can adaptively adjust their boundaries in response to reward contingencies. Finally, we discuss the implications to decision optimality and the nature of processes and representations in numerical cognition.

Consciousness without report: insights from summary statistics and inattention 'blindness'

We contrast two theoretical positions on the relation between phenomenal and access consciousness. First, we discuss previous data supporting a mild Overflow position, according to which transient visual awareness can overflow report. These data are open to two interpretations: (i) observers transiently experience specific visual elements outside attentional focus without encoding them into working memory; (ii) no specific visual elements but only statistical summaries are experienced in such conditions. We present new data showing that under data-limited conditions observers cannot discriminate a simple relation (same versus different) without discriminating the elements themselves and, based on additional computational considerations, we argue that this supports the first interpretation: summary statistics (same/different) are grounded on the transient experience of elements. Second, we examine recent data from a variant of 'inattention blindness' and argue that contrary to widespread assumptions, it provides further support for Overflow by highlighting another factor, 'task relevance', which affects the ability to conceptualize and report (but not experience) visual elements.This article is part of the theme issue 'Perceptual consciousness and cognitive access'.

Perceiving set mean and range: Automaticity and precision

To compensate for the limited visual information that can be perceived and remembered at any given moment, many aspects of the visual world are represented as summary statistics. We acquire ensemble representations of element groups as a whole, spreading attention over objects, for which we encode no detailed information. Previous studies found that different features of items (from size/orientation to facial expression/biological motion) are summarized to their mean, over space or time. Summarizing is economical, saving time and energy when the environment is too rich and complex to encode each stimulus separately. We investigated set perception using rapid serial visual presentation sequences. Following each sequence, participants viewed two stimuli, member and nonmember, indicating the member. Sometimes, unbeknownst to participants, one stimulus was the set mean, and or the nonmember was outside the set range. Participants preferentially chose stimuli at/near the mean, a "mean effect," and more easily rejected out-of-range stimuli, a "range effect." Performance improved with member proximity to the mean and nonmember distance from set mean and edge, though they were instructed only to remember presented stimuli. We conclude that participants automatically encode both mean and range boundaries of stimulus sets, avoiding capacity limits and speeding perceptual decisions.

Fast and effective: Intuitive processes in complex decisions

Is it possible to carry out complex multi-attribute decisions (which require an estimation of the weighted average) intuitively, without resorting to simplifying heuristics? Over the course of 600 trials, 26 participants had to choose the better-suiting job-candidate, a task requiring comparison of two alternatives over three/four/five dimensions with specified importance weights, with a time constraint forcing intuitive decisions. Participants performed the task fast (mean reaction time (RT) ~ 1.5 s) and with high accuracy (~86%). The participants were classified as users of one of three strategies: Weighted Additive Utility (WADD), Equal Weight rule and Take-The-Best heuristic (TTB). Fifty-nine percent of the participants were classified as users of the compensatory WADD strategy and 29% as users of the non-compensatory TTB. Moreover, the WADD users achieved higher task accuracy without showing time costs. The results provide support for the existence of an automatic compensatory mechanism in weighted average estimations.

Extraction of mean emotional tone from face arrays in social anxiety disorder

BACKGROUND

Social anxiety disorder (SAD) is characterized by intense fear when facing a crowd. Processing biases of crowd-related information have been suggested as contributing to the etiology and maintenance of the disorder. Here we tested whether patients with SAD display aberrant patterns of extracting the mean emotional tone from sets of faces.

METHODS

Twenty-one participants with SAD and 24 unanxious control participants had to determine the average emotion expression of sets of six different morphed faces ranging from happy to angry. In 20% of trials the six faces were randomly sampled from the entire happy-angry range. The remaining 80% of trials, considered the critical trials, had an emotional outlier: five faces were sampled from one-half of the emotional range, whereas the sixth face was sampled from the opposite emotional range.

RESULTS

Participants with SAD were less accurate than controls in extracting the mean emotional tone from sets of faces. Unanxious participants underweighted negative outliers and overweighed positive outliers when extracting the mean, whereas participants with SAD exhibited no such biases.

