A bistable stochastic model quantifies performance degradation during sleep deprivation

Abstract

Sleep deprivation impairs sustained attention, as measured on the psychomotor vigilance task. This is manifested in a general slowing of reaction times and an increase in periods of unresponsiveness, increasing the risk of accidents. However, the mechanisms are not fully understood. This study combines experiments and modeling to better explain and quantify the changes of sustained attention under sleep deprivation. A total of 317 male participants (age 22.1 2.7 y) underwent 40 h of sleep deprivation under a constant routine protocol. A 10-minute psychomotor vigilance task was performed at 2-h intervals, and saliva melatonin was sampled every hour to monitor circadian phase. We report a bimodal distribution of reaction speed in the data. An approximately normal primary peak characterizes typical performance (reaction time ≲0.5 s), while periods of unresponsiveness correspond to reaction times ≳1.5 s and are reflected in a secondary peak which emerges after ∼20 h of wakefulness. We developed a minimal, stochastic model that accurately reproduces the data, attributing the bimodality of the distribution to bistability in vigilance state. We find general response slowing to be subject to an ultradian oscillation (∼3 cycles per day), while periods of unresponsiveness are disproportionately affected during the wake maintenance zone. Our results attribute periods of unresponsiveness to the coexistence of two vigilance states in the sleep-deprived brain, enabling new approaches in understanding vulnerability to sleep loss.

Statement of Significance

Sleep deprivation is prevalent in modern society, leading to an increased risk of accidents due to lapses in attention. In many scenarios, like shiftwork, simply getting more sleep is not an option, so a better understanding of mechanisms is needed. Our study, for the first time, shows a bimodality of response rates during sleep deprivation. We explain this by the co-existence of two vigilance states in the brain. The first state corresponds to typical reaction times in all individuals, while the second state is linked to unresponsiveness with reaction times ≳1.5 s and is observed in ∼60% of individuals. Our model enables new approaches to predict and prevent accidents and new insights into the physiology of sustained attention.

Graphical Abstract

Graphical Abstract.

Introduction

Sleep deprivation and cumulative excess wakefulness are associated with severe degradation of various cognitive functions over the course of hours and days [1–3]. A reduction in sustained (or vigilant) attention, defined as the ability to maintain performance on repetitive or monotonous tasks over the course of minutes or tens of minutes [4], dramatically increases the risk of accidents in a variety of scenarios, such as in healthcare [5], construction work [6], the military [7, 8], or when driving [9, 10]. The circadian rhythm modulates sustained attention tasks, such that performance peaks in the evening and is worst in the early morning [11, 12], implying additional risk in shift work [13].

Sleep deprivation affects performance on sustained attention tasks in several ways. The psychomotor vigilance task (PVT) is the benchmark for measuring sleep deprivation-induced impairment of sustained attention [14] and consists of a series of simple reaction time (RT) trials performed in sequence over several minutes (commonly 10 minutes). A principal finding of sleep deprivation studies is that the proportion and length of longer RTs increases, dispersed among otherwise near-typical responses [14, 15]. This is reflected in the RT distribution as an increase in the heaviness of the right tail [16] and is usually quantified by the number of RTs longer than 0.5 s, referred to as lapses [14]. Importantly, the proportion of very long RTs, on the order of seconds or tens of seconds in length, is a substantial component of this change. Thus, diminished sustained attention is marked by an increase in the frequency and duration of periods of unresponsiveness which cause very long RTs. The exact causes of these periods of unresponsivenes are not fully understood and may vary, having been attributed to (e.g.) distractions [17, 18] and microsleeps [19]. Sleep deprivation also leads to changes that affect all responses, such as a subtle but general slowing of typical RTs [14, 20]. This overall slowing is reflected in a rightward skew of the primary peak in the RT distribution as time awake increases [16]. Other effects include an increase in the number of errors of commission and speeding up of the time-on-task effect [14, 16, 21].

A variety of physiological and behavioral phenomena are thought to contribute to or underlie periods of unresponsiveness and general response slowing. Periods of unresponsiveness can be attributed to microsleeps [14, 17, 19, 20], distractions [17, 18], visual inattention [17], or outright sleep episodes [14] which intrude into attentive task performance. Conceptually, lapses (as well as errors of commission) have been ascribed to “state instability”, where competing sleep and arousal-promoting drives alternate in dominance over time [15]. General response slowing is thought to reflect more gradual and less intermittent changes [14, 16] and is associated with a variety of physiological alterations in the brain [14].

Several mathematical models have been proposed to explain the observed performance decrement within preexisting quantitative frameworks [22–24]. The goal of these models is typically to predict the distribution of RTs on the basis of parameters that may reflect psychological or stimulus properties. Models of evidence accumulation have been used to explain a wide array of RT tasks [25], and physiological evidence suggests mechanisms highly reminiscent of these models may underlie responses [26]. Additionally, mathematical models of sleep–wake cycles instantiate competing wake and sleep-promoting processes [27, 28] which can give rise to microsleep-like transitions between the two states [29], recalling the state instability hypothesis.

The existing models predicting RT data under sleep deprivation do not incorporate the conceptual mechanisms supposed to underlie the performance decrement, and mechanistic models suggesting microsleep-like dynamics have not been applied to RT data. Moreover, all studies testing the former group to date have either aggregated all RTs 500 ms [23, 24] or RTs longer than a few seconds [22] for the purposes of model fitting. As a result, conceptual models such as the state instability hypothesis remain quantitatively untested, as does the broader intuition that long RTs are a result of an altered vigilance state. It remains unclear how suitable existing quantitative RT models are for predicting the specific profile of very long RTs. Additionally, the precise mechanisms and structure behind existing conceptualizations are not immediately apparent. To further our mechanistic understanding of the processes underlying the performance decrement, it is essential to link conceptual models like the state instability hypothesis [15] to the mathematical models used to predict RT data. It is equally important that those mathematical models be able to accurately handle very long RTs, as these are a core component of the sustained attention decrement.

Here we present a quantitative analysis of the decrease in sustained attention to prolonged sleep deprivation as encoded in PVT RT data, with emphasis on periods of unresponsiveness and general response slowing. As a cornerstone of our approach, we introduce a mathematical model that reproduces the main features of experimental data and provides a platform to characterize the cognitive processes underpinning them. We demonstrate that the trajectories of the model parameters over a 40 h period enable us to distinguish distinct processes driving the observed effects of sleep deprivation on PVT performance.

