Modelling changes in sleep timing and duration across the lifespan: Changes in circadian rhythmicity or sleep homeostasis?  Anne C. Skeldon, Gianne Derks, Derk-Jan Dijk  Sleep Medicine Reviews  Volume 28, Pages 96-107 (August 2016) DOI: 10.1016/j.smrv.2015.05.011 Copyright © 2015 The Authors Terms and Conditions

Fig. 1 Schematic presentation of the Phillips–Robinson model. The model consists of a core flip-flop system of mutually inhibitory sleep active (ventro-lateral preoptic area, VLPO) and wake active neurons (monaminergic (MA) neurons). The firing rate of wake active neurons (QMA) leads to an increase in homeostatic sleep pressure (H), where the parameter μ measures the rate of homeostatic rise during wake and is therefore a measure of sleep need. The VLPO is affected by a sleep drive that has homeostatic and circadian (C) components with strengths νvh and νvc respectively. Hence, νvh is a measure of the sensitivity to the homeostatic process, and νvc is a measure of the circadian amplitude. The circadian pacemaker's endogenous period Tc is 24.2 h but is synchronised to the external light–dark cycle via photoreceptors. Sleep Medicine Reviews 2016 28, 96-107DOI: (10.1016/j.smrv.2015.05.011) Copyright © 2015 The Authors Terms and Conditions

Fig. 2 Typical time courses resulting from simulating the Phillips–Robinson model. Neuronal firing rates of the monoaminergic (MA) nuclei and the ventro-lateral preoptic area (VLPO) are given by QMA and QVLPO respectively. During wake, the firing rate of the MA nuclei is high and low for the VLPO; whereas during sleep, firing rates are high for VLPO and low for MA (see Fig. 2 (a)). Wake is defined by a MA nuclei firing rate above a fixed threshold value. Timing of sleep is indicated by the shaded region. A flip-flop switching between wake and sleep occurs because a ‘drive’, DVLPO, see (d), consisting of homeostatic and circadian components, see (b), provides a time dependent input to the VLPO. The homeostatic sleep drive is νvhH, where H is the homeostatic sleep pressure and νvh is a constant which measures the sensitivity to the homeostatic process. The homeostatic sleep pressure is modelled in a similar way to the two-process model, with homeostatic sleep pressure (mostly) increasing during wakefulness and decreasing during sleep (the switch from wake to sleep and from sleep to wake occur within a few minutes of the maximum and minimum values of the homeostatic sleep pressure respectively). A distinctive feature of the Phillips–Robinson model is that the increase of the homeostatic sleep pressure during wakefulness is directly linked to high MA nuclei activity and is governed by two factors: the firing rate in the MA nuclei, and the parameter μ that measures the rate of homeostatic rise during wake. The circadian component of the drive is νvcC, where C is the output from the circadian pacemaker. As in [90], the circadian component of the drive is modelled as the approximately sinusoidal output from a van der Pol oscillator representing the endogenous circadian pacemaker with an intrinsic period Tc. The magnitude of the effect of the circadian signal on the VLPO is given by the parameter νvc and the phase of the circadian signal is determined by the interaction of the sleep-wake cycle with the light–dark cycle. The external light–dark cycle is modelled as a sinusoidal input where, to be specific, we have illustrated model behaviour with a maximal intensity (10000 lux) occurring at a clock time of noon, and minimum intensity (0 lux) at midnight. The external light–dark cycle acts as a forcing term on the endogenous circadian rhythm, through the interaction of light with photoreceptors. The effect of the light is to provide a strong pulse at the change from sleep to wake (c) that both adjusts the phase and entrains the period of the pacemaker to 24 h. The size of this pulse is dependent on the intensity of light at time of waking. The parameter δVLPO represents a background drive to the VLPO and takes the fixed value of 10.2. In addition to the drive to the VLPO, there is a drive to the MA nuclei, DMA motivated by the action of cholinergic and orexinergic nuclei on the MA nuclei. When considering age-related changes to sleep, Phillips and Robinson keep DMA fixed, although a time dependent DMA has been used to explain changes in sleepiness in sleep deprivation experiments [84,85] and to model the role of orexin in narcolepsy [98]. Sleep Medicine Reviews 2016 28, 96-107DOI: (10.1016/j.smrv.2015.05.011) Copyright © 2015 The Authors Terms and Conditions

Fig. 3 Sleep timing and sleep pressure as a function of model parameters. Left hand panels show sleep timing as a function of the parameters. Sleep times are shaded in blue with the timing of the circadian minimum () and the timing of mid-sleep () both marked. The right hand panels show the corresponding changes in sleep pressure, H (shaded magenta region) and sleep duration (). Panels (a) and (b) show changes for the parameter νvh that reflects sensitivity to the homeostatic process (ability to activate neuronal systems generating sleep). Panels (c) and (d) show changes for the parameter μ that describes the rate of homeostatic rise during wake (sleep need). Panels (e) and (f) show the effect of changing the circadian amplitude, νvc. Panels (g) and (h) show the effect of changing the period of the circadian pacemaker, Tc. Sleep Medicine Reviews 2016 28, 96-107DOI: (10.1016/j.smrv.2015.05.011) Copyright © 2015 The Authors Terms and Conditions

