Sleep physiology / 02

Less time in bed, more sleep. The arithmetic of a fragmented night.

Ten hours in bed. Five hours of sleep. The other five spent awake in the dark, which is a way of spending half your night in a room where nothing happens. Cut the window down and the sleep gathers itself into one block — immediately, and at a price. Getting the sleep back, and then some, takes four months.

Start with the baseline row. Bed at half past ten, up at half past eight, and barely half of it asleep — five hours of sleep bought with ten hours of lying down. The loss splits almost evenly in two. Two and a half hours go at the front, waiting: the pressure curve underneath is nowhere near the rose threshold at half ten, the gate is simply not open yet, and no amount of lying in the dark opens it early. Sleep starts around one in the morning. The other two and a half hours go in pieces, scattered through the small hours, once the curve has drained into the shaded zone and there is no longer enough pressure to hold sleep together.

This is the trap the therapy is named after. Sleeping badly, the obvious move is to spend longer in bed to catch more of it. But the extra hours are taken at the front of the night, where the gate is closed, and they are spent awake. More time in bed buys more wakefulness, almost none of it sleep.

Now week 1, and here the model does something worth pausing on. The window is prescribed from the sleep diary, not from a recording — and people with insomnia reliably under-report how much they slept. So the prescription lands below actual sleep, hits the five-hour floor the protocol keeps for safety, and total sleep falls off a cliff — an hour below baseline, though far less than the five hours of bed that were taken away, because most of what was cut was never sleep in the first place.

Watch those seven nights in order: 69, 85, 79, 86, 78, 84, 90. The first night on a new schedule is the worst of the fortnight — barely three and a half hours — and by the seventh the night is all but unbroken. Two things pull in opposite directions. Changing the schedule costs a night or two of settling in; the sleep debt those short nights build then raises the pressure each following night starts from. The debt wins, and the week tightens as it goes. Week 1 is not bad sleep. It is excellent, insufficient sleep, and it gets more excellent as it gets more insufficient. That is why the first fortnight is reported as the hard part — not broken nights, but a real sleep debt, paid out in daytime sleepiness and measurably slower reactions.

Every widening repeats it in miniature. Step the window out and the next night or two sag before the week climbs back — visible in almost every block of seven, and the reason this takes months rather than weeks. Watch the trace. Sleep dips below where it started, tracks under the dashed line for a month, crosses it at week five, and keeps climbing in a sawtooth all the way out: six and a half hours by week sixteen, against five at the start — an hour and twenty minutes more sleep, on two hours less time in bed. The window has ended up at eight hours, which is where an unremarkable night would have been all along. The waiting has gone from two and a half hours to a quarter of an hour.

That climb is worth being careful about, because the model cannot produce it from sleep pressure alone. With the alarm fixed and the body clock unmoved, sleep is bounded by the gap between the gate opening and the alarm going off. Deleting the two and a half hours of pre-gate waiting frees no sleep whatsoever — it was never sleeping time. Only the fragmented half is recoverable, and recovering it has a floor. Restriction can make a night solid; it cannot by itself make it longer than the gap. What lifts the line and keeps it rising is the arousal coming down — the bed slowly ceasing to be a place where one lies awake — which is the work of the other half of the therapy, and which this model assumes rather than derives, at a rate nobody has measured. Read the far end of that trace as an illustration, not a forecast.

Push the arousal slider up and the ceiling comes down: the same procedure converges on a shorter window and never reaches the same efficiency, because a second process keeps interrupting sleep independently of how much pressure is available. Drag the baseline window wider and the opposite emerges — the more of the night that was being spent awake, the more there is to recover. Set the baseline near seven hours and efficiency starts high enough that there is very little left for the procedure to do.

A model, not a protocol. Sleep pressure follows Daan, Beersma and Borbély (1984) with the parameters used in section 01; the fragmentation on top of it is phenomenological — a pressure-dependent fragility that accounts for the frayed tail of a long night, plus an arousal process whose rate and duration both fall as pressure rises. The titration rule is Spielman's (1987). Three further assumptions are bolted on rather than derived. The week-1 window is prescribed from a diary figure set 20% below actual sleep, which is what produces the early debt. A settling-in cost is applied for a night or two after any change of schedule — without it the model does the reverse, absorbing each new fifteen minutes as sleep immediately, because pressure under restriction sits far above the threshold. And arousal is assumed to fall by half across the ten weeks, which is the only thing that lets total sleep finish above where it began; nothing in Process S extinguishes a conditioned association, and a fixed alarm caps sleep at the interval between the gate and the morning. Spielman reported a rise in total sleep by week 8, though the pooled trial evidence finds no net effect. The misperception figure is calibrated against Kyle et al. (2014), who measured polysomnographic sleep losses of 91, 78 and 69 minutes on restriction nights 1, 8 and 22, alongside raised daytime sleepiness and slowed vigilance through the first three weeks, all resolved at three months. Effect sizes for the therapy itself — large for efficiency, onset latency and wakefulness after onset, and null for total sleep time — follow Maurer et al. (2021). Every night here is simulated, none is data, and nothing on this page is a recommendation or a stand-in for anyone's own sleep.