Methods

Experimental data

Participants and experimental procedure

The data used in this study are from a larger project assessing the genetic background of sleep regulation; details of the protocol can be found in [30–32]. Data collection took place from 2012 to 2015. A total of 364 young, healthy Caucasian males were recruited. For this analysis, we have included participants who had at least 19 complete PVT trials (out of 20), had well-defined melatonin profiles so their dim-light melatonin onset (DLMO) could be estimated, and did not have their data collection happen within a week after daylight saving time transitions. The final sample included 317 participants (age 22.1 ± 2.7 y). An additional fourteen participants without melatonin data were included in fitting and analysis aligned to constant routine (CR) time (full sample n = 331, age 22.1 2.8 y).

Participants underwent 40 h of total sleep deprivation under a CR protocol (dim light <5 lux, constant posture, and meals at regular intervals), each starting from their habitual wake time. During sleep deprivation, they completed cognitive assessments, including a 10-minute PVT session every 2 h, with the first session appearing at 85 10 minutes since awakening. Saliva melatonin was collected every hour. Prior to the CR, participants followed a 3-week fixed sleep schedule at home based on their habitual sleep times and underwent sleep saturation in the laboratory.

The research project was approved by the Ethics Committee of the Faculty of Medicine at the University of Liège (Belgium) and was performed in accordance with the Declaration of Helsinki. All participants signed informed consent before taking part in the study and received financial compensation.

Data processing

PVT RTs of 10 s and above were considered nonresponses and were removed from the analysis. We noticed a putative adaptation effect present in the first PVT session, resulting in much longer RTs for the first response than for any other response in the session. We removed the first response of the first session from all individuals to account for this effect. Additionally, one participant did not respond at all for the first 10 stimuli in the first PVT session, and since protocol required that participants be checked on after three successive non-responses, we removed all 10 of these RTs. Responses before a stimulus or shorter than 0.168 s were counted as false starts and were also removed.

The cutoff value of 0.168 s is higher than the typically used values of 0.1 s [33] or 0.15 s [23, 34] and was chosen based on the shape of the distribution of all RTs across the study (i.e. from all individuals and trials, n = 588 138). The RT distribution between 0 and ∼0.168 s is approximately uniform, which is indicative of a random process that is unlikely to be instigated by stimulus presentation, so the cutoff value was chosen as the lowest RT at which the distribution differed significantly from a discrete uniform distribution. To determine the cutoff value, the empirical distribution function was restricted to a sequence of intervals, , and for each interval the distance between the empirical distribution and the discrete uniform distribution on the same interval was calculated as the Cramér-von Mises statistic for discrete variables [35]. The exact cutoff point was identified as the endpoint of the first interval for which the distance became greater than the asymptotic cut-off value for a confidence level of 0.05. We found that values close to 0.168 s were obtained when the test was performed using the closely related Watson and Anderson-Darling test statistics for discrete distributions in place of the Cramér-von Mises statistic (critical values also reported in [35]). Reciprocal reaction times (rRT) were used in all subsequent analyses.

Saliva melatonin concentration data over the 40 h recording for each participant were fitted using the bimodal skewed baseline cosine function, and DLMO was calculated as the time of the start of the incline of the fitted function from baseline [36, 37]. The times at which PVT sessions took place for a given participant were indexed relative to their DLMO time to ensure that when aggregating samples across all participants, they were aligned and analyzed at the same circadian phase. PVT sessions across all participants were then binned in 2 h increments according to their assigned relative-to-DLMO time indices, with the bins centered on even hours from DLMO-16 h through DLMO after 28 h. The bin centers are referred to as the timepoints. At each timepoint, all rRT data (i.e. for every participant) were pooled and analyzed. The first and last timepoints were discarded due to insufficient data relative to other timepoints (, respectively). The median number of rRTs across timepoints was , with outliers at DLMO-14 h () and DLMO after 26 h ().

To determine peak locations from the empirical rRT distributions, we used kernel density estimation as implemented in SciPy’s Gaussian kde() class to estimate the probability density functions (PDF). To identify the peak locations, we found zero-crossings in the derivative of the PDF by linearly interpolating between neighboring points with opposite signs. The estimated PDF often exhibited several very small peaks alongside the larger peaks discussed in this study, so any peak with <1% of the total number of observations at a given timepoint was eliminated from consideration. Peak heights and the locations of the local minima between peaks were also obtained from the estimated PDFs.

Model fitting

Parameter constraints

To prevent different parameter sets corresponding to the same drift functions (i.e. to ensure that the model is identifiable), the real parts of the roots of were constrained by the inequality , and we required that . The former is necessary since otherwise, and are always interchangeable, and all three roots are interchangeable if . The latter is necessary since only appears in , so that positive and negative values of correspond to the same .

To prevent from being complex-valued, we enforced the constraint

(1)

This constraint divides the parameter space into three distinct regions and can be expressed as three separate conditional constraints, such that either , , or both. Each of these constraints defines a subspace of the whole parameter space:

• If , then is constrained only to be positive, and has only one real root at .

• If and , then has a real root at a_h and a double (real) root at ().

• If and , then , , and are all roots of .

The first and second subspaces combined define the monostable regime, as the system has only one stable fixed point (at ). The bistable regime corresponds to the third subspace, with and .

Parameter optimization

To obtain optimal model parameter estimates given the empirical data, we used the maximum likelihood method. The negative log-likelihood (NLL) function was minimized numerically using the SciPy package’s implementation of the Nelder–Mead simplex algorithm. The search space boundaries were chosen with reference to the empirical data for the parameters , , and . For the parameters less easily estimated directly from the data, boundaries were chosen by iterative narrowing on intervals initially chosen well outside expected behavior. To reduce the likelihood of selecting a non-global minimum, the minimization algorithm was run independently for 20 initial conditions sampled uniformly within the given boundaries. The optimal parameter set associated with the minimum NLL value found among all 20 runs was chosen as the estimate.

The unmodified Nelder–Mead simplex algorithm does not handle equality constraints on the parameter space. Since the monostable and bistable regimes can effectively be described as two distinct models, each with four free parameters, we ran the minimization procedure on both subspaces separately and chose as the maximum likelihood estimate the parameter set corresponding to the lower of the two returned minima. For both procedures, the number of model parameters and number of data points is the same, so this method is equivalent to using common model selection criteria to choose between the two subspaces (treated as different models), such as the Akaike information criterion, the Bayesian Information Criterion, or the likelihood ratio.

