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The authors have declared that no competing interests exist.

Conceived and designed the experiments: ACS DJD GD. Performed the experiments: ACS. Analyzed the data: ACS. Contributed reagents/materials/analysis tools: ACS. Wrote the paper: ACS DJD GD.

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.

Reduced or mis-timed sleep is increasingly recognized as presenting a significant health risk and has been correlated with increases in a diverse range of medical problems including all-cause mortality, cardio-vascular disease, diabetes and impaired vigilance and cognition

A review of early mathematical models of sleep is given in

Advances in neurophysiology have led to a proliferation of models that aim to extend the two-process model to a more physiological setting

In

First we give a summary of the main features of the two-process model and the PR model.

The two-process model considers a homeostatic pressure

The parameter

The switch between sleep and wake occurs when

With the parameters as in

At the core of the PR model are two groups of neurons: mono-aminergic (MA) neurons in the ascending arousal system that promote wake and neurons based in the ventro-lateral pre-optic (VLPO) area of the hypothalamus that promote sleep. Phillips and Robinson model the interaction between the MA and the VLPO as mutually inhibitory. In the absence of further effects, this would mean that the model would either stay in a state with the MA active (wake) or in a state with the VLPO active (sleep) and no switching between the states would occur. Switching between sleep and wake occurs because the model also includes a drive to the VLPO that is time dependent and consists of two components: a circadian drive,

(a) Diagrammatic description of the PR model showing the links between the VLPO, MA, the homeostatic and the circadian processes. (b), (c) and (d) show typical timeseries for the level of the homeostat,

The neurons are modelled at a population level and are represented by their mean cell body potential relative to rest,

The neuronal dynamics are represented by

The homeostatic component of the drive,

Typical results produced by the PR model are shown in

As recognised in

The lower threshold is therefore dependent on the mean drive to the VLPO and the threshold firing rate. The difference between the thresholds in the two-process model,

Using the standard parameters for the PR model, only a small part of the firing function (5) is used. This is illustrated in

(a) The dashed (black) line shows the firing function given by

Typical graphs of

(a) Comparison of the PR switch model with the two-process model. (b) Comparison of the PR model with the two-process model. Crosses show the two-process model; solid line the PR model and (blue) dashed line the PR switch model.

In

The link between the PR model and the two-process model not only gives us a physiological interpretation of the thresholds in the two-process model, it also allows us to gain a greater insight into the dynamics of the PR model, enabling understanding developed in the context of the two-process model to be interpreted in the physiological setting of the PR model. In the next sections, two different examples are discussed.

It is well-known that the two-process model can show a range of different sleep-wake cycles, including cycles that have multiple sleep episodes each day, see

First we introduce the one-dimensional map. Consider the two-process model and suppose we start on the upper threshold, at time

(a) A single trajectory of the two-process model showing successive times of sleep onset. (b) Trajectories of the two-process model for different initial sleep onset times. Each different sleep onset time results in a different sequence,

Phrasing the two-process model in these terms illustrates that it can be represented as a one-dimensional map. Probably the most well-known example of such maps is the logistic map

For the value of the clearance parameter

Using the two-process model with parameters as indicated in (15). Figures (a)–(d) give sleep-wake cycles for different values of the homeostatic time constant

The sleep-wake pattern for varying

Solutions of the two-process model showing periodicity on the period of two days. (a)

In

The upper and lower thresholds are moved simultaneously via

In

The quantitative agreement with

Sleep deprivation experiments involve keeping subjects awake for an extended period of time during which cognitive and behavioural tests are undertaken to measure sleepiness and performance. One measure of sleepiness is the Karolinska Sleepiness Scale (KSS) score

Wake effort in

(a) Sleep-wake cycle showing the MA firing rate

In sleep deprivation experiments, subjects are prevented from falling asleep at

where

In the two-process model, acute sleep deprivation is modelled as a continued increase in the homeostatic pressure. In

(a) The two-process model, showing the typical trajectory of the homeostatic pressure during a sleep deprivation experiment. Using the wake effort concept of

This resulting wake effort computed from the two-process model is shown by the solid line in

The close to linear relationship (the quadratic term has a very small coefficient) between wake effort in the PR model and

