Fig. 1: The two process model.

a The two processes in the two-process model (2pm) as depicted in Fig. 4 from¹. S represents the homeostatic sleep pressure, \(\overline{C}\) is the circadian drive for wakefulness. The grey shaded region highlights times when sleep occurs. b The 2pm including upper and lower thresholds, as described in². Homeostatic sleep pressure (black line) increases during wake and decreases during sleep. Switching between wake and sleep and sleep and wake occurs at upper and lower thresholds respectively (grey lines). The thresholds are modulated according to a circadian rhythm. Here the circadian rhythm is assumed to be entrained to 24 hours. The timing of sleep is shown by the horizontal grey bars with the orange triangles indicating the times of the circadian minimum. Parameter values: χ_s = 4.2 h, χ_w = 18.2 h, \({H}_{0}^{+}=\) 0.67, \({H}_{0}^{-}=\) 0.17, a = 0.12, μ = 1. c–e Simulations for different values of the mean lower threshold and χ_s = 4.2 h, χ_w = 18.2 h, \({H}_{0}^{+}=\) 0.75, a = 0.07, μ = 1. In each pair of panels the lefthand panel shows the 2pm and the righthand panel shows a raster plot indicating the timing of sleep relative to the circadian minima (orange triangles). c Shows a repeating rhythm consisting of three sleep periods every four days, \({H}_{0}^{-}=\) 0.09. d Shows a repeating rhythm consisting of three sleep periods every two days, \({H}_{0}^{-}=\) 0.50. e Shows a polyphasic sleep pattern consisting of two sleep periods every day, \({H}_{0}^{-}=\) 0.58.