Linear homeostatic and circadian model fits

The time series of the homeostatic () and circadian () drives during the 40-h sleep deprivation were generated using a biophysical model of arousal dynamics and are identical to those used for PVT prediction in [11]. We fit a linear model, to time series of and aligned to CR time. The simulations generating and were aligned to the and time series by assuming that the first PVT session took place 85 minutes after awakening, as this was the mean interval between awakening and the first PVT session across participants. The fits were performed by iterating the weighted least squares method, since variances at different times were found to differ substantially, particularly for . Initially, ordinary least squares was performed and the variance of the residuals within each timepoint was estimated by regressing against the corresponding means, since larger variances were most commonly associated with larger mean values. With the variances obtained, weighted least squares was then performed with the reciprocals of the estimated variances as weights, which were then updated using the new fit. This procedure was repeated until , , and converged within a tolerance of 10⁻⁶.

To fit the HC model, data were binned according to the time since waking during CR instead of DLMO. This was done since DLMO-aligned times span 42 h, whereas each individual only underwent 40 h of sleep deprivation. Thus, while alignment to DLMO is appropriate when each timepoint is considered separately, as a time series, the DLMO-aligned data cannot be expected to align with predictions made by the two-process model. The samples at each timepoint were split into 10 disjoint subsamples of equal size to estimate error at each CR time, but all other fitting processes and analyses were the same for these subsamples as for the DLMO-aligned data.

To determine whether the power spectral density (PSD) of the residuals of could be attributed to noise, we compared the peaks and peak heights of the residuals to noise-dominated synthetic data. We created synthetic residual time series by drawing sequences of 20 normally distributed random variables, one for each timepoint, each of which had 0 mean and the standard deviation of the residuals of at that timepoint. We generated 10⁶ of these, estimated their PSDs (following the procedure described below), and extracted the location and height of each peak in every PSD. We used this data to generate a joint distribution of the peaks and heights of the PSDs under the null hypothesis that the PSD of residuals could be explained by noise alone.

To determine whether the peaks of the PSD of residuals could be attributed to and alone, we fit a nonlinear combination of the PSD and cross-spectral densities (CSD) of and to the PSD of the residuals. We used ordinary least squares to perform this fit. The reasoning behind this process is as follows. The PSD of the residuals of is a quadratic combination of the PSDs and the CSDs of , , and a hypothetical omitted component, , which could in principle account for the 3.2 cycle per day oscillation:

Where are the coefficients of the “true” model, which incorporates and tildes with subscript denote Fourier transforms. The first bracketed term contains the component of the PSD of residuals due to only the PSDs and CSDs of and , while the second term contains contributions involving the unknown component. Our nonlinear model consisted only of the first bracketed term, to test whether the PSD of the residuals could be reproduced without an omitted term, and if not, whether and were sufficient to account for either of the two peaks alone.

PSD estimates for both sets of residuals were obtained as follows. For every time series of the 10 subsamples of rRT data, a periodogram was calculated on a zero-padded signal with SciPy’s periodogram function, which we used in favor of Welch’s method due to the small number of data points. The overall PSD was taken as the mean of these individual estimates. The estimates reported here used a Hann window for periodogram calculation, but similar estimates were obtained when a Hamming or Blackman window was used instead.

Goodness of fit

To quantify the goodness-of-fit of the bistable model, we calculated two distance metrics between the empirical cumulative distribution function (CDF) and model-fit CDF over the interval between the cut-off values of the lowest and highest RTs. Firstly, we calculated the Wasserstein metric, , which in one dimension is the area between the two CDF curves. Over a finite interval of length , , with only when the two distributions are equal. is proportional to a mean of the distance between the two functions over the interval. To ensure that averaging did not obscure any larger deviations between the two functions, we also computed the Kolmogorov–Smirnov statistic, , which is the magnitude of their maximum difference.

The overall goodness-of-fit of the linear homeostatic and circadian (HC) model was evaluated using the square root of the lack-of-fit sum-of-squares [38]. The lack-of-fit error is the component of the sum of squares resulting from deviations between the model and the mean of the data at each timepoint, with the sum-of-squares component due to variance of data at each timepoint around the mean value (the pure error) discarded. This allowed comparison of the deviation between model and data without accounting for the variances in the estimates, which substantially inflates the estimate for in particular. The intention here was not to artificially reduce the error in , but to account for the fact that the variance in (and thus the pure error component) is much larger than that in , particularly at early timepoints. The normalized root mean square error (NRMSE) (which retains the lack-of-fit error and the pure error) was also calculated for comparison with the lack-of-fit error alone, and to compare the fits to parameters with fits to other PVT performance metrics previously reported for different data sets [11].

Results

Reciprocal RT distribution exhibits secondary peak during sleep deprivation

The rRT distribution is nonstationary under sleep deprivation. Figure 1 shows the evolution of the group-aggregate rRT distribution over a 40 h sleep deprivation protocol for n = 317 participants. Data is aggregated according to time relative to each participant’s DLMO, a standard marker of circadian phase [39–41]. This time-varying distribution is characterized by four primary features, with numerical details given in Table 1:

Figure 1.

Empirical group-average and model rRT distributions at each time relative to DLMO. Empirical distributions have an approximately normal primary peak at all timepoints, the position of which shifts under circadian and homeostatic influences. The left tail of the distribution is heavier than the right, even at early timepoints, and becomes larger over time until a secondary peak appears at DLMO after 8 h. The secondary peak is most prominent around DLMO after 10 h to DLMO after 12 h and then declines until it disappears at DLMO after 24 h. The model distributions reproduce all of the described features. The black vertical line at rRT = 2 s⁻¹ (RT = 0.5 s) separates lapses (to the left) from non-lapses (to the right).

Table 1.

Empirical rRT Distribution Properties Over Time

Time relative to DLMO	Secondary peak loc. (s−1)	Secondary peak height	Main peak loc. (s−1)
<14 h	-	-	3.45
<14 h to ≥2 h	-	-	3.37
≥8 h	0.29	0.05	2.82
≥12 h	0.27	0.09	2.73
≥26 h	-	-	2.72

rRT distribution properties at various times relative to DLMO. The metrics shown reflect the distribution features described in the text. The second row contains means over DLMO before 14 to DLMO after 2 h (normal waking hours). Dashes indicate nonexistence of the secondary peak.

1. Bimodality. During normal wakefulness (DLMO before 14 h to DLMO after 2 h), there exists a single peak centered at rRT = 3.37 s⁻¹. A lower, secondary peak appears at DLMO after 8 h in Figure 1. It reaches a minimum rRT of 0.27 s⁻¹ and a maximum height of 0.09 at DLMO after 12 h. It then declines and disappears at DLMO after 22 h. A local minimum between the two peaks at ∼0.51 s⁻¹ defines their boundary.