The shape of the bistable region in the

The strengths of the two-process model have been its inclusion of the two fundamental processes that are believed to regulate the sleep-wake cycle along with its graphical simplicity. This has meant that it has been used extensively as a tool to understand the behaviour of the sleep-wake cycle, design experiments and interpret data

The PR model was developed with the same two governing processes in mind, but introduced some physiological basis for the switching that occurs between wake and sleep. In recent years, this model has been extensively tested in a range of scenarios, some of which depend on the fast dynamics within the model, like the role of disturbances during sleep

Here we have shown that the slow dynamics of the PR model can be explicitly related to the two-process model, which provides new perspectives on both the two-process model and the PR model. Using this relationship, new insight into the meaning of the two-process model has been gained. Specifically, the distance between the thresholds is related to the degree to which the MA inhibits the VLPO during wake and the values of the thresholds are related to the parameters associated with the modelling of the firing rates

Motivated by the strong relationship between the two-process model and the slow dynamics of the PR model, we have re-visited the two-process model in order to gain insight on the dynamics of the PR model. By using the fact that the two-process model can be represented as a one-dimensional map with discontinuities we are able to interpret the transitions from monophasic to polyphasic sleep as grazing bifurcations. This provides the dynamical underpinning for the observation that the PR model gives a systematic framework which encompasses many different mammalian species and confirms the hypothesis of

Varying the homeostatic time constant as shown in

The grazing bifurcations have been shown to occur as the clearance parameter

The two-process model has been compared with sleep deprivation experiments by assuming that the upper threshold is no longer present and that the sleep pressure continues to increase, with sleepiness linearly related to the difference between the homeostat and the circadian process. Here, we have demonstrated that the notion of ‘wake effort’ introduced in

Similarly, one could also imagine a ‘sleep effort’ that would be required to keep the model asleep when it would naturally wake. This could be achieved by reducing the lower threshold in the two-process model or, equivalently, decreasing the stimulation to the MA,

The equivalence between the PR model on the slow timescale and the two-process model is exact when the firing function is a hard switch, but when the firing function is sigmoidal the equivalence is more subtle. This is because, in the PR model, the upper/lower asymptotes of the homeostatic process are modelled as a a function of

However, the fact that implicit in the PR model is a non-constant asymptotic value for the homeostatic process has wider implications. Sleep deprivation experiments tend to show a leveling off of psychomotor vigilance test (PVT) scores over a period of a few days, similar to the levelling off seen in the wake effort shown in

The asymptotes and therefore the wake effort in the PR model are sensitive to the particular choice of the firing function and the functional dependence of the upper asymptote on

In order to better understand sleep/wake regulation it is essential that models that incorporate neurophysiology are developed, analysed and used. However, as models become more complex two problems arise. Firstly they become difficult to analyse systematically, with large numbers of numerical simulations becoming the principle method used to establish the behaviour of the system. Secondly, there is a proliferation of parameters which cannot be easily determined experimentally. One consequence is that it becomes difficult to establish the relative merits of different models. By demonstrating that the two-process model and the PR model are essentially the same for sleep-wake phenomena on the slow time-scale of hours we have not only gained insight on the interpretation of both models but also established the mechanism for transitions between different patterns of sleep and wake in the PR model. This link also suggests some interesting avenues for future extensions of the PR model based on recent insights and research on the two-process and related models.

The equations for the PR switch model are

Since

During wake, these have solution

During sleep, these have solution

Transitions between wake and sleep when

By comparison with

On the slow manifold, the PR model is

The values of

Parameter | PR | PR switch |

- | ||

The sleep-wake cycle corresponds to slowly changing

• First the identification between the threshold values and the saddle node bifurcations in the PR model is made, leading to

• Numerically integrating the PR model during monophasic sleep results in trajectories for the homeostat that increase to a maximum during wake and decrease to a minimum during sleep, this gives values for

Hence, taking

• One can do a similar matching for the decreasing

By integrating the PR model with the typical parameter values listed in

Going back to the PR switch model and its link to the two-process model, we can now find expressions for the parameters

• Comparing the expression for

• The relation for

• Considering

Hence for the typical values of the PR parameters listed in

It is also necessary to take

Following

Using the explicit relationships between the parameters in the PR model and the two-process model, the moving of the threshold in the two-process model corresponds to a modified value for

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