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 $\pm$ 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.

Keywords: cognitive function; mathematical modeling; sleep deprivation.

Graphical Abstract

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).

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.

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 .

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.