Discrete slot memory models

The main challenge that participants face in FtE is to avoid items where a target was already encountered and items that have already been selected in an ongoing search. Optimal performance, therefore, depends on the ability to store, retrieve, and update information about already tapped items. To introduce our model and its notation, we build upon Cowan’s model of VSWM [8], which posits that VSWM consists of a finite set of slots in which novel discrete information is stored. Below we summarize this model in a number of assumptions, several of which we will revise later:

1. (1). Items are stored in a discrete number of slots, jointly referred to as the memory pool.
2. (2). An item occupies exactly one slot in memory. Memory capacity M is defined as the number of slots in the memory pool.
3. (3). New items are stored in memory as long as there are free slots in the memory pool.
4. (4). Items are stored and retrieved deterministically from memory.
5. (5). Participants only search items that are not in memory. We refer to these items as the decision pool.
6. (6). Participants select items from the decision pool randomly.

These assumptions enable an estimation of the probability of an error conditional on the memory capacity M, the set size S and the items previously searched. First, these assumptions imply that errors are only possible when the optimal number of items that need to be remembered L (for load) is larger than the memory capacity M. Second, when L is larger than M, the probability of an error is the probability of selecting a searched item from the decision pool. The size of the decision pool is simply the set size S minus the memory capacity M. Thus, we write

where P(Error) is the probability of an error.

Here, it is important to note that with this approach we seek to predict the probability that a participant will make an error, and which type of error, but not the probability that a specific item is selected. This limitation of the model is captured by the assumption that items are selected randomly from the decision pool, as the model does not predict which specific item will be tapped. Non-random strategies like, for example, first select items from the top of the screen, are fundamentally compatible with our approach, as they provide a further level of granularity that we don’t seek to model here.

The assumption that players only search items in the decision pool can be relaxed by postulating that there is a small but positive probability of tapping any item on the screen. Thus, accounts for errors caused by, e.g., attentional slips or motoric errors, which we collectively refer to as ‘lapses’. The implementation details follow the same approach as in [30,31] and are explained in S1 Text. For the sake of simplicity, we will omit this aspect of the model from the rest of the exposition.

To enable the derivation of the probability of BSE and WSE separately, we add the assumption that:

1. (7). All items are left out of memory with the same probability, i.e., all items have the same chance or priority to be stored in memory.

Accordingly, on average, the conditional probability of a BSE is equal to the number of items that hid an egg in previous searches (i.e., between-search items) B divided by the load L. Therefore, we can write

Since the load L is simply the sum of B and W, it follows that

Fig 2A-2B shows predictions from this model. Notably, the marginal error rate (the sum of BSE and WSE rates) is a linear function of the load L. In S2 Text and S1 Fig, we present two variations of this model in which between-search items are stored before all within-search items or vice versa. Importantly, in both cases, the marginal error rate is a linear function of load.

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Fig 2. Model predictions in the set size S = 10 condition.

Single store-model. A) Marginal and within-search error rate (WSE) rate as a function of load. The marginal error rate increases linearly with the load, i.e., the sum of the within- and between- search items (W and B respectively). B) Between-search error (BSW) rate as a function of load, for fixed values of W, depicted as lines in different colors. Dual-store model. C) Similar to A for a dual-store model (). The marginal error rate is not a linear function of load. D) Similar to B. The dual-store model predicts that for a fixed W, the BSE rate increases linearly as a function of load. Stochastic multi-store model. Two simulations with , and (solid lines and circles) and (dashed lines and stars). E) The marginal error rate is a non-linear function of load. The marginal error rate is only slightly different between and . F) BSE rate similar to B and D. Contrary to a deterministic retrieval model (panel D), BSE is a non-linear function of load for fixed W.

https://doi.org/10.1371/journal.pcbi.1014535.g002

Multiple-store models

Above we assumed that between- and within-search items occupy and compete for the same slots in a single store. However, the demands imposed by the task on each type of item are very different. Within-search items are retained only briefly and are regularly removed from memory once a new target is discovered. By contrast, between-search items are stored for longer periods and should not be overwritten across searches. This suggests that different cognitive processes may be involved in managing these two types of information within the task. To enable our model to capture these discrepant demands, we can relax assumption (1) to:

(1a) Between- and within-search items are stored in independent memory pools of capacity and .

