Extracting temporal regularities and relations from experience/observation is critical for organisms' adaptiveness (communication, foraging, predation, prediction) in their ecological niches. Therefore, it is not surprising that the internal clock that enables the perception of seconds-to-minutes-long intervals (interval timing) is evolutionarily well-preserved across many species of animals. This comparative claim is primarily supported by the fact that the timing behavior of many vertebrates exhibits common statistical signatures (e.g., on-average accuracy, scalar variability, positive skew). These ubiquitous statistical features of timing behaviors serve as empirical benchmarks for modelers in their efforts to unravel the processing dynamics of the internal clock (namely answering how internal clock "ticks"). In this chapter, we introduce prominent (neuro)computational approaches to modeling interval timing at a level that can be understood by general audience. These models include Treisman's pacemaker accumulator model, the information processing variant of scalar expectancy theory, the striatal beat frequency model, behavioral expectancy theory, the learning to time model, the time-adaptive opponent Poisson drift-diffusion model, time cell models, and neural trajectory models. Crucially, we discuss these models within an overarching conceptual framework that categorizes different models as threshold vs. clock-adaptive models and as dedicated clock/ramping vs. emergent time/population code models.

Rahnev & Denison (R&D) argue against normative theories and in favor of a more descriptive "standard observer model" of perceptual decision making. We agree with the authors in many respects, but we argue that optimality (specifically, reward-rate maximization) has proved demonstrably useful as a hypothesis, contrary to the authors' claims.

Background: We assessed whether evidence accumulation could be observed in the BOLD signal during perceptual decision making. This presents a challenge since the hemodynamic response is slow, while perceptual decisions are typically fast.New method: Guided by theoretical predictions of the drift diffusion model, we slowed down decisions by penalizing participants for incorrect responses. Second, we distinguished BOLD activity related to stimulus detection (modeled using a boxcar) from activity related to integration (modeled using a ramp) by minimizing the collinearity of GLM regressors. This was achieved by dissecting a boxcar into its two most orthogonal components: an "up-ramp" and a "down-ramp." Third, we used a control condition in which stimuli and responses were similar to the experimental condition, but that did not engage evidence accumulation of the stimuli.Results: The results revealed an absence of areas in parietal cortex that have been proposed to drive perceptual decision making but have recently come into question; and newly identified regions that are candidates for involvement in evidence accumulation.Comparison with existing methods: Previous fMRI studies have either used fast perceptual decision making, which precludes the measurement of evidence accumulation, or slowed down responses by gradually revealing stimuli. The latter approach confounds perceptual detection with evidence accumulation because accumulation is constrained by perceptual input.Conclusions: We slowed down the decision making process itself while leaving perceptual information intact. This provided a more sensitive and selective observation of brain regions associated with the evidence accumulation processes underlying perceptual decision making than previous methods. (C) 2017 Elsevier B.V. All rights reserved.

Increased dopamine may cause our internal clock to slow down, creating a sense of speeding time

The fundamental assumption of pacemaker accumulator models of interval timing is that timed behavior relies on the accumulation of brain-derived clock signals. Following this theoretical tradition, a recent series of interval timing models has formulated the processing dynamics of timing behavior within the drift-diffusion decision theoretic framework, which has been traditionally applied to explain accuracy and response times in perceptual decision making. The generative processes assumed by these models and their key features can be implemented by neural populations given simple assumptions, and their predictions have received recent support from electrophysiological studies. This paper discusses the conceptual links of the diffusion model of interval timing to other prominent timing models and interprets recent electrophysiological evidence in relation to its predictions.

Weber's law is the canonical scale-invariance law in psychology: when the intensities of 2 stimuli are scaled by any value k, the just-noticeable-difference between them also scales by k. A diffusion model that approximates a spike-counting process accounts for Weber's law (Link, 1992), but there exist surprising corollaries of this account that have not yet been described or tested. We show that (a) this spike-counting diffusion model predicts time-scale invariant decision time distributions in perceptual decision making, and time-scale invariant response time (RT) distributions in interval timing; (b) for 2-choice perceptual decisions, the model predicts equal accuracy but faster responding for stimulus pairs with equally scaled-up intensities; (c) the coefficient of variation (CV) of decision times should remain constant across average intensity scales, but should otherwise decrease as a specific function of stimulus discriminability and speed-accuracy trade-off; and (d) for timing tasks, RT CVs should be constant for all durations, and RT skewness should always equal 3 times the CV. We tested these predictions using visual, auditory and vibrotactile decision tasks and visual interval timing tasks in humans. The data conformed closely to the predictions in all modalities. These results support a unified theory of decision making and timing in terms of a common, underlying spike-counting process, compactly represented as a diffusion process.

