Representing stimulus motion with waves in adaptive neural fields.

Traveling waves of neural activity emerge in cortical networks both spontaneously and in response to stimuli. The spatiotemporal structure of waves can indicate the information they encode and the physiological processes that sustain them. Here, we investigate the stimulus-response relationships of traveling waves emerging in adaptive neural fields as a model of visual motion processing. Neural field equations model the activity of cortical tissue as a continuum excitable medium, and adaptive processes provide negative feedback, generating localized activity patterns. Synaptic connectivity in our model is described by an integral kernel that weakens dynamically due to activity-dependent synaptic depression, leading to marginally stable traveling fronts (with attenuated backs) or pulses of a fixed speed. Our analysis quantifies how weak stimuli shift the relative position of these waves over time, characterized by a wave response function we obtain perturbatively. Persistent and continuously visible stimuli model moving visual objects. Intermittent flashes that hop across visual space can produce the experience of smooth apparent visual motion. Entrainment of waves to both kinds of moving stimuli are well characterized by our theory and numerical simulations, providing a mechanistic description of the perception of visual motion.

Fast decisions reflect biases, slow decisions do not.

Decisions are often made by heterogeneous groups of individuals, each with distinct initial biases and access to information of different quality. We show that in large groups of independent agents who accumulate evidence the first to decide are those with the strongest initial biases. Their decisions align with their initial bias, regardless of the underlying truth. In contrast, agents who decide last make decisions as if they were initially unbiased, and hence make better choices. We obtain asymptotic expressions in the large population limit that quantify how agents' initial inclinations shape early decisions. Our analysis shows how bias, information quality, and decision order interact in non-trivial ways to determine the reliability of decisions in a group.

Learning efficient representations of environmental priors in working memory.

Experience shapes our expectations and helps us learn the structure of the environment. Inference models render such learning as a gradual refinement of the observer's estimate of the environmental prior. For instance, when retaining an estimate of an object's features in working memory, learned priors may bias the estimate in the direction of common feature values. Humans display such biases when retaining color estimates on short time intervals. We propose that these systematic biases emerge from modulation of synaptic connectivity in a neural circuit based on the experienced stimulus history, shaping the persistent and collective neural activity that encodes the stimulus estimate. Resulting neural activity attractors are aligned to common stimulus values. Using recently published human response data from a delayed-estimation task in which stimuli (colors) were drawn from a heterogeneous distribution that did not necessarily correspond with reported population biases, we confirm that most subjects' response distributions are better described by experience-dependent learning models than by models with fixed biases. This work suggests systematic limitations in working memory reflect efficient representations of inferred environmental structure, providing new insights into how humans integrate environmental knowledge into their cognitive strategies.

Bed nucleus of the stria terminalis GABA neurons are necessary for changes in foraging behaviour following an innate threat.

Foraging is a universal behaviour that has co-evolved with predation pressure. We investigated the role of the bed nucleus of the stria terminalis (BNST) GABA neurons in robotic and live predator threat processing and their consequences in post-threat encounter foraging. Both robotic and live predator interactions increased BNST GABA neuron activity. Mice were trained to procure food in a laboratory-based foraging apparatus in which food pellets were placed at incrementally greater distances from a nest zone. After mice learned to forage, they were exposed to a robotic or live predator threat, while BNST GABA neurons were chemogenetically inhibited. Post-robotic threat encounter, mice spent more time in the nest zone, but other foraging parameters were unchanged compared with pre-encounter behaviour. Inhibition of BNST GABA neurons had no effect on foraging behaviour post-robotic threat encounter. Following live predator exposure, control mice spent significantly more time in the nest zone, increased their latency to successfully forage, and significantly altered their overall foraging performance. Inhibition of BNST GABA neurons during live predator exposure prevented changes in foraging behaviour from developing after a live predator threat. BNST GABA neuron inhibition did not alter foraging behaviour during robotic or live predator threats. We conclude that these results demonstrate that while both robotic and live predator encounters effectively intrude on foraging behaviour, the perceived risk and behavioural consequences of the threat are distinguishable. Additionally, BNST GABA neurons may play a role in the integration of prior innate predator threat experience that results in hypervigilance during post-encounter foraging behaviour.

Multiscale motion and deformation of bumps in stochastic neural fields with dynamic connectivity.

