Neural variability in the default mode network compresses with increasing belief precision during Bayesian inference

Abstract

To make optimal decisions, intelligent agents must learn latent environmental states from discrete observations. Bayesian frameworks argue that integration of evidence over time allows us to refine our state belief by reducing uncertainty about alternate possibilities. How is this increasing belief precision during learning reflected in the brain? We propose that temporal neural variability should scale with the degree of reduction of uncertainty during learning. In a sample of 47 healthy adults, we found that BOLD signal variability (SD_BOLD, as measured across independent learning trials) indeed compressed with successive exposure to decision-related evidence. Crucially, more accurate participants expressed greater SD_BOLD compression primarily in default mode network regions, possibly reflecting the increasing precision of their latent state belief during more efficient learning. Further, computational modeling of behavior suggested that more accurate subjects held a more unbiased (flatter) prior belief over possible states that allowed for larger uncertainty reduction during learning, which was directly reflected in SD_BOLD changes. Our results provide first evidence that neural variability compresses with increasing belief precision during effective learning, proposing a flexible mechanism for how we come to learn the probabilistic nature of the world around us.

Introduction

To make optimal decisions, animals often need to learn about “states” (ie decision-relevant properties) of the environment, such as the availability of fish in a nearby pond based on previous catch successes. These states are often not directly observable, so optimal agents need to combine the uncertainty about their beliefs with the evidence available to them. Research suggests that humans and other animals often take into account uncertainty during learning and decision-making (Bach and Dolan 2012; Ma and Jazayeri 2014; Bruckner et al. 2022), while suboptimal use of uncertainty has been linked to maladaptive behavior observed in clinical and aging populations (Bruckner et al. 2022). Bayesian decision theory prescribes how agents should combine evidence with their internal beliefs under uncertainty to arrive at optimal estimates of external states and provides a benchmark to assess rational behavior in the context of learning (Jacobs and Kruschke 2011; Bach and Dolan 2012; Bruckner et al. 2022). In this framework, an agent must represent a probability distribution over possible states. The variance of this distribution reflects the uncertainty of the agent’s state belief. As evidence accumulates and the agent refines its representation of the “correct” world state, the variance of the belief distribution is thought to reduce. How this increasing belief precision during learning is reflected in the brain remains unclear.

A potential candidate for tracking the degree of state uncertainty is the temporal variability of neural responses. At the level of sensory cortex, computational modeling and non-human animal work suggests that neural variability signals uncertainty about peripheral inputs, such that neural population activity over time reflects sampling from a probability distribution of potential stimulus features; the higher perceptual uncertainty, the higher the neural variability (Fiser et al. 2010; Berkes et al. 2011; Orbán et al. 2016; Echeveste et al. 2020; Festa et al. 2021). This idea is in line with theoretic work suggesting that neural systems should maintain an appropriate degree of instability in the face of uncertainty to permit exploration of alternative causes of incoming sensory input (Friston et al. 2012). With regard to human data, a study by Kosciessa et al. (2021) showed that increasing uncertainty about the task-relevance of different perceptual stimulus features is accompanied by an increase in cortical “excitability” (heightened desynchronization of alpha rhythms and increased entropy in the EEG signal), which may facilitate neural network state switches for different feature activations. However, it is unknown whether these uncertainty-based shifts in neural variability in early perceptual areas translate to uncertainty about abstract latent environmental states during Bayesian inference/learning and how it relates to task accuracy.

In the context of learning, larger temporal neural variability under high uncertainty may not just represent exploration of potential world states but may allow more flexible belief updating once new information becomes available. Indeed, previous human work suggests that higher brain signal variability affords larger cognitive flexibility (Garrett et al. 2013; Waschke et al. 2021). For example, several studies have found that brain signal variability increases with increasing task demands (at least until processing limits are reached) and that the ability to upregulate variability predicts task performance (Palva et al. 2011; Rouhinen et al. 2013; Garrett et al. 2013; Grady and Garrett 2014; Garrett et al. 2015; Honkanen et al. 2015; Armbruster-Genç et al. 2016; Grundy et al. 2019). These studies commonly argue that neural variability supports performance under increasing task demand by allowing the brain to maintain flexible responding to stimulus information. In line with this idea, Armbruster-Genç et al. (2016) observed better performance on a task switching paradigm under higher BOLD signal variability, whereas cognitive stability during distractor inhibition was related to lower brain signal variability. During learning, task demands are highest early on, when the brain needs to maintain high flexibility to incorporate incoming evidence for belief updating. This should allow it to converge on the “correct” state representation with learning. We therefore hypothesize that temporal brain signal variability should compress with increasing belief precision (ie decreasing uncertainty) during learning (Fig. 1).

Fig. 1.

Hypothetical model of uncertainty and brain dynamics. In this example, the agent has to infer the proportion of blue birds in a world made up of red and blue birds by combining information across observations over time. Under high state uncertainty early on in the learning process, more moment-to-moment brain signal variability (BSV) allows the brain to entertain a larger variety of possible states (ie proportions of blue birds in the world). With each new observation, the precision of the belief distribution increases and moment-to-moment brain signal variability compresses (as highlighted for one learning trial t in the brain signal plots). This compression of (within-trial) BSV can also be captured by slow neuroimaging methods that sparsely sample the brain signal during the learning period (as illustrated by the colored dots for each learning trial t plotted as separate lines). In this case, within-trial BSV is approximated by BSV measured across trials for a given time window of the learning period.

The (changing) belief distribution over possible environmental states may be represented in ventromedial prefrontal cortex (vmPFC). Several lines of evidence point to a general function of this brain region in maintaining a compact internal model of the environment (ie state belief) by extracting information across individual experiences to guide goal-directed behavior, such as the value of different choice options (Levy and Glimcher 2012; Averbeck and O’Doherty 2022; Klein-Flügge et al. 2022), cognitive maps (Boorman et al. 2021; Klein-Flügge et al. 2022; Schuck et al. 2016; Wilson et al. 2014), or schemas (Gilboa and Marlatte 2017; Bein and Niv 2025). Studies employing probabilistic learning tasks furthermore show that neural activity in vmPFC also reflects uncertainty about external states, which were linked to adaptive exploration behavior and learning-rate adjustments (Karlsson et al. 2012; McGuire et al. 2014; Starkweather et al. 2018; Domenech et al. 2020; Trudel et al. 2021). Notably, Karlsson et al. (2012) found that trial-to-trial neural population spiking variability in the medial PFC of mice peaked around transitions from exploitation to exploration periods following changes in reward structure when state uncertainty is highest, which may reflect more variable belief states. While ours is the first study to link human brain signal variability to belief precision, a previous study by Grady and Garrett (2018) observed increased BOLD signal variability while subjects performed externally- versus internally-oriented tasks; an effect spanning the vmPFC and other nodes of the canonical default mode network (DMN; Yeo et al. 2011). Since learning an abstract world model reflects a shift towards an internal cognitive mode, we tentatively expected brain signal variability compression over the course of learning to be (partly) expressed in the vmPFC.

We assume that uncertainty-related neural dynamics unfold on a fast temporal scale, as suggested by electrophysiological evidence in human and nonhuman animals (Berkes et al. 2011; Palva et al. 2011; Rouhinen et al. 2013; Honkanen et al. 2015; Orbán et al. 2016; Grundy et al. 2019). However, within-trial dynamics should also affect neural variability across independent learning trials (see Fig. 1). A more variable system should have a higher probability of being in a different state every time it is (sparsely) sampled. Conversely, when a system is in a less stochastic state, the within-trial variance is expected to reduce, yielding less across-trial variance at the same time. This argument aligns with work by Orbán et al. (2016), who showed that a computational model of the sampling account of sensory uncertainty captures empirically observed across-trial variability of neural population responses in primary visual cortex. In the case of human research, this means that neuroimaging methods with slower sampling rates, such as functional MRI (fMRI), may be able to approximate within-trial neural variability from variability observed across trials. Indeed, the majority of previous fMRI studies reporting within-region, within-subject modulation of brain signal variability by task demand have exclusively employed block designs, necessitating that the main source of variability be between- rather than within-trial (Garrett et al. 2013; Grady and Garrett 2014; Garrett et al. 2015; Armbruster-Genç et al. 2016).