Hasty decisions often lead to poor choices, whereas accurate decisions are ineffective if they take too long. Thus, good choices require cognitive mechanisms to determine the appropriate balance between speed and accuracy, and to control decision processing accordingly. This balance is referred to as the speed-accuracy trade-off (SAT) and the mechanisms by which it is determined and imposed are the subject of this Frontiers Research Topic. Given the near-ubiquity of the SAT across species and experimental tasks, it is not surprising that a wide range of methods have been used to investigate it. Our aim is to provide a unified view of the SAT in light of this diverse methodology. Computationally, decision making and the SAT are well characterized by the framework of bounded integration, providing a solid foundation for this view. Under this framework, noisy evidence for the available choices is added up (integrated) until the running total for one of them reaches a criterion (the bound). The SAT is readily controlled by the bound, where a higher bound favors accuracy at the expense of speed and vice versa. In this collection, we use bounded integration as a reference point for considering the factors that determine the optimal balance between speed and accuracy, the interpretation of behavior by different models from this general class, and the neural implementation of the computations captured by these models. Articles herein further consider conditions under which the above descriptions of the SAT and bounded integration do not explain behavior, and the utility of the SAT for manipulating the context of decisions.

We present new findings that distinguish drift diffusion models (DDMs) from the linear ballistic accumulator (LBA) model as descriptions of human behavior in a two-alternative forced-choice reward maximization (Rmax) task. Previous comparisons have not considered Rmax tasks, and differences identified between the models' predictions have centered on practice effects. Unlike the parameter-free optimal performance curves of the pure DDM, the extended DDM and LBA predict families of curves depending on their additional parameters, and those of the LBA show significant differences from the DDMs, especially for poorly discriminable stimuli that incur high error rates. Moreover, fits to behavior reveal that the LBA and DDM provide different interpretations of behavior as stimulus discriminability increases. Trends for threshold setting (caution) in the DDMs are consistent between fits, while in the corresponding LBA fits, thresholds interact with distributions of starting points in a complex manner that depends upon parameter constraints. Our results suggest that reinterpretation of LBA parameters may be necessary in modeling the Rmax paradigm.

Many perceptual decision making models posit that participants accumulate noisy evidence over time to improve the accuracy of their decisions, and that in free response tasks, participants respond when the accumulated evidence reaches a decision threshold. Research on the neural correlates of these models' components focuses primarily on evidence accumulation. Far less attention has been paid to the neural correlates of decision thresholds, reflecting the final commitment to a decision. Inspired by a model of bistable neural activity that implements a decision threshold, we reinterpret human lateralized readiness potentials (LRPs) as reflecting the crossing of a decision threshold. Interestingly, this threshold crossing preserves signatures of a drift-diffusion process of evidence accumulation that feeds in to the threshold mechanism. We show that, as our model predicts, LRP amplitudes and growth rates recorded while participants performed a motion discrimination task correlate with individual differences in behaviorally-estimated prior beliefs, decision thresholds and evidence accumulation rates. As such LRPs provide a useful measure to test dynamical models of both evidence accumulation and decision commitment processes non-invasively.

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.

The analysis of response rates has been highly influential in psychology, giving rise to many prominent theories of learning. There is, however, growing interest in explaining response rates, not as a global response to associations or value, but as a decision about how to space responses in time. Recently, researchers have shown that humans and mice can time a single response optimally; that is, in a way that maximizes reward. Here, we use the well-established differential reinforcement of low rates (DRL) timing task to show that humans and rats come close to optimizing reinforcement rate, but respond systematically faster than they should.

The majority of two-alternative forced choice (2AFC) psychophysics studies have examined speed-accuracy trade-offs either in free-response or fixed viewing time paradigms with no hard time constraints on responding. Under response deadlines, reward maximization requires participants to modulate decision thresholds over the course of a trial such that when the deadline arrives a response is ensured despite the possible reduction of accuracy to the chance level. Importantly, this normative threshold collapsing process should take account not only of the deadline time but also the participants level of timing uncertainty about the deadline. For instance, for a given level of timing uncertainty participants facing a more stringent deadline and for a given deadline participants with higher timing uncertainty should start collapsing thresholds earlier within a trial(Frazier, P., & Yu, A. J. (2008). Sequential hypothesis testing under stochastic deadlines.Advances In Neural Information Processing Systems, 20, 465-472). We tested human participants in random dot motion discrimination task in three different conditions: free-response paradigm and with two different response deadlines (800 and 1000 ms). We also quantified the level of timing uncertainty using two different temporal reproduction tasks. Micro speed-accuracy trade-off curves suggested that participants might have collapsed thresholds within a trial when facing a response deadline; however, observed performance diverged markedly from optimality. Participants did not take account of either response deadline times or their timing uncertainty. Our findings define a limit on the optimal temporal risk assessment performance of humans. We also consider other possible sources of observed reduction in accuracy as response times approach the deadline, through simulations of data using empirical model parameters.