The distinct timescales of synaptic plasticity and neural activity dynamics play an important role in the brain's learning and memory systems. Activity-dependent plasticity reshapes neural circuit architecture, determining spontaneous and stimulus-encoding spatiotemporal patterns of neural activity. Neural activity bumps maintain short term memories of continuous parameter values, emerging in spatially-organized models with short term excitation and long-range inhibition. Previously, we demonstrated nonlinear Langevin equations derived using an interface method accurately describe the dynamics of bumps in continuum neural fields with separate excitatory/inhibitory populations. Here we extend this analysis to incorporate effects of slow short term plasticity that modifies connectivity described by an integral kernel. Linear stability analysis adapted to these piecewise smooth models with Heaviside firing rates further indicate how plasticity shapes bumps' local dynamics. Facilitation (depression), which strengthens (weakens) synaptic connectivity originating from active neurons, tends to increase (decrease) stability of bumps when acting on excitatory synapses. The relationship is inverted when plasticity acts on inhibitory synapses. Multiscale approximations of the stochastic dynamics of bumps perturbed by weak noise reveal the plasticity variables evolve to slowly diffusing and blurred versions of that arising in the stationary solution. Nonlinear Langevin equations associated with bump positions or interfaces coupled to slowly evolving projections of plasticity variables accurately describe the wandering of bumps underpinned by these smoothed synaptic efficacy profiles.

Bed Nucleus of the Stria Terminalis GABA neurons are necessary for changes in foraging behavior following an innate threat.

Foraging is a universal behavior that has co-evolved with predation pressure. We investigated the role of bed nucleus of the stria terminalis (BNST) GABA neurons in robotic and live predator threat processing and their consequences in post-threat encounter foraging. Mice were trained to procure food in a laboratory-based foraging apparatus in which food pellets were placed at discrete and incrementally greater distances from a nest zone. After mice learned to forage, they were exposed to either a robotic or live predator threat, while BNST GABA neurons were chemogenetically inhibited. Post-robotic threat encounter, mice spent more time in the nest zone, but other foraging parameters were unchanged compared to pre-encounter behavior. Inhibition of BNST GABA neurons had no effect on foraging behavior post-robotic threat encounter. Following live predator exposure, control mice spent significantly more time in the nest zone, increased their latency to successfully forage, and their overall foraging performance was significantly a ltered. I nhibition o f BNST GABA neurons during live predator exposure prevented changes in foraging behavior from developing after live predator threat. BNST GABA neuron inhibition did not alter foraging behavior during robotic or live predator threat. We conclude that while both robotic and live predator encounter effectively intrude on foraging behavior, the perceived risk and behavioral consequence of the threats are distinguishable. Additionally, BNST GABA neurons may play a role in the integration of prior innate predator threat experience that results in hypervigilance during post-encounter foraging behavior.

An emergent temporal basis set robustly supports cerebellar time-series learning.

The cerebellum is considered a "learning machine" essential for time interval estimation underlying motor coordination and other behaviors. Theoretical work has proposed that the cerebellum's input recipient structure, the granule cell layer (GCL), performs pattern separation of inputs that facilitates learning in Purkinje cells (P-cells). However, the relationship between input reformatting and learning has remained debated, with roles emphasized for pattern separation features from sparsification to decorrelation. We took a novel approach by training a minimalist model of the cerebellar cortex to learn complex time-series data from time-varying inputs, typical during movements. The model robustly produced temporal basis sets from these inputs, and the resultant GCL output supported better learning of temporally complex target functions than mossy fibers alone. Learning was optimized at intermediate threshold levels, supporting relatively dense granule cell activity, yet the key statistical features in GCL population activity that drove learning differed from those seen previously for classification tasks. These findings advance testable hypotheses for mechanisms of temporal basis set formation and predict that moderately dense population activity optimizes learning.NEW & NOTEWORTHY During movement, mossy fiber inputs to the cerebellum relay time-varying information with strong intrinsic relationships to ongoing movement. Are such mossy fibers signals sufficient to support Purkinje signals and learning? In a model, we show how the GCL greatly improves Purkinje learning of complex, temporally dynamic signals relative to mossy fibers alone. Learning-optimized GCL population activity was moderately dense, which retained intrinsic input variance while also performing pattern separation.

Representing stimulus motion with waves in adaptive neural fields.

