Dissociating the intensity and phase origins of sleepiness through a threshold-distance model of sleep-wake dynamics

<jats:p> Sleepiness is a leading proximate cause of drowsy-driving fatalities, medical errors and industrial accidents, yet it has resisted mechanistic prediction; although it arises from well-characterized sleep-wake physiology, it is experienced as a subjective state and has lacked a quantitative link to the underlying dynamics. We previously showed that subjective sleepiness maps linearly, with a protocol-invariant form, onto the signed distance H - H^+ between the homeostatic pressure H and the circadian-modulated sleep-onset threshold H^+. This single quantity predicts sleepiness accurately but is mechanistically ambiguous: the same value can arise either because H sits far from the boundary or because the threshold H^+(t) has shifted with circadian phase, and these two origins call for entirely different interpretations and interventions. Here we resolve this ambiguity by decomposing H - H^+ into two mechanistically separable axes-- <jats:italic>intensity</jats:italic> and <jats:italic>phase</jats:italic> . The <jats:italic>intensity</jats:italic> axis is the time-averaged margin ⟨ H - H^+ ⟩, set by how far, on average, H sits from the sleep boundary: slowed homeostatic accumulation accounts for the paradoxically blunted sleepiness of older adults, and pharmacological suppression of H accounts for the dose-dependent alerting effect of caffeine. The <jats:italic>phase</jats:italic> axis is set by the circadian modulation of H^+(t): under a forced-desynchrony protocol, in which the pacemaker free-runs and the homeostatic and circadian processes are experimentally decoupled, sleepiness tracks the circadian profile of H^+(t) across all phases while the intensity mapping itself remains unchanged--a clean dissociation of the two axes. By resolving felt sleepiness into these two physiological degrees of freedom, this framework renders previously isolated phenomena--aging, caffeine and circadian misalignment--commensurable within a single theory and provides a physiologically interpretable basis for prospective fatigue-risk prediction. </jats:p>