Adolescent decision-making has been described as impulsive and suboptimal in the presence of incentives. In this study we examined the neural substrates of adolescent decision-making using a perceptual discrimination task for which small and large rewards were associated with correctly detecting the direction of motion of a cloud of moving dots. Adults showed a reward bias of faster reaction times on trials for which the direction of motion was associated with a large reward. Adolescents, in contrast, were slower to make decisions on trials associated with large rewards. This behavioral pattern in adolescents was paralleled by greater recruitment of fronto-parietal regions important in representing the accumulation of evidence sufficient for selecting one choice over its alternative and the certainty of that choice. The findings suggest that when large incentives are dependent on performance, adolescents may require more evidence to accumulate prior to responding, to be certain to maximize their gains. Adults, in contrast, appear to be quicker in evaluating the evidence for a decision when primed by rewards. Overall these findings suggest that rather than reacting hastily, adolescents can be incentivized to take more time to make decisions when large rewards are at stake. A video abstract of this article can be viewed at http://youtu.be/1g4F5vzFDl0

Pacemaker-accumulator (PA) systems have been the most popular kind of timing model in the half-century since their introduction by Treisman (1963). Many alternative timing models have been designed predicated on different assumptions, though the dominant PA model during this period — Gibbon and Church’s Scalar Expectancy Theory (SET) — invokes most of them. As in Treisman, SET’s implementation assumes a fixed-rate clock-pulse generator and encodes durations by storing average pulse counts; unlike Treisman’s model, SET’s decision process invokes Weber’s law of magnitude-comparison to account for timescale-invariant temporal precision in animal behavior. This is one way to deal with the ‘Poisson timing’ issue, in which relative temporal precision increases for longer durations, contrafactually, in a simplified version of Treisman’s model. First, we review the fact that this problem does not afflict Treisman’s model itself due to a key assumption not shared by SET. Second, we develop a contrasting PA model, an extension of Killeen and Fetterman’s Behavioral Theory of Timing that accumulates Poisson pulses up to a fixed criterion level, with pulse rates adapting to time different intervals. Like Treisman’s model, this time-adaptive, opponent Poisson, drift–diffusion model accounts for timescale invariance without first assuming Weber’s law. It also makes new predictions about response times and learning speed and connects interval timing to the popular drift–diffusion model of perceptual decision making. With at least three different routes to timescale invariance, the PA model family can provide a more compelling account of timed behavior than may be generally appreciated.

The model consists of units that represent the firing rate V of entire neural populations. We assume the inputs to a population have Gaussian white noise added to them (Wiener process W ), which models the input variability introduced by stochastic spike-generation in the input populations, and is weighted by a factor c. V (t) can thus be interpreted as the dynamic rate parameter of a nonhomogeneous Poisson process; it is defined as follows, with the subscript i denoting ‘units’ representing distinct populations, and wij denoting entries of the connection matrix linking units into a network:

Most psychological models of perceptual decision making are of the accumulation-to-threshold variety. The neural basis of accumulation in parietal and prefrontal cortex is therefore a topic of great interest in neuroscience. In contrast, threshold mechanisms have received less attention, and their neural basis has usually been sought in subcortical structures. Here I analyze a model of a decision threshold that can be implemented in the same cortical areas as evidence accumulators, and whose behavior bears on two open questions in decision neuroscience: (1) When ramping activity is observed in a brain region during decision making, does it reflect evidence accumulation? (2) Are changes in speed-accuracy tradeoffs and response biases more likely to be achieved by changes in thresholds, or in accumulation rates and starting points? The analysis suggests that task-modulated ramping activity, by itself, is weak evidence that a brain area mediates evidence accumulation as opposed to threshold readout; and that signs of modulated accumulation are as likely to indicate threshold adaptation as adaptation of starting points and accumulation rates. These conclusions imply that how thresholds are modeled can dramatically impact accumulator-based interpretations of this data.

Optimal performance and physically plausible mechanisms for achieving it have been completely characterized for a general class of two-alternative, free response decision making tasks, and data suggest that humans can implement the optimal procedure. The situation is more complicated when the number of alternatives is greater than two and subjects are free to respond at any time, partly due to the fact that there is no generally applicable statistical test for deciding optimally in such cases. However, here, too, analytical approximations to optimality that are physically and psychologically plausible have been analyzed. These analyses leave open questions that have begun to be addressed: (1) How are near-optimal model parameterizations learned from experience? (2) What if a continuum of decision alternatives exists? (3) How can neurons' broad tuning curves be incorporated into an optimal-performance theory? We present a possible answer to all of these questions in the form of an extremely simple, reward-modulated Hebbian learning rule by which a neural network learns to approximate the multihypothesis sequential probability ratio test.