About the Author

Mathematics of the Nap

By Cecilia Diniz Behn

Don’t wake the baby!

Though we all know that sleep is important, caring for a newborn or young child can bring a heightened sense of urgency to the need for it (for both child and caregiver). This situation also underscores the differences in sleep patterns at various ages. Adult humans typically sleep in one consolidated nighttime sleep episode, but human sleep is very different at other life stages. Infants sleep in a polyphasic pattern that involves multiple sleep episodes per day, and most children continue napping regularly until they reach school age. School-aged children generally have a single nighttime sleep episode that is qualitatively similar to that of adults, but the duration and timing of that sleep continue to evolve across adolescence.

In early childhood, the greatest changes in overall sleep behavior occur in daytime sleep. Napping is strongly tied to sleep need. Young children require 10-14 hours of sleep per every 24 hours, and the ability to nap represents a key adaptive strategy that allows them to fulfill part of this sleep need during the day. As sleep needs decrease during the toddler and preschool years, children transition from two naps to one nap, and then drop daytime naps altogether. The transition from napping to non-napping status usually occurs around age four or five, although this varies from child to child. However, maturing physiology is not the only factor mediating this transition. Sleep schedules for young children are often determined by caregivers, and these schedules may reflect both practical and cultural considerations. For example, schedules and staffing are often based on assumptions about napping requirements, and it is common to apply the same schedule to all children of a given age.

Based on the anatomy and physiology of the neuronal network governing sleep, we developed a physiologically-based mathematical model of sleep/wake regulation and are using it to investigate the mechanisms driving the transition from napping to non-napping behavior. This high-dimensional, differential equations-based model translates interactions in a network of specific neuronal populations to the dynamics of sleep/wake behavior. Network interactions are modulated by a circadian process that reflects changing sleep propensity over the 24-hour day, and a homeostatic process that reflects sleep propensity and increases with time awake. Biologically, the circadian drive is an approximately 24-hour rhythm that is generated by coordinated molecular clocks and entrains exactly to the 24-hour day. It is typically modeled as a self-sustaining oscillator that modulates the network on a 24-hour time scale. The biology of the homeostatic sleep drive has been more challenging to identify; however, slow wave activity (SWA) provides a reliable marker of homeostatic sleep drive and has enabled experimental characterization of this process. SWA is a feature of deep sleep marked by slow (0.5 to 4.0 Hertz), synchronized brain activity, and the growth and decay of SWA reflect sleep need and recovery, respectively. The homeostatic sleep drive is typically modeled by a variable that grows exponentially during wake and decays exponentially during sleep. In our recent work, the model homeostatic sleep drive \(h\) is governed by the following equation:

\[h' = \begin{cases}
        \frac{(H_{max}-h)}{\tau_{hw}} & \text{during wake}\\
       -\frac{h}{\tau_{hs}} & \text{during sleep,}

where the units are \(\%\) mean SWA and the maximum is \(H_{max}\), and time constants \(\tau_{hw}\) and \(\tau_{hs}\) are calibrated to the measured dynamics of adult SWA. Under baseline circadian and homeostatic modulation, our model describes a stereotypical pattern of adult human sleep/wake behavior with a single \(\sim\)8-hour nighttime sleep episode per day (see Figure 1).

Figure 1. Hypnograms summarize simulated sleep/wake behavior. 1a. Under baseline conditions (χ=1) of the circadian and homeostatic (% SWA) modulation, our model produces stereotypical adult human sleep/wake behavior with a single ∼8-hour nighttime sleep episode every 24 hours. 1b. For scaling parameter χ=0.58, the homeostatic modulation increases and decrease more quickly, resulting in two sleep episodes every 24 hours: a nap and a nighttime sleep. Black and white bars indicate periods of dark and light, respectively.