In the current study, we acquired fMRI while participants performed a “marble task”. In this task, participants had to learn the probability of drawing a blue marble from an unseen jar (ie urn) based on five samples (ie draws from the urn with replacement). In a Bayesian inference framework, the jar marble ratio can be considered a latent state that participants must infer. We hypothesized that (i) across-trial variability in the BOLD response (SD_BOLD) would compress over the sampling period, thus mirroring the reduction in state uncertainty, and that (ii) subjects with greater SD_BOLD compression would show smaller estimation errors of the jars’ marble ratios as an index of more efficient belief updating. A secondary aim of the current study was to directly compare the effect of uncertainty on SD_BOLD with a more standard General Linear Modeling (GLM) approach, which looks for correlations between average BOLD activity and uncertainty. This links our findings directly to previous investigations of neural uncertainty correlates, which disregarded the magnitude of BOLD variability (Huettel et al. 2005; Grinband et al. 2006; Behrens et al. 2007; Bach et al. 2011; Bach and Dolan 2012; Badre et al. 2012; Vilares et al. 2012; Payzan-LeNestour et al. 2013; McGuire et al. 2014; Michael et al. 2015; Meyniel and Dehaene 2017; Nassar et al. 2019; Meyniel 2020; Tomov et al. 2020; Trudel et al. 2021; Walker et al. 2023). We hypothesized (iii) that SD_BOLD would uniquely predict inference accuracy compared to these standard neural uncertainty correlates.

Our results showed that SD_BOLD closely tracked individual differences in the reduction of state uncertainty during learning, which related to task accuracy in a unique manner that was not captured by a more standard GLM approach. As such, we identified a novel neural signal reflecting uncertainty during learning.

Materials and methods

Participants and procedure

Fifty-one young adults (24 female, 19–40 yr [mean age = 26.37, SD = 3.74]) participated in the study. They were recruited from the participant database of the MPI for Human Development (Berlin, Germany) and gave written informed consent according to the guidelines of the German Psychological Society ethics board, who approved the study (approval number: RH 062015). All participants were right-handed, had normal or corrected-to-normal vision, did not concurrently participate in any clinical trials/pharmacological studies, had no metal implants, did not take any centrally active medications, were not pregnant, and reported to be in good health with no known history of neurological, psychiatric or other severe (cancer, cardiac insufficiency, chronic liver/kidney disease, a severe hematopoietic condition, severe allergic/toxic reactions, brain/heart operation) health condition. Participants underwent fMRI while performing a perceptual gambling task. They received 10 Euros per hour in addition to a variable performance bonus of up to 2 Euros.

The marble task

The task was divided into four blocks of 18 trials each. Participants performed three blocks during fMRI acquisition, while the final block was performed outside the scanner. We will only consider the three MR blocks for our analyses. Participants received instructions and completed nine practice trials prior to the first scanning block. Each trial of the task consisted of three phases: a sampling, an estimation, and a gambling phase. To investigate brain signal variability changes during learning, we focused on the sampling and estimation phases in this study (referred to as the “marble task”). A description of the gambling phase, which was unrelated to the learning process, is provided in the Supplementary Methods. The task design also included a between-trial reward manipulation. Here, we collapsed trials across the two levels of the reward manipulation given that we assumed those would only affect the choice and not the learning part. The task was programmed using Presentation software (Version 14.9, Neurobehavioral Systems Inc., Albany, CA, USA).

In the marble task, participants were asked to estimate the proportion of blue marbles in an unseen “jar” (ie urn), containing a total of 100 blue and red marbles, based on five successive samples (ie draws from the urn with replacement) presented successively during the sampling phase (Fig. 2A). In total, 18 different jars with different proportions of blue marbles (ranging from 0.1 to 0.91) were presented across all trials (Supplementary table S1). Each jar was presented four times across the four experimental blocks in random order. Each sample from a given jar could contain either one, five or nine marbles, which manipulated the informativeness of the samples (ie one marble was least informative for inferring the proportion of blue marbles whereas nine marbles was most informative). Samples were presented in 3x3 grids with gray marbles serving as placeholders to ensure similar visual inputs across samples. Additionally, within each sample the order of the marbles (ie the location in the 3x3 grid) was permuted. The samples for each jar were randomly drawn from a binomial distribution with the corresponding probability of drawing a blue marble. This draw was performed once so each jar was associated with a consistent set of five samples with varying sizes (note that the total number of marbles across samples thus varied between jars). This ensured that all participants received identical sample information. For each trial with the same unseen jar, the order of the associated samples was varied randomly. Each sample was presented for 1 s followed by a fixation cross presented for 2 to 6 s. Following the sampling phase, participants were asked to indicate their estimate of the proportion of blue marbles in the jar by adjusting the ratio of blue to red marbles in a 100-marble grid using an MR-compatible five-button box. The starting value of the grid was always set to 50 red and 50 blue marbles. With the upper and lower buttons, the participants were able to adjust the grid in steps of five marbles, and with the left and right button in a one marble step, respectively. With the middle button participants confirmed their estimation. The maximum reaction time for the estimation phase was set to 7 s. Participants received a bonus payment based on the average accuracy of their marble ratio estimates (performance-dependent bonus).

Fig. 2.

The marble task. A) Task design. Participants observed five samples from an unseen jar (sampling phase) and then entered their estimate of the proportion of blue marbles in the jar into a 10x10 grid (estimation phase). B) Schematic illustration of the Bayesian observer model with varying priors. A narrower prior belief centered on a default 50:50 marble ratio predicts more misestimation of extreme jar proportions (top) and less uncertainty reduction during the sampling phase (bottom; s1 to s5 correspond to the five samples presented on a given trial). C) Illustration of SD_BOLD computation. The standard deviation is computed over separate GLM β estimates for each trial t for a given sample period (see main text for details). HRF = hemodynamic response function.

Behavioral modeling

We modeled participants’ responses using variants of a Bayesian observer model. The Bayesian observer represents the probability of drawing a blue marble from the unseen jar as a beta distribution with parameters and , which is updated after each draw according to Bayes’ rule (Fig. 2B). Because the beta distribution is a conjugate prior for the binomial distribution (the likelihood function in this task), the posterior belief distribution is also a beta distribution with α_s + 1 = α_s + B_s and β_s + 1 = β_s + R_s, where B_s and R_s are the number of blue and red marbles for a sample s respectively. The prior for the first draw of each trial is given by , representing a flat prior. The best estimate for the probability of drawing a blue marble is given by the expectation of the beta distribution .To model empirical response patterns affecting subjects’ task accuracy, we considered two variants of this Bayesian observer model. In one model variant, we fit a parameter for each subject, which described the initial setting of and on each trial prior to observing samples from a given jar (where ; Fig. 2B). A higher value for ρ describes a narrower prior around the default belief of 0.5. In a second Bayesian observer model variant, we fit a parameter , which exponentially weighs the evidence of each sample: α_s + 1 = α_s + (B_s)^δ and _s + 1 = _s + (R_s) ^δ. If , the evidence of larger draws is underweighted, while for the evidence of larger draws is overweighted. Comparing these models allowed us to test whether deviations from the unbiased Bayesian observer model were related to prior representation or evidence weighting respectively.

