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.

Sampling-based Bayesian inference in recurrent circuits of stochastic spiking neurons.

Two facts about cortex are widely accepted: neuronal responses show large spiking variability with near Poisson statistics and cortical circuits feature abundant recurrent connections between neurons. How these spiking and circuit properties combine to support sensory representation and information processing is not well understood. We build a theoretical framework showing that these two ubiquitous features of cortex combine to produce optimal sampling-based Bayesian inference. Recurrent connections store an internal model of the external world, and Poissonian variability of spike responses drives flexible sampling from the posterior stimulus distributions obtained by combining feedforward and recurrent neuronal inputs. We illustrate how this framework for sampling-based inference can be used by cortex to represent latent multivariate stimuli organized either hierarchically or in parallel. A neural signature of such network sampling are internally generated differential correlations whose amplitude is determined by the prior stored in the circuit, which provides an experimentally testable prediction for our framework.

Tunable Dynamics in a Multistrain Transcriptional Pulse Generator.

One challenge in synthetic biology is the tuning of regulatory components within gene circuits to elicit a specific behavior. This challenge becomes more difficult in synthetic microbial consortia since each strain's circuit must function at the intracellular level and their combination must operate at the population level. Here we demonstrate that circuit dynamics can be tuned in synthetic consortia through the manipulation of strain fractions within the community. To do this, we construct a microbial consortium comprised of three strains of engineered Escherichia coli that, when cocultured, use homoserine lactone-mediated intercellular signaling to create a multistrain incoherent type-1 feedforward loop (I1-FFL). Like naturally occurring I1-FFL motifs in gene networks, this engineered microbial consortium acts as a pulse generator of gene expression. We demonstrate that the amplitude of the pulse can be easily tuned by adjusting the relative population fractions of the strains. We also develop a mathematical model for the temporal dynamics of the microbial consortium. This model allows us to identify population fractions that produced desired pulse characteristics, predictions that were confirmed for all but extreme fractions. Our work demonstrates that intercellular gene circuits can be effectively tuned simply by adjusting the starting fractions of each strain in the consortium.

Inferring delays in partially observed gene regulation processes.

MOTIVATION: Cell function is regulated by gene regulatory networks (GRNs) defined by protein-mediated interaction between constituent genes. Despite advances in experimental techniques, we can still measure only a fraction of the processes that govern GRN dynamics. To infer the properties of GRNs using partial observation, unobserved sequential processes can be replaced with distributed time delays, yielding non-Markovian models. Inference methods based on the resulting model suffer from the curse of dimensionality. RESULTS: We develop a simulation-based Bayesian MCMC method employing an approximate likelihood for the efficient and accurate inference of GRN parameters when only some of their products are observed. We illustrate our approach using a two-step activation model: an activation signal leads to the accumulation of an unobserved regulatory protein, which triggers the expression of observed fluorescent proteins. With prior information about observed fluorescent protein synthesis, our method successfully infers the dynamics of the unobserved regulatory protein. We can estimate the delay and kinetic parameters characterizing target regulation including transcription, translation, and target searching of an unobserved protein from experimental measurements of the products of its target gene. Our method is scalable and can be used to analyze non-Markovian models with hidden components. AVAILABILITY AND IMPLEMENTATION: Our code is implemented in R and is freely available with a simple example data at https://github.com/Mathbiomed/SimMCMC.

Rapid compensatory plasticity revealed by dynamic correlated activity in monkeys in vivo.

To produce adaptive behavior, neural networks must balance between plasticity and stability. Computational work has demonstrated that network stability requires plasticity mechanisms to be counterbalanced by rapid compensatory processes. However, such processes have yet to be experimentally observed. Here we demonstrate that repeated optogenetic activation of excitatory neurons in monkey visual cortex (area V1) induces a population-wide dynamic reduction in the strength of neuronal interactions over the timescale of minutes during the awake state, but not during rest. This new form of rapid plasticity was observed only in the correlation structure, with firing rates remaining stable across trials. A computational network model operating in the balanced regime confirmed experimental findings and revealed that inhibitory plasticity is responsible for the decrease in correlated activity in response to repeated light stimulation. These results provide the first experimental evidence for rapid homeostatic plasticity that primarily operates during wakefulness, which stabilizes neuronal interactions during strong network co-activation.

Hierarchical Modular Structure of the Drosophila Connectome.

