The two-process model for sleep–wake regulation: A nonsmooth dynamics perspective

Since its inception four decades ago the two-process model introduced by Borbély has provided the conceptual framework to explain sleep–wake regulation across many species, including humans. At its core, high level notions of circadian and homeostatic processes are modelled with a low dimensional description in the form of a one dimensional nonautonomous and nonsmooth flow, with the rate of change of homeostatic sleep pressure switching at specific times. These events in time can be described by an implicit map from one switching time to another and have given rise to an elegant mathematical description of periodic orbits and their instabilities using the theory of iterated maps. In this paper we show that an equivalent description can be obtained from a direct analysis of the underlying nonsmooth flow. We further show how to construct the Lyapunov exponent of the nonsmooth flow and use this to uncover a more detailed picture of the Arnol'd tongue structure of the model. Given the growing interest in studying networks of sleepers, where interactions may occur continuously throughout the day–night cycle and not just at event times, we advocate for the future use of techniques from nonsmooth dynamical systems in studying networks of the two-process model