2. General response slowing. The location of the primary peak shifts to lower values as sleep deprivation progresses. It declines from a maximum of 3.49 s⁻¹ at DLMO before 8 h to a minimum of 2.70 s⁻¹ at DLMO after 8 h.

3. Normality of typical responses. The distribution around the primary peak is approximately normal. However, it is negatively skewed, with a mean skewness of -0.42 0.10 between DLMO before 14 h and DLMO after 0 h, and deviates from normality for rRT ≳ 5.0 s⁻¹ and rRT ≲ 2.5 s⁻¹. For more details, see Figure S1.

In addition to the secondary peaks observed during sleep deprivation, peaks at 1.36 s⁻¹ are observed at DLMO before 14 h and DLMO before 8 h. As these are much higher than the other secondary peaks, we consider them distinct and do not analyze them further here. Similarly, at DLMO after 16 h, a tertiary peak at 0.65 s⁻¹ appears alongside the larger secondary peak at 0.28 s⁻¹. Of the two, the latter is closer to the other secondary peaks observed during sleep deprivation, and since the former is the only tertiary peak observed at any timepoint, we do not analyze it further in this study.

Importantly, the RTs comprising the second peak are not the same as typical PVT lapses (RT > 0.5 s; rRTs to the left of the vertical line in Figure 1). Lapses include all responses to the left of the black dashed line, combining rRTs from the primary and secondary peaks. The secondary peak is a qualitative feature of the distribution that may be utilized to provide insight into the processes underlying response generation and their impairment due to sleep deprivation.

The trends outlined here reflect group aggregate phenomena, and a given individual will exhibit them to a greater or lesser extent. To determine how well the group-aggregate trends represent individuals and to verify that none of the reported trends are spurious consequences of data aggregation, we have also analyzed properties of individuals’ rRT distributions. We find that 88.6% of individuals contribute to a secondary peak at least one timepoint during the sleep deprivation period (DLMO after 2 h onward). Additionally, 60.9% of individuals exhibit a secondary peak of their own, which is located under the group-aggregate secondary peak at least one timepoint during the same period. This shows that secondary peaks are relatively common across individuals, though not universal; see Figure S2 for examples.

The trajectory of individuals’ primary peaks is also highly correlated with that of the group aggregate (r = 0.745 0.161), demonstrating that the group aggregate data is a good representation of a typical response to sleep deprivation, see Figure S3 for details. The primary peaks in individual distributions also appear normal in Q-Q plots, though the range over which this holds varies between distributions. A subset of 20 of these is plotted in Supplemental Figure S4. Normality of the group aggregate peak follows from normality of individuals’ peaks provided that, over the population of individuals, the locations of the peaks are also normally distributed. Q-Q plots of the latter distributions (one for each DLMO time) are shown in Supplemental Figure S5, and the Shapiro–Wilk test yielded a mean p-value of .36 ± .27 across timepoints. Therefore, the approximate normality of the group aggregate primary peak is probably a result of the normality of both individuals’ primary peaks and their locations over the population.

Bistable rate model

Model formulation

We propose that the two peaks in the empirical rRT distribution can be explained by bistability in a psychomotor vigilance state of the brain. To test this hypothesis, we have developed a quantitative model which assumes that, following stimulus presentation, an individual’s psychomotor vigilance state influences the dynamics of a decision-making process, acting as a dynamic input driving the process to initiate a motor response. Following a similar class of models [25], we describe the process as one of “evidence accumulation” toward a threshold sufficient to make a decision, Figure 2. We assume that the psychomotor vigilance state determines the instantaneous rate of evidence accumulation, . The dynamics of follow an Ito stochastic differential equation with additive noise, , and state-dependent drift, :

Figure 2.

Components of bistable rate model in the monostable and bistable regimes. (A) The drift function of the rate process, , is the deterministic component of dynamics of . Conceptually, is the expected change in over a small time interval . In the monostable regime, moves toward on average. In the bistable regime, is attracted toward or when it is above or below , respectively. (B) Sample paths of . In the bistable regime, noise can drive from the high-performance state to the low-performance state, causing to decrease over some portion of a response trial. (C) The instantaneous rate of evidence accumulation, . When drops into the low-performance state, the slope of decreases accordingly, resulting in a longer RT (). (D) rRT distributions output by the model, which can be fit and compared to the empirical rRT distributions. The main peak is located at , while the secondary peak and local minimum between the two peaks are located at and , respectively.

(2)

where , , and are the higher, lower, and middle roots of (with respect to their real parts; is the imaginary unit), and is a scalar noise parameter. > 0 is included to eliminate a singularity at , but it is held constant for the purposes of parameter optimisation. The timescale constant is introduced so that the drift term on the RHS of Eq. 1 has units of evidence per second, which are the units of , and is fixed. The other model parameters are constrained such that is always real (see Methods).

The system occupies either a monostable or a bistable regime, shown in the left and right columns of Figure 2. The model is parameterized according to the roots of because they are the equilibria of the drift component of ’s dynamics, and their values determine which regime the system is in (Figure 2A). In the monostable regime, the drift term has a single stable equilibrium at the higher root, , (Figure 2A, left) and is driven around by the external driving noise (Figure 2B, left). In the bistable regime, a second stable equilibrium exists at , separated from by an unstable equilibrium at (Figure 2A, right). Gaussian noise with instantaneous variance can drive across between the two stable equilibria’s basins of attraction. This causes to oscillate stochastically between two distinct regions around each stable point (Figure 2B, right). The basin of attraction of is taken to represent an attentive, task-oriented psychomotor vigilance state, which we call the high-performance state. The basin of attraction of is called the low-performance state and corresponds to disengagement from the task.

The specific form of is chosen to satisfy two conditions. First, the system must be capable of bistability. This motivates the cubic term in the numerator, which allows to have two stable points simultaneously. Second, should be approximately linear around the higher stable point (). This is suggested by the approximate normality of the primary peak in the empirical rRT distribution, since linearity leads to a normal distribution for (i.e. an Ornstein–Uhlenbeck process), and for sufficiently fast responses this means that the rRT distribution is also approximately normal (see below). This condition motivates the quadratic term in the denominator. Note that the form of is empirically motivated.

