6 days ago
New Sleep Model Reveals Why Babies Just Won't Nap Some Days
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If you've ever wondered why your baby naps inconsistently, the answer may lie not in parenting guides or alarm clocks—but in mathematics.
A new study by researchers from the University of Surrey, England, has applied mathematical modeling to explain how and why sleep patterns change over a lifetime.
The research focused on refining a long-standing concept known as the two-process model ("2PM") of sleep regulation, originally introduced in the 1980s, which considers both biological sleep pressure (how the longer you are awake, the more sleepy you get) and our internal body clocks to understand when and how we sleep.
But this latest study takes the model further, incorporating the mathematics of light exposure to show how environmental cues—like indoor lighting or late-night screen time—interact with biology to shape our sleep.
The result is an expanded "2PM plus light" model that offers a unified explanation for a host of sleep mysteries—from unpredictable baby naps to the early morning habits of older adults.
High angle shot of little Black child dressed in knitted jumpsuit of blue color serenely sleeping in cozy wooden crib with baby mobiles hanging above, while mother checking on daughter in bedroom.
High angle shot of little Black child dressed in knitted jumpsuit of blue color serenely sleeping in cozy wooden crib with baby mobiles hanging above, while mother checking on daughter in bedroom.
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"This model gives us hope that sleep problems can be better understood and tackled," said paper author and mathematician professor Anne Skeldon of the University of Surrey in a statement.
"By using math, we can see how small changes in light, routine or biology shift our sleep—and test practical ways to support better sleep for everyone."
One of the more curious phenomena explained by the model is why babies nap on some days but not others, even when their schedule seems consistent.
Oscillator theorists refer to this behavior as the Devil's staircase—a mathematical term for sudden jumps or skips in patterns.
The researchers showed how the interplay between a baby's rising sleep pressure and immature circadian rhythms can create seemingly erratic nap behavior that, in reality, follows a mathematically predictable pattern.
The same model explains why teenagers tend to go to bed and wake up later than younger children.
The mathematical simulations revealed that adolescents experience a slower buildup of sleep pressure, allowing them to stay awake longer.
Combined with bright light exposure in the evening—from devices, indoor lighting or nighttime socializing—this can delay their sleep even further, shifting their rhythms into a cycle that is out of sync with early school start times.
With models like 2PM plus light, scientists can simulate the impact of small changes in lighting, bedtime routines or social schedules—potentially leading to personalized interventions for people struggling with sleep in our increasingly 24/7 world.
"This work shows how math can bring clarity to something as complex and personal as sleep," said paper co-author and Surrey sleep researcher professor Derk-Jan Dijk.
"With the right data and models, we can give more tailored advice and develop novel interventions to improve sleep patterns for those whose rest is affected by modern routines, ageing or health conditions," he concluded.
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Reference
Skeldon, A. C., & Dijk, D.-J. (2025). The complexity and commonness of the two-process model of sleep regulation from a mathematical perspective. npj Biological Timing and Sleep, 2(24).