Traveling waves of neural activity emerge in cortical networks both spontaneously and in response to stimuli. The spatiotemporal structure of waves can indicate the information they encode and the physiological processes that sustain them. Here, we investigate the stimulus-response relationships of traveling waves emerging in adaptive neural fields as a model of visual motion processing. Neural field equations model the activity of cortical tissue as a continuum excitable medium, and adaptive processes provide negative feedback, generating localized activity patterns. Synaptic connectivity in our model is described by an integral kernel that weakens dynamically due to activity-dependent synaptic depression, leading to marginally stable traveling fronts (with attenuated backs) or pulses of a fixed speed. Our analysis quantifies how weak stimuli shift the relative position of these waves over time, characterized by a wave response function we obtain perturbatively. Persistent and continuously visible stimuli model moving visual objects. Intermittent flashes that hop across visual space can produce the experience of smooth apparent visual motion. Entrainment of waves to both kinds of moving stimuli are well characterized by our theory and numerical simulations, providing a mechanistic description of the perception of visual motion.

Normative decision rules in changing environments.

Models based on normative principles have played a major role in our understanding of how the brain forms decisions. However, these models have typically been derived for simple, stable conditions, and their relevance to decisions formed under more naturalistic, dynamic conditions is unclear. We previously derived a normative decision model in which evidence accumulation is adapted to fluctuations in the evidence-generating process that occur during a single decision (Glaze et al., 2015), but the evolution of commitment rules (e.g. thresholds on the accumulated evidence) under dynamic conditions is not fully understood. Here, we derive a normative model for decisions based on changing contexts, which we define as changes in evidence quality or reward, over the course of a single decision. In these cases, performance (reward rate) is maximized using decision thresholds that respond to and even anticipate these changes, in contrast to the static thresholds used in many decision models. We show that these adaptive thresholds exhibit several distinct temporal motifs that depend on the specific predicted and experienced context changes and that adaptive models perform robustly even when implemented imperfectly (noisily). We further show that decision models with adaptive thresholds outperform those with constant or urgency-gated thresholds in accounting for human response times on a task with time-varying evidence quality and average reward. These results further link normative and neural decision-making while expanding our view of both as dynamic, adaptive processes that update and use expectations to govern both deliberation and commitment.

How do we make good choices? Should I have cake or yoghurt for breakfast? The strategies we use to make decisions are important not just for our daily lives, but also for learning more about how the brain works. Decision-making strategies have two components: first, a deliberation period (when we gather information to determine which choice is 'best'); and second, a decision 'rule' (which tells us when to stop deliberating and commit to a choice). Although deliberation is relatively well-understood, less is known about the decision rules people use, or how those rules produce different outcomes. Another issue is that even the simplest decisions must sometimes adapt to a changing world. For example, if it starts raining while you are deciding which route to walk into town, you would probably choose the driest route ­ even if it did not initially look the best. However, most studies of decision strategies have assumed that the decision-maker's environment does not change during the decision process. In other words, we know much less about the decision rules used in real-life situations, where the environment changes. Barendregt et al. therefore wanted to extend the approaches previously used to study decisions in static environments, to determine which decision rules might be best suited to more realistic environments that change over time. First, Barendregt et al. constructed a computer simulation of decision-making with environmental changes built in. These changes were either alterations in the quality of evidence for or against a particular choice, or the 'reward' from a choice, i.e., feedback on how good the decision was. They then used the computer simulation to model single decisions where these changes took place. These virtual experiments showed that the best performance ­ for example, the most accurate decisions ­ resulted when the threshold for moving from deliberation (i.e., considering the evidence) to selecting an option could respond to, or even anticipate, the changing situations. Importantly, the simulations' results also predicted real-world choices made by human participants when given a decision-making task with similar variations in evidence and reward over time. In other words, the virtual decision-making rules could explain real behavior. This study sheds new light on how we make decisions in a changing environment. In the future, Barendregt et al. hope that this will contribute to a broader understanding of decision-making and behavior in a wide range of contexts, from psychology to economics and even ecology.

Stochastic dynamics of social patch foraging decisions.

