Fig. 7: Entrainment of the sleep-wake oscillator by the circadian oscillator: Arnold tongues.
a Number of sleeps per day as a function of the mean level of the upper threshold, \({H}_{0}^{+}\) (upper horizontal axis) and corresponding natural period of the sleep-wake oscillator (lower horizontal axis) and the circadian amplitude, generated by simulating the 2pm for approximately 110,000 different parameter values. The remaining parameters are: χs = 4.2 h, χw = 18.2 h, \({H}_{0}^{-}=\) 0.17, μ = 1. Regions are shaded according to the number of sleeps per day. The black lines are contour lines. Both the shading and the contours are on a logarithmic scale (base 2). The largest region corresponds to monophasic sleep. In principle there are regions of n sleeps per day for m circadian periods for every integer pair (n, m) for every n/m. In practice, the upper and lower limits of the values for n/m are bounded by the choices of the other parameters. For the example shown 0.068 < n/m < 2.54 days. Each region forms a tongue with the tip of the tongue occurring at zero circadian amplitude for the value of \({H}_{0}^{+}\) which gives a natural period of m/n. b–g Examples of simulations for different values of \({H}_{0}^{+}\) at a fixed circadian amplitude of a = 0.08, i.e. taking a horizontal line across panel a. In each case the two-process model along with a raster plot is shown. The orange triangles mark the time of the circadian minimum. b Polyphasic sleep pattern. c Two sleeps a day. d Monophasic sleep with the circadian minimum occurring near the end of the night. e Monophasic sleep with the circadian minimum occurring near the beginning of the night. f Four sleeps every five days. g One sleep every two days.
