The two-process model considers a homeostatic pressure that decreases exponentially during sleep, (1)and increases during wake, (2)

The parameter is known as the ‘upper asymptote’ [13], [ 14], this is the value that the homeostatic pressure would reach if no switch to sleep occurred. Similarly there is a ‘lower asymptote’ of zero. Switching between wake and sleep occurs when the homeostatic pressure reaches an upper threshold, , that consists of a mean value modulated by a circadian process , (3)

The switch between sleep and wake occurs when reaches a lower threshold, , (4)where is a periodic function of period 24 hours. In the simplest cases but more complicated forms that include higher harmonics, such as have also been used [11]. Typical results of this model illustrating its rich dynamics are shown in Figure 1.

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, , and a homeostatic drive . The structure of the PR model is shown in Figure 2(a).

The neurons are modelled at a population level and are represented by their mean cell body potential relative to rest, for , where represents the VLPO group and represents the MA. The potential is related to the firing rates of the neurons by the firing function, , (5)where is the maximum firing rate and is the mean firing threshold relative to resting. The function is a sigmoid function, which is close to zero for all negative values of and then saturates exponentially fast to .

The neuronal dynamics are represented by (6)where the drive to the VLPO, and to the MA, are given by

The homeostatic component of the drive, is modelled by (7)and the circadian drive, , is approximated by where and is a phase shift that specifies the distance from the circadian maximum. Typically, is chosen so that the switch from sleep to wake occurs at an appropriate clock time.

Typical results produced by the PR model are shown in Figure 2(b)–(d). During wake, the firing rate of the MA neurons is high (), that of the VLPO is low and the homeostatic pressure tends to increase, while during sleep the firing rate of the MA neurons is low (), that of the VLPO is high and the homeostatic pressure tends to decrease. Note that in the PR model switching between wake and sleep is defined to occur when reaches the threshold value of one; this differs from the timing of the maximum and minimum homeostatic pressure by a few minutes. Obviously, the exact choice of the threshold value does not play an important role in the dynamics of the system, but does change the regions that are labelled as sleep or wake.

As recognised in [23], since changes in neuronal potentials happen much faster than changes associated with the homeostatic pressure, , there is a strong separation of timescales in the PR model. This strong separation of timescales means that the dynamics of the PR model is well approximated by two separate models: one on the ‘slow’ timescale that is appropriate when considering changes on the timescale of the circadian and homeostatic processes such as the timings of sleep and wake; and the other, the ‘fast’ timescale, which is appropriate when considering changes on the timescale of the neuronal potentials such as the response to a night time disturbance. If the firing switching function given in equation (5) in the PR model is replaced by a hard switch, (8)where is the mean maximum firing rate of the neuronal population and is the value at which the switch occurs, we show in the Methods section that the parameters for the slow dynamics of the PR model with a switch can be exactly mapped to parameter values in the two-process model, specifically, (9)

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, can then be interpreted physiologically as the amount by which the MA inhibits the firing of the VLPO during wake. This makes intuitive sense: there is hysteresis in the switch between wake and sleep because of the mutual inhibition between the MA and the VLPO. In the wake state, the VLPO requires a large drive to fire to counteract the inhibitory effects of the MA. Once in the sleep state, less drive is needed to maintain firing because the MA is quiescent.

Using the standard parameters for the PR model, only a small part of the firing function (5) is used. This is illustrated in Figure 3(a), where the firing function is shown by the dashed line and the typical range of values for is shown by the thick (red) line. We show in the Methods section that there is a systematic way to relate the parameters for the original PR model to equivalent parameters for the two-process model that retain the timings and values at the extrema of the homeostat. In keeping with the fact that the mean firing rate across the neuronal population is much less than the maximum possible firing rate , the value for is significantly less than but close to the mean firing rate across the population in the PR model: in fact the actual firing function needed in the PR-switch model is shown by the blue line in Figure 3(b).

Typical graphs of and for both the original PR model and the PR switch model are shown in Figure 3(c)–(e) demonstrating the close agreement between the two cases. Graphs comparing timeseries computed from the two-process model and numerical integrations of the corresponding PR/PR switch model are shown in Figure 4. The extremely good agreement of the two models is a result of the very large disparity in timescales between the fast and slow systems. Consequently, solutions of the PR model converge to solutions on the slow manifold on the timescale of minutes. Once on the slow manifold, solving the PR model is essentially equivalent to solving the two-process model.

In [31] it was recognised that the PR model could be plotted in a similar way to the two-process model, but the explicit connection between parameters was not made. It is stated that a key difference is that in the two-process model the value of remains between the thresholds at all times, as in Figure 1. However, we note that this could be regarded as a matter of parameter choice rather than a fundamental difference between the two models: whether the two-process model remains between the thresholds depends on the relative gradients of the circadian and homeostatic processes at each wake/sleep or sleep/wake transition. Figure 4 shows that, with the PR parameters used to model sleep regulation in humans, the two thresholds in the two-process model are very close, hence the circadian oscillation is the dominant sleep regulator and the two thresholds merge almost into one.

