During sleep, we experience a loss of consciousness coupled with decreased muscle tone and increased sensory thresholds. Yet sleep is not a uniform state. For most mammals, sleep involves alternations between the rapid eye movement (REM) sleep in which we have our most vivid dreams and non-rapid eye movement (NREM) sleep including deep slow wave sleep. These two sleep states are very different physiologically. NREM sleep is a state of reduced brain activity, consistent with notions of sleep as rest, and it has been found that slow-wave activity during NREM sleep is homeostatically regulated with respect to the duration of the preceding wake episode. During REM sleep on the other hand, brain metabolism is as high as during wake or even higher. REM is distinguished from both NREM and wake by an absence of muscle tone and a suspension of thermoregulation. The ultradian rhythm of REM/NREM cycling is often described as a 90-minute sleep cycle in humans, but actually the data is much more variable than that description suggests. In fact, many mathematical models of REM/NREM dynamics in mammals have been purely stochastic [1,5,6]. These stochastic models capture many features of the data but are phenomenological, not taking into account the biological substrate. Recent mathematical models focusing on the biological mechanisms underlying REM/NREM cycling have been deterministic [3,11]. This project aims to develop a model for REM/NREM dynamics within sleep/wake cycling that carefully combines deterministic and stochastic elements. We will build on recent neurophysiological models (such as [7,10,12]) by considering how the stochastic dynamics may arise naturally in the biological mechanisms. We will test the model against data used for stochastic models, such as the statistics of state transitions and durations. We will then use the model to explore some current hypotheses concerning the regulation of REM sleep such as that REM sleep propensity increases primarily during NREM sleep [2], that REM sleep propensity increases during both wakefulness and NREM sleep [4], that REM cycling drives the REM/NREM cycles [9], and that thermoregulation and energy management determine REM bout durations [8]. While the functions of sleep remain mysterious, progress is being made in understanding the mechanisms by which sleep and its substates are regulated. This project aims to contribute to that understanding with respect to REM/NREM cycling, providing insights that bring us closer to identifying the purpose of REM sleep. This project will use methods from dynamical systems and stochastic processes, some basic statistics, and numerical simulation. References [1] Bassi A, Vivaldi E, Ocampo-Garces A. The time course of the probability of transition into and out of REM sleep. Sleep 32(5):655-669, 2009. [2] Benington JH. Debating how REM sleep is regulated (and by what). J Sleep Res 11:29-33, 2002. [3] Diniz Behn C, Ananthasubramaniam A, Booth V. Contrasting existence and robustness of REM/non-REM cycling in physiologically based models of REM sleep regulatory networks. SIAM J App Dyn Sys 12(1):279-314, 2013. [4] Franken P. Long-term vs. short-term processes regulating REM sleep. J Sleep Res 11:17-28, 2002. [5] Gregory G and Cabeza R. A two-state stochastic model of REM sleep architecture in the rat. J Neurophysiol 88:2589-2597, 2002. [6] Kim JW, Lee J-S, Robinson PA, Jeong D-U. Markov analysis of sleep dynamics. Physical Review Letters 102:178104, 2009. [7] Kumar R, Bose A, Mallick BN. Mathematical model towards understanding the mechanism of neuronal regulation of wake-NREMS-REMS states. PLOS One 7(8):e42059, 2012. [8] Kumar VM. Body temperature and sleep: are they controlled by the same mechanism? Sleep and Biol Rhythms 2:103-124, 2004. [9] LeBon O. Which theories on sleep ultradian cycling are favored by the positive links found between the number of cycles and REMS? Biol Rhythm Res DOI:10.1080/09291016.2012.721590. [10] Phillips AJK, Robinson PA. A quantitative model of sleep-wake dynamics based on the physiology of the brainstem ascending arousal system. J Biol Rhythms 22(2):167-179, 2007. [11] Phillips AJK, Robinson PA, Klerman EB. Arousal state feedback as a potential physiological generator of the ultradian REM/NREM sleep cycle. J Theoretical Biol 319:75-87, 2013. [12] Rempe M, Best J, Terman D. A mathematical model of the sleep/wake cycle. J Math Biol 60:615-644, 2010.