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MBW:Modelling Sleep Regulation

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Authors: Sara Bessman and Ben Brown

The purpose of this wiki article is to explore a model that describes the timing of sleep and wakefulness in adult humans. The model was originally theorized by Alexander A. Borbely in 1982[1], and quantified two years later in collaboration with Daan et al. (1984).[2] For a review of Daan et al. (1984), see APPM4390:Sleep Regulation. Here we extend their model to include ultradian dynamics of the sleep-wake cycle, and we explore the model in various contexts of sleep regulation. Specifically, we investigate the relationship between sleep pressure and sleep duration, phase shifting as an adaptation to jet lag, and plasticity of the circadian period.


Mathematics Used

This paper explores an application that uses oscillatory ordinary differential equations. The implementation in MATLAB uses the built-in ode15s solver. Specifically, the van der Pol oscillator is used to model the circadian rhythm in adult sleep cycle. In addition, a modification of the limit cycle interaction model, proposed by McCarley and Massaquoi, is used to model the oscillatory nature of the ultradian dynamics of REM sleep. A threshold activation function is used to determine when the wake cycle is activated or deactivated. Analysis of the steady-states and bifurcations is included in the results.

Type of Model

The model explored by the authors looks at the neuroscience phenomena of sleep patterns and predicts when, and for how long, a person sleeps. Specifically, this paper looks at the relationship between sleep pressure and sleep duration, where sleep pressure refers to need for sleep. The hypothesis that increased sleep pressure results in a smaller sleep duration is examined by observing the effect of varying the parameters in the model. Using the model developed in this paper, one can predict when a person will sleep, and how sleep patterns are effected by lighting.

Biological System

The model examined in this paper provides insight on the duration of sleep, how sleep is affected by light, and how to measure the need for sleep. It specifically studies sleep patterns in adult humans in terms of the circadian rhythm, REM sleep and sleep homeostasis.


Biological Context

Sleep is a phenomenon that has been characterized in all mammals, and demonstrated as essential to survival in many species.[3] In adult humans, sleep is pervasive in life, comprising roughly one-third of the day. Despite the essential and pervasive nature of sleep, there are still many unanswered questions about its nature: Why do we sleep? What makes some individuals need more sleep than others? What causes the timing of sleep and wakefulness? The model presented in this article can be used to investigate the latter question. In addition, this model may explain inter-individual variations in sleep patterns and provide insight into the physiological processes responsible for the onset and maintenance of sleep and wakefulness.


According to Daan et al. (1984),[2] the mechanisms behind sleep and wakefulness can be studied with two main approaches: circadian and physiological. The circadian approach views sleep and wakefulness as spontaneous phenomena that occur due to the brain's sense of time of day. In contrast, the physiological approach focuses on the changes taking place within the body during sleep and wakefulness and how these changes effect the body's drive for sleep. A commonly used tool for monitoring changes in cortical neuronal activity during sleep and wakefulness is the electroencephalogram (EEG). The EEG can be used to examine changes in brain wave activity, such as transitions between sleep stages, or changes in sleep intensity.

In 1937, Blake and Gerard[4] characterized a well-known feature of sleep with EEG. They recognized that sleep intensity is directly related to EEG power between frequencies of 0.75 and 4.5 Hz. This "slow-wave" activity is represented by the variable A in the following. Many studies since 1937 have validated the relationship between A and sleep intensity, in addition to revealing that increased duration of wakefulness leads to increased slow-wave sleep (sleep EEG in the 0.75-4.5Hz range) and decreased duration of wakefulness leads to decreased slow-wave sleep. Borbely (1980)[5] coined the term sleep homeostasis to describe this compensatory mechanism.

The original sleep model, quantified by Daan et al. (1984)[2], was composed of two processes: S and C. Process S, the homeostatic process, is used to describe the compensatory mechanism that increases the drive for sleep while awake and decreases the drive for sleep while sleeping. Process C, the circadian process, is incorporated to describe the influence of the circadian timing system. Many extensions of the model have been proposed since the original two-process model. The ultradian dynamics of A due to rapid eye movement sleep (REMS), i.e. the cyclic variation between non-REMS and REMS, have since been incorporated. [6][7][8][9] In addition, process C has been modified to account for the effects of light intensity on the circadian pacemaker.[10] We incorporated these modifications into the original model by adapting a new model proposed by Achermann and Borbeley (1992)[9](see Figure 1).

