Mathematical modeling of sleep state dynamics in a rodent model of shift work

Abstract

Millions of people worldwide are required to work when their physiology is tuned for sleep. By forcing wakefulness out of the body's normal schedule, shift workers face numerous adverse health consequences, including gastrointestinal problems, sleep problems, and higher rates of some diseases, including cancers. Recent studies have developed protocols to simulate shift work in rodents with the intention of assessing the effects of night-shift work on subsequent sleep (Grønli et al., 2017). These studies have already provided important contributions to the understanding of the metabolic consequences of shift work (Arble et al., 2015; Marti et al., 2016; Opperhuizen et al., 2015) and sleep-wake-specific impacts of night-shift work (Grønli et al., 2017). However, our understanding of the causal mechanisms underlying night-shift-related sleep disturbances is limited. In order to advance toward a mechanistic understanding of sleep disruption in shift work, we model these data with two different approaches. First we apply a simple homeostatic model to quantify differences in the rates at which sleep need, as measured by slow wave activity during slow wave sleep (SWS) rises and falls. Second, we develop a simple and novel mathematical model of rodent sleep and use it to investigate the timing of sleep in a simulated shift work protocol (Grønli et al., 2017). This mathematical framework includes the circadian and homeostatic processes of the two-process model, but additionally incorporates a stochastic process to model the polyphasic nature of rodent sleep. By changing only the time at which the rodents are forced to be awake, the model reproduces some key experimental results from the previous study, including correct proportions of time spent in each stage of sleep as a function of circadian time and the differences in total wake time and SWS bout durations in the rodents representing night-shift workers and those representing day-shift workers. Importantly, the model allows for deeper insight into circadian and homeostatic influences on sleep timing, as it demonstrates that the differences in SWS bout duration between rodents in the two shifts is largely a circadian effect. Our study shows the importance of mathematical modeling in uncovering mechanisms behind shift work sleep disturbances and it begins to lay a foundation for future mathematical modeling of sleep in rodents.

Fig. 1

Overview of the experimental design. All rats underwent a 24-h baseline recording in an 12/12 LD cycle. The animals were then split into two groups: Active-workers (AW) and Rest-workers (RW). Each animal underwent 8 h of forced ambulation on a motorized running wheel. ZT: Zeitgeber time.

Fig. 2

A homeostatic model (Process S) quantifies differences in sleep slow wave dynamics between baseline days and work days. A When optimally fit to data points representing median delta power (1–4 Hz) during 5-minute episodes of slow wave sleep, both the rising time constant (Ti) and the decay time constant (Td) were significantly different between the baseline day and the workdays for the Rest-Workers (RW) group. Only Td was significantly different between baseline and work days for the Active-Workers (AW) group. B The lower asymptote values (LA) were the same for both AW and RW groups during baseline and during the workdays. The upper asymptote values (UA) was significantly larger during the work period for the AW group, but not the RW group. Also, UA was significantly larger during the work period for the AW group as compared to the RW group. Representative fits of the Process S model to data from one recording from the AW group (panel C) and one recording from the RW group (panel D). In the lower two panels data were normalized to the mean value during the 24 h baseline. Optimal values for Ti and Td were found using the Nelder-Mead method. Alternating yellow and black bands indicate the timing of light and dark intervals, respectively. *p < 0.05, **p < 0.01 unpaired t-test.

Fig. 3

The Markov Chain generates hypnograms that are similar to recorded hypnograms. Shown are individual recordings and individual simulations for active-workers (AW) (A) and resting-workers (RW) (C) during a 24 h baseline and a 4-day work protocol. The upper row in each panel shows recorded data and the lower panel shows the simulation output. Note the gaps representing work periods starting in the first active phase (ZT14–22) after baseline for the AW group (panel A), and during the first inactive phase (ZT2–10) after baseline for the RW group (panel C). Vertical lines indicate sleep state changes. Yellow and black bars denote times of lights-on and lights-off respectively. Panels B and D show 10 h of the same data shown in panels A and C. The black rectangles at the bottom of the simulation output in panels A and C indicate the location of the expanded sections shown in panels B and D respectively. Note the occurrence of long Wake (W) episodes in the recordings and the simulations shown in panels B and D. REMS=rapid eye movement sleep, SWS= slow wave sleep. Panels E, F, and G show the cumulative probability distributions for the durations of Wake, SWS and REMS episodes respectively. Panels E,F,G were generated using experimental data recorded during the baseline. Wake episode durations were better fit with a power law than an exponential (red curve and blue curve respectively), while SWS and REMS episode durations were better fit with a Burr type XII distribution rather than an exponential distribution. In each case the blue curve represents the optimized fit of the exponential model to the data. Panels H, I, and J show changes in episode duration over the course of the baseline day for wake, SWS, and REMS respectively. Data are binned into 2-hour bins. Error bars indicate standard error of the mean.

Fig. 4

Sleep/Wake profile of baseline and four work days. Plots of sleep state timing for the Active-workers (AW) (panel A) and Rest-workers (RW) (panel B) cases. The upper plot in each panel shows the experimentally recorded data and the lower plot shows the output of the model. Alternating yellow and black bands at the top of each panel indicate the timing of light and dark intervals, respectively. Dark brown bands in each panel indicate the time spent in forced ambulation. REMS= rapid eye movement sleep, SWS = slow wave sleep. The upper plots in both panels are reproduced from (Grønli et al. 2017).

