Table 1.
Modeling framework.
Fig 1.
Circadian drive u with and without Process L.
Fig 2.
Illustrations of sleep state (gray region) and sleepiness (blue curve) under the spontaneous sleep schedule in Eq (9).
The subject wakes up spontaneously when B = Lm (red nodes) and falls asleep spontaneously when B = Hm (green nodes).
Table 2.
Two-process models.
Table 3.
Entrainment formulation for two-process models.
Fig 3.
Illustrations of sleep state (gray region) and sleepiness (blue curve) under the controllable sleep schedule in (16) and (17).
The two horizon pale red bars demonstrate the range of sleepiness that the subject could fall asleep and wake up.
Fig 4.
Entrainment time comparison of different entrainment strategies with Imax = 10000 lux (left), Imax = 1000 lux (right) and different time shifts Δshift.
The black curve shows the open-loop entrainment time with spontaneous sleep; the blue curve shows the minimum entrainment time with spontaneous sleep; the red curve shows the minimum entrainment time with controllable sleep; the green dash line shows the minimum entrainment time of circadian rhythm without Process S and sleep, solved by the method in [15].
Fig 5.
Sleepiness comparison of the minimum-time entrainment with spontaneous sleep (blue curves) and with optimal (controllable) sleep (black curves).
In this figure, we only plot the sleepiness value during the entrainment process (i.e., before reaching the terminal condition in Eq (14)). The black curves terminate earlier than the blue curves, emphasizing that the entrainment times with optimal sleep are shorter than those with spontaneous sleep.
Fig 6.
(Left) Scheduled sleep with Bsleep = 0.77, Bwake = 0.17. (Right) Scheduled sleep with Bsleep = 0.67, Bwake = 0.27. The red region represents that the reference light is present between 6 am and 10 pm, the gray region in each figure shows the spontaneous sleep period, the blue line shows the value of phase response curve. The black region in the left panel shows the scheduled sleep period that the subject wakes up when B = 0.17 and falls asleep when B = 0.77; the black region in the right panel shows the sleep period that the subject wakes up when B = 0.27 and falls asleep when B = 0.67.
Fig 7.
The periodic circadian state of the JFK model with (upper) and without (lower) Process S and sleep-wake cycle.
The red region in the upper panel shows the reference daily light is present between 6 am and 10 pm, the black region in the upper panel indicates the reference sleep period is from about 11:10 pm to 7:30 am. The red region in the lower panel shows the actual daily light that the reference subject receives is from 7:30 am to 10 pm.
Fig 8.
Minimum-time entrainment cases with the tuned sleep schedule in Eq (18) (left column) and controllable sleep schedule (right column).
The upper panel of every sub-figure shows the reference states (blue curves), entraining states (black curves) as well as the light input. Note that the optimal lights from the gradient descent process in this paper are all bang-off controls, i.e., I*(t) = 0 or Imax. In the following figures, we represent the light-on region in the form of a vertical pale red bar. The lower panel of every sub-figure demonstrates the sleep period (black region) and sleepiness (blue curves) during entrainment. The green dashed lines in left subfigures indicate the entrainment time.
Fig 9.
Periodic circadian and sleep homeostasis states in three models.
The blue lines of x3rd, xc3rd, H3rd, θ3rd represent the periodic circadian and sleep homeostasis states of the S+C3 model, the red lines of x2nd, xc2nd, H2nd, θ2nd represent those of the S+C2 model, the black lines of θ1st and H1st represent the periodic circadian phase and sleep homeostasis of the S+C1 model.
Fig 10.
Entrainment time by applying the optimal light and sleep schedule of the simplified models on the full model with Imax = 10000 lux (left) and Imax = 1000 lux (right).
Fig 11.
Comparison of the minimum-time (red) and feedback entrainment (blue) time on the cross-validation entrainment cases.
We can observe that, in most cases, the feedback entrainment time is close or equal to the minimum entrainment time.
Fig 12.
Illustrations of the entrainment process of shift workers under constant light during the night shift from 8 pm to 8 am.
The entrainment process starts at the end of the night shift, i.e., t0 = 0 corresponds to 8 am. The red region represents the time when the light is on for the shift worker, the light red region represents the time when the reference daylight is present, the black region shows the sleep period of shift workers and the gray region shows the sleep period of the reference subject.
Fig 13.
Open-loop entrainment with different night shift light intensity Ishift.
The solid curves show the time courses of the entraining state of night shift workers, and the dashed curves show the time courses of the reference state. The open-loop entrainment time is 3.46 hours for Ishift = 0 lux, 6.29 hours for Ishift = 100 lux and 18.32 hours for Ishift = 1000 lux. The blue dot line shows the phase response curve (PRC) of the entraining state.
Fig 14.
Minimum-time optimal entrainment with controllable sleep schedule and different night shift light intensity Ishift, with maximum entrainment light intensity Imax = 1000 lux (left) or Imax = 10000 lux (right).
The minimum entrainment time of these cases is given as 3.46 hours for Ishift = 0 lux and Imax = 1000 lux, 5.80 hours for Ishift = 100 lux and Imax = 1000 lux, 17.62 hours for Ishift = 1000 lux and Imax = 1000 lux, 3.46 hours for Ishift = 0 lux and Imax = 10000 lux, 4.80 hours for Ishift = 100 lux and Imax = 10000 lux and 6.96 hours for Ishift = 1000 lux and Imax = 10000 lux, respectively.
Fig 15.
Open-loop entrainment with different lighting condition Ishift during night shift between 10 pm to 8 am, where the open-loop entrainment time is 44.06 hours for Ishift = 0 lux, 13.25 hours for Ishift = 100 lux and 18.32 hours for Ishift = 1000 lux.
Fig 16.
Minimum-time optimal entrainment of shift workers with night shift from 10 pm to 8 am with controllable sleep schedule and different working light intensity Ishift, maximum light intensity Imax = 1000 lux (left) and Imax = 10000 lux (right).
The minimum entrainment time of these cases is given as 17.68 hours for Ishift = 0 lux and Imax = 1000 lux, 6.47 hours for Ishift = 100 lux and Imax = 1000 lux, 17.62 hours for Ishift = 1000 lux and Imax = 1000 lux, 11.38 hours for Ishift = 0 lux and Imax = 10000 lux, 5.60 hours for Ishift = 100 lux and Imax = 10000 lux and 6.96 hours for Ishift = 1000 lux and Imax = 10000 lux, respectively.
Fig 17.
Illustrations of the entrainment process of the shift worker with controllable light during the night shift.
The entrainment process starts at the beginning of the night shift.
Fig 18.
Minimum-time entrainment with optimal light input during the night shift from 8 pm to 8 am with Imax = 1000 lux (upper) and Imax = 10000 lux (lower).
The entrainment time of these two cases is both 3.28 hours (12 hours night shift is excluded from the entrainment time).
Fig 19.
Minimum-time entrainment with optimal light input during the night shift from 10 pm to 8 am with Imax = 1000 lux (upper) and Imax = 10000 lux (lower).
The entrainment time of these two cases is both 3.46 hours (10 hours night shift is excluded from the entrainment time).