A Combined Model for Sleep Regulation

A schematic representation for the sleep regulation model of interest can be found in Figure 1. The three key components of this model include: 1) Circadian Oscillator, 2) REMS Oscillator, and 3) Sleep Homeostat. The circadian oscillator is defined by a basic circadian variable, C, and a complementary circadian variable, Cc, and influences both the sleep homeostat and the REMS oscillator through C. There are countless external factors that can influence the circadian oscillator, but this model accounts only for the primary zeitgeber (time-giver), light. The REMS oscillator can be described in terms of REM-on activity, x, and REM-off activity, y, and influences the sleep homeostat through the REMS trigger, R. Finally, the sleep homeostat is defined by slow-wave activity during sleep, A, and Process S. The three components interact to influence the timing of sleep and wakefulness, as well as the architecture of sleep.

Figure 1. Combined model for sleep regulation, adapted from Achermann and Borbely (1992).[9]C = basic circadian variable, Cc = complementary circadian variable, x = REM-on activity, y = REM-off activity, R = REM sleep trigger, A = slow-wave activity, S = Process S

Circadian Oscillator: Processes C and Cc

The circadian oscillator represents the output of the master biological clock, the suprachiasmatic nucleus (SCN), in the hypothalamus. (For more information on circadian oscillators and biological clocks, please see APPM4390:Biological Clocks and Switches). As shown in Figure 1, this oscillator is directly affected by light, and it influences both the REMS oscillator and sleep homeostat. The circadian oscillator directly influences S in the sleep homeostat through manupulation of the upper and lower thresholds, H and L, respectively (see the next section). Since the circadian pacemaker behaves as a limit cycle oscillator, Kronauer (1990)[10] showed that it can be modelled with a van der Pol oscillator.[11] We present here the model developed by Kronauer (1990)[10], which also incorporates the effects of acute light exposure.

The following three equations describe how C and Cc oscillate under the influence of external light of intensity I.

Circadian DEs.jpg

Here, μ is the "stiffness" of the circadian oscillator, and κ is the sensitivity of the circadian oscillator to the influence of light intensity. In our model, μ and κ were set to 0.26 and 0.025, respectively. The parameter τc represents the natural period of the circadian oscillator. The natural period of the average adult human's circadian oscillator is approximately 24.2 hours[12].

The effect of light on the circadian oscillator is to augment of the rate of change of C. Consequently, if external lighting occurs when C is increasing, the amplitude of C will increase. However, if light is applied when C is falling, then the amplitude of C decreases. In general, we used a 50-50 light-dark cycle that mimicked the variation of sunlight intensity throughout the day: On Earth, the intensity of the light varied as a quadratic function, fitted to 400 lux at sunrise and sunset and 20000 lux at midday. On Mars, the intensity of the light throughout the day was reduced from the Earth case by approximately 40%. In practice, we forced the value of B (or I) to zero during sleep. This was done to more accurately estimate the behaviour of a real person, who would seek darkness and shelter when they wished to sleep.

Sleep Homeostat: Processes S and A

Process S of the sleep homeostat is derived from the original two-process model quantified by Daan et al. (1984)[2] and reviewed in APPM4390:Sleep Regulation. The new component, A, accounts for ultradian dynamics of slow-wave activity during sleep[6][7]. Specifically, the greatest proportion of slow-wave activity is found during the first half of sleep.

The following two equations describe the behavior of the sleep homeostat, processes S and A:

Homeostat DEs.jpg

Where α, β, γ, δR, and δW are all positive parameters that help fit the S and A solutions. We set each of the above parameters to the following values, respectively: 0.84, 0.055, 16.98, 14.16, and 60. The parameter AL is simply a small positive number used to attenuate A when a person is awake or in REMS.

In this model, R and W represent the "REMS" and "wake" triggers, which attenuate A when activated. Each of R and W can be alternatively equal to 0 or 1, but they may not both equal 1 at the same time. The REMS trigger is activated when the REM-on activity x exceeds xR (see the following section on the REMS oscillator). The wake trigger is activated when S falls below a low threshold L, and the trigger is deactivated when S exceeds a high threshold H. The upper and lower thresholds are calculated from the circadian process C in the following manner:

H and L.jpg

We varied the "average sleep pressure" by raising or lowering the H and L thresholds. The typical amplitude of oscillation used for both the upper and lower thresholds was aH=aL=0.12. It was typical for the thresholds to have a separation H-L of 0.33.