CONCLUSIONS

Results suggest a possible mechanism associated with the anxiety experienced by socially anxious individuals when facing a crowd.

A Perceptual-Like Population-Coding Mechanism of Approximate Numerical Averaging

Humans possess a remarkable ability to rapidly form coarse estimations of numerical averages. This ability is important for making decisions that are based on streams of numerical or value-based information, as well as for preference formation. Nonetheless, the mechanism underlying rapid approximate numerical averaging remains unknown, and several competing mechanism may account for it. Here, we tested the hypothesis that approximate numerical averaging relies on perceptual-like processes, instantiated by population coding. Participants were presented with rapid sequences of numerical values (four items per second) and were asked to convey the sequence average. We manipulated the sequences' length, variance, and mean magnitude and found that similar to perceptual averaging, the precision of the estimations improves with the length and deteriorates with (higher) variance or (higher) magnitude. To account for the results, we developed a biologically plausible population-coding model and showed that it is mathematically equivalent to a population vector. Using both quantitative and qualitative model comparison methods, we compared the population-coding model to several competing models, such as a step-by-step running average (based on leaky integration) and a midrange model. We found that the data support the population-coding model. We conclude that humans' ability to rapidly form estimations of numerical averages has many properties of the perceptual (intuitive) system rather than the arithmetic, linguistic-based (analytic) system and that population coding is likely to be its underlying mechanism.

Continuous track paths reveal additive evidence integration in multistep decision making

Multistep decision making pervades daily life, but its underlying mechanisms remain obscure. We distinguish four prominent models of multistep decision making, namely serial stage, hierarchical evidence integration, hierarchical leaky competing accumulation (HLCA), and probabilistic evidence integration (PEI). To empirically disentangle these models, we design a two-step reward-based decision paradigm and implement it in a reaching task experiment. In a first step, participants choose between two potential upcoming choices, each associated with two rewards. In a second step, participants choose between the two rewards selected in the first step. Strikingly, as predicted by the HLCA and PEI models, the first-step decision dynamics were initially biased toward the choice representing the highest sum/mean before being redirected toward the choice representing the maximal reward (i.e., initial dip). Only HLCA and PEI predicted this initial dip, suggesting that first-step decision dynamics depend on additive integration of competing second-step choices. Our data suggest that potential future outcomes are progressively unraveled during multistep decision making.

Selective overweighting of larger magnitudes during noisy numerical comparison

Humans are often required to compare average magnitudes in numerical data; for example, when comparing product prices on two rival consumer websites. However, the neural and computational mechanisms by which numbers are weighted, integrated and compared during categorical decisions are largely unknown^1,2,3,4,5. Here, we show a systematic deviation from 'optimality' in both visual and auditory tasks requiring averaging of symbolic numbers. Participants comparing numbers drawn from two categories selectively overweighted larger numbers when making a decision, and larger numbers evoked disproportionately stronger decision-related neural signals over the parietal cortex. A representational similarity analysis⁶ showed that neural (dis)similarity in patterns of electroencephalogram activity reflected numerical distance, but that encoding of number in neural data was systematically distorted in a way predicted by the behavioural weighting profiles, with greater neural distance between adjacent larger numbers. Finally, using a simple computational model, we show that although it is suboptimal for a lossless observer, this selective overweighting policy paradoxically maximizes expected accuracy by making decisions more robust to noise arising during approximate numerical integration². In other words, although selective overweighting discards decision information, it can be beneficial for limited-capacity agents engaging in rapid numerical averaging.

