Summary: A new mathematical study explains why sleep patterns vary across the lifespan — why babies nap unpredictably, teenagers stay up late, and many older adults wake earlier. By extending the classic two-process model of sleep to include light exposure, researchers reveal how internal sleep drives and environmental cues combine to shape when we sleep.
Computer simulations show that today’s typical indoor lighting and evening light exposure can disturb the balance among sleep pressure, the circadian body clock, and external light signals. The results point toward targeted, practical strategies for improving sleep across different ages and lifestyles.
Key Facts:
- Integrated model: A unified framework that links sleep pressure, the circadian clock, and light effects.
- Across the lifespan: Offers clear explanations for sleep behaviors in infants, adolescents, and older adults.
- Actionable insights: Identifies how adjusting light exposure, routines, and timing can improve sleep quality.
Source: University of Surrey
Why do babies nap on some days and not others? Why do older adults often wake earlier than they used to?
A mathematical analysis of sleep regulation sheds light on these common questions, according to a new study from the University of Surrey. The researchers revisited the two-process model (2PM) of sleep, a foundational framework developed in the 1980s that explains sleep timing as the result of two interacting influences: a homeostatic sleep drive that increases with wakefulness and decreases during sleep, and a near-24-hour circadian rhythm.
Using mathematical tools, the team showed how the 2PM maps onto brain dynamics that control transitions between sleep and wakefulness. The model helps explain developmental patterns such as why infants may switch between napping and not napping across days — a behavior known in oscillator theory as the “Devil’s staircase.” The same mathematical structure also accounts for sleep patterns observed in other species.
The researchers extended the 2PM by adding a mathematical description of how light affects the circadian pacemaker. This combined model creates a layered system of coupled oscillators: a sleep–wake oscillator, the circadian clock, and the daily light–dark signal reaching the brain via the eyes. The interactions among these oscillators produce entrainment, the process that aligns internal rhythms to the external day–night cycle.
This integrated approach clarifies common observations. For example, teenagers typically fall asleep and wake later than younger children because their sleep pressure builds more slowly during the day, allowing them to stay awake longer. When that slower rise in sleep pressure coincides with bright evening light — from screens or indoor lighting — the circadian clock is pushed later, delaying sleep further.
A surprising implication of the model is that earlier waking in older adults may not stem mainly from shifts in the circadian clock itself. Instead, age-related changes in how sleep pressure, circadian signals, and light interact can lead to earlier wake times. In other words, sleep timing differences may reflect altered system interactions rather than a single broken clock.
The team ran simulations with the 2PM-plus-light model to explore how modern environments influence sleep. These simulations indicate that spending most of the day indoors and exposing ourselves to light in the evening can disrupt the coupled oscillator system, making it harder to fall asleep at regular times, shifting sleep later after late-night light exposure, and reducing alignment with social schedules.
Those simulation results help explain why some people find it difficult to wake early or to sleep at socially expected times: their sleep propensity is not necessarily abnormal but is shaped by a combination of biology and the light environment they create. That reframing suggests practical interventions — altering evening light, adjusting routines, or timing exposure to daylight — could help many people realign sleep without assuming a dysfunctional circadian clock.
Professor Anne Skeldon, Head of the School of Mathematics at the University of Surrey and lead author of the study, said:
“This model gives us hope that sleep problems can be better understood and tackled. By using maths, we can see how small changes in light, routine or biology shift our sleep, and test practical ways to support better sleep for everyone. It’s a step towards more personalised, effective solutions that improve people’s daily lives.”
Professor Derk‑Jan Dijk, co-author and Director of the Surrey Sleep Research Centre, added:
“This work shows how maths can bring clarity to something as complex and personal as sleep. 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.”
About this math modeling and sleep research news
Author: Dalitso Njolinjo
Source: University of Surrey
Contact: Dalitso Njolinjo – University of Surrey
Image: Image credited to Neuroscience News
Original Research: Open access. “The complexity and commonness of the two-process model of sleep regulation from a mathematical perspective” by Anne Skeldon et al., published in npj Biological Timing and Sleep.
Abstract
The complexity and commonness of the two-process model of sleep regulation from a mathematical perspective
The two-process model (2PM) of sleep regulation is a conceptual framework represented by mathematical equations. It shares characteristics with models for cardiac, respiratory and neuronal rhythms and belongs to the broader class of coupled oscillator systems. The 2PM relates closely to neuronal mutual inhibition models of sleep–wake control.
The mathematical form of the 2PM, in which the sleep–wake cycle becomes entrained to the circadian pacemaker, accounts for sleep patterns observed without strong 24 h time cues, across species, and during early childhood development.
Extending the 2PM to include a process describing the circadian response to light yields a hierarchical entrainment system with feedback. This extension enables quantitative modeling of how self-selected light exposure alters sleep timing and circadian phase.
The extended 2PM offers new interpretations of common sleep phenotypes and produces quantitative predictions for the effects of sleep and light interventions. These predictions can help support sleep and circadian alignment in individuals, including those living with neurodegenerative disorders.