REMS Oscillator: Processes x and y

The REMS oscillator accounts for the ultradian dynamics of REM sleep. We use an adaptation of the limit cycle interaction model proposed by McCarley and Massaquoi (1986)[8]. This model accurately depicts the increasing duration of REMS episodes throughout the night. The modification of the Massaquoi model, from Achermann and Borbely (1992),[9] allows for high levels of REM-off activity, y, during wakefulness, more closely resembling the occurrence of REM sleep in humans.

The following two equations describe the behavior of the x and y processes in the REMS oscillator:

REMS DEs.jpg

Here, σ is a positive constant used to denote the time scale of the REMS oscillation. We set σ to 6.4486 in order to replicate the typical 90 minute human REMS cycle. The remaining parameters are defined as follows:

More REMS Es.jpg

CSO and tSO represent the level of C and the time, t, of the last sleep onset. In our model, we set the parameters d0 and ew to 0.08 and 0.95, respectively. The level of x required to turn on the REMS trigger, R, is xR = 1.4.

A different type of bio-oscillator can be found here. Oscillatory behavior in enzymatic control processes, rights and wrongs

Implementation in MATLAB

The original combined model was integrated by the authors on a uniformly spaced time grid with time step equal to 1 minute. In our implementation in MATLAB, we used a variable time step (typical mean time step of ~3.5 minutes) in conjunction with a root-finding "event" locator associated with the built-in MATLAB ode integrators. The event locator was used to more accurately solve for the exact sleep and wake times (when S crosses H and L, respectively). Because of the nature of the functions C, Cc, S, A, x, and y, we used the ode15s integrator. The implementation of the proposed model in MATLAB in the above manner allowed for better control over the accuracy of the simulation, increased resolution of the sleep and wake times, and decreased calculation time due to the longer average time step.

Validation of the Model

Below we provide both analytic and computational validation for the sleep regulation model. Analytically, we provide a steady state and bifurcation analysis of the sleep homeostat. Computationally, we vary the parameters described above for each component of the model to study the relationship between sleep pressure and sleep duration, phase shifting of the circadian oscillator, and adaptability of the circadian period.

Figure 2. Plot of the sleep regulation model. The upper and lower dotted red lines correspond to the upper and lower thresholds, H and L, respectively. The dark blue line represents process S. The cyan line represents slow wave activity, A.

Steady State and Bifurcation Analysis

The homeostatic process S and slow wave activity A are effectively analyzed using steady state analysis. Solving the differential equations for S and A, we can show that the stationary A values are given by the following:

A Stability.jpg

These equations indicate that during non-REMS, slow wave activity A is repelled from an unstable stationary point at 0 and attracted to a stable stationary point equal to the current level of S. In contrast, during REMS or wakefulness A is attracted to a stable stationary point at some small positive value (while being repelled from an unstable stationary point at some large negative value).

The stationary S values are given by the following equations:

S Stability.jpg

The equations above indicate that S is constantly being attracted to a stable stationary point that varies with the current level of slow wave activity, A. Generally, if A is small (e.g. in REMS or waking), S will monotonically grow to approximately 1. As A increases, the stable stationary point for S decreases, causing S to fall.

The bifurcations of this system are described by the sudden change in the steady states and stability of A when the REMS and wake triggers switch on and off. Switching from non-REMS to REMS or from sleeping to waking results in the attenuation of A, while non-REMS sleep allows A to approach the current level of S. While the behaviour is relatively easy to describe, the mathematical properties of these bifurcations are not well understood since the parameters R and W do not vary smoothly. The proposed model is adept at providing meaningful predictions about the timing of sleeping and waking in conjunction with external lighting. However, it is clear that the discontinuities in the stability of A are an undesirable trait. Changing to a simpler, better understood system with well studied bifurcations may be the necessary next step for mathematical biologists intending to model slow wave activity during sleeping and waking.

Sleep Pressure and Sleep Duration

In this section, our aim is to examine the relationship between sleep pressure and sleep duration in the model. We hypothesize that individuals under a greater amount of sleep pressure (measured by the amplitude of S) will have shorter daily sleep durations, and individuals under less sleep pressure will have longer daily sleep durations. This is based on data by Aeschbach et al. (1996)[13] who showed that habitual short sleepers are under greater homeostatic pressure for sleep compared to habitual long sleepers (see Figure 3). Our results are in agreement with Aeschbach et al. (1996) (see Figure 4), which indicate the inverse proportionality between overall sleep pressure and sleep duration. The average sleep pressure of an individual was raised or lowered by alternatively raising or lowering the upper and lower thresholds, H and L.