Robust sampling of decision information during perceptual choice

Humans move their eyes to gather information about the visual world. However, saccadic sampling has largely been explored in paradigms that involve searching for a lone target in a cluttered array or natural scene. Here, we investigated the policy that humans use to overtly sample information in a perceptual decision task that required information from across multiple spatial locations to be combined. Participants viewed a spatial array of numbers and judged whether the average was greater or smaller than a reference value. Participants preferentially sampled items that were less diagnostic of the correct answer ("inlying" elements; that is, elements closer to the reference value). This preference to sample inlying items was linked to decisions, enhancing the tendency to give more weight to inlying elements in the final choice ("robust averaging"). These findings contrast with a large body of evidence indicating that gaze is directed preferentially to deviant information during natural scene viewing and visual search, and suggest that humans may sample information "robustly" with their eyes during perceptual decision-making.

Transcranial Direct Current Stimulation over the Parietal Cortex Improves Approximate Numerical Averaging

The parietal cortex has been implicated in a variety of numerosity and numerical cognition tasks and was proposed to encompass dedicated neural populations that are tuned for analogue magnitudes as well as for symbolic numerals. Nonetheless, it remains unknown whether the parietal cortex plays a role in approximate numerical averaging (rapid, yet coarse computation of numbers' mean)—a process that is fundamental to preference formation and decision-making. To causally investigate the role of the parietal cortex in numerical averaging, we have conducted a transcranial direct current stimulation (tDCS) study, in which participants were presented with rapid sequences of numbers and asked to convey their intuitive estimation of each sequence's average. During the task, the participants underwent anodal (excitatory) tDCS (or sham), applied either on a parietal or a frontal region. We found that, although participants exhibit above-chance accuracy in estimating the average of numerical sequences, they did so with higher precision under parietal stimulation. In a second experiment, we have replicated this finding and confirmed that the effect is number-specific rather than domain-general or attentional. We present a neurocomputational model postulating population-coding underlying rapid numerical averaging to account for our findings. According to this model, stimulation of the parietal cortex elevates neural activity in number-tuned dedicated detectors, leading to increase in the system's signal-to-noise level and thus resulting in more precise estimations.

Decisions reduce sensitivity to subsequent information

Behavioural studies over half a century indicate that making categorical choices alters beliefs about the state of the world. People seem biased to confirm previous choices, and to suppress contradicting information. These choice-dependent biases imply a fundamental bound of human rationality. However, it remains unclear whether these effects extend to lower level decisions, and only little is known about the computational mechanisms underlying them. Building on the framework of sequential-sampling models of decision-making, we developed novel psychophysical protocols that enable us to dissect quantitatively how choices affect the way decision-makers accumulate additional noisy evidence. We find robust choice-induced biases in the accumulation of abstract numerical (experiment 1) and low-level perceptual (experiment 2) evidence. These biases deteriorate estimations of the mean value of the numerical sequence (experiment 1) and reduce the likelihood to revise decisions (experiment 2). Computational modelling reveals that choices trigger a reduction of sensitivity to subsequent evidence via multiplicative gain modulation, rather than shifting the decision variable towards the chosen alternative in an additive fashion. Our results thus show that categorical choices alter the evidence accumulation mechanism itself, rather than just its outcome, rendering the decision-maker less sensitive to new information.

Non-monotonic Temporal-Weighting Indicates a Dynamically Modulated Evidence-Integration Mechanism

Perceptual decisions are thought to be mediated by a mechanism of sequential sampling and integration of noisy evidence whose temporal weighting profile affects the decision quality. To examine temporal weighting, participants were presented with two brightness-fluctuating disks for 1, 2 or 3 seconds and were requested to choose the overall brighter disk at the end of each trial. By employing a signal-perturbation method, which deploys across trials a set of systematically controlled temporal dispersions of the same overall signal, we were able to quantify the participants' temporal weighting profile. Results indicate that, for intervals of 1 or 2 sec, participants exhibit a primacy-bias. However, for longer stimuli (3-sec) the temporal weighting profile is non-monotonic, with concurrent primacy and recency, which is inconsistent with the predictions of previously suggested computational models of perceptual decision-making (drift-diffusion and Ornstein-Uhlenbeck processes). We propose a novel, dynamic variant of the leaky-competing accumulator model as a potential account for this finding, and we discuss potential neural mechanisms.