To accommodate probabilistic responding of subjects, we compared these models with two alternative choice rules. One model family assumes that subjects’ reported estimates represent draws from the final beta distribution on each trial. Another model family assumes that subjects’ choices are draws from a truncated normal distribution () that is centered on the expectation of the final beta distributions: , where is a free model parameter that captures response noise around the model prediction. This parameter may reflect imprecision in entering one’s marble ratio estimate into the response grid, but it can also capture any unmodeled sources of (biased and random) errors in jar estimates beyond those explained by the model parameters of interest. This parameter may thus also capture model misfit.

Bayesian observer models assume that participants track the full belief distribution over potential marble ratios and consider uncertainty when updating their estimate. To test whether subjects indeed behave in a Bayesian manner, we also fitted a simpler Rescorla–Wagner model that only updates a point estimate of the marble ratio belief: , where is the marble ratio point estimate after observing sample s, B_s and R_s are the number of blue and red marbles respectively, and is the free learning rate parameter. To model subjects’ responses, we included a free noise parameter modeling Gaussian response noise around the final point estimate, similar to the noisy choice rule described previously for the Bayesian observer models.

Model fitting and comparison

We fitted behavioral models to participants’ responses on all trials of the three MR blocks by minimizing the joint negative log likelihoods under each model using the fmincon routine implemented in MATLAB (version 2017b). We ensured model convergence by fitting each model 10 times and used the fitting iteration with the minimal negative log likelihood across subjects. We compared model fits by computing the Bayesian Information Criterion (BIC): , where k is the number of parameters, n is the number of datapoints, and L is the maximized model likelihood. We report BICs for each subject and across all subjects.

Model simulations

We simulated behavior using the empirical parameter estimates and the trial sequences of each subject. This resulted in one synthetic dataset for each model with 51 simulated subjects each. We used these simulated datasets to perform model and parameter recovery checks. See the Supplementary Material for the results of these checks.

Image acquisition and preprocessing

Participants underwent fMRI scanning at the Max Planck institute for Human Development (Berlin, Germany) in a 3 Tesla Siemens TrioTim MRI system (Erlangen, Germany) using a multi-band EPI sequence (factor 4; TR = 645 ms; TE = 30 ms; flip angle 60°; FoV = 222 mm; voxel size 3 × 3 × 3 mm; 40 transverse slices). This amounted to 1010 T2^*-weighted functional images per scanner run per subject (except for one subject, who had 1020 acquired images per run). A T1-weighted structural scan was also acquired (MPRAGE: TR = 2500 ms; TE = 4.77 ms; flip angle 7°; FoV = 256 mm; voxel size 1 × 1 × 1 mm; 192 sagittal slices).

T1-weighted images were brain extracted using ANTs software (version 2.3.5, http://stnava.github.io/ANTs/) using population level templates from the OASIS dataset (https://figshare.com/articles/dataset/ANTs_ANTsR_Brain_Templates/915436). The functional T2^*-weighted scans were preprocessed in FSL (version 5.0.11; Jenkinson et al. 2012; Smith et al. 2004) FEAT separately for each run. The pipeline includes motion correction, brain extraction of the functional images, and spatial smoothing using a Gaussian 5 mm kernel. Following FEAT, the functional images were first detrended using Statistical Parametric Mapping's (SPM) detrend function (at a third order polynomial) and then high-pass filtered using a standard eight order butterworth filter implemented in MATLAB (version 2017b) with a cut-off of 0.01 Hz. We then performed Independent Component Analysis (ICA) on the resulting data using FSL MELODIC (Beckmann and Smith 2004) to identify residual artifacts. We manually labeled rejected components for 19 subjects (~37% of the total data; for details on the component rejection criteria see Garrett et al. 2014) and then used these labels to train FSL’s ICA classifier FIX (Griffanti et al. 2014; Salimi-Khorshidi et al. 2014) to automatically label artifactual components for the remaining subjects. We tested different classification thresholds and validated classification accuracy for several randomly selected subjects from the test set. We accepted a threshold of 20 after this manual inspection. Rejected components were then regressed out of the data using FSL’s fsl_regfilt function. Lastly, functional images were first registered to the brain-masked T1-weighted images and then to MNI space (using the MNI152 template provided in FSL) via linear rigid-body transformation as implemented in FSL’s flirt function.

Statistical analysis

Behavioral analysis

All behavioral analyses reported were run in SPSS (version 25). To limit the effect of univariate outliers on the reported results, we winsorized the estimation error, extreme jar bias, and behavioral model parameter variables. Outliers were defined as datapoints that were 1.5 times the interquartile range beyond the 25th or 75th percentile respectively. We used the highest/lowest score of the non-outlying datapoints to impute outlying values. Between zero and three values were imputed for each variable.

Functional MRI analysis

To quantify across-trial BOLD response variability (SD_BOLD), we adapted a “least squares—single” (LS-S) approach as described by Mumford et al. (2012). In this approach, a trial-wise General Linear Model (GLM) beta parameter is estimated for each event of interest (ie each sampling period) by iteratively fitting (mass-univariate) GLMs including one regressor modeling a single trial event and a second regressor modeling all other events (of the same and all other trials). Subsequently, we computed the standard deviation over the resulting trial-wise beta estimates for each sampling period separately to yield our measure of SD_BOLD (Fig. 2C). Compared to previous operationalizations of SD_BOLD, which compute the standard deviation of the BOLD signal over entire task blocks (see eg Garrett et al. 2013; Grady and Garrett 2014; Garrett et al. 2015), this method allowed us to parse SD_BOLD for neighboring events/time points, while accounting for hemodynamic delays.

Technically, this approach was implemented in an in-house version of variability toolbox for SPM (http://www.douglasdgarrett.com/#software) developed by our research group. This toolbox takes as input a standard (subject-level) GLM design matrix as implemented in SPM12 (Wellcome Department of Imaging Neuroscience, London, UK). We included the following regressors in each subject’s design matrix: Five regressors for the onsets of each successive sample presentation (with a duration of 1 s), one regressor for the estimation phase onsets (with a duration of subjects’ RTs), and one regressor for the gambling phase onsets (with a duration of subjects’ RTs). All regressors were modeled as stick functions that were convolved with the canonical hemodynamic response function (HRF) and its first and second derivatives resulting in a total of 21 regressors per scanner run plus one constant regressors for each run. The toolbox then proceeds to iteratively fit GLMs, which include one regressor modeling a single event of a given task condition and a second regressor modeling all other events of the same and all other conditions (this is done separately for each task run). To avoid issues with beta estimation close to the end of a timeseries (caused by truncation of the HRF), the toolbox discards any onsets that occur within the final 20s of the timeseries. Finally, the toolbox computes the standard deviation over the resulting (trial-wise) beta estimates for each task condition.

To compare our SD_BOLD results to standard analysis approaches, we also obtained restricted maximum likelihood (ReML) GLM beta estimates for each task condition. The design matrix of this GLM included the same regressors as the one passed to variability toolbox for SD_BOLD estimation. We defined contrasts to summarize condition effects across task runs. However, this approach ignores the variance in uncertainty trajectories across trials that result from our sample size manipulation (see Fig. S1B). To account for this variance, we ran another GLM modeling the modulation of the BOLD signal by the posterior variance (ie uncertainty) of the unbiased Bayesian observer model during the sampling process. The design matrix of this GLM included the following regressors for each scanner run: one regressor for the sample onsets (with a duration of 1 s), one regressor for the parametric modulation of the sample onsets regressor by model-derived posterior variances, one regressor for the estimation phase onsets (with a duration of subjects’ RTs), and one regressor for the gambling phase onsets (with a duration of subjects’ RTs). We mean-centered the parametric modulation regressor prior to model estimation. All regressors were modeled as stick functions (except for the parametric modulation regressor) that were convolved with the canonical HRF and its first and second derivatives resulting in a total of 12 regressors per scanner run. We also included three additional constant regressors modeling the mean of each scanner run. To look at the effect of the parametric modulation regressor, we defined a contrast that picked out the beta estimates of this regressor for each scanner run (more precisely, the parametric modulation regressors convolved with the canonical HRF).