Using the same logic as before, BSE are only possible when the number of between-search items B exceeds the between-search memory capacity :

In other case, the probability of a BSE is given by

The denominator corresponds to the size of the decision pool, which is the set size S minus the total number of items in memory .

Importantly, the marginal probability of an error is no longer a linear function of the load (Fig 2C), yet this dual-store model predicts that for a fixed number of within-search items W, the probability of a BSE is a linear function of the number of between-search items B, as displayed by Fig 2D.

Stochastic memory retrieval

The models above share the assumption that participants can always correctly retrieve items stored in memory. However, on occasion, participants might not be able to retrieve an item from memory. Here, we can distinguish between failures in retrieval and failures in encoding or storage, in that the latter imply irreversible information loss, which on average would manifest as reduced memory capacity. By contrast, retrieval failures are reversible as information is not forgotten – rather, on every individual trial, there is a chance that information stored in memory is not correctly retrieved.

To enable the model to distinguish between retrieval and encoding/storage failures, we relax our assumption of deterministic retrieval to the following:

(4a) Participants can always correctly retrieve a limited number of items from memory . The remaining items are stored in slots, from which retrieval is stochastic and fails with a constant rate , , independently of the set size or difficulty. Items not retrieved correctly from memory are part of the decision pool and therefore can lead to errors.

This assumption can also be understood as enabling non-discrete (i.e., partial) memory capacity. Specifically, the capacity of a slot can be formally understood as r, and its variability to be equal to r(1-r), in analogy to non-discrete models [5].

Unlike the more restrictive models articulated earlier, this model predicts neither a linear marginal error rate as a function of load nor linear increases in BSE rate as a function of load for fixed W Fig 2E-2F). Rather, it predicts three different phases as a function of B. Initially, errors can only be explained as lapses captured by parameter , as items are perfectly stored and retrieved. Following this phase, items are kept in a store of maximum capacity , from which items are correctly retrieved at rate . In the third phase, no more memory is available () and the error rate quickly increases as a function of B (Fig 2F). The formalization of this model is presented at length in the Methods section.

An important challenge in estimating the parameters of this model is that and are linearly correlated since the expected total number of remembered items , takes the form

when , as is the expected value of the underlying binomial process. To mitigate the effects of this collinearity, we focus the inference on the total memory capacity , rather than on its constituent elements.

Measuring visuospatial working memory in a clinical population

The models of VSWM presented here are elaborations of classical models of working memory. Critically, we have shown how a hybridization of classical assumptions entails distinguishable predictions. Next, we evaluate these competing models of VSWM in clinical data, collected from different populations and age ranges. First, we evaluate the models presented above on the data from two clinical trials in ASD (oRBiting and V1aduct) described in detail in the methods section. Here, we briefly summarize both studies.

oRBiting (NCT03611075) was a non-drug study seeking to characterize clinical scales to measure repetitive and restrictive behaviors across ASD sub-populations. Participants were stratified in three age groups: children (5–12 years old), adolescents (13–17) and adults (18–45). oRBiting included both individuals with intellectual disability (IQ < 70) and IQ ≥ 70. Demographics breakup is reported in S2 Table. The study enrolled 144 participants, from which 22 were excluded because less than 4 sessions of FtE were available. FtE was administered every five days. The mean BSE and WSE rates were 19%(±8) and 2.5%(±2.7) respectively. After controlling for age, IQ was significantly correlated with BSE rate () but not with the mean WSE rate (). Performance in the task is summarized in S2-S3 Figs and S2 Table.

V1aduct (NCT03504917) was a double blind, placebo controlled, 24-week phase III clinical trial investigating the safety and efficacy of balovaptan —an antagonist of the AVPR1A receptor— on adaptive behavior in autism [7,18]. After completing the 24 weeks, participants were offered to participate in an open- label study extending for up to 104 weeks. V1aduct was stopped after around half of the participants completed 24 weeks because of futility, as the study was deemed unlikely to reach its primary endpoint [18]. Data from the placebo and treatment arms were combined in all the following analyses, as there is no evidence of a treatment effect across several secondary analyses [18].

