Analyzing Perturbations of Sleep-Wake Dynamics Using Bifurcation Theory and Circle Maps

Abstract

Sleep patterns and timing can be influenced by gradual developmental changes or more acute perturbations such as sleep deprivation. In this thesis, we employ physiologically-based mathematical models of neural sleep-wake regulatory networks to analyze 1) biological factors that influence the developmentally-mediated transition from polyphasic to monophasic sleep, and 2) recovery responses to sleep deprivation.

In the first project, we utilize a sleep-wake flip-flop (SWFF) model to analyze how developmentally-mediated transitions in sleep-wake dynamics are affected by homeostatic and circadian modulation. Specifically, we show that varying the rates at which the homeostatic sleep drive evolves leads to the transition from polyphasic to monophasic sleep in a period adding bifurcation structure of the average number of sleeps per day. We numerically construct circle maps that capture sleep onset phases, and find that saddle-node and border collision bifurcations in these maps result in the gain or loss of stable solutions. Moreover, we show that imposing a steeper circadian temporal profile reduces the variability in sleep patterns and promotes the persistence of specific sleep behaviors during the polyphasic to monophasic transition.

In the second project, we consider a physiologically-based model that produces wake, rapid eye movement (REM) and non-REM (NREM) sleep states to investigate how NREM-REM cycling influences the types of sleep patterns obtained under a similar homeostatic variation. We conduct a computationally-based analysis, including numerical construction of sleep onset circle maps, and find a disrupted, non-monotonic period adding bifurcation structure in the average number of sleeps per day. Our analysis shows that NREM-REM cycling, resulting in more complex sleep onset map structures in this three-state model, allows for both higher order cycles and bistability to occur. The structure of the circle map reflects variation in the number of REM bouts per sleep, and saddle-node, border collision and period-doubling bifurcations causing the transition to different sleep patterns, whose characteristics can be highly variable due to the homeostatic dynamics, ultradian dynamics of NREM-REM cycling and their interactions.

In the third project, we focus on the transition from napping (biphasic) to non-napping (monophasic) sleep behavior observed in early childhood (between ages of 2 and 5 years). Using the SWFF model, we set values for the parameters governing the evolution of the homeostatic sleep drive to data estimated in preschool children. We then identify other model parameters to generate the timing of experimentally measured sleep patterns in 2 and 5 year old children. We show that the homeostatic parameters and the sensitivity of the model to the sleep homeostat are sufficient for the generation of the transition from napping to non-napping sleep behaviors. We consider different variations of these parameters across development that lead to distinct sleep transition behaviors that may account for interindividual differences. Finally, we investigate the effect of forced light schedules that promote napping or maintain wakefulness during daytime on the transition from biphasic to consolidated sleep.

In the last project, we use the three-state model to show that sleep onset circle maps can be employed to predict recovery from acute sleep deprivation. We compare map predictions with experimental data and numerical simulations of the model when behaviorally-gated light schedules are incorporated. The map predictions reproduce trends in the durations of recovery sleep observed in experimental data and simulations of sleep deprivation using the full model, thus validating its use as a predictive method.