Mathematical Models for Sleep-Wake Dynamics: Comparison of the Two-Process Model and a Mutual Inhibition Neuronal Model

Sleep is essential for the maintenance of the brain and the body, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the mutual inhibition of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations. This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the mutual inhibition model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is valuable because not only does it allow some features of the two-process model to be interpreted physiologically but it also means that knowledge gained from study of the two-process model can be used to inform understanding of the behaviour of the mutual inhibition model. This is important because the mutual inhibition model and its extensions are increasingly being used as a tool to understand a diverse range of sleep-wake phenomena such as the design of optimal shift-patterns, yet the values it uses for parameters associated with the circadian and homeostatic processes are very different from those that have been experimentally measured in the context of the two-process model.

The two-process model as a one-dimensional map As discussed in the Results section, the two-process model can be represented as a one dimensional map with discontinuities. The map is given by where G(T n 0 ) is defined as follows. For t ∈ (T n 0 , T n w ] where C(t) = sin(ω(t − α)); H ± 0 are the mean values for the upper/lower thresholds; µ is the upper asymptote; χ s and χ w are the time constants for the homeostatic process during sleep and wake; a, ω and α are the amplitude, frequency, phase shift, of the circadian process C(t).
In [27] a detailed analysis of piecewise-linear discontinuous maps is presented. It is argued that these represent the normal forms for many systems in the neighbourhood of the discontinuity and that three different bifurcation scenarios are observed in such systems. Of the three scenarios, the particular case that is relevant for the two-process model parameters used to match the PR model in this paper is the scenario labelled as period adding bifurcations. It is not yet clear whether the two other scenarios, which [27] label as period increment with coexistence of attractors and pure period increment scenarios, can also occur.
The sequence of bifurcations shown in Figure 6(e) is typical of period adding bifurcations. A more conventional way to present the bifurcation diagram is shown in Figure S1 where the values of T n 0 are plotted against the bifurcation parameter. The term 'period-adding' refers to the fact that the period of the iterated map changes as a function of the parameter. So, for example, when the number of daily sleep episodes changes from one to two, the map repeats itself after two iterations.
In the main body of the text it is highlighted that between parameter values where solutions with N daily sleep episodes and solutions with N + 1 daily sleep episodes there are solutions which alternate between N and N + 1 sleep episodes, as shown in Figure 7. The same sequence occurs in the PR model, as illustrated in Figure S2.
Yet another way of presenting the bifurcation diagram for the iterated map is to plot the length of the daily sleep episodes. This is shown in Figure S3. On this diagram is also plotted the mean daily total sleep. This shows that for the two-process model, as for the PR model, the mean total daily sleep is approximately independent of the homeostatic time constant.

The upper asymptote
In the Methods section, it is shown how to fit the parameter µ of the two-process model such that the homeostatic switching happens with the same timing as in the PR model. This fitting is dependent on the sleep-wake cycle and if χ varies, this sleep wake cycle will slightly vary and lead to different timings of the homeostat. In its turn, this would lead to a correction for the paremeter µ. In Figure S4 the dependence of the parameter µ is depicted for monophasic sleep.