Here, we present data from the digital biomarker sub-study comprising 29 individuals who consented to use Roche’s ASD digital biomarker suite. Participants were provisioned with a smartphone configured to administer FtE and other cognitive tests every five days. At any moment, participants could opt out of this sub-study, which led to large differences in the number of FtE sessions available per participant (S2 Table). Three individuals were not included in the final analysis, as they participated in fewer than 4 sessions. The mean IQ of the rest of the participants (21 male) was 104 ± 16 (range 73–137) and mean age of 27 ± 16 years (range 18–62).

In V1aduct, we analyzed 33644 individual trials, ranging from as few as 131 trials per participant to as many as 3905. The longer duration of V1aduct and the factors mentioned above explain the large amount of data available for certain participants. IQ was correlated with mean BSE () and mean WSE () rates.

Modeling.

Were any of the models proposed above able to capture participants’ performance in FtE? To answer this question, we first examined the group level marginal error rate as a function of load across set sizes. Fig 3A demonstrates that, in contrast to the predictions of the single store models, the marginal error rate did not follow a linear trend. This was true at the level of single subjects (Fig 3B) and thereby this pattern is not an artifact of averaging across participants.

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Fig 3. A) Mean marginal error rate from all participants in oRBiting.

Error bars represent the standard error of the mean. The error rate is a nonlinear function of load. Dotted lines display the fits of the dual compartment stochastic memory retrieval model. B) Marginal error rate of two representative participants and model estimates, similar to A. Each row displays data from a representative participant. Clearly, the error rate is a nonlinear function of the load, ruling out the predictions of the single-store models. C) Between-search error rate as a function of load for fixed within-search item at different set sizes in oRBiting and model predictions averaged across all participants. W: Within-search items.

https://doi.org/10.1371/journal.pcbi.1014535.g003

Given the very low number of WSE in both studies (less than 3% of all trials), in the following we assumed that participants’ memory capacity for within-search items was larger than the max set size, i.e., . This implies that all WSE and a fraction of all BSE can be explained as lapses, quantified by parameter .

The multi-store model makes a further prediction: the number of between-search items should not affect the WSE rate. The data overwhelmingly conforms to this prediction across loads (see Fig 4), with the exception of the highest loads, in which the WSE sharply increases with the number of between-search items for a fixed number of within-search items.

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Fig 4. Within-search error (WSE) rate as a function of load for different set sizes (color lines) and model predictions (black, dotted lines).

A key prediction of the multi-store model is that the WSE rate should be constant with increasing load, when W stays constant. Our analysis suggests that only in the most challenging conditions, the WSE rate increased with load for fixed W. WSE errors constituted only 3% of all the data. W=Within-search items.

https://doi.org/10.1371/journal.pcbi.1014535.g004

Qualitatively, we closely replicated our findings in V1aduct, as shown in S4 Fig.

S5 Fig displays a pooled analysis of both studies.

The previous analyses indicate that between- and within-search items are stored independently but leave open whether a distinction between deterministic and stochastic retrieval may better capture participants’ behavior in FtE. There are two main alternatives to the stochastic retrieval model proposed above: Items are always retrieved without error ( or no items are retrieved perfectly ().

Models were fitted quantitatively using the Hamiltonian Monte Carlo (HMC) algorithm as implemented in TensorFlow Probability [47]. HMC is a hybrid algorithm that uses the gradient of a target distribution to efficiently sample from it. The gradient computation is automatically handled by TensorFlow, using automatic differentiation. The target probability of the models is the trial-by-trial likelihood multiplied by the prior probability of the model parameters, as specified in the Methods section. The full model has two discrete parameters and , which could range from 0 to the largest set size possible in the task in addition to two continuous parameters: the probability of a slip and the retrieval rate . Since the HMC algorithm can only be used on continuous parameters, the discrete parameters can be easily marginalized out by computing

Once samples from the marginal posterior probability of parameters and were drawn using the HMC algorithm, these were used to estimate and to make posterior model predictions. Data from all sessions of a single participant were pooled together to obtain individual posterior estimates that were used to predict participant’s response on a trial-by-trial basis. Details of the inference are presented in the Methods and the code used for model fitting is openly available at https://osf.io/3ufg5.