To relate within-person SD_BOLD or standard GLM beta estimates to the task design or behavior, we use a partial least-squares approach (PLS; Krishnan et al. 2011). PLS finds latent factors that express maximal covariance between brain and behavior/design matrices using singular value decomposition. Brain scores are defined as the dot product of a subject’s brain data matrix and the latent variable weights (saliences) estimated within PLS, thus providing a whole-brain (weighted) summary measure over all voxels. We will refer to brain scores in our SD_BOLD analyses as “latent SD_BOLD” and to the brain scores of our parametric modulation analyses as “latent uncertainty β”. We performed 1000 permutation tests to assess the significance of the brain-design/behavior relationship against a null model. We then divided each voxel’s salience by its bootstrapped standard error, at +/− 3 (approximating a 99.9% confidence interval), which serves as a pseudo-normalized measure of voxel robustness. We determined clusters of voxels with robust saliences by applying a cluster mask with a minimum threshold of 25 voxels. Prior to all PLS analyses, brain measures images were gray matter masked using the tissue prior provided in FSL at a probability threshold of > 0.37, and restrained to voxels with nonzero values for the respective brain measure across subjects.

To account for univariate outliers and non-linear relationships, we ran all reported brain-behavior analyses on ranked brain and behavioral variables in line with previous research (Waschke et al. 2025). The results should thus be interpreted as monotonic rather than strictly linear relationships. Furthermore, we used a criterion of Cook’s distance > 4/n to identify multivariate outliers in all PLS models, removing where present. We nevertheless report the results of all analyses for the full dataset (n = 51) without outliers removed in the Supplementary Results.

Data availability

The code to perform all behavioral and neuroimaging analyses is available at: https://github.com/LNDG/Skowron_etal_2023. The research data required to replicate all reported results can be obtained by contacting the research data management unit at the Max Planck Institute for Human Development (Lentzeallee 94, 14,195 Berlin, Germany) via this e-mail address: rdm@mpib-berlin.mpg.de. Interested parties will be required to provide a signed data handling agreement stipulating the conditions for data sharing before receiving access to the anonymized dataset.

Results

Subjects’ estimation errors are explained by individual differences in prior belief width

Participants performed the “marble task” during fMRI scanning (Fig. 2A). On each trial, they observed five sequential sample draws from a hypothetical “jar” containing a certain proportion of red and blue marbles (sampling phase). The marble ratio of the unseen jar varied from trial to trial and sample draws could contain one, five or nine marbles. Afterwards, participants indicated their estimate of the proportion of blue marbles in the jar by adjusting the number of blue marbles in a 10 by 10 response grid representing the unseen jar (estimation phase; see Materials and Methods for further details). Our primary measure of interest was estimation error, defined as , where θ is the experienced proportion of blue marbles on trial t (ie the mean across samples; note that this quantity differs from the true proportion of the unseen jar but constitutes the best (unbiased) estimate given the available samples) and is the subject’s reported estimate of the blue marble proportion of the jar on trial t (Fig. 3A). We computed the median estimation error across trials as an individual difference measure of estimation accuracy. Subjects’ median estimation error ranged from 0.01 to 0.20 (Median = 0.07, SD = 0.035). Previous research suggests that people often underestimate probabilities of frequent events and overestimate probabilities of rare events in similar decision-from-experience tasks (Spiliopoulos and Hertwig 2022; Spiliopoulos and Hertwig 2023). To see whether this effect contributed to estimation accuracy in our task, we examined whether subjects made more errors for jars with more extreme marble ratios. We ran a linear mixed model predicting trial-wise estimation error from the marble ratio distance from 50:50 for a given jar. To account for individual differences in intercepts, we entered subject ID as a covariate in the model.

Fig. 3.

Latent sources of individual estimation errors. A) Left: Reported blue marble proportion estimates across trials and subjects against the experienced blue marble proportion across samples on a given trial. Groups are based on a median split on the fitted prior width parameter ρ of the Bayesian observer model (ie the narrow prior group had higher parameter estimates). Right: Schematic example of our trial-wise estimation error measure. B) Individual differences in median estimation error related to estimation accuracy for jars with extreme marble ratios. C–D) The fitted prior width and response noise parameters of the Bayesian observer model predicted differences in median estimation error and extreme jar bias. Plotted parameters are residualized by the other model parameter respectively.

There was a significant positive relationship between trial-wise estimation-error and jar marble ratio distance from 50:50 (F(1,2702) = 154.608, P = , semi-partial η² = 0.048). The more extreme the jar marble ratio, the larger subjects’ estimation error on a given trial. In the following sections, we will refer to this effect as “extreme jar bias”. Next, we examined whether individual differences in this extreme jar bias explained individual differences in median estimation error. We computed individuals’ extreme jar bias by running regression models predicting trial-wise estimation error from marble ratio distance from 50:50 for each subject separately. Individuals’ extreme jar bias was defined as the fitted (unstandardized) slopes of these regression models (Mean = 0.198, SD = 0.230). We found that individual differences in extreme jar bias correlated strongly and positively with subjects’ median estimation error (Pearson’s r(49) = 0.63, P = ; Fig. 3B). Biased estimation of extreme jar proportions thus contributed to individual differences in task accuracy.

We next fitted a set of Bayesian observer models that could account for this estimation bias in terms of inadequate uncertainty representations. The model with the overall best BIC value was a Bayesian observer model which included a fitted prior parameter ρ, describing individual differences in the width of the belief distribution over marble ratios before observing any samples, and a response noise parameter σ, which captures variability of subjects’ reported marble ratio estimates around model predictions (Table 1). The second-best overall model included an exponential evidence weight and a noisy choice rule parameter σ, followed by an exponential evidence weight and fitted prior model assuming responses were sampled from the final trial posterior of the belief distribution (see Materials and Methods for details). A Rescorla–Wagner model had the worst fit out of all models, which suggests that people indeed keep track of and leverage uncertainty estimates to update their marble ratio belief in a Bayesian manner. Parameters of the winning model were recoverable from simulations and model recovery was acceptable at the group level. Note that theoretically biased prior beliefs and belief updating could jointly contribute to participants’ responses. However, including this model variant negatively affected model recovery and provided poorer fit to behavior (BIC = ) compared to the more parsimonious, winning Bayesian observer model and was thus dropped from the final model set (see Supplementary Results for details on model and parameter recovery checks).

Table 1.

Model comparison. BIC for each fitted behavioral model.