The structure of neural circuitry plays a crucial role in brain function. Previous studies of brain organization generally had to trade off between coarse descriptions at a large scale and fine descriptions on a small scale. Researchers have now reconstructed tens to hundreds of thousands of neurons at synaptic resolution, enabling investigations into the interplay between global, modular organization, and cell type-specific wiring. Analyzing data of this scale, however, presents unique challenges. To address this problem, we applied novel community detection methods to analyze the synapse-level reconstruction of an adult female Drosophila melanogaster brain containing >20,000 neurons and 10 million synapses. Using a machine-learning algorithm, we find the most densely connected communities of neurons by maximizing a generalized modularity density measure. We resolve the community structure at a range of scales, from large (on the order of thousands of neurons) to small (on the order of tens of neurons). We find that the network is organized hierarchically, and larger-scale communities are composed of smaller-scale structures. Our methods identify well-known features of the fly brain, including its sensory pathways. Moreover, focusing on specific brain regions, we are able to identify subnetworks with distinct connectivity types. For example, manual efforts have identified layered structures in the fan-shaped body. Our methods not only automatically recover this layered structure, but also resolve finer connectivity patterns to downstream and upstream areas. We also find a novel modular organization of the superior neuropil, with distinct clusters of upstream and downstream brain regions dividing the neuropil into several pathways. These methods show that the fine-scale, local network reconstruction made possible by modern experimental methods are sufficiently detailed to identify the organization of the brain across scales, and enable novel predictions about the structure and function of its parts.Significance Statement The Hemibrain is a partial connectome of an adult female Drosophila melanogaster brain containing >20,000 neurons and 10 million synapses. Analyzing the structure of a network of this size requires novel and efficient computational tools. We applied a new community detection method to automatically uncover the modular structure in the Hemibrain dataset by maximizing a generalized modularity measure. This allowed us to resolve the community structure of the fly hemibrain at a range of spatial scales revealing a hierarchical organization of the network, where larger-scale modules are composed of smaller-scale structures. The method also allowed us to identify subnetworks with distinct cell and connectivity structures, such as the layered structures in the fan-shaped body, and the modular organization of the superior neuropil. Thus, network analysis methods can be adopted to the connectomes being reconstructed using modern experimental methods to reveal the organization of the brain across scales. This supports the view that such connectomes will allow us to uncover the organizational structure of the brain, which can ultimately lead to a better understanding of its function.

Signaling in microbial communities with open boundaries.

Microbial communities such as swarms or biofilms often form at the interfaces of solid substrates and open fluid flows. At the same time, in laboratory environments these communities are commonly studied using microfluidic devices with media flows and open boundaries. Extracellular signaling within these communities is therefore subject to different constraints than signaling within classic, closed-boundary systems such as developing embryos or tissues, yet is understudied by comparison. Here, we use mathematical modeling to show how advective-diffusive boundary flows and population geometry impact cell-cell signaling in monolayer microbial communities. We reveal conditions where the intercellular signaling lengthscale depends solely on the population geometry and not on diffusion or degradation, as commonly expected. We further demonstrate that diffusive coupling with the boundary flow can produce signal gradients within an isogenic population, even when there is no flow within the population. We use our theory to provide new insights into the signaling mechanisms of published experimental results, and we make several experimentally verifiable predictions. Our research highlights the importance of carefully evaluating boundary dynamics and environmental geometry when modeling microbial cell-cell signaling and informs the study of cell behaviors in both natural and synthetic systems.

Signaling in microbial communities with open boundaries.

Microbial communities such as swarms or biofilms often form at the interfaces of solid substrates and open fluid flows. At the same time, in laboratory environments these communities are commonly studied using microfluidic devices with media flows and open boundaries. Extracellular signaling within these communities is therefore subject to different constraints than signaling within classic, closed-boundary systems such as developing embryos or tissues, yet is understudied by comparison. Here, we use mathematical modeling to show how advective-diffusive boundary flows and population geometry impact cell-cell signaling in monolayer microbial communities. We reveal conditions where the intercellular signaling lengthscale depends solely on the population geometry and not on diffusion or degradation, as commonly expected. We further demonstrate that diffusive coupling with the boundary flow can produce signal gradients within an isogenic population, even when there is no flow within the population. We use our theory to provide new insights into the signaling mechanisms of published experimental results, and we make several experimentally verifiable predictions. Our research highlights the importance of carefully evaluating boundary dynamics and environmental geometry when modeling microbial cell-cell signaling and informs the study of cell behaviors in both natural and synthetic systems.

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.

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.

Emergent spatiotemporal population dynamics with cell-length control of synthetic microbial consortia.

The increased complexity of synthetic microbial biocircuits highlights the need for distributed cell functionality due to concomitant increases in metabolic and regulatory burdens imposed on single-strain topologies. Distributed systems, however, introduce additional challenges since consortium composition and spatiotemporal dynamics of constituent strains must be robustly controlled to achieve desired circuit behaviors. Here, we address these challenges with a modeling-based investigation of emergent spatiotemporal population dynamics using cell-length control in monolayer, two-strain bacterial consortia. We demonstrate that with dynamic control of a strain's division length, nematic cell alignment in close-packed monolayers can be destabilized. We find that this destabilization confers an emergent, competitive advantage to smaller-length strains-but by mechanisms that differ depending on the spatial patterns of the population. We used complementary modeling approaches to elucidate underlying mechanisms: an agent-based model to simulate detailed mechanical and signaling interactions between the competing strains, and a reductive, stochastic lattice model to represent cell-cell interactions with a single rotational parameter. Our modeling suggests that spatial strain-fraction oscillations can be generated when cell-length control is coupled to quorum-sensing signaling in negative feedback topologies. Our research employs novel methods of population control and points the way to programming strain fraction dynamics in consortial synthetic biology.