To compare these two alternatives, we fixed the corresponding model parameter and compared the resulting Watanabe-Akaike Information Criterion (WAIC), a measure of model parsimony [14] that penalizes models for their complexity. In our two data sets, the full model explained participants’ data the best (Table 1). Removing the ability of the model to retrieve some of the items perfectly (by setting ), or to retrieve some items imperfectly (by setting deeply hurt the parsimony of the model. S6 Fig shows how the full model yielded qualitatively the best fits. We also calculated the relative probability of each model on the individual participant level. The mean relative probability of a model across participants represents the expected fraction of individuals in which that model is favored. Overall, the full model was favored by between 61 and 65% of participants, with the model without deterministic storage being favored by between 29 and 34% of participants. Thus, most participants used some form of non-deterministic retrieval, and the majority also relied on perfectly reliable storage.

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Table 1. Quantitative model comparison.

https://doi.org/10.1371/journal.pcbi.1014535.t001

To investigate if there were differences between the models across groups in oRBiting, we applied MANOVA on the participants’ model probabilities transformed to the real interval. Participants with IQ below 70 were excluded, as this group was composed exclusively of autistic people. We found no effect of diagnostic group (P = 0.15), or age (P = 0.92).

In summary, our qualitative analysis showed that sequential recalling of items in FtE is mediated by two types of memory: one for within-search items and one for between-search items. These two types of memory are largely independent and only in the most challenging conditions are there signs of interference between them. Moreover, the recall of between-search items is characterized by both deterministic and stochastic retrieval.

Individual differences

Can total memory capacity shed light on participants’ overall cognitive skill? Since working memory is closely related to fluid intelligence [9,13,43], we hypothesized that total memory capacity should be associated with IQ across groups, independently of age. Indeed, in both studies we found a significant and similar correlation between IQ and (Fig 5, oRBiting: ; V1aduct: ).

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Fig 5. Total memory capacity correlated with IQ in both oRBiting (blue) and V1aduct (yellow).

https://doi.org/10.1371/journal.pcbi.1014535.g005

Alzheimer’s disease progression is associated with loss of memory capacity

Above, we demonstrated that , our model-based index of VSWM, is well associated with differences in IQ across age and neurodevelopmental diagnostic groups. More important from a clinical perspective is whether the model is sensitive to disease progression or treatment. To test this hypothesis, we analyzed longitudinal data from Scarlet RoAD (NCT01224106), a large, 104 weeks, Phase 3 clinical trial that assessed the safety and efficacy of gantenerumab in slowing the progression of AD in early stages of the disease [25]. The most relevant inclusion criteria were age between 50 and 85 years, classification of prodromal AD and presence of amyloid pathology (CSF . Among others, participants were excluded in case of history of abnormal brain MRI and history of a neurological disease other than AD. Scarlet RoAD was terminated early for futility. At the time of termination, there was no significant difference in the primary endpoint between the placebo group and any of the active treatment groups. Hence, data from participants on active treatment and placebo were combined in the present analysis. Details about Scarlet RoAD have been published elsewhere [25].

Since this study started much before FtE was developed, we instead fitted the model to data from the CANTAB Spatial Working Memory task (CSWM) collected during the trial. While these two tests are structurally similar, there are several differences between them. Saliently, CSWM was administered during site visits and not remotely. Moreover, CSWM follows a fixed schedule of assessment, comprising of two rounds of 4, 6 and 8 items. Other differences between tasks are explained in the Methods section.

CSWM was administered at screening, baseline and weeks 24, 52, 76 and 104, in combination with several neuropsychological tests, including the ADAS-COG 13, the Mini Mental State Examination (MMSE)–two instruments designed to measure cognitive impairments–, and the Clinical Dementia Rating Scale Sum of Boxes (CDR-SB), a clinical endpoint used to measure both loss of function and cognitive impairment in dementia. In addition, volumetric T1 weighted images were collected (see Methods), from which hippocampal volume, expressed as percentage of intracranial volume, were extracted. We used these clinical instruments, as well as changes in hippocampal volume, to assess the ability of to quantify changes in cognition in prodromal AD.

