Homeostatic forces underlying the daily pattern of sleep propensity.

The mismatch between rising sleep need and the fluctuating ability to fall asleep underlies insomnia-the most common sleep disorder-yet remains poorly understood. While sleep need increases steadily with time awake, sleep propensity-the likelihood of transitioning from wake to sleep-follows a bimodal pattern, peaking in the mid-afternoon, dipping in the evening, and rising again near bedtime. Building on our previously developed wave model of sleep dynamics, we extend this homeostatic framework to the waking period and show that it predicts the observed bimodal sleep propensity curve. This pattern emerges from two interacting factors: wake-state instability, which increases exponentially across the day, and interaction strength between states, which follows a biphasic trajectory. Together, they produce a daily profile of sleep propensity that closely aligns with experimental data. Notably, the empirical curve demonstrates a deeper evening dip than the model alone predicts-reflecting the known circadian modulation of sleep propensity. The model reveals that the mid-afternoon peak reflects maximal interaction at the homeostatic equilibrium threshold, while the evening dip results from minimal coupling between sleep and wake states, counteracting high instability. A late-day rise in both factors facilitates sleep onset at bedtime and beyond. Experimental data from sleep deprivation further support these predictions. This work provides a mechanistic foundation for understanding daily sleep propensity and may inform strategies to improve sleep and performance in both health and disease.