We define a response trial as the event following stimulus presentation up to the recording of a response. As a first approximation, we ignore non-decision times originating from processes such as axonal conduction delays and stimulus encoding [22, 26]. This choice is based on the assumption that non-decision times are constant and the fact that is already chosen to meet the empirical observation of normality around the primary peak, which is the only distributional feature affected if constant nondecision times are introduced to the model. Hence, may be altered to account for inaccuracies introduced by constant nondecision times. A response trial in the model, therefore, consists only of the evidence accumulation process, , starting at 0 and climbing toward a threshold, (Figure 2C). If a trial starts at time , then the random variable representing RTs, which we call , is defined by:

(3)

Since is the time integral of , the dynamics of are reflected in ’s slope. In the monostable regime, closely follows a straight line during a response trial since varies relatively evenly around the fixed point at a_h (Figure 2B and C, left). In the bistable regime, however, oscillations between the high- and low-performance states cause the average slope of to change, and longer RTs occur as a consequence of falling into the low-performance state (Figure 2B and C, right).

Simplification of the model

During a PVT session, rRTs are accumulated and can be thought of as being drawn from the distribution of , shown in Figure 2D. The exact distribution of requires simulations to obtain, but it can be approximated by that of under the following assumptions: (a) the ratio ; (b) the distribution of reaches its stationary form much faster than the inter-stimulus interval; and (c) the stationary distribution of is approximately 0 for all . Conceptually, assumption (a) causes the RHS of Eq. 1 to go to 0, so that the integral in Eq. 2 is approximately constant and equal to ; assumption (b) ensures that is drawn from the stationary distribution of ; and assumption (c) ensures that RTs cannot be negative. Assumption (c) is imposed as a condition by making the parameter small.

Following the above assumptions, we set and approximate the distribution of with the stationary distribution, , of . We use in place of to denote the state space of , as opposed to the process itself. This distribution can be found by solving the steady-state Fokker–Planck equation, yielding

(4)

where is the potential associated with the system, equal to the negative of the integral over , and is a normalizing constant determined by numerical integration. Note that is ergodic [42], a necessary property because the model predicts distributions over the state space of rRTs, whereas the empirical distributions are collected over time.

We must distinguish between the rate distribution and the rRT distribution in the interpretation of our results because they are conceptually distinct, and because they are only equal if the simplifying assumptions made above are valid. The low- and high-performance states and their associated rate distribution peaks are properties of , while the primary and secondary peaks are features of the rRT distribution. However, since we utilize the approximation, we only refer to “the rRT distribution” except where it is important to differentiate them.

The model parameters directly relate to features of the rRT distribution. In both the monostable and bistable regimes, is the location of the main peak (Figure 2D). In the monostable regime, controls the weight of the left tail of the distribution (smaller values of lead to a heavier tail). In the bistable regime, is the location of a lower secondary peak, and is the location of the local minimum between the two peaks (Figure 2D, right column). The parameter controls dispersion, determining the extent to which the distribution is concentrated around the peaks and therefore the variability of responses within each state. Larger values of cause the peaks to blend together, while smaller values lead to concentration around the peaks. Note that the distributions shown in Figure 2D are obtained under the approximation, and the relationships between model parameters and distribution features shown in the figure only hold under this approximation.

Separation of typical and long RTs

The shape of the empirical rRT distribution suggests that we may be able to treat and quantify “typical” and “low” rRTs separately. Specifically, we either observe an approximately normal primary peak with a heavy tail on its left, or a primary peak along with a secondary peak in place of the left tail. Normality of rRTs for saccadic tasks under non-sleep deprivation conditions in non-human primates is well documented [43–46]. Together, these findings suggest that typical rRTs may be defined by a generating process that leads to a normal rRT distribution, while the left tail and secondary peak can be attributed to a distinct, attention-disrupting process.

To represent this in the bistable model, we explicitly separate the potential defined in Eq. 3 into a quadratic component, , centered at , a secondary component, , and a constant remainder (Figure S6):

(5)

accounts for the normality of the primary peak of the rRT distribution around , while we view as a perturbation which interferes with typical performance. exhibits a single metastable equilibrium, , regardless of whether the model is in the monostable or bistable regime. Note that is not an equilibrium of the full potential, only of the component of . However, in this framing, is responsible for the left tail/secondary peak of the rRT distribution, and the metastable equilibrium at persists in both the monostable and bistable regimes (whereas the equilibrium at only exists in the bistable regime). Thus, we view the metastable equilibrium as a counterpart to the stable equilibrium at and take as the low-rRT counterpart metric to .

Bistable rate model reproduces key features of the reciprocal RT distributions

The fits of the bistable model to the data are illustrated as solid red lines in Figure 1. The time series of model-derived quantities corresponding to the key features of the distributions are plotted against the corresponding data-derived quantities in Figure 3. With respect to each of these features, we find that:

Figure 3.

Time series of the estimated empirical distribution quantities (markers) and their model counterparts (dotted lines). (A) Estimates of the location of the secondary peak (circles) and the trough between peaks (triangles) where a secondary peak was detected. In the bistable regime of the model, parameters and quantify these features. (B) Location of the primary peak of the rRT distribution, matching the model parameter . (C) Proportion of rRTs less than or equal to .

1. The model and data estimates agree on the presence or absence of a secondary peak at all-time points except DLMO after 6 h, where a small secondary peak appears in the model but not in the empirical rRT distribution. The locations of the secondary peak and the local minimum separating the peaks in the data closely align with the positions of and fit by the model, Figure 3A.

2. The location of the main rRT peak in the model, , closely tracks the location of the main peak in the empirical data, as seen in Figure 3B.

3. The approximate normality of the main peak in the model, particularly to the right of its peak location, a_h, is guaranteed by the quadratic term in Eq. 1. Specifically, as , the drift term in Eq. 1 approaches . As a result, the system behaves like an Ornstein–Uhlenbeck process at large , the stationary distribution of which is normal.

Importantly, fits made with a cubic alone were not able to reproduce the secondary peak. The quadratic term in the denominator of was necessary for all of the fits to match the empirical distributions qualitatively. This indicates that approximate normality of the primary peak is an essential feature of the data.

Additionally, we compare the probability of sampling rRTs smaller than the local minimum between the peaks in the model, to the proportion of rRTs in the data, shown in Figure 3C. These match closely, demonstrating the agreement between the model and data, specifically under the secondary peak, when one exists, or simply at low rRTs otherwise.

We use two complementary metrics to quantify the deviations between model fits and empirical distributions. The Wasserstein metric, , measures the difference between the empirical CDFs and corresponding model predictions as the total area between the two curves. The average of across all timepoints is 0.009 0.003 (mean STD) and ranges from 0.004 to 0.014, indicating that the amount by which either distribution has to be deformed to equal the other is small for all times relative to DLMO. The average of the Kolmogorov–Smirnov statistic, , is 0.009 ± 0.003, and it ranges from 0.004 to 0.015 across timepoints, showing that no substantial deviations at specific rRT values were obscured by . This indicates a good fit of the model to data across the rRT range.