Fig. 4 Effects on sleep timing and duration of simultaneous changes in the rate of homeostatic rise during wake, μ, and the circadian amplitude νvc. Data for sleep duration and mid-sleep on free days (MSF) across the lifespan are given in [4] and [3] respectively and are reproduced in panel (a). These show a monotonic decrease in sleep duration and an ‘n’-shaped curve for MSF values, peaking at around age 20. The same changes in sleep duration and MSF can be obtained from the extended Phillips–Robinson model if the two parameters μ and νvc, representing rate of homeostatic rise during wake and the circadian amplitude respectively, both reduce with age, as shown in panel (b). The derivation of the relationship between μ and νvc and age is explained below with reference to panels (c) and (d). Running a simulation of the model results in a sleep-wake cycle with a sleep duration and MSF that is dependent on the values of μ and νvc. Contour plots for sleep duration and MSF as a function of μ and νvc are shown in panels (c) and (d) respectively. Each age shown in panel (a) corresponds to a particular sleep duration and MSF. Taking each age in turn, the corresponding values of (μ,νvc) that result in the same value of sleep duration and MSF in the model can be found—the black crosses on panels (c) and (d) show the positions of these points. This then gives the direct relationship between μ and νvc and age shown in panel (b). The red cross on panels (c) and (d) indicate the position of the ‘standard’ parameters used in the extended Phillips–Robinson model. Sleep Medicine Reviews 2016 28, 96-107DOI: (10.1016/j.smrv.2015.05.011) Copyright © 2015 The Authors Terms and Conditions

Fig. 5 The effect of perturbations on the Phillips–Robinson model. The mutual inhibition between the ventro-lateral preoptic area (VLPO) and the monoaminergic (MA) nuclei results in hysteresis between wake and sleep states, leading to a region where both sleep and wake states exist. This can be seen by plotting the firing rate of the MA nuclei, QMA, against the sleep drive, as in Fig. 5 (a). As the drive to the VLPO, DVLPO, oscillates the model switches from wake to sleep and back again, as indicated by the arrows. The region where both wake and sleep states occur for the same drive value is shown in red. Plotting the Phillips–Robinson model in the form of the two-process model illustrates that the hysteretc region corresponds to the region between the upper and lower thresholds, see Fig. 5 (b). Sleep in the hysteretic region is vulnerable, in the sense that a sufficiently strong stimulus will result in spontaneous waking. Since the time spent in the hysteretic region is dependent on the parameters, for different parameter values the same perturbation can have a significantly different effect. This is illustrated in (c) and (d) where the effect of applying a stimulus that momentarily increases the firing rate of the MA nuclei at 04:30 h every morning is shown for two different values of the parameter μ that describes the rate of homeostatic rise during wake. One value corresponds to the ‘normal’ value (μ=4.4) and the other value corresponds to an ‘older’ value (μ=3.8). In the ‘normal’ case, the model rapidly falls back asleep following the perturbation. For the reduced value, the perturbation is large enough to shift the model from the sleep to the wake state and no immediate return to sleep is possible. Sleep Medicine Reviews 2016 28, 96-107DOI: (10.1016/j.smrv.2015.05.011) Copyright © 2015 The Authors Terms and Conditions

Fig. 6 Vulnerability of sleep and ageing. As described in Fig. 5, the flip-flop switch results in sleep at the end of the night being ‘vulnerable’ in the sense that perturbations can result in early awakening. In panel (a), the shaded region shows sleep times as a function of the parameter μ, the rate of homeostatic rise during wake. This is the same as the shaded region in Fig. 3 (a), but now the vulnerable sleep region is distinguished in red. The size of the sleep vulnerability region increases as μ decreases. The average mid-sleep on free days (MSF) is also marked (). The presence of the sleep vulnerability region means that when simulations are performed with random perturbations to the firing rate of the monoaminergic (MA) neurons, QMA, the model frequently ‘wakes up’ earlier than when no perturbations are applied. This earlier wake time results in earlier light exposure, shifting the phase of the circadian clock, and resulting in earlier bedtimes. The inclusion of random perturbations means that each simulation yields slightly different results. In panel (b), grey crosses indicate the results of 50 simulations for each of a sequence of values of μ: the solid grey line indicates the mean bedtimes and wake times. For comparison, the unperturbed sleep region is shown in blue. The effect of perturbation is more pronounced for smaller μ because of the increased size of the sleep vulnerability region. Consequently, the shape of the MSF () is no longer monotonic. Since the computed MSF now has an ‘n’-shape like the data in [3], the computed MSF data can be mapped to the measured MSF data (black points in panel (c)), resulting in the fit (black line) shown in panel (c). This mapping then gives a relationship between the value of μ and age, as shown in panel (d). This suggests that, if perturbations are included, age-related change in the MSF can be explained by reducing the single parameter μ. Sleep Medicine Reviews 2016 28, 96-107DOI: (10.1016/j.smrv.2015.05.011) Copyright © 2015 The Authors Terms and Conditions