Sleep physiology / 01

Two processes, one sleeper. Why 6 am feels better than 4 am.

Sleepiness is not a single quantity that fills up while you are awake and empties while you sleep. It is the distance between two independent things: a pressure that builds the longer you have been awake, and a clock that decides how much pressure it takes to put you under. Almost everything strange about sleep — why an evening nap costs more than it gives, why a lie-in on Sunday costs you Sunday night, why an all-nighter feels easier at breakfast than it did at four in the morning — falls out of the gap between them.

Start with the navy curve. It is the whole of Process S: a pressure that climbs from the moment you wake and drains only while you are asleep. It has no idea what time it is. Left to itself it would be a perfectly boring sawtooth, and if sleepiness were only this, the longer you stayed awake the worse you would feel, without exception, forever.

Now the rose curve — Process C, and the part people miss. This is not sleepiness. It is the height of the bar: how much pressure it takes to put you under, set by the circadian pacemaker in the hypothalamus and reset each morning by daylight through a direct pathway from the eye. The bar is highest in the early evening and lowest around dawn. There are two of them, and Borbély called the arrangement a somnostat: sleep begins where the navy curve crosses the upper threshold going up, and ends at the dashed lower threshold, or when the alarm gets there first. The moment of crossing is what Peretz Lavie named the sleep gate, after finding that sleep propensity does not drift upward through the evening but opens abruptly — preceded by a couple of hours in which sleep is unusually hard to come by, the forbidden zone. Nothing else in this model decides when you sleep.

This is why the evening is such a strange time to be tired. Drag bedtime back to nine o'clock. You have been awake fourteen hours, pressure is high and still climbing — and you lie there for the better part of three hours, inside Lavie's forbidden zone. Then look at the column headed “asleep at” and notice what has not moved. Bed at nine, bed at ten, bed at a quarter to midnight: sleep starts within a minute or two of the same time in every case, because what decides it is where the two curves cross, and going to bed earlier does not move either curve. The extra hours are not sleep gained. They are wakefulness, taken lying down.

Now the nap slider, which is the control that turns this diagram into advice. Give the sleeper thirty minutes at three in the afternoon. The navy curve takes a visible bite out of itself, and every night that follows starts from further down. Sleep now begins three quarters of an hour later than it did, and because the alarm has not moved, the night is three quarters of an hour shorter. Thirty-one minutes of nap against forty-nine minutes of night: the day ends eighteen minutes down on where it would have been with no nap at all.

Make the nap longer and it gets worse rather than better — ninety minutes costs three quarters of an hour of net sleep. Move it later and it gets worse faster: those same ninety minutes cost about twenty minutes net at one in the afternoon, and closer to a hundred at seven in the evening. Pressure spent near bedtime is precisely the pressure bedtime needed. This is the whole argument against napping while repairing insomnia, and it is not a rule handed down from anywhere — it is arithmetic, and the sliders will do it in front of you. It also carries a condition the model makes plain: the loss is this clean only because the alarm is fixed. Let the morning float and the night reclaims some of what the nap took.

Which brings us to the strip at the bottom, the gap itself — the model's own measure of how sleepy you are, positive whenever sleep would begin if you let it. Set night 2 to a night without sleep and read that strip across the following morning. Sleepiness peaks around seven or eight in the morning, at about twenty-four hours awake. Then it falls — and keeps falling all morning, so that by early afternoon, thirty-one hours awake, it is a third of what it was at breakfast. Nothing has been repaid. Sleep pressure is higher than at any point in the night. What changed is that the clock woke up and started pushing again, lifting the bar out from under a curve that never stopped rising. That is the second wind, and it is not a metaphor or a matter of willpower: it is two curves moving in opposite directions, and it is the single cleanest piece of evidence that one process could never have produced this.

The last scenario is the one that catches people every weekend. Give night 2 a three-hour lie-in. The extra sleep is real — nearly nine hours, and it ends on its own before the alarm, because pressure has drained all the way to the lower gate. But look at the night after. Bedtime has not changed and the clock has not moved; the only thing different is that the tank was emptied more thoroughly than usual, so it takes longer to refill past the bar. Lights out at the usual time, and the model has the sleeper lying awake — an hour and a half of it, against twelve minutes on an ordinary night — then sleeping under six hours. Sunday night insomnia is not anxiety about Monday. It is Saturday morning, arriving late.

Once the shape is in view, a lot of ordinary advice stops sounding arbitrary. A fixed wake time matters because it is the only lever that reliably pins the clock, and the clock sets the bar. Morning light matters because it is what the pacemaker reads. Naps are not forbidden so much as expensive, because pressure spent in the afternoon is not available at midnight. Caffeine works on this diagram in a very specific way: adenosine accumulating in the brain across waking is the best-understood physical correlate of the navy curve, and caffeine blocks its receptors — so it does not lower sleep pressure, it hides it, and the curve you were not feeling is still exactly where it was when the drug wears off.

And the therapy in section 02 is this diagram used as a tool. Sleep restriction works by driving the navy curve up until it clears the bar quickly and stays clear — buying consolidation with pressure, on purpose. The sleeper in that exhibit is this sleeper, with the same constants.

A model, not a measurement. Process S rises toward saturation with a time constant of 18.2 h and decays with 4.2 h; the thresholds sit at 0.67 and 0.17 with a circadian excursion of 0.12, following Daan, Beersma and Borbély (1984) — the quantitative version of the model Borbély first set out in 1982, and which he retraced in Journal of Sleep Research in 2022. Process C is the skewed sine of that paper. Sleep onset, spontaneous waking, sleep latency and the failure of daytime naps are all consequences of those constants rather than choices made here, which is the reason to trust the shapes and not the decimal places — with one deliberate exception. Naps here are taken, not gated: the sleeper lies down and sleeps, where nights have to wait for pressure to reach the threshold. Held strictly to these constants the model would refuse almost every daytime nap, because the lowest the upper threshold ever falls is 0.563 while a rested sleeper carries about 0.43 at three in the afternoon — there would be no hour of the day at which they could drop off. People plainly do nap, and Daan, Beersma and Borbély said why: bedrest, warmth, darkness and the absence of social stimuli all lower the upper threshold, and none of that is modelled here. Rather than fake a threshold to get the right answer, the nap is treated the way the alarm already is — a thing the reader decides, whose cost the model then works out. The nap durations and the resulting delays are real consequences of Process S; only the decision to fall asleep is imposed. Relatedly, the classic waveform has no post-lunch dip, so the mid-afternoon lull many people feel is absent from the diagram, and the circadian trough lands nearer 7 am here than the 5 to 6 am usually measured from core body temperature. On vocabulary: Process S, Process Cand the upper and lower thresholds are Borbély's and Daan's own terms; sleep gate, forbidden zone and the closely related wake-maintenance zonecome from Lavie's ultrashort sleep–wake experiments (1986) and describe the crossing rather than the threshold itself. The neurobiology referred to in passing — the suprachiasmatic nucleus as the pacemaker, the retinohypothalamic tract as its light input, adenosine as the somnogen accumulating across waking, and caffeine acting at its receptors — follows Scammell, Arrigoni and Lipton's review in Neuron(2017). No night on this page is anyone's data, and nothing here is medical advice or a description of any individual's sleep.