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 Figure 1(a), and cycles that have a period greater than one day, see Figure 1(c). Indeed in [10], the authors postulate that the two-process model can explain the polyphasic sleep of many animals. In [24], it is shown that the sleep-wake cycles of many different mammals can be understood by varying two parameters in the PR model: the homeostatic time constant and the constant component to the VLPO drive, . In the previous sections, we have demonstrated how the parameters of the PR model relate to those of the two-process model, specifically, the homeostatic time constant is present in both models and varying the drive to the VLPO corresponds to varying the upper and lower thresholds without changing the distance between them. In [29], [ 30] it is shown that the two-process model can be understood as a one-dimensional map with discontinuities. In this section, we use this map to show how the observations in [24] and the postulate in [10] are linked and clarify how the transition between different numbers of daily sleep episodes occurs.

First we introduce the one-dimensional map. Consider the two-process model and suppose we start on the upper threshold, at time , where the model switches from wake to sleep. The dynamics of the two-process model takes this starting point and, propagating it forward through one sleep and one wake episode, results in the next wake to sleep time, , and then through a further sleep-wake episode to and so on, generating a sequence of sleep onset times . This is illustrated for in Figure 5(a). Different starting values generate different sequences of sleep times, as illustrated in Figure 5(b). For the parameter values chosen here, all sequences converge rapidly to the same monophasic periodic cycle. A graphical way of understanding this sequence is to plot modulo 1 day against modulo 1 day (the first return map). For any particular starting value, the sequence of iterates can then be found by drawing the cobweb diagram, as shown in Figure 5(d). A monophasic sleep pattern corresponds to modulo 1 day and so corresponds to the intersection of the diagonal line with the map. The fact that the sequences converge rapidly is related to the fact that the gradient of the map is close to zero for most values of . This rapid convergence means that a temporary change to timing of sleep will revert to the regular sleep-wake cycle within a few days.

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 [32] which has been widely used to show that simple rules can lead to very complex dynamics. A distinctive feature of the two-process model is the fact that the map contains a discontinuity. For the parameter values shown in Figure 5(d) this discontinuity occurs at The discontinuity is a consequence of the fact that there exist neighbouring starting values that lead to trajectories that follow very different paths. These occur whenever there are points that result in trajectories that become tangent to the thresholds. For example, starting at the first sleep just misses the wake threshold at so remains asleep until resulting in a sequence , as shown in Figures 5(b) and (c); whereas starting at the nearby value of the trajectory hits, rather than misses, the sleep threshold and the resulting sequence is .

For the value of the clearance parameter used in Figure 5, the discontinuity does not have a significant impact on the dynamics and all trajectories converge rapidly to the same periodic cycle. However, the presence of the discontinuity is key to understanding the transition from monophasic to polyphasic sleep. This is illustrated in Figure 6(a)–(d), where a sequence of converged solutions to the two-process model are shown for decreasing . For the sleep-wake cycle is monophasic, but in the wake episode the trajectory comes close to, but does not touch, the upper threshold (Figure 6(a)). If distance from the upper threshold is a measure of sleepiness during wake, this would correspond to a dip in alertness. If is reduced further, say to as shown in Figure 6(b), then the wake trajectory does not only come close to, it touches the upper threshold resulting in a short nap and a sleep-wake cycle that is bi-phasic with one longer sleep and one short sleep. Decreasing further results in a sequence of further tangencies each of which adds one additional sleep-wake episode. Such transitions are known as grazing bifurcations, tangent bifurcations, or border collision bifurcations and are characteristic of one-dimensional maps with discontinuities [33]–[35]. In the return map, a grazing bifurcation occurs when the discontinuity in the map coincides with the diagonal line. They are responsible for period-adding transitions in the context of electronic circuits and here, we see, are responsible for sleep-episode-adding transitions. Such transitions have also been observed and analysed using one-dimensional maps in the context of understanding the dynamics of neurons [36], [ 37].