Hierarchical Bayesian models of transcriptional and translational regulation processes with delays.

MOTIVATION: Simultaneous recordings of gene network dynamics across large populations have revealed that cell characteristics vary considerably even in clonal lines. Inferring the variability of parameters that determine gene dynamics is key to understanding cellular behavior. However, this is complicated by the fact that the outcomes and effects of many reactions are not observable directly. Unobserved reactions can be replaced with time delays to reduce model dimensionality and simplify inference. However, the resulting models are non-Markovian, and require the development of new inference techniques. RESULTS: We propose a non-Markovian, hierarchical Bayesian inference framework for quantifying the variability of cellular processes within and across cells in a population. We illustrate our approach using a delayed birth-death process. In general, a distributed delay model, rather than a popular fixed delay model, is needed for inference, even if only mean reaction delays are of interest. Using in silico and experimental data we show that the proposed hierarchical framework is robust and leads to improved estimates compared to its non-hierarchical counterpart. We apply our method to data obtained using time-lapse microscopy and infer the parameters that describe the dynamics of protein production at the single cell and population level. The mean delays in protein production are larger than previously reported, have a coefficient of variation of around 0.2 across the population, and are not strongly correlated with protein production or growth rates. AVAILABILITY AND IMPLEMENTATION: Accompanying code in Python is available at https://github.com/mvcortez/Bayesian-Inference. CONTACT: kresimir.josic@gmail.com or jaekkim@kaist.ac.kr or cbskust@korea.ac.kr. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Balanced networks under spike-time dependent plasticity.

The dynamics of local cortical networks are irregular, but correlated. Dynamic excitatory-inhibitory balance is a plausible mechanism that generates such irregular activity, but it remains unclear how balance is achieved and maintained in plastic neural networks. In particular, it is not fully understood how plasticity induced changes in the network affect balance, and in turn, how correlated, balanced activity impacts learning. How do the dynamics of balanced networks change under different plasticity rules? How does correlated spiking activity in recurrent networks change the evolution of weights, their eventual magnitude, and structure across the network? To address these questions, we develop a theory of spike-timing dependent plasticity in balanced networks. We show that balance can be attained and maintained under plasticity-induced weight changes. We find that correlations in the input mildly affect the evolution of synaptic weights. Under certain plasticity rules, we find an emergence of correlations between firing rates and synaptic weights. Under these rules, synaptic weights converge to a stable manifold in weight space with their final configuration dependent on the initial state of the network. Lastly, we show that our framework can also describe the dynamics of plastic balanced networks when subsets of neurons receive targeted optogenetic input.

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.

Majority sensing in synthetic microbial consortia.

As synthetic biocircuits become more complex, distributing computations within multi-strain microbial consortia becomes increasingly beneficial. However, designing distributed circuits that respond predictably to variation in consortium composition remains a challenge. Here we develop a two-strain gene circuit that senses and responds to which strain is in the majority. This involves a co-repressive system in which each strain produces a signaling molecule that signals the other strain to down-regulate production of its own, orthogonal signaling molecule. This co-repressive consortium links gene expression to ratio of the strains rather than population size. Further, we control the cross-over point for majority via external induction. We elucidate the mechanisms driving these dynamics by developing a mathematical model that captures consortia response as strain fractions and external induction are varied. These results show that simple gene circuits can be used within multicellular synthetic systems to sense and respond to the state of the population.

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.

Bayesian inference of distributed time delay in transcriptional and translational regulation.

MOTIVATION: Advances in experimental and imaging techniques have allowed for unprecedented insights into the dynamical processes within individual cells. However, many facets of intracellular dynamics remain hidden, or can be measured only indirectly. This makes it challenging to reconstruct the regulatory networks that govern the biochemical processes underlying various cell functions. Current estimation techniques for inferring reaction rates frequently rely on marginalization over unobserved processes and states. Even in simple systems this approach can be computationally challenging, and can lead to large uncertainties and lack of robustness in parameter estimates. Therefore we will require alternative approaches to efficiently uncover the interactions in complex biochemical networks. RESULTS: We propose a Bayesian inference framework based on replacing uninteresting or unobserved reactions with time delays. Although the resulting models are non-Markovian, recent results on stochastic systems with random delays allow us to rigorously obtain expressions for the likelihoods of model parameters. In turn, this allows us to extend MCMC methods to efficiently estimate reaction rates, and delay distribution parameters, from single-cell assays. We illustrate the advantages, and potential pitfalls, of the approach using a birth-death model with both synthetic and experimental data, and show that we can robustly infer model parameters using a relatively small number of measurements. We demonstrate how to do so even when only the relative molecule count within the cell is measured, as in the case of fluorescence microscopy. AVAILABILITY AND IMPLEMENTATION: Accompanying code in R is available at https://github.com/cbskust/DDE_BD. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

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.