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 about eight weeks.

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 and hits the five-hour floor the protocol keeps for safety. Five hours of bed are taken away and three quarters of an hour of sleep goes with them, because almost all of what was cut was never sleep in the first place. What is left behind is a different night altogether: the two and a half hours of waiting collapse to four minutes, the scattered awakenings fall from six to two, and efficiency goes from 51% to 88% across the week — and to 94% the week after.

Watch those seven nights in order: 61, 92, 88, 96, 90, 93, 93. The first one is genuinely bad — three hours of sleep, six separate awakenings, nearly two hours awake in the dark. That night is worth expecting rather than being ambushed by. It is the night the schedule changes: the bed is still a cue for lying awake, and the pressure that will later carry the night straight through has not been built yet. It is also, on this model, the worst night of the whole program — and by the following night sleep has closed to 92%. The first night is not the therapy failing. It is the last night of the old pattern.

After that the nights are solid, and every one of them is better held together than any night of baseline. What stays hard is that they are short. Four and a half hours of nearly unbroken sleep is still four and a half hours, and the shortfall is paid out in daytime sleepiness and measurably slower reactions across the first fortnight. Week 1 is not bad sleep. It is excellent, insufficient sleep. The therapy's whole bet is that the excellence is the part worth keeping, and that the insufficiency can be bought back a quarter of an hour at a time.

And it is. That first rough night pulls the week's average down to 88%, which would put it under the widening threshold — but the rule does not read that night. The night the schedule changed says nothing about whether the window is the right size; it says the schedule changed. Read across nights 2 to 7, where the sleeper is actually living on the new window, week 1 comes in at 92%, and the first quarter of an hour is given back. From there the window widens by the same quarter of an hour every week, each step absorbed rather than resisted: 5h15, 5h30, 5h45, 6h00, 6h15, 6h30, 6h45. Efficiency holds in the low-to-mid nineties the whole way out, which is what tells the rule it is safe to keep going. Total sleep passes baseline in week 3 and keeps climbing. Then in week 8 the night cannot quite carry another quarter of an hour — efficiency slips just under ninety — and that is the answer. The program settles there rather than cutting back, because the sleeper's own window is what it has just found. Six and three quarter hours in bed, just over six hours asleep — an hour more sleep than baseline, on three and a quarter hours less time in bed. The waiting has gone from two and a half hours to four minutes.

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. It is assumed here to happen faster and more completely than it does in a clinic, so that the figure resolves in nine rows instead of forty. A real course of this runs longer than what you are looking at, and moves less tidily. 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 settles on a shorter window and finishes with less sleep, because a second process keeps interrupting the night independently of how much pressure is available. Drag the baseline window wider and something worth noticing happens — the program ends in almost exactly the same place regardless, a little over six hours asleep, because a fixed alarm and an unmoved body clock cap it there. What changes is how much waste is removed on the way: an hour and a half of pointless lying down at a seven-and-a-half hour baseline, six hours of it at a twelve-hour one. Set the baseline near seven hours and efficiency starts high enough that there is very little left for the procedure to do — drag it there and watch the whole program flatten into nothing.

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 adapted from Spielman's (1987), and the adaptation matters: Spielman's rule also narrowsthe window after a poor week, and so cycles — widen, overshoot, narrow, recover, widen again — for as long as the program runs. This figure runs a one-directional version instead. The window only widens, and the first week that cannot carry a step is treated as having arrived, so the program settles there for good. That buys a legible shape — a cut, a climb, a plateau — at the cost of the real protocol's ability to recover from a window set too wide, and it is the largest liberty taken on this page. A course run to Spielman's actual rule takes longer and moves in both directions. The rule here also skips the first night on the restricted schedule when it decides whether to widen, on the grounds that the night a schedule changes measures the change rather than the window — a clinician reading the diary would do the same, but Spielman's rule as published does not say so. 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. Arousal is assumed to fall by 86% within the first two or three weeks — faster and far more completely than a clinic would see — 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. And each widening is absorbed immediately, with no settling-in cost for the change of schedule; that is what the unforced model does, but it flatters the first nights after every step. 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 — a cost this figure understates, because it shows the nights and not the days. 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.