Given the longitudinal nature of our question, we extended the model to capture changes in working memory using a hierarchical approach, explained in detail in the Methods section. This hierarchical approach is essentially a random effects model that captures the correlations between sessions from a single individual while allowing for changes across sessions. Since the data from any individual session is limited, this extension of the model improves our ability to capture longitudinal changes in memory capacity. S3 Text and S8 and S9 Figs demonstrate the model’s accurate estimation of with parameter recovery tests. We fitted the model to trial-by-trial data from participants with at least five visits. Group and individual level fits are displayed in S10 and S11 Figs. respectively, showing qualitative agreement in participants’ behavior across tasks and accurate model fits. In total, 371 participants were included in the final analysis (mean age 70 ± 7; 261 females). Additionally, we fitted this hierarchical model to data from oRBiting and V1aduct, obtaining estimates of for each session. No significant longitudinal change was detected (S12 Fig).

Once the model was estimated from the longitudinal data of every participant, an estimate of was extracted for each session. To test the longitudinal changes in and compare it with other clinical scales and hippocampal volume, we fitted mixed effect models with random intercepts and slopes for each participant, setting age and sex as covariates (see Fig 6A-6E). Exemplary individual longitudinal trajectories are displayed in S13 Fig. Following an uptick in after the screening session—likely due to familiarization with the test—there was a decline in over the next two years (Fig 6; ). Since this effect was clearly attenuated by the familiarization with the test after the screening session, we repeated this test after excluding the screening session, which demonstrated a much larger effect size ( Quantitatively, the decline in working memory was small, but this was to be expected based on the early stage of the disease. The clinical scales and the hippocampal volume revealed a progressive decline in cognition and increased neurodegeneration in the same period.

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Fig 6. A-D) Mean value of , ADAS-COG-13, CDR SOB, MMSE and fractional hippocampal volume over Screening, Baseline, weeks 24, 52, 76 and 104.

F-I) Estimated individual mean over all sessions except screening and mean individual values of clinical scales and hippocampal volume. Solid read lines depict moving average fit. L-O) Estimated individual slopes of change of , clinical scales and hippocampal volume. Slope of change of the hippocampal volume was scaled by a factor of 100 for clarity. Solid read lines depict moving average fits. ADAS-COG 13: Alzheimer’s Disease Assessment – Cognitive Subscale 13 Questions, CDR-SOB: Clinical Dementia Rating Scale, Sum of Boxes. MMSE: Mini Mental State Examination. S: Screening. B: Baseline. : Total memory capacity.

https://doi.org/10.1371/journal.pcbi.1014535.g006

Not only were the random intercepts of significantly correlated with other clinical endpoints (S3 Table; P < 10^-5) but also their rate of change (Fig 6; P < 0.001). Interestingly, when we extended this analysis to the hippocampal volume (Fig 6H), intercepts were not significantly associated with (), although there was a nominally significant but weak association between their slopes (). By contrast, hippocampal volume was associated with the rest of the clinical scales, both in terms of participants’ intercepts and slopes (S4 Table). This suggests that while hippocampal integrity is associated with functional and cognitive changes in AD, these changes are independent of the loss of VSWM in early stages of the disease.

In summary, our model of VSWM can be extended to structurally similar tasks. Furthermore, it can characterize nontrivial aspects of working memory performance in both neurodevelopmental and aging populations. Moreover, it is sensitive to small longitudinal changes in VSWM—in this case, to changes caused by neurodegeneration —, which are well correlated with cognitive changes measured by the MMSE and ADAS-COG13. Nevertheless, loss of VSWM does not seem to be associated with reduction of hippocampal volume, suggesting a more secondary role for hippocampal function in the VSWM needed for this task.

Ethics statement

Written informed consent was obtained from all subjects and/or their legal guardian(s) (in case of minors). A list of institutional review boards or ethics committee that approved each of the studies analyzed here is provided in S4 Text.

V1aduct and oRBiting

Data from FtE were collected in two clinical trials: oRBiting (NCT0361107) and V1aduct (NCT03504917). Orbiting was a 12-week observational study seeking to characterize different scales to measure repetitive and restrictive behaviors in different ASD sub-populations over time. The most relevant inclusion criteria were age between 5 and 45 years, IQ above 50 as assessed by the Stanford-Binet Intelligence Scale Fifth Edition [19] and a parent or study companion that could accompany participants to the clinical visits. For autistic individuals, the ASD diagnosis was confirmed by the Autism Diagnostic Observation Schedule-2 (ADOS-2; [15]). Written Informed consent or assent was obtained during screening. Participants were excluded among other reasons if a disease or condition could interfere with the study in opinion of the principal investigators, and for autistic participants, if a diagnosis of a syndromic form of ASD was present (e.g., fragile-X syndrome, Angelman Syndrome, Prader-Willi Syndrome).