Figure 3. From Aeschbach (1996)[13]Sleep regulation in habitual short (5.5hr) and long (9.9hr) sleepers. The y-axis is the level of S on a scale from 0 to 1. The x-axis is time of day, with black bars representing sleep episodes. BL = baseline sleep episode, R = recovery sleep episode following 40hr sleep deprivation.
Figure 4. Results from the simulation. Blue curve corresponds to "normal" sleep (~8 hours per night). Red and cyan curves correspond to short (~6 hours) and long (~10 hours) sleep, respectively.

Phase Shifting the Circadian Oscillator

One feature of the circadian oscillator that is interesting to analyze is its phase with respect to the daily rhythm of light intensity, I. In modern times, with the increased prevalence of methods for traveling long distances in short times, it is not uncommon for people to experience a form of circadian phase misalignment known as jet lag. The proposed model is effective at depicting the entrainment of the circadian oscillator after experiencing a phase shift of several hours. The following two plots illustrate results from the model of a "westward" and "eastward" phase shift, respectively. Note that in the simulation it takes longer to entrain eastward than westward. This has been shown to be true in individuals with intrinsic periods slightly longer than 24 hours.

Figure 5. Recovery from jet lag due to a westward circadian phase shift of approximately 8 hours. The raster plot is read forward in time from bottom to top in days and from left to right in time of day. Time on the x-axis is the new time zone. Black bars indicate sleep episodes.
Figure 6. Recovery from jet lag due to a eastward circadian phase shift of approximately 6 hours. The raster plot is read forward in time from bottom to top in days and from left to right in time of day. Time on the x-axis is the new time zone. Black bars indicate sleep episodes.

Plasticity of the Circadian Period

In addition to studying the response of our model to a circadian phase shift, we were interested in assessing the plasticity of the circadian period itself. This aspect of the model piqued our interest because it has direct application to the human space exploration of other planets in the solar system (e.g. Mars). Scheer et al. (2007)[14] recently showed that moderately bright lighting (~450 lux) applied for either the first or second half of the natural waking time, resulted in the entrainment of the circadian rhythm of all studied subjects to a 23.5 hour or 24.65 hour day, respectively. See Figure 7.

Figure 7. Laboratory protocol from Scheer et al. (2007)[14]. The raster plot is read forward in time from top to bottom in days and from left to right in time of day. Black bars indicate sleep episodes. First and last 3 days served as habituation to the lab and the external day, respectively. All subjects entrained to the longer and shorter day lengths, as assessed by the forced desynchrony protocols following light treatment.

In an attempt to recreate the results of this study, the following two raster plots were created using the proposed model representing a person with a natural 24.2 hour intrinsic period:

Figure 8. Simulated entrainment to a 24.65 hour lighting schedule. Moderately bright light of ~450 lux applied for the later half of the subject's waking period effectively entrained the subject to the longer day. The first and last several days were used to mimic the habituation days of the aforementioned study.
Figure 9. Simulated entrainment to a 23.5 hour lighting schedule. Moderately bright light of ~450 lux applied for the early half of the subject's waking period effectively entrained the subject to the shorter day. The first and last several days were used to mimic the habituation days of the aforementioned study.

The above plots are in many ways similar to those provided in the previous section. Effectively, by applying light during the later half of the waking period, the circadian phase is shifted "westward" daily which leads to entrainment to the longer lighting schedule. The opposite effect is seen when applying light during the early half of the waking period, which results in a daily phase shift "eastward" and leads to entrainment to the shorter lighting schedule.

In addition, we wanted to take a broader look at the ability of adult humans to entrain to a Martian day. We attempted to entrain natural circadian periods (23.0hr - 25.0hr) and average sleep pressures (0.35-0.65) to a lighting schedule mimicking the Martian sol. The first 30 days of the simulation served as the synchronization period, and we used the second 30 days to calculate the mean and standard deviation sleep onset times and durations. The following three plots compile the results of this analysis. The first shows the standard deviation of the daily sleep onset time, which provides a sense of how well people may entrain to the Martian day (i.e. entrainment results in a low standard deviation of daily sleep onset time). The standard deviation of the daily sleep onset time will grow only when a person is unable to entrain to the Martian day.

Figure 10. Results from the simulation depicting entrainment to a Martian lighting schedule. colorbar = Standard deviation of sleep onset time (min), x = circadian periods (23.0hr - 25.0hr) y = sleep pressure (0.35-0.65)

This plot demonstrates that the vast majority natural circadian periods and average sleep pressures in the described ranges are able to entrain to the Martian day. Only extraordinarily low sleep pressures and natural circadian periods far from the Martian day length (24.65 hours) were unable to synchronize to the alien lighting conditions.