Animals typically forage in groups. Social foraging can help animals avoid predation and decrease their uncertainty about the richness of food resources. Despite this, theoretical mechanistic models of patch foraging have overwhelmingly focused on the behavior of single foragers. In this study, we develop a mechanistic model that accounts for the behavior of individuals foraging together and departing food patches following an evidence accumulation process. Each individual's belief about patch quality is represented by a stochastically accumulating variable, which is coupled to another's belief to represent the transfer of information. We consider a cohesive group, and model information sharing by considering both intermittent pulsatile coupling (only communicate decision to leave) and continuous diffusive coupling (communicate throughout the deliberation process). Groups employing pulsatile coupling can obtain higher foraging efficiency, which depends more strongly on the coupling parameter compared to those using diffusive coupling. Conversely, groups using diffusive coupling are more robust to changes and heterogeneities in belief weighting and departure criteria. Efficiency is measured by a reward rate function that balances the amount of energy accumulated against the time spent in a patch, computed by solving an ordered first passage time problem for the patch departures of each individual. Using synthetic departure time data, we can distinguish between the two modes of communication and identify the model parameters. Our model establishes a social patch foraging framework to identify deliberative decision strategies and forms of social communication, and to allow model fitting to field data from foraging animal groups.

Suboptimal human inference can invert the bias-variance trade-off for decisions with asymmetric evidence.

Solutions to challenging inference problems are often subject to a fundamental trade-off between: 1) bias (being systematically wrong) that is minimized with complex inference strategies, and 2) variance (being oversensitive to uncertain observations) that is minimized with simple inference strategies. However, this trade-off is based on the assumption that the strategies being considered are optimal for their given complexity and thus has unclear relevance to forms of inference based on suboptimal strategies. We examined inference problems applied to rare, asymmetrically available evidence, which a large population of human subjects solved using a diverse set of strategies that varied in form and complexity. In general, subjects using more complex strategies tended to have lower bias and variance, but with a dependence on the form of strategy that reflected an inversion of the classic bias-variance trade-off: subjects who used more complex, but imperfect, Bayesian-like strategies tended to have lower variance but higher bias because of incorrect tuning to latent task features, whereas subjects who used simpler heuristic strategies tended to have higher variance because they operated more directly on the observed samples but lower, near-normative bias. Our results help define new principles that govern individual differences in behavior that depends on rare-event inference and, more generally, about the information-processing trade-offs that can be sensitive to not just the complexity, but also the optimality, of the inference process.

Strategy-dependent effects of working-memory limitations on human perceptual decision-making.

Deliberative decisions based on an accumulation of evidence over time depend on working memory, and working memory has limitations, but how these limitations affect deliberative decision-making is not understood. We used human psychophysics to assess the impact of working-memory limitations on the fidelity of a continuous decision variable. Participants decided the average location of multiple visual targets. This computed, continuous decision variable degraded with time and capacity in a manner that depended critically on the strategy used to form the decision variable. This dependence reflected whether the decision variable was computed either: (1) immediately upon observing the evidence, and thus stored as a single value in memory; or (2) at the time of the report, and thus stored as multiple values in memory. These results provide important constraints on how the brain computes and maintains temporally dynamic decision variables.

Working memory, the brain's ability to temporarily store and recall information, is a critical part of decision making ­ but it has its limits. The brain can only store so much information, for so long. Since decisions are not often acted on immediately, information held in working memory 'degrades' over time. However, it is unknown whether or not this degradation of information over time affects the accuracy of later decisions. The tactics that people use, knowingly or otherwise, to store information in working memory also remain unclear. Do people store pieces of information such as numbers, objects and particular details? Or do they tend to compute that information, make some preliminary judgement and recall their verdict later? Does the strategy chosen impact people's decision-making? To investigate, Schapiro et al. devised a series of experiments to test whether the limitations of working memory, and how people store information, affect the accuracy of decisions they make. First, participants were shown an array of colored discs on a screen. Then, either immediately after seeing the disks or a few seconds later, the participants were asked to recall the position of one of the disks they had seen, or the average position of all the disks. This measured how much information degraded for a decision based on multiple items, and how much for a decision based on a single item. From this, the method of information storage used to make a decision could be inferred. Schapiro et al. found that the accuracy of people's responses worsened over time, whether they remembered the position of each individual disk, or computed their average location before responding. The greater the delay between seeing the disks and reporting their location, the less accurate people's responses tended to be. Similarly, the more disks a participant saw, the less accurate their response became. This suggests that however people store information, if working memory reaches capacity, decision-making suffers and that, over time, stored information decays. Schapiro et al. also noticed that participants remembered location information in different ways depending on the task and how many disks they were shown at once. This suggests people adopt different strategies to retain information momentarily. In summary, these findings help to explain how people process and store information to make decisions and how the limitations of working memory impact their decision-making ability. A better understanding of how people use working memory to make decisions may also shed light on situations or brain conditions where decision-making is impaired.