Fig. 5

Comparison of Experimental Data to model output: numbers of episodes of each state. In the experimental data (left panels) both the Active-work (AW) and Rest-work (RW) groups showed fewer wake episodes (upper left panel), fewer slow-wave sleep (SWS) episodes (middle left panel), and fewer rapid eye movement sleep (REMS) episodes (lower left panel) between workdays (W1 to W4) as compared to baseline, with no difference between the two groups except SWS episodes on W4 and REMS episodes on W1. The simulation output (right column) showed very similar numbers of episodes of each state and the same statistical differences between baseline and workdays. Only REMS episodes on W1 showed a significant difference between AW and RW. Error bars in the simulation panels represent SEM from 50 simulations. Asterisks in all panels indicate significant differences between RW (n = 15 in experimental data) and AW (n = 12 in experimental data); * p < 0.05, ** p < 0.01, *** p < 0.001, and hashes denote significant differences compared to baseline: # p < 0.05, ## p < 0.01 and ### p < 0.001. When AW and RW data points overlay, asterisks above the data point refer to RW and asterisks below the data refer to AW. Open and filled triangles above graphs in right hand panels represent significant differences compared to experimental data for the AW and RW groups respectively. One triangle: p < 0.05, two triangles: p < 0.01, three triangles: p < 0.001.

Fig. 6

Comparison of Experimental Data to Model Output: Episode Duration. A In the experimental data (left panel), Wake episode durations changed only slightly with no significant difference between baseline (B) and workdays (W1 to W4) except Rest-work (RW) at W2 which was significantly lower than baseline. There was no significant difference between RW and Active-work (AW) until W4. In the simulation output (right panel), wake episode durations were similar to those in the experimental data, but the RW group changed slightly from baseline in all four workdays and there was a small but significant difference between work groups for each of the work days. B In the experimental data, both groups showed an increase in slow-wave sleep (SWS) episode duration between workdays as compared to baseline (left panel), with the AW group experiencing longer durations compared to the RW group. Significant differences between baseline and workdays and between AW and RW was captured with the mathematical model (right panel). C In the experimental data (left panel) REM sleep episodes became significantly longer between workdays as compared to baseline with no difference between AW and RW until W4. In the simulations the differences between baseline and workdays was captured accurately, although a difference between AW and RW was introduced. Error bars in the simulation panels represent SEM from 50 simulations. Asterisks in all panels indicate significant differences between RW (n = 15 in experimental data) and AW (n = 12 in experimental data); * p < 0.05, ** p < 0.01, ***p < 0.001, and hashes denote significant differences compared to baseline: # p < 0.05, ## p < 0.01 and ### p < 0.001. Open and filled triangles above graphs in right hand panels represent significant differences compared to experimental data for the AW and RW groups respectively. One triangle: p < 0.05, two triangles: p < 0.01, three triangles: p<0.001.

Fig. 7

Comparison of Experimental Data to Model Output: Percentage of time spent in each state. A In the experimental data (left panel), the percentage of time spent in wakefulness (over each workday, W) increases modestly but significantly in both Active-work (AW) and Rest-work (RW) groups compared to baseline (B). The experimental data showed a significant drop in the percentage of time spent in slow-wave sleep (SWS) as compared to baseline for both groups (B left panel) and a lower percentage of time in SWS for the RW group on days W2, W3, and W4. The simulation reproduced the significant drop in the RW group, but not the AW group and the model demonstrates a significantly longer SWS episodes for all four work days, rather than the W2-W4. The experimental data showed modest yet significant drop in time spent in rapid eye movement sleep (REMS) for both AW and RW groups as compared to baseline with AW showing a significantly lower value than RW on days W1 and W2. In the simulation RW was significantly lower than AW for all four work days while both groups showed a slight increase with respect to the baseline value. Asterisks in all panels indicate significant differences between RW (n = 15 in experimental data) and AW (n = 12 in experimental data); * p < 0.05, ** p < 0.01, ***p < 0.001, and hashes denote significant differences compared to baseline: # p < 0.05, ## p < 0.01 and ### p < 0.001. When AW and RW data points overlay, asterisks above the data point refer to RW and asterisks below the data refer to AW. Open and filled triangles above graphs in right hand panels represent significant differences compared to experimental data for the AW and RW groups respectively. One triangle: p < 0.05, two triangles: p < 0.01, three triangles: p < 0.001.

Fig. 8

Homeostat, sleepiness, and alertness in simulations. A The simulated homeostat for both groups (upper panel: Active-work, AW; lower panel: Rest-work, RW) Shown in the average homeostat value for each epoch averaged over 50 simulations. B Sleepiness (blue curves) and Alertness (red curves) for both groups (upper panel: AW, lower panel: RW). Shown are average values, averaged over 50 simulations. In all panels, yellow and black bars represent times of lights-on and lights-off respectively. Vertical gray bands represent work shifts. Long-wake episodes (indicated with red dots in panel B) tend to happen when sleepiness is low. Long-wake episodes are less common than regular wake episodes and tend to happen when alertness is high.