Model	Rescorla–Wagner	Bayesian observer
Parameters	α, σ	ρ	δ	ρ, σ	δ, σ
BIC

We next tested whether the prior width parameter ρ of the winning Bayesian observer model captured individual differences in extreme jar bias that contributed to subjects’ estimation error while controlling for other unmodelled sources of error captured by the response noise parameter . The fitted prior parameter ρ of the winning Bayesian observer model had a median of 2.45 (SD = 4.23), indicating that subjects on average had a narrower prior than the unbiased Bayesian observer (ρ = 1). Moreover, the fitted noise parameter had a median of 0.10 (SD = 0.05), highlighting that subjects’ reported marble ratios deviated from model predictions by 10 marbles on average. We ran multiple regression analyses predicting median estimation error and extreme jar bias from the fitted prior parameter ρ and noise parameter σ respectively. In the first regression model, a narrower prior parameter ρ (t(48) = 13.152, P = , semi-partial η² = 0.771; Fig. 3C) and a higher noise parameter σ (t(48) = 4.501, P = , semi-partial η² = 0.090; Fig. 3D) uniquely predicted larger estimation errors (full model: (F(2,48) = 88.135, P = , R² = 0.786). Importantly, the prior width parameter ρ accounted for most of the explained variance in estimation errors suggesting that our model captured individual differences in subjects’ response behavior well. A narrower parameter ρ reflects a stronger prior belief in a 50:50 marble ratio, which should bias estimates for more extreme jars towards this (incorrect) default belief. In line with this expectation, our second regression model revealed that subjects’ extreme jar bias was positively related to prior width ρ (t(48) = 11.780, P = , semi-partial η² = 0.499; Fig. 3C) but also negatively to response noise σ (t(48) = −2.983, P = , semi-partial η² = 0.032; full model: F(2,48) = 114.968, P = , R² = 0.827; Fig. 3D). The significant (although smaller) effect of the noise parameter σ on extreme jar bias was driven by a small number of subjects with negative extreme jar bias slopes (ie subjects who made larger errors for jars with even rather than extreme marble ratios), which the Bayesian observer model could not account for (see Supplementary Results for further details). Taken together, these analyses indicate that the width of the prior belief substantially impacted task performance by capturing subjects’ tendency to overestimate jars with blue marble proportions close to 0 and underestimate jars with blue marble proportions close to 1 (Fig. 3A).

BOLD signal variability tracks individual differences in prior uncertainty and compresses during learning

We next investigated our hypothesis that (across-trial) BOLD signal variability should compress with successive sample presentations. We quantified BOLD signal variability for each sample period as the standard deviation of the BOLD response over trials in each voxel (SD_BOLD; see Fig. 2C and Materials and Methods for details of this measure). We then ran a task PLS analysis (Krishnan et al. 2011) relating these SD_BOLD estimates to the five sample presentation periods. This analysis returned a significant latent effect (permuted P = 0) showing that SD_BOLD reduced with successive exposure to marble samples across a distributed set of cortical brain regions spanning parietal, prefrontal and temporal lobes (Fig. 4A, see Fig. S2A for full axial brain plots and Table S3 for peak voxel coordinates in robust clusters). For reference, this constituted a median reduction of ~ 7.36% in SD_BOLD from the first to final sample presentation across robust voxels (see Fig. 4A) and subjects. We further investigated this trend by fitting orthogonal polynomial contrasts in a linear mixed model entering subject ID as a covariate. The pattern of latent SD_BOLD change was best described by a linear contrast (F(1,201) = 40.444, P = , semi-partial η² = ) over a quadratic (F(1,201) = 3.066, P = , semi-partial η² = ) and cubic (F(1,201) = 3.416, P = , semi-partial η² = ) one.

Fig. 4.

SD_BOLD tracks idiosyncratic uncertainty representations and predicts inference accuracy. A) Task PLS revealed average SD_BOLD compression over the sampling phase. Error bars represent bootstrapped 95% confidence intervals of the mean. B) Change in SD_BOLD over the sampling phase correlated with estimation accuracy. Greater SD_BOLD collapse related to smaller median estimation errors. The inset shows the distribution of median SD_BOLD change over positive effect voxels (BSR > 3). The diamond marker indicates the distribution median. C) Regions showing performance-related SD_BOLD change (see panel B) largely overlapped with the canonical DMN (Yeo et al. 2011). D) Fitted prior width parameter estimates (residualized by noise parameter σ) of the Bayesian observer model positively correlated with performance-related SD_BOLD change. Note: All variables are rank-transformed.

Next, we asked whether the ability to compress SD_BOLD over the sampling period is related to individual differences in task accuracy. To this end, we quantified SD_BOLD change (ΔSD_BOLD) by fitting linear regression slopes to the SD_BOLD estimates for the five sample presentations in each voxel. We then ran a behavioral PLS analysis relating these SD_BOLD slopes to subjects’ median estimation error. Four Cook’s d (ie multivariate) outliers were removed from this and all subsequent reported analyses on ΔSD_BOLD resulting in n = 47 (we refer to the Supplementary Material for PLS results ran on the full dataset). We found a significant latent relationship (permuted P = , Fig. 4B, cf. Fig. S4A for n = 51) revealing that subjects who decreased SD_BOLD more also produced smaller estimation errors (Spearman’s r = 0.612). For reference, subjects showed a median reduction in SD_BOLD of ~ 5.84 (SD = 9.17%) from the first to final sample presentation across robust voxels (see Fig. 4B). Effect regions largely overlapped with the canonical DMN (Yeo et al. 2011) that is typically observed in resting-state fMRI, including parietal (precuneus and lateral parietal cortex spanning the angular gyrus), prefrontal (frontopolar and orbitofrontal cortex, dorso- and ventro-medial PFC, superior frontal gyrus) and medial temporal brain regions (BSR > 3; Fig. 4C, see Fig. S2B for full axial brain plots and Table S4 for peak voxel coordinates in robust clusters).

Notably, SD_BOLD compression was most pronounced for the final sample presentation period immediately preceding the estimation phase (Fig. 4A). To ensure that SD_BOLD effects were not solely driven by the final sample, we reran all PLS models excluding this trial period. The resulting task PLS still revealed robust SD_BOLD compression over this shortened sampling period (permuted P = ) and the behavioral PLS relating subjects’ estimation error to (linear) SD_BOLD change from the first to fourth sample also found a robust latent relationship (permuted P = ). Furthermore, we looked at the correlation between SD_BOLD estimated separately for the first and second half of task trials to ensure that our SD_BOLD estimates (including all trials) were not confounded by task exposure/practice effects. This control analysis showed that first-half and second-half SD_BOLD estimates were strongly and positively correlated (r > 0.85 for each sample period, Fig. S5).

Next, we investigated our hypothesis that individual differences in state uncertainty representations could explain the relationship between ΔSD_BOLD and estimation error. Our winning behavioral model suggested that subjects with a narrower prior belief distribution over marble ratios (ie large ρ parameter) showed little decrease in state uncertainty during sampling, because they started out with less uncertainty to begin with, and made more errors because they remained inflexible about their (incorrect) prior belief of a 50:50 marble ratio. In contrast, more accurate subjects (ie ρ close to 1) represented maximal state uncertainty at the start of a trial and reduced uncertainty more with each presented sample (Fig. 2B). We expected ΔSD_BOLD to mirror these individual differences in state uncertainty change during the sampling phase. To test this hypothesis, we ran a multiple regression analysis predicting latent ΔSD_BOLD (ie the whole-brain pattern of ΔSD_BOLD, which relates to median estimation error; see Fig. 4B) from the prior width ρ parameter of the winning behavioral model. We also entered the response noise parameter σ as a control variable, which captures all unmodeled sources of estimation error and thus serves as a measure of model misfit (see Materials and Methods). This model explained a significant amount of variance (F(2,44) = 4.489, P = , R² = 0.169). There was a significant main effect only for prior width ρ (t(44) = 2.703, P = , semi-partial η² = 0.138) but not for noise parameter σ (t(44) = 0.425, P = 0.673). Therefore, the wider a participant’s prior belief (ie smaller ρ), the more they compressed SD_BOLD over the sampling phase (Fig. 4D). To ensure that that these effects were not due to the specific parameterization of the winning behavioral model, we replicated this analysis also for the parameters of the (second-best) evidence weight model (see Supplementary Results).