We also fit the model to each PVT session individually (i.e. for every participant and time awake). The percentage of individuals exhibiting a secondary peak reaches a maximum of 60.0% at DLMO after 12 h, while 66.6% of individuals exhibit the secondary peak in five or more PVT sessions. This supports the idea that the secondary peak in the group-aggregated data is reproducible across individuals, and not a result of a small number of particularly poor-performing individuals.

Model-based metrics display sensitivity to ultradian fluctuations in performance and the wake maintenance zone

The time series of and in Figure 3A suggest influences other than the HC drives, due to an apparent oscillation prior to DLMO after 0 h. Given that the oscillation is not present or at least substantially smaller in the time series of , we hypothesized that it is primarily a result of an increase and subsequent decrease in low rRTs and is independent of the dynamics of the main peak. To test this, we have simulated the homeostatic () and circadian () drives on the basis of a physiologically motivated model of arousal dynamics [11, 47]. We have then fit a linear combination of , and a constant term, the HC model, to two performance metrics, representing typical performance (or typical RTs) and representing poor performance (long RTs); Figure 4.

Figure 4.

Linear HC model fits to bistable model performance metrics. Timepoints at which the magnitude of the residual is greater than 4 times the SEM of the bistable model metric are highlighted in green or red when the metric is above or below the HC prediction, respectively. Observations are aligned by CR time, not time relative to DLMO, see methods. (A) Fit to primary peak location, . (B) Fit to metastable equilibrium location, . (C) Estimates of the power spectra of the residuals of both fits.

For this result, the data is aggregated according to CR time (time since waking) as opposed to time relative to DLMO. This is because the data must be properly aligned with the HC model predictions, as explained in the Methods.

Figure 4A and B show fits of the HC model to the means SEM of and . The NRMSE values are 0.05 and 0.17 s⁻¹ for the HC fits to and , comparable to values reported for fitting PVT metrics over several datasets in [11]. These values indicate that the linear HC model captures the majority of the influences on both parameters and fits more closely than . To account for the larger variance in as compared to , we have also calculated the normalized square root of the sum of squares due to lack of fit, removing the “pure error” component from the NRMSE. Their values are 0.05 and 0.16 for and respectively, confirming that the difference in variance does not substantially affect the outcome that model fits are better for than for .

We have examined the residuals of each fit to identify potential origins of disagreement between the HC model predictions and the bistable model parameter time series. Times at which the residuals are large (>4 times SEM) are highlighted in Figure 4A and B. We have also performed a spectral analysis to identify periodicity in the residuals, with the resulting PSDs shown in Figure 4C.

For , the negative deviation at the first timepoint in Figure 4A might be a result of sleep inertia (which is not accounted for by the HC model), but other times highlighted are not reminiscent of any known effects on performance or alertness. However, there are two distinct peaks in the PSD of the residuals at 1.2 and 3.2 cycles per day, Figure 4C (teal line). The lower peak suggests a near-circadian rhythmicity of the residuals, while the latter indicates an ultradian rhythm with a period of ∼8 h.

These peaks may reflect genuine periodicities not captured in the HC model or may be artifactual consequences of model-fitting bias or noise. To determine whether they could result from noise, we compared the PSD peaks and peak heights to those of synthetic data with the same statistical properties but no periodic trends. Both peaks are observed consistently across non-synthetic samples, while the synthetic data generally exhibits peaks which are much more uniformly distributed and much smaller (Figure S7). To investigate whether the peaks could be artifacts due to model bias, we fit a quadratic model of the PSD and CSD of and to the PSD of the residuals of (see Methods). The closest fit produces a 1.2 cycle per day peak but is unable to reproduce the 3.2 cycle per day peak (Figure S8). This indicates that a process other than or is necessary for the 3.2 cycle per day peak to appear. However, the 1.2 cycle per day peak is consistent with artifactual circadian and homeostatic components in the residuals. That is, the 1.2 cycle per day peak emerges only because of the circadian rhythmicity present in the model, whereas the 3.2 cycle per day peak indicates a phenomenon that is missing from the HC model entirely. This helps to explain the timings of deviations as seen in Figure 4A in terms of a systematic, ultradian discrepancy between the HC model and the data.

Inspecting in Figure 4B, a clear positive deviation occurs at 14–16 h, corresponding roughly to DLMO before 3 h to DLMO before 1 h (~1900–2100 h in clock time). This suggests that the deviation results from improved performance during the wake maintenance zone (WMZ). The WMZ is a period of reduced sleep propensity due to the circadian drive, and has been associated with increased PVT performance previously [48]. Another positive deviation is observed 40 h into the CR (24–26 h after the first deviation), which suggests that the effect is circadian, consistent with expectations for the WMZ. Negative deviations observed for at 20 and 22 h into the CR do not correspond to any known effects on performance, but seem to be a consequence of the emergence of the secondary peak between the fits at 18 and 20 h, corresponding roughly to the appearance of the secondary peak in the DLMO-aligned data at DLMO after 8 h, as described in the first results section. Thus, the deterioration of performance associated with the appearance of the secondary peak is more rapid than predicted by the linear HC model. The PSD of the residuals of (Figure 4C, purple line) does not indicate any periodic component, consisting of a broad peak over about 1–2 cycles per day, consistent with artifactual contributions from the PSDs and CSDs of and .

Discussion

In this study, we identified and quantified three features of PVT performance under sleep deprivation and developed a bistable rate model of the decision-making process accounting for these features. Specifically, we show that: (1) bimodality in the rRT distribution is predicted by bistability in the model, and reflects periods of unresponsiveness which interrupt typical task performance; (2) the location of the primary peak, corresponding to typical response speed, shifts over time under homeostatic, circadian, and potentially ultradian influences; and (3) normality of the primary peak in the rRT distribution characterizes typical responses.

Sustained attention decrement in features of the PVT rRT distribution

Periods of unresponsiveness leading to long RTs and a general slowing of typical responses are two defining characteristics of the sustained attention decrement on the PVT. [14, 21] We found that these phenomena correspond to distinct features of the rRT distribution: the increasing frequency and duration of periods of unresponsiveness correspond to a secondary peak emerging at low rRTs, and response slowing corresponds to a shift of the primary peak.

