Dynamical mechanism for the interplay of circadian, homeostatic, and ultradian rhythm in normal human sleep.

The 90-minute ultradian rhythm is a hallmark of healthy human sleep, yet its governing mechanisms remain elusive. In this study, we develop a biologically grounded sleep model to unravel the complex dynamics underlying this rhythm. Our model integrates both circadian and ultradian drives, which collectively shape sleep architecture, along with bidirectional "flip-flop" switches that control transitions between wakefulness, nonrapid eye movement (NREM) sleep, and rapid eye movement (REM) sleep. Calibrated with empirically derived neurophysiological parameters, the model successfully reproduces core sleep features, including the 24-hour circadian rhythm and the 90-minute ultradian rhythm. To dissect state transition mechanisms, we employ potential landscape analysis to quantify how global stability is modulated by three key factors: circadian drive, homeostatic drive, and REM pressure. Our results reveal that the ultradian rhythm emerges from an interplay between a weak ultradian drive and REM pressure. In a reduced model focusing on NREM-REM interactions, we demonstrate that the periodic transitions between NREM and REM sleep arise from a saddle-node bifurcation on an invariant circle (SNIC) induced by REM sleep pressure. Additionally, the ultradian drive entrains the rhythmic NREM-REM system to exhibit the stable 90-minute ultradian rhythm, as characterized by the Arnold tongue. Our work provides the mechanistic explanation of the 90-minute ultradian rhythm, identifying REM pressure as its core regulator and highlighting the SNIC bifurcation together with the Arnold tongue as its dynamical mechanisms. This framework establishes testable neurophysiological requirements for experimental validation, thereby bridging theoretical models with empirical sleep neuroscience.