It is possible that the observed within-person effects of ΔSD_BOLD are not independent of between-subject differences in SD_BOLD levels, which have also been linked to task performance in previous studies (Waschke et al. 2021). To account for this, we reran all regression models involving ΔSD_BOLD controlling for SD_BOLD at first sample exposure in spatially-matched brain regions (see Supplementary Results for details). In all models, the inclusion of this control variable did not diminish the effect size of ΔSD_BOLD and thus effects were independent of between-subject differences in initial SD_BOLD levels.

Overall, these results indicate that the degree to which subjects compressed SD_BOLD with learning predicted individual differences in inference accuracy, which was best explained in terms of idiosyncratic uncertainty representations.

Standard GLM neural uncertainty correlates are distinct from BOLD variability effects

We next investigated whether SD_BOLD effects were spatially distinct and uniquely predictive of inference accuracy as compared to average BOLD signal modulation by uncertainty. First, we tested whether mean BOLD changed over the sampling period. We obtained voxel-wise GLM beta estimates for each sampling period (MEAN_BOLD) and entered them into a task PLS. This analysis revealed a significant latent effect (permuted P = 0) showing that MEAN_BOLD increased from the first to last sample presentation in large parts of the cortex (including occipital, parietal, and pre-/frontal regions) and subcortex (mainly thalamus and striatum; Fig. S3, Table S5). In contrast to the linear changes in SD_BOLD, the pattern of latent MEAN_BOLD change was best described by a cubic contrast (F(1,201) = 17.984, P = , semi-partial η² = ).

To examine whether interindividual differences in MEAN_BOLD change related to task performance (ie subjects’ median estimation errors), we next performed a behavioral PLS analysis. In line with the change trajectory suggested by our task PLS results, we quantified MEAN_BOLD change by fitting a cubic regression coefficient to the MEAN_BOLD estimates for the five sample presentations in each voxel and subject (note that all subsequently reported results also hold for alternative quantifications MEAN_BOLD change; see Supplementary Results). Unlike for SD_BOLD (see Fig. 4B), there was no significant latent relationship between sample-related MEAN_BOLD change and subjects’ estimation error (permuted P = 0.173 for n = 48 with Cook’s d outliers removed; permuted P = 0.755 for n = 51). An additional control analysis further verified that the association between ΔSD_BOLD and task performance still held when directly controlling for MEAN_BOLD change in the same brain regions (see Supplementary Results). Together, these analyses confirmed that only SD_BOLD but not MEAN_BOLD modulation predicted subjects’ inference accuracy.

However, change in MEAN_BOLD over the sampling period does not account for more nuanced BOLD response modulation by uncertainty resulting from the varying sample sizes in our task design (see Fig. S1B). In other words, an unbiased Bayesian observer model would predict more uncertainty reduction after observing a larger compared to a smaller sample in addition to the overall reduction in uncertainty with each successive sample presentation. Our SD_BOLD metric required discrete uncertainty conditions (with sufficient trial counts) and we thus only considered the main effect of sample presentation on uncertainty. However, in a standard GLM approach we could model the parametric modulation of the BOLD signal by the posterior variance of the belief distribution (ie uncertainty) for the unbiased Bayesian observer model, which accounts for both the number of presented samples as well as sample size effects on uncertainty (see Materials and Methods).

Again, we used a multivariate PLS approach to relate individuals’ voxel-wise GLM beta estimates (for a parametric uncertainty regressor) to median estimation error. This revealed a significant latent association (permuted P = , Spearman’s r = 0.675, Fig. 5A, cf. Fig. S4B for n = 51). Robust positive clusters were mainly located in the lateral parietal cortex, PFC (dorsomedial, ventrolateral, and rostrolateral parts), and in the insula. Robust negative clusters were present in the vmPFC and in the left hippocampus (see Fig. S2C for full axial brain plots and Table S6 for peak voxel coordinates in robust clusters). This latent relationship was related to the prior width parameter ρ of the winning Bayesian observer model, similar to the effect of ΔSD_BOLD (Fig. 5B). For further details on these results, please refer to the Supplementary Results.

Fig. 5.

Standard GLM neural uncertainty correlates are distinct from SD_BOLD effects. A) BOLD signal modulation by uncertainty (GLM β estimates) correlated with average estimation accuracy. More downmodulation with increasing state uncertainty across the whole brain related to smaller estimation errors. The inset shows the distribution of β estimates over positive (BSR > 3, top) and negative (BSR < 3, bottom) effect voxels. The diamond marker indicates the distribution median. B) Fitted prior width parameter values (residualized by noise parameter σ) of the Bayesian observer model positively correlated with performance-related BOLD signal modulation by uncertainty (GLM β estimates). C) Changes in SD_BOLD over the sampling phase uniquely predicted task performance when controlling for BOLD signal modulation by uncertainty (ie latent ΔSD_BOLD residualized by latent uncertainty GLM β estimates). Note: All PLS results are presented for rank-transformed variables.

Qualitatively, these effect regions seem to show little overlap with those showing an effect for ΔSD_BOLD. To test the spatial specificity of ΔSD_BOLD effects, we computed a new latent variable by extracting the brain saliences from the behavioral PLS analysis relating latent ΔSD_BOLD to estimation error and matrix-multiplied them with the uncertainty beta estimates in each voxel from our standard parametric GLM analysis. We then ran multiple regression analysis predicting median estimation error from latent ΔSD_BOLD and this spatially-matched latent parametric uncertainty modulation variable. The regression model was significant (F(2,44) = 13.901, P = ) with an R² of 0.387. The main effect of latent ΔSD_BOLD was significant (t(44) = 5.265, P = , semi-partial η² = 0.386) but not for the new latent control variable (t(44) = 0.975, P = 0.335). This indicates that ΔSD_BOLD effects and the parametric uncertainty modulation of the BOLD signal on performance were spatially separated in the brain.

We next checked whether ΔSD_BOLD would still relate to task performance when controlling for standard GLM neural uncertainty correlates (in distinct brain regions). We ran multiple regression analysis predicting estimation errors from both latent ΔSD_BOLD and latent uncertainty modulation. The full model was significant (F(2,44) = 31.883, P = , R² = 0.592). We found a significant main effect of both latent ΔSD_BOLD (t(44) = 3.829, P = , semi-partial η² = 0.136) and latent uncertainty modulation (t(44) = 4.844, P = , semi-partial η² = 0.218). This shows that ΔSD_BOLD accounted for unique variance in task accuracy (Fig. 5C). Note, however, that the explained variance in estimation error by latent ΔSD_BOLD dropped from ~ 38% to ~ 14% when including latent uncertainty modulation as a predictor (cf. Fig. 4B and Fig. 5C). Indeed, there was a significant correlation between latent ΔSD_BOLD and latent uncertainty modulation (Spearman’s r(45) = 0.407, P = ) demonstrating some shared variance. This suggests that some accurate performers displayed both types of (spatially-distinct) uncertainty representations in the brain (Fig. S6).

Overall, these analyses revealed that SD_BOLD effects were spatially distinct and uniquely related to task accuracy compared to neural correlates revealed by a more standard GLM approach.

Discussion

In this study we show that reductions in state uncertainty over the course of learning co-occur with a compression in BOLD signal variability. Moreover, individuals who decreased state uncertainty to a greater extent showed more pronounced SD_BOLD compression and made smaller estimation errors on average. Behavioral modeling suggested that better performers reduced state uncertainty to a greater extent because they began with a wider (more uncertain and flexible) prior belief distribution before observing any samples, resulting in more unbiased marble ratio estimates.