There is evidence that the time scales of the homeostatic sleep drive change over the course of development, with the need for sleep increasing more quickly in young children compared to adults. To model this effect, we introduced a scaling parameter \(\chi\) into the equation of the homeostatic sleep drive to modulate \(\tau_{hw}\) and \(\tau_{hs}\), the rates at which the homeostat increased and decreased, respectively. We found that faster dynamics of the homeostatic sleep drive interacted with the circadian modulation to produce two regular sleep episodes per day (see Figure 1B). To investigate the dependence of sleep/wake behavior on \(\chi\), we varied \(\chi\) and looked for qualitative changes in model behavior. We identified intervals of \(\chi\) values associated with distinct patterns of sleep/wake behavior, including one sleep episode per day, two sleep episodes per day, alternation between one and two sleep episodes per day, and higher-order patterns (see Figure 2). These simulations showed that the same network could produce many different patterns of behavior for different values of \(\chi\). However, what was occurring in the transition regions was not clear.

Figure 2. Patterns of simulated sleep/wake behavior vary with scaling parameter χ. Plotting simulated sleep (blue) and wake (white) behavior over four days for different values of χ reveals that patterns of behavior change with χ. Certain intervals of χ produce regular patterns of one or two sleep episodes per day. Between these intervals, χ values produce stable, higher-order patterns of sleep/wake behavior that contribute to the complexity of the transition from napping to non-napping status.

To better understand the dynamics of the transition regions, we numerically constructed a 1-dimensional discrete map that captured much of the dynamics of the full, high-dimensional model. We found that the \(\chi\)-dependence of these patterns exhibited an interesting mathematical structure suggesting that transitions from biphasic (napping) to monophasic (non-napping) sleep involve stable, higher-order patterns of behavior. These higher-order patterns may correspond to physiological states experienced by children, but it is unlikely that such patterns would be observed in practice. Instead, for a child in the transition region, caregivers’ efforts to establish a regular daily schedule would constantly perturb the child away from the stable higher-order behavior consistent with their physiology. Such a mismatch between physiology and actual sleep schedules may contribute to challenges associated with the transition from biphasic to monophasic sleep, including nighttime settling difficulties and effects on emotion processing, stress reactivity, and cognition.

To date, our analysis has focused primarily on the dependence of sleep/wake behavior to changes in the homeostatic sleep drive. However, the maturing physiology of the circadian system in young children likely also contributes to developmentally-mediated changes in sleep/wake behavior. Our current work aims to incorporate these changes into our sleep/wake network model in order to examine relative contributions of the circadian and homeostatic system to the pattern of sleep/wake behavior across the lifespan. We hope that the resulting novel insights into the changes that occur as children transition from napping to non-napping behavior will help to inform best practices for caregivers of young children and help everyone get a good night’s sleep.

The author presented this research as part of a minisymposium entitled "Novel Applications of Discrete Maps in Neuroscience" during the 2017 SIAM Conference on the Applications of Dynamical Systems, which took place in May in Snowbird, Utah.

Acknowledgments: This work was supported by the National Science Foundation DMS 1412571.

Further reading:
[1] Booth, V., Xique, I., & Diniz Behn, C.G. (2017). One-dimensional map for the circadian modulation of sleep in a sleep-wake regulatory network model for human sleep. SIAM J .App. Dyn. Syst., 16(12), 1086-1112.
[2] Jenni, O., & LeBourgeois, M. (2006). Understanding sleep-wake behavior and sleep disorders in children: the value of a model. Curr. Opin. Psych., 19,  282-287.
[3] Kalmbach, K., Booth, V., & Diniz Behn, C.G. (2017). REM sleep complicates period adding bifurcations from monophasic to polyphasic sleep behavior in a sleep-wake regulatory network model from human sleep, arXiv:1710.05494.

Cecilia Diniz Behn is an assistant professor in the Department of Applied Mathematics and Statistics at Colorado School of Mines. She also holds an appointment as an adjoint assistant professor in the Department of Pediatrics at the University of Colorado School of Medicine. Her research focuses on multiscale mathematical modeling with applications in metabolism, sleep, and circadian rhythms. Recent work investigating sleep in early childhood was done in collaboration with colleagues Victoria Booth (University of Michigan) and Monique LeBourgeois (University of Colorado), and graduate student Kelsey Kalmbach (Colorado School of Mines). Diniz, Behn, and Booth have also developed a freely-available module to introduce high school students to mathematical modeling using a simpler model of the interaction between circadian and homeostatic processes. The module includes guided exploration of the model using an interactive, online simulator.