V1aduct was an interventional, 24-week Phase 3 clinical trial evaluating the efficacy and safety of balovaptan, an antagonist of the AVPR1A receptor. Details about V1aduct have been published elsewhere [18]. All participants were adults with a confirmed ASD diagnosis by the ADOS-2 and IQ equal to 70 or above as assessed by the Wechsler Adult Intelligence Scale-II [46]. The most relevant exclusion criteria were the intake of any drug that could cofound balovaptan’s hypothesized effect, pregnancy or breast feeding, substance use disorder as per DSM-5, epilepsy, seizures or an unstable psychotic or neurological disorder. After completing the 24-week study, participants could enroll in a 104-week open label study. Written Informed consent was obtained from all participants during screening. V1aduct was stopped after 322 participants were enrolled because of futility [18]. A small group of participants (35) were invited to participate in a digital biomarker sub-study including the FtE that extended into the open label study.

At the start of both trials, a study coordinator handed out a smartphone to participants or their study companion and instructed them on how to complete the tasks delivered through the phone. According to the schedule, FtE was administered every fifth day. At every point, participants had the option to deactivate FtE, which led to variable amounts of data collected from each participant.

Experimental procedure: Find the egg

Fig 1 summarizes Find the Egg. Non-spatially overlapping items (chickens) appeared arranged on a regular grid on the screen of a smartphone in sets of 4, 6, 8, 10, or 12 items. Participants’ instructions were to find the hidden targets (eggs) by tapping on the items (chickens). Once a target was discovered, subjects had to drag it to the basket in the bottom of the screen. To complete a block, participants had to find all the hidden items. At no moment, participants received explicit feedback about their performance except for the total number of targets found so far.

FtE incorporates several changes with respect to [26] task, aimed at keeping the test challenging and interesting despite its dense administration schedule. First, when participants successfully completed a round, the difficulty of the task increased by displaying two extra items on the screen in the next round. Sessions ended after participants made more than three consecutive errors or after the most difficult condition (12 items) was completed. In the next session, the number of items on the screen was reduced by two compared to the highest accomplished level in the previous session. In case a participant failed to complete the easiest condition (4 items), the next session started with the same number of items.

Scarlet RoAD

Scarlet RoAD was a multicenter, randomized, double-blind, placebo-controlled, Phase 3 clinical trial evaluating the safety and efficacy of gantenerumab, an anti-amyloid-beta monoclonal antibody, in slowing the progression of AD in prodromal patients. Details about this trial have been published before [25] and here we provide only a short summary of the most relevant aspects of Scarlet RoAD. Prodromal AD was asserted based on the criteria proposed by an International Working Group [11], and included an MMSE score 24, a CDR score of 0.5 together with memory box score of 0.5 or 1 and abnormal memory based on the Free and Cued Selective Reminding Test [16] and a Hachinski Ischemic Score ≤ 4 [29]. In addition, as evidence of amyloid pathology, cerebrospinal fluid levels of amyloid beta 1–42 below 600 ng/L were required. Scarlet RoAD was stopped because of futility after 797 individuals had been randomized.

Clinical scales (MMSE, CDR-SB and ADAS-COG 13) and the CANTAB were collected during screening, baseline and weeks 24, 52, 76 and 104. The CANTAB Spatial Working Memory Component (CSWMC) consisted of two training rounds of set size 4, followed by six games of set size 4, 6 and 8 items. The target (a yellow square) was randomly hidden behind one of the items (a different color square). As an additional rule, after a participant had searched two empty items, the location of the target was modified to be the next empty item searched. Thus, participants never had to remember more than two within-search items, reducing the length and complexity of the task. In contrast to FtE, there was no limit on the number of WSE and BSE so that a game only stopped after all targets were found. No further specific constraint applied to between-search items.

