Zzzzzzzz: The Mathematics of a Good Night’s Sleep

This Behind the Scenes article was provided to LiveScience in partnership with the National Science Foundation.

We boast when our infant finally sleeps through the night. We bemoan the teenager who requires a canon shot to arise from his bed before noon. And in our “golden” years, we wonder why sleep is so fleeting, yet napping seems to come as easily as breathing. Such are the mysteries of sleep.

Troubled Sleep

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But the mysteries of sleep are more than just a source of passing wonder or inconvenience for many people. In fact, the Centers for Disease Control (CDC) reports that 70 million Americans suffer from chronic sleep problems that range from insomnia and sleep apnea  to narcolepsy, restless legs syndrome , and circadian rhythm disorders. In addition, “sleep deprivation is associated with injuries, chronic diseases, mental illnesses, poor quality of life and well-being, increased health care costs and lost work productivity,” according to CDC’s Sleep and Sleep Disorders Team, which evaluates the prevalence and impacts of sleep insufficiency and sleep disorders.

To help address such problems, biologists, behavioral scientists, neuroscientists and even mattress makers have, for years, been studying the mysteries of sleep and wakefulness and sleep disorders. But more recently, researchers have recognized that another necessary discipline that should be included in collaborative approaches to sleep-related issues is good ol’ fashioned mathematics. 

Working to Understand Sleep

Contributing to such collaborative approaches is Janet Best — a mathematician at Ohio State University whose research is funded by the National Science Foundation (NSF). Also affiliated with the University’s NSF-funded Mathematical Biosciences Institute, Best has spent the past 10 years studying sleep-wake cycles using mathematical models.

“To understand sleep, we try to reformulate biological questions in terms of mathematics, typically systems of differential equations,” she explained. “Sleep is both regular and random. It’s regular in that we go to sleep generally at the same time of day. The randomness occurs in infants who seem to have no pattern to their sleep cycles and in the variability of when we might wake up during the night. I’ve been investigating how neural structures in the brain affect the random and regular transitions between sleep and wake.”

By describing through equations the properties of neurons involved in sleep-wake brain circuitry, Best develops mathematical models that represent ways in which neurons interact and influence each other. She validates her models by checking their predictions against data that biologists gather in studies involving both humans and rats. (Surprisingly, baby rats’ sleep patterns go through similar changes as human infants’ sleep patterns, but it is not clear how similar adult rat sleep is to human sleep.) Once validated, Best’s models can be used to test ideas about sleep and wake patterns.

“The idea is to see how people sleep normally, so we can understand when things go wrong,” Best said. “Throughout the night we experience ‘bouts’ of sleep and wakefulness. There’s variability that we’re aware of, but actually even more variability is occurring – we only recall longer wake episodes. However, both short and long episodes occur, and that’s something I’m trying to understand. Experimentalists collect data on these wake/sleep bouts. Since the length of sleep and wake bouts and the transitions between them show some regular and some random behavior, the differential equations must capture both of these facets.”

A Personal Interest

Best became interested in studying sleep when — while working on her doctorate in mathematics — she was involved in a bike accident in which she sustained a serious head injury. After the accident, she began to experience simultaneous sleep and wakeful moments. In other words, while awake, she had dreams that were not daydreams. Also, after Best’s accident, her brain began to store memories and dreams in almost the same way, and so it became difficult for her to distinguish one from the other. The medical literature of the time, however, said her experience was impossible. 

“In 10 years, there have been a lot of changes in this field,” she said. “Ten years ago, the emphasis was on regular patterns. Now the random aspects of sleep are getting more attention. Models are now based on the real underlying physiology.”

Collaborative Approaches

Best and her collaborators work to develop such models based, in part, on collaborations with non-mathematicians. To this end, Best reads papers by biologists and neuroscientists that present new data and new ideas related to specific challenges in people’s sleep cycles. For example, a paper by a biologist or neuroscientist might present new data on a subgroup of people with a specific challenge in with their sleep cycles that Best may plug into her models. Best’s research also involves working directly with sleep/wake researchers who conduct experiments on rodents or who see patients clinically.

“You need a lot of interaction with biologists and medical scientists, and you have to have conversations with the people who generate the data,” Best said. “If I relied just on reading the papers, I would not be able to understand all of the underlying hypotheses and the ways in which the data was collected, and that could significantly affect how I formulate the mathematical models.”

Best’s research also benefits from her affiliation with Ohio State’s Mathematical Biosciences Institute, which hosts 12 workshops a year, drawing world-renowned bioscience experts and providing important opportunities for cross-fertilization between biologists and mathematicians.

Modeling the complexities of the brain

“The understanding of sleep-wake cycles can have enormous impact on developing a better knowledge of the dynamics of the brain and, in turn, how systems within an entire physiological organism interact and function,” said Mary Ann Horn, an NSF Division of Mathematical Sciences program director. “Research that involves collaboration between mathematical and biological scientists gives rise to results for which not only does the biology inform the modeling and analysis, but also spurs new mathematical developments as novel techniques are developed to address these challenging questions.”

 “It’s enormously difficult to figure out how the brain works,” Best said. “We’re talking about 200 million neurons, all this chemistry, hormones — so many variables. We have to infer how brains accomplish their tasks. And there are always multiple ways that a particular task can happen, so the challenge comes in teasing apart information, and in my case, building a good model that helps fill in the missing pieces.”

So far, models of sleep/wake cycles developed by Best and her collaborators indicate that the longer a “wake bout” lasts during the night, the less likely it is to be interrupted by sleep. But the models also indicate that the same pattern does not appear to apply to a “sleep bout” — which seems to be equally prone to interruption at any moment. In addition, the models have helped reveal the structure of the neuronal network affects the timing of the sleep/wake bouts. 

Findings such as these about quirky sleep phenomena may, bit by bit, help advance our understanding of the underlying sleep/wake mechanism — and thereby support the development of models of this mechanism. Ultimately, such models may help researchers develop insomnia treatments, effective remedies for medical condition-induced sleep disorders, or strategies to reduce jet lag more quickly. 

“There are a lot of data from sleep studies,” Best said, “but data by itself does not give understanding. To gain understanding, one must understand the underlying neural mechanisms. The sleep/wake field is growing very rapidly now and this is providing new data for us to interpret and understand. The mathematical analysis and the comparison with new data should enable us to formulate a new understanding of how sleep-wake functions.”

The researchers depicted in Behind the Scenes articles have been supported by the National Science Foundation, the federal agency charged with funding basic research and education across all fields of science and engineering. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. See the Behind the Scenes Archive.