Distinct Excitatory and Inhibitory Bump Wandering in a Stochastic Neural Field.

Localized persistent cortical neural activity is a validated neural substrate of parametric working memory. Such activity "bumps" represent the continuous location of a cue over several seconds. Pyramidal (excitatory (E)) and interneuronal (inhibitory (I)) subpopulations exhibit tuned bumps of activity, linking neural dynamics to behavioral inaccuracies observed in memory recall. However, many bump attractor models collapse these subpopulations into a single joint E/I(lateral inhibitory) population and do not consider the role of interpopulation neural architecture and noise correlations. Both factors have a high potential to impinge upon the stochastic dynamics of these bumps, ultimately shaping behavioral response variance. In our study, we consider a neural field model with separate E/I populations and leverage asymptotic analysis to derive a nonlinear Langevin system describing E/I bump interactions. While the E bump attracts the I bump, the I bump stabilizes but can also repel the E bump, which can result in prolonged relaxation dynamics when both bumps are perturbed. Furthermore, the structure of noise correlations within and between subpopulations strongly shapes the variance in bump position. Surprisingly, higher interpopulation correlations reduce variance.

Uncertainty drives deviations in normative foraging decision strategies.

Nearly all animals forage to acquire energy for survival through efficient search and resource harvesting. Patch exploitation is a canonical foraging behaviour, but there is a need for more tractable and understandable mathematical models describing how foragers deal with uncertainty. To provide such a treatment, we develop a normative theory of patch foraging decisions, proposing mechanisms by which foraging behaviours emerge in the face of uncertainty. Our model foragers statistically and sequentially infer patch resource yields using Bayesian updating based on their resource encounter history. A decision to leave a patch is triggered when the certainty of the patch type or the estimated yield of the patch falls below a threshold. The time scale over which uncertainty in resource availability persists strongly impacts behavioural variables like patch residence times and decision rules determining patch departures. When patch depletion is slow, as in habitat selection, departures are characterized by a reduction of uncertainty, suggesting that the forager resides in a low-yielding patch. Uncertainty leads patch-exploiting foragers to overharvest (underharvest) patches with initially low (high) resource yields in comparison with predictions of the marginal value theorem. These results extend optimal foraging theory and motivate a variety of behavioural experiments investigating patch foraging behaviour.

Hive geometry shapes the recruitment rate of honeybee colonies.

Honey bees make decisions regarding foraging and nest-site selection in groups ranging from hundreds to thousands of individuals. To effectively make these decisions, bees need to communicate within a spatially distributed group. However, the spatiotemporal dynamics of honey bee communication have been mostly overlooked in models of collective decisions, focusing primarily on mean field models of opinion dynamics. We analyze how the spatial properties of the nest or hive, and the movement of individuals with different belief states (uncommitted or committed) therein affect the rate of information transmission using spatially-extended models of collective decision-making within a hive. Honeybees waggle-dance to recruit conspecifics with an intensity that is a threshold nonlinear function of the waggler concentration. Our models range from treating the hive as a chain of discrete patches to a continuous line (long narrow hive). The combination of population-thresholded recruitment and compartmentalized populations generates tradeoffs between rapid information propagation with strong population dispersal and recruitment failures resulting from excessive population diffusion and also creates an effective colony-level signal-detection mechanism whereby recruitment to low quality objectives is blocked.

Heterogeneity Improves Speed and Accuracy in Social Networks.

How does temporally structured private and social information shape collective decisions? To address this question we consider a network of rational agents who independently accumulate private evidence that triggers a decision upon reaching a threshold. When seen by the whole network, the first agent's choice initiates a wave of new decisions; later decisions have less impact. In heterogeneous networks, first decisions are made quickly by impulsive individuals who need little evidence to make a choice but, even when wrong, can reveal the correct options to nearly everyone else. We conclude that groups comprised of diverse individuals can make more efficient decisions than homogenous ones.