There are practical implications of being able to easily differentiate between and distinctly quantify these two manifestations of the performance decrement. Understanding the effects of sleep deprivation on sustained attention is useful because it facilitates precautionary measures in scenarios where impairment of sustained attention is both likely and potentially catastrophic, such as when operating vehicles. It is well established that inter-individual variability in vulnerability to sleep deprivation is large [49], and that vulnerability of individuals should be accounted for in professions where sleep deprivation is an inherent issue. It is likely to be useful to determine not only who is vulnerable to sleep deprivation and who is not, but specifically how that vulnerability manifests. For example, an increase in periods of unresponsiveness indicates a greater risk of catastrophe in scenarios where sustained attention is vital, while general response slowing may reduce overall efficiency in some tasks over longer periods of time. It is not obvious that existing metrics, such as the number of lapses, differentiate between the two cases, because general response slowing without an increase in very long RTs nonetheless shifts a considerable proportion of responses across the 0.5 s threshold. We suggest that precautionary measures can benefit from taking into account the specific nature of the sustained attention decrement in any given individual, and the bistable rate model offers a first step toward doing so comprehensively.

Normality of the primary peak

The upper part of the rRT distribution is characterized by approximate normality, in particular between ∼2.5 and ∼5 s⁻¹ (RTs between about 0.2 and 0.4 s). This has been previously described for simple saccadic tasks [45, 46] as well as for a visual lexical decision task [50], though more focus appears to have been placed on this property in the former context than for any other task type. Our result shows that the normal approximation to the rRT distribution is also applicable in PVT.

Characterization of the rRT distribution where it corresponds to typical RTs is useful in its own right and as contrast against its left tail. For the practical purpose of accurately quantifying the major components of the rRT distribution, approximate normality greatly constrains modeling efforts and indicates that a relatively simple description (in terms of a normal distribution’s mean and variance) suffices for typical responses. More importantly, in the present context, it suggests that the heavy left tail of the rRT distribution, which exists in the absence of the secondary peak, is nonetheless capturing periods of unresponsiveness, and thus can be leveraged to quantify their frequency and severity in the absence of the secondary peak. This is what we have done with the quantity .

Bistable rate model in the context of evidence accumulation models

The bistable rate model is part of the evidence accumulation model class. These models are common in the decision-making literature and successfully predict many features of empirical RT and accuracy distributions [25]. They are unified by the idea that a large part of any given RT is due to a decision process, in which evidence regarding the presence and nature of a stimulus is integrated toward a threshold, and a response is initiated upon the threshold being reached.

In practice, the bistable rate model is most similar to the Linear Approach to Threshold with Ergodic Rate (LATER) model [46]. In its simplest form [43] LATER proposes that evidence accumulation rates are constant within a trial, and for each trial are drawn from a normal distribution. The simplified bistable rate model effectively proposes the same thing, with the crucial difference in the sleep-deprivation context that the rate distribution is of the form given in Eq. 3, rather than being normal. Thus, while there are substantial conceptual differences between the two models, in implementation, they differ only in the proposed distribution from which evidence accumulation rates are drawn.

There are a few conceptual distinctions between the bistable rate model and other evidence accumulation models. First, we take the instantaneous rate of evidence accumulation, , to be the fundamental variable of interest, as opposed to the evidence itself, . This change in focus makes our dynamical systems approach possible. Second, we interpret as being determined primarily by a psychomotor vigilance state of the brain. This contrasts with the existing understanding of drift rates in evidence accumulation as being determined by properties of the stimulus [26], instead emphasizing the role of endogenous cognitive processes. Third, our justification for assuming approximately constant evidence accumulation during a trial differs from previous models. In the past, constant evidence accumulation has been assumed for the sake of simplicity [51] or justified by observations of neurophysiological data [44]. Here, we assume that the dynamics of are slow relative to the length of a typical RT, so that rRT distribution can be approximated by the distribution of . Finally, this same assumption leads to a distinct explanation for inter-trial variability in RTs, namely, that the process accounting for the bulk of that variability varies on a timescale comparable to a typical inter-stimulus interval.

Bistable rate model in the context of dynamic alertness models

Non-pathological sleep disturbances are often due to sleep restriction or sleep not aligned with the circadian rhythm, for example, in shiftwork. In real-world scenarios, vigilance is predicted by sleep history and circadian zeitgebers [13, 52], and alertness can be maximized by following specific sleep schedules according to the requirements of an individual [52]. Dynamic and biomathematical models predicting alertness on the timescale of hours and days [11, 53] can flexibly deal with such real-world situations, as specific sleep schedules and interventions can be modeled directly and their effects predicted by the model dynamics. However, these models do not handle task or trial-level dynamics, instead assuming that typical metrics (such as PVT lapses or mean rRT) are correlated with alertness. By contrast, the bistable rate model deals with RT dynamics directly, but provides no means of relating the model parameters (which govern performance) to alertness. Integration of biomathematical models predicting alertness with the bistable rate model would enhance the applicability of both, enabling flexible and specific predictions of performance.

Physiological basis of bistability and the low-performance state

In this study, we have linked conceptual understandings of periods of unresponsiveness directly to RT data. By representing vigilance states as distinct stable equilibria of an evidence accumulation rate, we have shown that periods of unresponsiveness may emerge due to stabilization of a task-inattentive state we call the low-performance state. This is conceptually reminiscent of the state instability hypothesis [15], where competing sleep and wake-promoting drives lead to alternating periods of fast and slow RTs. The model’s bistable structure is also shared with a one-dimensional formulation of an influential, biophysical sleep–wake cycle model [54]. While it is not clear whether the bistable model’s low-performance state is identical to the sleep state (see below), our results indicate that the mechanisms underlying periods of unresponsiveness result from the same basic structure. As such, they represent the first direct link between the conceptual framework explaining periods of unresponsiveness and PVT data.

Possible physiological origins of the low-performance state in the bistable rate model are constrained by the length of the RTs occurring during that state. Most RTs falling under the secondary peak, which we can broadly attribute to the presence of a low-performance state, are probably a result of microsleeps [55], which have durations on the order of seconds to tens of seconds [56–58]. Additionally, distinct fMRI BOLD networks are associated with typical performance and lapses, as reduced inactivation of the default-mode network [59, 60] and reduced activation in attention-controlling regions [59] are associated with longer RTs. However, it is unclear how closely these network dynamics are related to the low-performance state since precise RT lengths were not reported.

