Two processes, one changing day: sources & model

Change a fictional sleep history, catch the moment two curves meet, and separate a local clock change from a biological clock. Then follow light information from eye to brain.

What supports the explanation?

What this model assumes

1. A mathematical model with one published parameter set and a specified initial condition. It is not fitted to a child, adult or patient.
2. S is a dimensionless state. No alertness score, diagnostic result, driving-safety judgment, exam-performance prediction or personal sleep recommendation is calculated.
3. Prescribed blocks mean fictional modeled sleep or wake, not time in bed and not a guarantee that a person can fall asleep on command.
4. The fixed circadian wave has no light-dose input or adaptation dynamics. Clock-label changes, assumed phase changes and elapsed time remain separate operations.
5. The added daytime sleep block increases total modeled sleep. The separate order experiment holds total durations equal. These answer different comparison questions.
6. The model does not simulate sleep stages, melatonin concentrations, caffeine, medication, long-term sleep-restriction effects or individual sleep requirements.
7. The at-home activity observes an ordinary routine or uses supplied fictional cards. It asks for no altered sleep, alarm, caffeine, medication or light behavior.
8. The declared mathematical model: We use the two-process presentation in Skeldon, Dijk & Derks (2014), Figure 1b. Wake: dS/dt = (1 − S)/18.2 h. Sleep: dS/dt = −S/4.2 h. C = sin[2π(t − α)/24 h]. The boundaries are U = 0.60 + 0.10C and L = 0.17 + 0.10C. S and the thresholds are dimensionless. The reset S = 0.17, awake at t = 0, is an authored initial condition.
9. What the computation does: Each branch uses its exact exponential solution. A bracketed first-crossing search switches states without resetting pressure. The plot and downloaded table sample that same trajectory. Removing circadian modulation is an intervention on this model’s structure; it is not a simulated treatment or brain lesion.
10. A fixed wave does not model light adaptation: The quantitative circadian signal is prescribed. It has no light-response or entrainment equation. An assumed phase shift restarts a comparison; it is not the calculated result of turning on a lamp. The anatomy and light-experiment views explain separate biological evidence.
11. Pressure, measurements and experience differ: EEG slow-wave activity has been used to estimate homeostatic dynamics. Parameters can vary with individuals, normalization and modeling choices. Our S is not measured EEG voltage, blood adenosine concentration, subjective sleepiness, accumulated brain damage or a health score.
12. Sleep has stages this model does not generate: Human sleep includes REM and three NREM stages. A binary wake/sleep trace does not tell us those stages, dream content, learning performance or a person’s exact sleep requirement. Those questions need other observations and models.
13. Light timing can change shift direction: Khalsa et al. studied a laboratory protocol with 6.7 hours of bright-light exposure and melatonin-phase measurements. Exposure centered before the critical phase produced delays; after it, advances. That biological phase convention is distinct from the arbitrary zero of our sine model. It is not a household-light schedule or a rule tied to one civil-clock hour.
14. Anatomy and hormonal output are separate: The SCN region helps coordinate timing. Clock-related signaling also influences the pineal gland, which produces melatonin. The SCN is not itself the melatonin-producing gland. Added anatomical callouts are magnified teaching locations, not measured segmented nuclei.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.

About the cover illustration

Original plot of the declared two-process sleep model. Curves are calculations, not physiological measurements from a person. Model parameters and independently verified event times are sourced in the lesson. The separate eye and brain references are credited to Servier Medical Art, CC BY 4.0.