The sleep-wake pattern for varying is shown in Figure 6(e). For larger values of there is one episode of sleep each day: the model falls asleep exactly once and always at roughly the same time (). A grazing bifurcation occurs at around and results in a region between where sleep is bi-phasic with one longer and one shorter sleep each day (). A succession of further grazing bifurcations take place as is reduced, resulting in increasing numbers of daily sleep episodes. From Figure 6(e) we see there are intermediate regions between each value of . For example, between the monophasic and biphasic region there is a small region around where the sleep pattern has a period of two days. This corresponds to a region where a grazing bifurcation has taken place, causing an extra sleep period on one day, but this extra sleep period is enough to mean that no additional sleep is needed on the following day. The sleep wake trajectory in this case is shown in Figure 7(a). Similar behaviour is seen at each transition between different numbers of daily sleep episodes and is characteristic of such transitions in one-dimensional discontinuous maps [38]: this is illustrated for the transition between two and three sleep episodes in Figures 7(b) and is a similar pattern of sleep to that shown in Figure 1(a) using parameters as in [10]. In fact, as shown for one-dimensional discontinuous maps in [38], the situation is even more complicated: in Figure 1 of [27] the first few layers of an infinite adding scheme are set out. This shows that, for example, the sequence of transitions from sleeping once a day to sleeping twice a day is , etc. Further discussion of the map is given in the Information S1.

In [24], the behaviour of the PR model is examined both as the time constant and the mean drive to the VLPO, are varied. Our parameter equivalences between the PR model and the two-process model (9) show that increasing is equivalent to increasing the upper and lower thresholds without changing the distance between them. One can then deduce for the two-process model that for low , the homeostat will never reach the lower threshold and no wake will occur. Similarly, for high no sleep will occur. For large values of ( greater than approximately the amount of daily sleep varies approximately linearly with the mean drive to the VLPO as shown in Figure 8(a) and observed in [24]. As seen before, the sleep-wake cycle is monophasic and is largely independent of in this range. The actual transition between monophasic sleep and no sleep (or no wake) occurs through grazing bifurcations, where this time the grazing bifurcations result in periodic cycles that have wake (sleep) episodes of greater than 24 hours: examples of such cycles are evident in Figure 8(a) at the extremes of the values of that are shown. For smaller values of where polyphasic sleep exists, varying shows that, as the no sleep (or no wake) threshold are approached, grazing bifurcations result in ever decreasing numbers of sleep (wake) episodes until no sleep (no wake) occurs, see Figure 8(b).

In [24], it was shown that the sleep of many mammalian species could be understood in the context of the PR model by varying just two, physiologically plausible, parameters: and . Their results show: a sequence of transitions from monophasic to polyphasic sleep as the time constant is reduced but where total sleep daily sleep remains approximately constant; for fixed and varying mean drive to the VLPO a sequence of transitions from a state with no wake to a state with no sleep. By using the relationship between the PR model and the two-process model we see that reducing results in a sequence of transitions from monophasic to polyphasic sleep through grazing bifurcations that successively add sleep episodes; at the transition between episodes of sleep and episodes of sleep, there are regions where sleep alternates between and daily episodes (examples of such trajectories for the PR model are shown in Information S1). The identified parameter equivalences show that changing the mean drive to the VLPO is equivalent to simultaneously shifting the upper and lower thresholds of the two-process model. The relation between the PR model and the two-process model shows how this inevitably leads to grazing bifurcations and ultimately cycles with either no sleep or no wake.

The quantitative agreement with [24] is close, but not exact: this is because we have chosen a fixed value for , the upper asymptote, in the two-process model, the value to match the PR model for Varying in the PR model results in a small change to the precise region of the switching function that is used, which in turn induces some change in the value of . Since this results in some dependence of on in the equivalent two-process model. One consequence is that the switch from monophasic sleep to biphasic sleep occurs at around for the two-process model instead of for the PR model. More details can be found in the Information S1.

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 [39]. In [25], the concept of ‘wake effort’ is introduced for the PR model and good agreement between wake effort and experimental data on KSS scores is found. Wake effort corresponds to a change in the drive to the MA and is interpreted as a need to provide the MA with greater stimulation in order to maintain wake. Here, we show how this can be re-interpreted in the context of the two-process model.

Wake effort in [25] is presented by considering the graph of the MA firing rate (or equivalently, ), against the drive to the VLPO, . In a regular sleep-wake cycle, follows a hysteretic loop, see Figure 9(a), where the transition from wake to sleep occurs close to and the transition from sleep to wake occurs close to . During sleep deprivation, it is argued in [25] that by increasing , rather than switch from wake to sleep, it is possible to stabilise the ‘ghost’ of the wake state: the extent to which is increased is known as the wake effort. An alternative view of the same idea is to consider the -plane as shown in Figure 9(b) and recognise that are curves that divide the parameter plane into regions where only the wake state exists, only the sleep state exists, and a bistable region where both wake and sleep exist. There are also regions for low and high , where the two states cannot readily be distinguished. The region of relevance for the parameters used in [25] is close to the bottom of the bistable region, and is shown in blow-up in Figure 9(c). The horizontal line represents the normal sleep-wake cycle: the time dependence of the homeostatic and circadian processes result in oscillating backwards and forwards along the line, switching from wake to sleep for increasing when and from sleep to wake for decreasing when .