There are two relevant physiological systems exhibiting bistability, to which we could attribute the bistability in our model. First, the coexistence of sleep and waking states is the core feature of the sleep–wake “flipflop switch” [27, 61]. The simultaneous stability of the two states can be effectively maintained during sleep deprivation by a “wake effort” drive [28] with presumed cortical or orexinergic origins. Microsleeps may be a result of noise-driven fluctuations or transitions into the sleep state when the waking state begins to lose stability [29, 62]. Note that we are not talking about full wake-to-sleep transitions, which occur over several minutes [27, 61].

Second, sleep deprivation increases the prevalence of cortical bistability during wakefulness, specifically leading to increased frequency and magnitude of local cortical “OFF” states [63]. These events and their correlates have been observed in rodents [63, 64], non-human primates [65], and humans [66–68], and they are associated with impaired task performance. OFF states during wakefulness are usually reported as having durations within about 50–200 ms, and so seem at first unlikely to be responsible for the very long RTs which fall under the secondary peak. However, OFF states during sleep (which are thought to be mechanistically similar to those observed during wakefulness) are associated with a loss of “memory” in the affected neuronal groups [69, 70], which may cause cognitive disruptions to extend beyond the duration of a given OFF state. Alternatively, OFF states may factor into the flip-flop switch framework outlined above. If the wake effort drive originates in part from the cortex, then OFF states may constitute a transient loss of input to the subcortical ascending arousal network. This, in turn, would bring the waking state closer to destabilization, briefly increasing the probability of microsleeps or other transient sleep-like events.

Performance factors influencing time series of typical and long RT metrics

The fit of the linear two-process (HC) model to is quite good, indicating that the majority of the variance in typical PVT performance is captured by a combination of HC drives. However, our findings indicate the presence of ultradian rhythmicity (with an 8 h period) in the bistable model that is not accounted for in the HC model. This suggests that models designed to predict performance decrement due to sleep deprivation [11, 53, 71] may be improved by the addition of an ultradian component with a period of 8 h. This said, the physiological origin of such a component remains unclear. Additionally, the only similar previous result of which we are aware is an oscillation in performance on a sentence verification task with a period of about 8 h [72], but the total length of the experiment was only 9 h and thus not considered long enough to draw conclusions. Moreover, PVT and the sentence verification task have quite distinct cognitive underpinnings. Thus, further verification of this effect is required.

Positive deflections in (indicating better than predicted performance) at 14, 16, and 40 h of wakefulness are consistent with vigilance-promoting effects in the WMZ [73], during which PVT performance has been shown to improve [48]. The lack of any counterparts in suggests that the frequency of very long RTs decreases during the WMZ, without any effect on the speed of typical responses. This result is consistent with the definition of the WMZ as a period of time in which sleep propensity is reduced [73, 74]. Specifically, if longer RTs are a result of microsleeps, and the frequency of microsleeps decreases with reduced sleep propensity, then a commensurate reduction in the frequency of long RTs is to be expected.

Differential effects of sleep deprivation on typical responses compared to very slow responses have been shown before [21], suggesting that the PVT performance decrement is multifaceted. By developing metrics that sharply distinguish between the two response types, we expand on this by revealing specific effects on each of them. The fact that the effects differ—i.e. typical responses undergo an ultradian fluctuation while long responses are reduced during the WMZ—supports the idea that the two response types are in fact mechanistically distinct. The fact that different mechanisms may affect performance is not in itself surprising, but our findings show that if those effects manifest in responses characterized by different timescales, then they may be identified and categorized on the basis of behavioral data alone.

Limitations

There are four main limitations in the data and our analysis. First, data were aggregated over individuals, and individuals are known to vary in their baseline PVT performance and response to sleep deprivation. However, we have shown that the three features under study are reproducible across individuals and thus believe that they serve as a useful baseline for future investigations. Such an investigation would likely benefit from a hierarchical modeling approach, where the dynamics and individual variation of bistable model parameters are specified in a higher-level model. Second, the study participants were from a homogeneous background (all young, Caucasian males). Thus, similar analyses must be performed on data drawn from larger demographics to assess the generalizability of our results. Third, the simplification of the model may only be strictly accurate for short RTs, limiting the usefulness of the secondary peak for directly inferring properties of the low-performance state. In future, simulations may be necessary to relax the simplifying assumptions and thus learn more about the properties of the low-performance state. Fourth, we ignore nondecision times as a component of responses. Our focus is on very long RTs, of which nondecision times constitute a smaller fraction than otherwise, but a more complete RT model may help to elucidate typical PVT response mechanisms more clearly.

Conclusion

We have shown that attentional lapsing and general response slowing correspond to a secondary peak and a shift in the primary peak of PVT rRT distribution. Additionally, variability in typical RTs increases with time awake, and the primary peak of the rRT distribution is approximately normal. With our bistable rate model, we have proposed that the instantaneous rate at which evidence accumulates toward a decision threshold can be attributed in part to the vigilance state of an individual. The model suggests that particularly long RTs can be attributed to bistability in vigilance state. The model also provides natural quantities corresponding to attentional lapses and general response slowing. The quantities associated with lapsing and response slowing are sensitive to distinct influences, and in particular, long RTs are considerably less frequent during the WMZ, consistent with the known reduction in sleep propensity. The model reveals important new insights into the manifestations of sleep deprivation-induced sustained attention decrement, and that the corresponding metrics will likely prove useful in practical applications.

Funding

This research is supported by an Australian Government Research Training Program (RTP) Scholarship. This study was supported by the Australian Research Council Discover Program (DP230101113). CS is supported by the Fonds de la Recherche Scientifique—FNRS-Belgium. VM is supported by the University of Liège (ULiège), the European Research Council (COGNAP-GA-757763) and the University of Molise. The study was supported by the Wallonia-Brussels Federation (Actions de Recherche Concertées—ARC—09/14-03), WELBIO/Walloon Excellence in Life Sciences and Biotechnology Grant (WELBIOCR- 2010-06E), University of Liège (ULiège), Fondation Simone et Pierre Clerdent, European Regional Development Fund (Radiomed project), Fonds Léon Fredericq, Fondation Médicale Reine Elisabeth.

Disclosure statement

Financial disclosure: None.

Non-financial disclosure: None.

Data availability

Anonymized-processed data are available at https://zenodo.org/records/14752106 and analysis scripts supporting the results of this study are available at https://github.com/SR393/PVT-SD-vulnerability. However, the data are not publicly available due to ethical and privacy restrictions, as they are part of a dataset containing sensitive genetic information. Data may be made available upon reasonable request to the authors, subject to compliance with relevant ethical guidelines and data-sharing agreements. Requests will undergo evaluation by the local Research Ethics Board, and data sharing will require the completion of a Data Transfer Agreement.