Fig 1.
Schematic of the mathematical model.
Sleep/wake state determines the dynamics of the total adenosine concentration, Atot, with Atot increasing in wake and decreasing in sleep. This pool fractionates into the concentrations of those unbound, Au, those that are bound to A1 receptors, A1,b, and those that are bound to A2A receptors, A2,b. The model also includes pools of A1 and A2A receptors, with total concentrations of Rn,tot, n = 1 or 2, which fractionate into bound, Rn,b, and unbound Rn,u,. The total concentration of A1 receptors, R1,tot = R1,b + R1,u, changes in response to adenosine concentration by attempting to achieve a homeostatic level of fractional occupancy, R1,b/R1,tot. For modeling sleep and performance, R1,b is used as the sleep homeostatic process. This is added to a circadian process to give the overall sleep drive, D. A sigmoid function is then used to convert D into PVT lapses, P.
Table 1.
Fit values and units for the 15 model parameters.
Fig 2.
Dynamics of the adenosine model under two simulated conditions: Acute sleep deprivation for 4 days (red line), and chronic sleep restriction with 4 h sleep per night for 8 days (black line).
Four model variables are shown as functions of time: (A) Unbound adenosine concentration, Au. (B) Total adenosine concentration, Atot. (C) Bound A1 receptor concentration, R1,b, which is used as a proxy for sleep homeostatic pressure in the model. (D) Total A1 receptor concentration, R1,tot. Red horizontal lines represent the level of each variable at the end of the 4 days of acute sleep deprivation.
Fig 3.
PVT data (open triangles) and simulations (filled circles) for Experiment 1, with time in bed (TIB) illustrated as gray bars.
Panels show four experimental conditions: (A) Acute sleep deprivation with 0 h TIB, (B) 4 h TIB for 13 nights, then 2 nights of 8 h TIB, (C) 6 h TIB for 13 nights, then 2 nights of 8 h TIB, (D) 8 h TIB for 15 nights. PVT data were collected every 2 h during wakefulness, beginning 4 h after morning awakening, which was simulated to occur at 8am in all conditions (schedules from experimental data were shifted by 0.5 h for convenience). Zero time on the x-axis corresponds to midnight on the last baseline night. Data points are adapted from [17].
Fig 4.
Daily sleep durations and sleep patterns during recovery from sleep chronically restricted to 7 h per night, with sleep allowed for 14 h per day from 18:00 to 8:00.
(A) Daily sleep durations over 28 days for experimental data [40] and the model’s best fit. (B) Sleep patterns exhibited by the model across 50 days, displayed as a raster diagram, with black bars corresponding to sleep. The data are double plotted. During different stages of recovery, the model predicts both monophasic sleep (one sleep episode per day) and biphasic sleep (two sleep episodes per day). Vertical blue lines indicate the start and end of allowed sleep times. (C) and (D) show the sleep drive as a function of time across days 1–20 and 21–40 respectively, with the blue dashed lines representing the upper and lower sleep/wake thresholds.
Fig 5.
Mathematical structure of (A) previous mathematical models based on the two-process model, and (B) the physiologically based adenosine model developed here. Arrows show functional dependences between model variables.