In sleep deprivation experiments, subjects are prevented from falling asleep at . At this point, in order to remain awake the only alternatives that keep the system in the wake region are: decrease the drive to the VLPO, ; increase the drive to the MA, or some combination of both of these. In [25], it is argued that in order to maintain wake it is necessary to stimulate the MA, and therefore is increased to remain on the ‘ghost state’, but this is equivalent to following the line . The additional amount by which the MA is stimulated, the wake effort, is then

where is a function of and is the solution of equations (14) in the Methods section.

In the two-process model, acute sleep deprivation is modelled as a continued increase in the homeostatic pressure. In [10] this is interpreted as a suspension of the upper threshold, but with insight gained from the the PR model, we see that an alternative interpretation is that the upper threshold is continuously moved to keep the model in the wake state, as shown in Figure 10(a). The wake effort is then related to the extent to which the threshold has to be moved, that is the quantity with the upper threshold as given by (3). This quantity is shown in Figure 10(b). In the Methods section it is shown that this relation is (10)

This resulting wake effort computed from the two-process model is shown by the solid line in Figure 10(c) and agrees very well with the calculation of the wake effort from the PR model in [25] (crosses).

The close to linear relationship (the quadratic term has a very small coefficient) between wake effort in the PR model and , which is essentially the difference between the homeostatic pressure and the circadian oscillator, demonstrates that the wake effort used in [25] is fundamentally similar to previous measures used to compare performance and sleepiness scores. The precise scaling relationship and the degree of nonlinearity is dependent on the shape of the bistable region in the -plane shown in Figure 9(b), and on the choice of function for the dependence of the homeostatic process on the firing rate of the MA. In [17] and for many of the subsequent papers, the upper asymptote is give by . However, in [25] the functional form is used in order to “limit the unrealistically high production rate at high ”. This change in functional form has the effect of keeping approximately constant during wake, which is why the agreement between the wake effort as defined by [25] agrees well with our analogous computation from the two-process model. This is illustrated in Figure 10(c) and (d) where the wake effort and the dependence of on time are shown for the two-process model and for the PR model with the two different functional forms for .

The shape of the bistable region in the plane shows that for larger than about there is a transition from relatively small changes in needed to maintain wake to very large changes in needed to maintain wake; eventually it becomes impossible to maintain wake at all. While for typical parameters used in the PR model this transition occurs for infeasibly large values of and , we note that the shape of the bistable region is dependent on the parameters within the firing function and the choice of firing function itself. Once fixed in [31] these parameters have largely been left unchanged: we will return to this point in the discussion.

The equations for the PR switch model are (11)where

Since we introduce the small parameter , the fast time and the slow time , and . Then, at (slow time) equations (11) become (12)where

During wake, these have solution

During sleep, these have solution

Transitions between wake and sleep when , so the switch from wake to sleep occurs when and from sleep to wake when

By comparison with equations (1)–(4) we see that the two-process model and the dynamics of the PR switch model on the slow manifold are equivalent if (13)

On the slow manifold, the PR model is where is given by equation (5). For a fixed value of these have one or three solutions, with the transition between one and three solutions happening at saddle-node bifurcations, that satisfy (14)

The values of depend on and , and for the values commonly used in the PR model and listed in Table 1 give and

The sleep-wake cycle corresponds to slowly changing , tracing out a path on the slow manifold as shown in Figure 9(a). Transitions from wake to sleep and from sleep to wake occur close to and respectively. In order to find parameter values for the two-process model that retain the maximum and minimum values and timings for the homeostatic process for monophasic sleep. Away from bifurcation points the following algorithm is followed:

• 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 , , and . These maximum and minimum values occur close to the switches from wake to sleep and sleep to wake respectively. During wake, the two-process model gives

Hence, taking results in a trajectory for the two-process model that passes through the required values at the required times.

• One can do a similar matching for the decreasing phase to find a value for the lower asymptote. For the simulations presented here, the value of zero was taken for the lower asymptote.

By integrating the PR model with the typical parameter values listed in Table 1, it is found that the minimum occurs at and the maximum at This implies that in this case the parameter values for the two-process model are (15)

Going back to the PR switch model and its link to the two-process model, we can now find expressions for the parameters and such that it is close to the full PR model:

• Comparing the expression for above with that for the PR switch model in (13), gives

• The relation for in (13) gives

• Considering as given above with and as in (13), leads to

Hence for the typical values of the PR parameters listed in Table 1 and used in Figure 4,

It is also necessary to take in the PR switch model, otherwise no switching occurs.

Following [25], the additional amount by which the MA has to stimulated to follow the “wake ghost”, the wake effort, is where is a function of and is the solution of transition equations (14) above. For the region of relevance shown in Figure 9 (c) and (d), the relationship is close to linear with a small quadratic term and is well-approximated by

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 is given by , as and , the value of if no wake effort is applied, so the wake effort for the two-process model is (16)