Our findings establish SD_BOLD as a novel within-person neural correlate of uncertainty, which reduces over the course of Bayesian inference/learning, and relates to inference accuracy. The current findings thus add to a growing literature demonstrating that within-person SD_BOLD modulation in the face of varying task demands facilitates adaptive behavior across different cognitive domains (see Waschke et al. 2021). Our study is the first to directly show that SD_BOLD tracks reductions in uncertainty. The bulk of past empirical work arguing for such a link showed modulation of brain signal variability across disparate task conditions that varied along several dimensions beyond uncertainty, eg cognitive demand, bottom-up sensory input or processing requirements (Garrett et al. 2013; Grady and Garrett 2014; Grady and Garrett 2018; Garrett et al. 2020). In contrast, here we employed a learning paradigm, which enabled us to operationalize uncertainty in a precise manner; specifically, one’s uncertainty regarding the latent state of an environmental variable, a property that can be reduced through learning from observations. We used this paradigm in conjunction with a novel event-related method to quantify SD_BOLD, which allowed us to track the evolution of SD_BOLD with changes in uncertainty on a short time scale (several seconds). This method was inspired by the machine-learning fMRI literature (Mumford et al. 2012) and is a stark departure from previous fMRI work, which usually computed SD_BOLD over entire task blocks (Garrett et al. 2013; Waschke et al. 2021). Moreover, our GLM-based analyses revealed that SD_BOLD effects are still held when statistically controlling for average BOLD changes in the same brain regions. This result was expected given that previous studies have reported only modest associations between SD_BOLD and mean BOLD response magnitude (Garrett et al. 2010; Garrett et al. 2011; Garrett et al. 2014; Garrett et al. 2015).

Due to the limited temporal resolution of fMRI (which was one time point for each sample period on a given trial in our study design), we defined SD_BOLD as the across-trial variability in the BOLD signal for each within-trial sampling period, and subsequently computed the change in SD_BOLD across sampling periods as our primary neural measure of interest for each subject. We have argued that despite each SD_BOLD estimate being derived from across-trial data, this nevertheless can serve as a direct proxy for within-trial neural dynamics (see Fig. 1). We support this argument with several pieces of evidence. First, our task design ensured that bottom-up visual input and difficulty were matched across trials. We also showed that separate SD_BOLD estimates for the first and second half of task trials were strongly and positively correlated (r > 0.85 for every sampling period), suggesting that SD_BOLD computed over all trials for a given sample period likely does not reflect task exposure/practice in any appreciable way. Further, SD_BOLD compression cannot easily be explained by other potential between-trial sources of variability (eg arousal, attention), as such sources would likely contribute similarly to all within-trial sampling periods in any given trial. Therefore, we believe the key source of variability contributing to across-trial BOLD variability is within-trial neural variability related to stimulus processing (ie learning/uncertainty). Our findings thus align well with recent evidence showing that (within-trial) Electroencephalogram (EEG) signal entropy decreases with decreasing uncertainty on a perceptual decision-making task, which also ensured matched bottom-up visual inputs between task conditions (Kosciessa et al. 2021).

Taken together, our results support the longstanding idea that more brain signal variability enables more cognitive flexibility (Garrett et al. 2013; Waschke et al. 2021), which we show is required under high uncertainty early on in the learning process to efficiently update one’s internal belief about the environmental state.

Individual differences in uncertainty representations track inference accuracy

We found that low performing individuals made most estimation errors for extreme marble ratios, such that jars with blue marble proportions close to one were underestimated and those with proportions close to zero were overestimated. This effect has been reported previously for tasks where participants are asked to judge outcome probabilities from experience, sometimes referred to as under-extremity (Griffin and Brenner 2004; Spiliopoulos and Hertwig 2022; Spiliopoulos and Hertwig 2023). Notably, some studies found that this effect is abolished when participants were asked to give verbal judgments instead (Gottlieb et al. 2006). This may suggest that individual differences in extreme jar bias result from people’s (in)ability to accurately enter their marble ratio estimates into the grid in the estimation phase of our task. However, we found that brain activity during sampling (prior to the estimation phase) predicted individual differences in peoples’ (average) estimation error. It is thus likely that the observed bias reflects people’s internal representations rather than being a mere consequence of our chosen response modality.

Our winning Bayesian observer model suggested that such estimation errors could best be explained by individual differences in the representation of the initial prior belief over blue marble proportions rather than in belief updating; low performers held a narrower prior belief distribution over the default belief of a 50:50 marble ratio. It is possible that this prior was induced by our task design given the default position of the response grid. The fitted prior could thus reflect individual differences in anchoring to this task feature (Camilleri and Newell 2009). Alternatively, this bias could reflect bona fide trait-like individual differences in expectations about extreme outcomes. In line with this explanation, Glaze et al. (2018) found that performance differences on a change-point inference task were best explained by participants’ prior width over potential change rates, which cannot simply be explained by anchoring. Note that the distinction between prior beliefs and belief updating is also pertinent to other approximately-Bayesian and non-Bayesian cognitive models that have been used to model the inference process in similar tasks, which we did not consider here (Olschewski and Scheibehenne 2023; Spiliopoulos and Hertwig 2023). These models account for cognitive limitations in human information processing and may provide an even better fit to behavioral data. While not the main aim of the current study, future modeling and experimental work is required to pin down the source of individual differences in under-extremity observed in the current study.

We showed that less SD_BOLD compression (independent of initial SD_BOLD levels) in poorer performers could best be explained by more limited uncertainty reduction, based on parameter estimates for the winning behavioral model. Still, our results rely on the assumption that participants represent and utilize uncertainty akin to an (approximate) Bayesian observer. Future work should reduce reliance on model assumptions and obtain subjective uncertainty estimates. For example, one could regularly ask people to report their state belief together with their belief confidence throughout the learning process to directly track updates in their subjective belief distribution.

Finally, it is interesting to note that SD_BOLD compression was most pronounced after the final sample was presented, just prior to the onset of the estimation phase (Fig. 4A). Arguably, belief uncertainty (and their cognitive flexibility demands) is maximally reduced at this stage: Subjects can anticipate the (certain) onset of the subsequent estimation phase and may already prepare a specific response. In other words, BOLD variability reduction at the end of the sampling period may not only reflect uncertainty reductions due to learning, but may also capture decision-related processes. Our modeling results tentatively support this interpretation; subjects’ responses best reflected a point-estimate from the final belief distribution rather than a random sample, which would require maintenance of the full belief distribution for decision-making. Unfortunately, our paradigm is not designed to dissociate these potential sources of uncertainty reduction during the final sample presentation. Future work may directly test differing accounts by only probing a response after a random number of samples has been presented (to control for decision-related processes).

Performance-related SD_BOLD compression in the default mode network

We observed performance-related compression in SD_BOLD in a network of brain regions that largely overlapped with regions commonly ascribed to the DMN in resting-state fMRI connectivity analyses (see Fig. 4C; Damoiseaux et al. 2006; Fox et al. 2005; Greicius et al. 2003; Raichle 2013; Yeo et al. 2011). Previous work has consistently reported deactivation of this brain network during externally-cued, demanding cognitive tasks compared to rest (Shulman et al. 1997; Raichle et al. 2001; Greicius et al. 2003; Fox et al. 2005; Raichle 2013; Utevsky et al. 2014). In contrast, internally-oriented tasks that call for self-referential processing and memory recollection demonstrate increased activity in key regions of the DMN (Gusnard et al. 2001; Buckner and Carroll 2007; Spreng and Grady 2010; Raichle 2013). With respect to brain signal variability, one study by Grady and Garrett (2018) found increased SD_BOLD in DMN regions in externally- as compared to internally-guided tasks. Therefore, SD_BOLD compression during learning may reflect the gradual build-up of an internal world model; larger SD_BOLD under high state uncertainty early on may afford more flexible incorporation of incoming information to update one’s internal state belief. As state uncertainty reduces and SD_BOLD compresses, the brain arrives at a stable internal belief representation, which informs one’s jar marble ratio estimate in the subsequent task phase.