Hippocampal volume as a percentage of total intracranial volume was computed from 3D T1-weighted gradient images acquired mostly from 1.5 Tesla MRI scanners, and in exceptional cases, from 3.0 Tesla scanners. Segmentation was performed with the MIPAV software package [6].

The average clinical scales, hippocampal volume, and total memory capacity and their rate of change were estimated using mixed effect models with age and sex as fixed effects. Each participants’ behavior was modeled through a random intercept and a random slope modeling the change over the course of the trial. Correlations between outcomes were calculated from participants estimated random intercepts and slopes. S8 Fig displays representative data from six participants.

Stochastic retrieval

Formally, we divide between-search memory into a deterministic store with capacity and a stochastic store from which items can be retrieved. Again, the conditional probability of a BSE when is zero. When , we have that

The probability of correct retrievals depends on the failure rate and the number of slots in stochastic memory . Assuming that retrievals are independent events, it follows that

Moreover, we assume that unused capacity cannot be reallocated to increase the precision of the memory slots already in use, as noted before. Thus, the marginal probability of a BSE is

While somewhat complicated, this formula corresponds to the marginal probability of a BSE for different memory capacities , where items succeed at retrieval at rate .

Model estimation

The most complex model we introduced here comprises 5 parameters: and (see Table 1). Although , the memory capacity for within-search items, is in principle an important parameter, WSE were very uncommon at around 3% of all trials and its estimation was not feasible. Thus, we fixed its value to be larger than the maximum set size possible (in FtE, 12 items). This implies that all WSE were explained as lapses by the parameter.

For all other parameters, we used Bayesian model inversion to compute their posterior probability. As and are discrete parameters, their prior was a categorical distribution, whereas the logit transform of and were assumed to be Gaussian.

The Markov Chain Monte Carlo method was used to sample from the posterior distribution of the model parameters. Since the model has only two discrete parameters ( and ) with a small range, we sampled from the unnormalized marginal posterior distribution

In other words, we marginalized discrete parameters and and sampled exclusively the continuous parameters and . This allowed us to use the Hamiltonian Monte Carlo algorithm, a hybrid method that leverages the gradient of the posterior to propose samples. We ran four different chains and used the criterion to assess convergence. All routines were implemented using tensor flow probability 0.24.

The prior of the model consists of the distribution of the discrete parameters and and continuous parameters . We defined a prior for individual and , such that

For and , we assumed that

The same prior distribution was used for . This model was used to fit data from oRBiting and V1aduct.

For model comparison, in addition to the WAIC, we computed the expected relative model probability of each model , for each participants’ data Y such that

where the expected value is over all participants and is a normalization constant such that

Thus, for each participant, all three models were compared as per their model evidence and normalized up to 1. We report the expected value of the normalized model evidence across all participants.

Longitudinal models

The models presented before are well suited to represent the resources available to participants to store and retrieve visuospatial information but are not well suited to be sensitive to small longitudinal changes, typical in clinical applications. Thus, we made several modifications to the model above to capture the correlation between different sessions from a single participant, while allowing for different estimates of across sessions. This modified model was applied to data from Scarlet RoAD.

First, we simplified the model’s likelihood, by reducing it to a mixture model in which all possible memory capacities for between-search items are considered. The memory capacity for within-search items was assumed to be larger than the maximum set size in the game, and thus .

The probability of an action in session at trial is given by the expression

Where represent the maximum set size in the experiment minus 1 and is the vector of probabilities of different memory capacities . Ignoring again the effect of any lapse, the probability of a between-search error if given by

If

In other words, this is a mixture model with mixing probabilities for each memory capacity. Note that we have assumed that all sessions share a single parameter.

We modeled the correlations between sessions of the same participant by setting a hierarchical prior for the mixing probabilities . Specifically, we assumed that the probability of the mixture probabilities is given by the Dirichlet distribution

Finally, the prior of was set such that

Where is a vector of shape with all values fixed at 2 and L representing the number of sessions.

The central goal of this model is to estimate the posterior probability of . Assuming, posterior expected values

This corresponds to the total memory capacity in session .

This model allowed us to fit all sessions jointly, using to represent the correlation between sessions of the same participant, while generating an estimate of on each session.

In S3 Text, we show that this model can be used to estimate under a wide variety of conditions with high accuracy and minimal bias.