A Mathematical Model of the Sleep-Wake Cycle
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The daily sleep-wake cycle usually consists of three distinct states: wakefulness, non-rapid-eye-movement (NREM) and rapid-eye-movement (REM). The process of switching between different states is complex, but a common assumption is that it is regulated primarily by two processes (the circadian and the homeostatic process) via reciprocal interactions of several downstream neuron groups. These interactions not only result in often rapid transitions from one state to another, but also allow for a certain degree of bi-stability that locks the organism in a given state for some while before it switches back. In order to better understand how the behavioral states are regulated by different neuron groups, I describe how to use the S-system method for the development of a mathematical model consisting of two phases. The first phase covers the switch between wakefulness and sleep, which is controlled by the interactions between wake- and sleep-promoting neurons, whereas the second phase addresses the generation of NREM-REM alternation, which is believed to be regulated by REM-OFF and REM-ON neurons. In this set-up I interpret the circadian rhythm as external input and homeostatic regulation as a feedback controller. Both open-loop and closed-loop forms of the two-phase model are investigated and implemented. Discharging activities of the corresponding neuron groups and the switches of behavioral states are shown in the simulation results, from which we can easily identify the basic roles of wake- and sleep-promoting neurons, REM-OFF and REM-ON neurons. The special regulatory function of the neuropeptide orexin is also tested by simulation.