A hierarchical model of perceptual multistability involving interocular grouping.

Ambiguous visual images can generate dynamic and stochastic switches in perceptual interpretation known as perceptual rivalry. Such dynamics have primarily been studied in the context of rivalry between two percepts, but there is growing interest in the neural mechanisms that drive rivalry between more than two percepts. In recent experiments, we showed that split images presented to each eye lead to subjects perceiving four stochastically alternating percepts (Jacot-Guillarmod et al. Vision research, 133, 37-46, 2017): two single eye images and two interocularly grouped images. Here we propose a hierarchical neural network model that exhibits dynamics consistent with our experimental observations. The model consists of two levels, with the first representing monocular activity, and the second representing activity in higher visual areas. The model produces stochastically switching solutions, whose dependence on task parameters is consistent with four generalized Levelt Propositions, and with experiments. Moreover, dynamics restricted to invariant subspaces of the model demonstrate simpler forms of bistable rivalry. Thus, our hierarchical model generalizes past, validated models of binocular rivalry. This neuromechanistic model also allows us to probe the roles of interactions between populations at the network level. Generalized Levelt's Propositions hold as long as feedback from the higher to lower visual areas is weak, and the adaptation and mutual inhibition at the higher level is not too strong. Our results suggest constraints on the architecture of the visual system and show that complex visual stimuli can be used in perceptual rivalry experiments to develop more detailed mechanistic models of perceptual processing.

Bayesian Evidence Accumulation on Social Networks.

To make decisions we are guided by the evidence we collect and the opinions of friends and neighbors. How do we combine our private beliefs with information we obtain from our social network? To understand the strategies humans use to do so, it is useful to compare them to observers that optimally integrate all evidence. Here we derive network models of rational (Bayes optimal) agents who accumulate private measurements and observe the decisions of their neighbors to make an irreversible choice between two options. The resulting information exchange dynamics has interesting properties: When decision thresholds are asymmetric, the absence of a decision can be increasingly informative over time. In a recurrent network of two agents, the absence of a decision can lead to a sequence of belief updates akin to those in the literature on common knowledge. On the other hand, in larger networks a single decision can trigger a cascade of agreements and disagreements that depend on the private information agents have gathered. Our approach provides a bridge between social decision making models in the economics literature, which largely ignore the temporal dynamics of decisions, and the single-observer evidence accumulator models used widely in neuroscience and psychology.

Analyzing dynamic decision-making models using Chapman-Kolmogorov equations.

Decision-making in dynamic environments typically requires adaptive evidence accumulation that weights new evidence more heavily than old observations. Recent experimental studies of dynamic decision tasks require subjects to make decisions for which the correct choice switches stochastically throughout a single trial. In such cases, an ideal observer's belief is described by an evolution equation that is doubly stochastic, reflecting stochasticity in the both observations and environmental changes. In these contexts, we show that the probability density of the belief can be represented using differential Chapman-Kolmogorov equations, allowing efficient computation of ensemble statistics. This allows us to reliably compare normative models to near-normative approximations using, as model performance metrics, decision response accuracy and Kullback-Leibler divergence of the belief distributions. Such belief distributions could be obtained empirically from subjects by asking them to report their decision confidence. We also study how response accuracy is affected by additional internal noise, showing optimality requires longer integration timescales as more noise is added. Lastly, we demonstrate that our method can be applied to tasks in which evidence arrives in a discrete, pulsatile fashion, rather than continuously.

Optimizing sequential decisions in the drift-diffusion model.

To make decisions organisms often accumulate information across multiple timescales. However, most experimental and modeling studies of decision-making focus on sequences of independent trials. On the other hand, natural environments are characterized by long temporal correlations, and evidence used to make a present choice is often relevant to future decisions. To understand decision-making under these conditions we analyze how a model ideal observer accumulates evidence to freely make choices across a sequence of correlated trials. We use principles of probabilistic inference to show that an ideal observer incorporates information obtained on one trial as an initial bias on the next. This bias decreases the time, but not the accuracy of the next decision. Furthermore, in finite sequences of trials the rate of reward is maximized when the observer deliberates longer for early decisions, but responds more quickly towards the end of the sequence. Our model also explains experimentally observed patterns in decision times and choices, thus providing a mathematically principled foundation for evidence-accumulation models of sequential decisions.