We speculate that SD_BOLD compression in the DMN alongside learning tracks changes in belief state that reflect belief updating processes across the brain. The vmPFC node of the DMN in particular is a candidate region for locally reflecting the (changing) belief distribution over possible environmental states. Evidence suggests that this region encodes an internal world model by extracting information across individual experiences, such as the value of different choice options (Levy and Glimcher 2012; Averbeck and O’Doherty 2022; Klein-Flügge et al. 2022), cognitive maps (Boorman et al. 2021; Klein-Flügge et al. 2022; Schuck et al. 2016; Wilson et al. 2014), or schemas (Gilboa and Marlatte 2017; Bein and Niv 2025). Moreover, neural activity in this region also tracks uncertainty about external states, which has been related to adaptive behavior on probabilistic learning tasks (Karlsson et al. 2012; McGuire et al. 2014; Starkweather et al. 2018; Domenech et al. 2020; Trudel et al. 2021). These prefrontal neural dynamics in turn may be shaped by (bottom-up) sampling representations in a hierarchical way. SD_BOLD changes in parietal and temporal regions of the DMN may thus reflect functional coupling to prefrontal regions to support belief updating processes, eg via memory-based information sampling (Vincent et al. 2006; Sestieri et al. 2011; Kaefer et al. 2022).

Standard analytic approaches reveal different neural uncertainty correlates compared to SD_BOLD

Our standard GLM analyses revealed that BOLD modulation by uncertainty, as derived from an unbiased Bayesian observer model, also related to estimation accuracy in our task. On one hand, negative coupling between the BOLD signal and state uncertainty predicted better task performance. In other words, high performers showed an increase in BOLD activity as they observed more and larger samples and became more certain about the jar marble ratio on a given trial. Brain areas showing this effect overlapped with fronto-parietal control and dorsal attention brain networks, which support goal-directed behavior in externally-driven tasks (Corbetta and Shulman 2002; Pessoa et al. 2003; Dosenbach et al. 2007; Vincent et al. 2008; Yeo et al. 2011). Conversely, positive coupling between the BOLD signal and state uncertainty (mainly in the vmPFC and a cluster in the left hippocampus) also predicted better performance in our task, which has previously been linked to confidence representations (Vaccaro and Fleming 2018; please refer to the Supplementary Discussion for further details).

Importantly, we found that uncertainty representations in SD_BOLD (in DMN regions) and parametric BOLD signal modulation (mainly in fronto-parietal control/attention regions) were spatially distinct and uniquely predicted estimation accuracy. Why would the brain track uncertainty in these different neural signals and across different networks? One explanation could be that SD_BOLD compression reflects changes in belief state representations across the brain, while parametric BOLD modulation may reflect the read-out of this uncertainty in down-stream brain areas relevant for attention-based visual sampling (Radulescu et al. 2019) and decision-making. In support of this view, previous research has shown modulation of brain signal variability by stimulus features even when no action was required. For example, in the human neuroimaging literature, changes in neural variability have been observed in response to varying complexity of visual stimuli during passive viewing, which were linked to offline cognitive performance (Garrett et al. 2020; Waschke et al. 2025). In a similar vein, it has recently been argued that early perceptual uncertainty, which has been the main focus of sampling accounts of neural variability, is tracked irrespective of task demands but can be flexibly utilized in higher-order decision-making (Fiser et al. 2010; Berkes et al. 2011; Orbán et al. 2016; Koblinger et al. 2021). In contrast, parametric BOLD modulation by uncertainty has been reported for learning and decision-making tasks that require participants to make choices on every trial (Huettel et al. 2005; Grinband et al. 2006; Behrens et al. 2007; Bach et al. 2011; Bach and Dolan 2012; Badre et al. 2012; Vilares et al. 2012; Payzan-LeNestour et al. 2013; McGuire et al. 2014; Michael et al. 2015; Meyniel and Dehaene 2017; Nassar et al. 2019; Meyniel 2020; Tomov et al. 2020; Trudel et al. 2021; Walker et al. 2023). Interestingly, we also found that some high-performing subjects displayed both neural uncertainty signatures (see Fig. S6), and a previous study by Spreng et al. (2010) suggests that internally-guided cognitive tasks evoke a coupling between the DMN and the frontoparietal-control network to support task performance. As such, functional coupling between the DMN and the fronto-parietal network may have scaffolded attention and decision-making based on learned state representations in our task.

Limitations and next steps

One potential limitation of our task design was that our stimulus sample size manipulation introduced trial-to-trial variance in how much uncertainty could be reduced with each presented sample (Fig. S1B). Because our SD_BOLD measure was computed for each sample period across trials, uncertainty levels were by definition not equivalent within each condition bin (although uncertainty always decreased across sample periods). However, our results showed that SD_BOLD effects could not be explained by parametric BOLD signal modulation by trial-to-trial differences in uncertainty trajectories in the same brain regions, which could have accounted for this variance. Ideally, future studies should keep sample size consistent between trials so that SD_BOLD can be computed over comparable uncertainty levels.

Another potential limitation is that uncertainty was collinear with within-trial time in our task; the more samples were presented, the more uncertainty could be reduced in general. However, this is an inevitable consequence of decision-making and learning in stable environments, in which one can simply accumulate evidence over time. Although within-trial time could have introduced arousal- or attention-related effects, which have previously been linked to human brain signal variability (Waschke et al. 2021), we observed performance-related interindividual variability in SD_BOLD compression over the sampling period. Such an effect cannot, by definition, be accounted for by the fixed factor of trial time. Nevertheless, future work could investigate the coupling between brain signal variability and uncertainty in nonstationary learning environments (see eg Bruckner et al. 2022), within which uncertainty can increase or decrease over time.

Our observations also call for future (systems neuroscience) investigations of candidate neural mechanisms that may jointly shape both local variability and belief precision. Past work has linked a variety of candidate neurochemical/−modulatory mechanisms of neural variability to broad-scale cognition, eg local E/I balance (Kosciessa et al. 2021; Waschke et al. 2021; Lalwani et al. 2025) or dopamine (Garrett et al. 2015), but none yet to belief precision directly.

Conclusion

We provide first evidence that temporal brain signal variability compresses with increasing belief precision during learning. Whether brain signal compression is directly proportional to the informativeness of the available evidence is an important prediction that should be investigated in future work.

Author contributions

Alexander Skowron (Conceptualization, Data curation, Formal analysis, Visualization, Project administration, Writing—original draft, Writing—review & editing, Funding acquisition), Julian Q. Kosciessa (Writing—review & editing), Robert Lorenz (Investigation, Data curation, Methodology, Resources, Software), Ralph Hertwig (Writing—review & editing), Wouter van den Bos (Methodology, Writing—review & editing), and Douglas D. Garrett (Conceptualization, Methodology, Formal analysis, Project administration, Supervision, Writing—review & editing, Funding acquisition).

Funding

This work was supported by the International Max Planck Research School on the Life Course and the International Max Planck Research School on Computational Methods in Psychiatry and Ageing Research (A.S.); Emmy Noether Programme grant from the German Research Foundation (to D.D.G.); Max Planck UCL Centre for Computational Psychiatry and Ageing Research (D.D.G.).

Conflict of interest statement. The authors declare no conflict of interest.