The next plot shows how the average duration of nightly sleep varies with intrinsic circadian period and average sleep pressure. We can clearly see that average sleep duration is most closely linked to average sleep pressure. Those who sleep longer every night tend to have a generally lower sleep pressure, while the opposite is true for the short sleepers.

Figure 11. Results from the simulation depicting entrainment to a Martian lighting schedule. colorbar = Mean daily sleep duration(min), x = circadian periods (23.0hr - 25.0hr) y = sleep pressure (0.35-0.65).

The final plot shows how the standard deviation of the duration of nightly sleep varies with intrinsic circadian period and average sleep pressure. The most noticeable characteristic of this plot is that there seem to be diagonal "stripes" of elevated deviation in sleep duration. The stripes run perpendicular to the gradient of the previous plot, showing that variation in sleep time goes up only when the average sleep time does not fit an integer number of 90 minute sleep cycles. In this case, the duration of sleep will change by approximately the length of one sleep cycle from night to night.

Figure 12. Results from the simulation depicting entrainment to a Martian lighting schedule. colorbar = Standard deviation daily sleep duration(min), x = circadian periods (23.0hr - 25.0hr) y = sleep pressure (0.35-0.65).

Concluding Remarks

This study has shown that the proposed combined model for sleep regulation is very effective at predicting the timing of sleeping and waking in adult humans under a large variety of lighting conditions. As described in our section on the implementation in MATLAB, our model improves on the one proposed by Achermann and Borbely (1990)[6] by adding control over the accuracy of the prediction, increasing the resolution of the sleep and wake times, and reducing the number of required time steps. Analytical and Computational tools were used to show that the model is relatively well understood and captures the desired relationship between external lighting and the circadian rhythm.

It is important to note that the model presented here is simplified from Achermann and Borbely (1990)[6]. Their model and other more sophisticated models incorporate additional processes by haphazardly adding systems of differential equations. This often results in further complications of the combined system, and in fact may be the ultimate downfall of this type of modelling (see Occam's Razor). As previously mentioned, it is our recommendation that further studies aim to simplify the model while retaining the desired behaviour.

In our steady state analysis, we showed that the sleep homeostat equations are built to describe the ultradian oscillation of A during sleep and the attenuation of A during wakefulness. One type of simple system that captures similar behaviour is a combined subcritical Hopf bifurcation and saddle node. This system of two coupled oscillators provides a stable limit cycle and a stable stationary point, separated by an unstable limit cycle. The subcritical Hopf bifurcation allows for the sudden appearance of large oscillations at sleep onset (moving from the stationary point to the stable limit cycle), the decrease in oscillation amplitude throughout the sleep episode, and the sudden disappearance of oscillations upon waking (moving from the limit cycle back to the stable stationary point). While the subcritical Hopf bifurcation is a promising candidate for a simpler model of the variation of slow wave activity during sleep and wakefulness, it is unclear whether this system is easily married to the existing concept of the van der Pol circadian oscillator. Further studies into the application of the subcritical Hopf bifurcation (common in modelling signal propagation in communities of cortical neurons) may prove vital to improving the way we model and understand the behaviour and function of human sleep.


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Further Reading

  • The following is a summary of the paper "Circadian clock genes and sleep homeostasis" by P. Franken and D.-J Dijk (2009) which cites the paper by Alexander A. Borbely (1982).

In this paper, Franken & Dijk reassert the "two-process" model introduced by Borberly in which sleep is regulated by circadian rhythm (C) and the homeostasis process (S). They also emphasize, though, that these processes work in opposition. That is, when S increases our need for sleep throughout the day, C serves to counter this need so we are able to stay awake during the daytime. Likewise, we are able to stay asleep at night even though the effect of S decreases during the nighttime.

Although there is a well-established notion that S and C are independent processes (the absence of one does not change the effect of the other), this paper discusses what are called "clock-genes", or circadian genes, and explores the idea that the molecular structure behind C can be used to predict S. In studies conducted on mice lacking these clock genes, the mice exhibited an increase in sleep pressure proving that clock genes do influence the need for sleep. This is a similar result to mice with sleep deprivation. A marked increase in sleep pressure was also observed in fruit flies lacking clock genes, and the fruit flies actually died after a sleep deprivation greater than 10 hours. In humans, these patterns can be observed by studying people with different sleep patterns ("night owls" vs. "early birds"), or short-sleepers vs. long-sleepers. In fact, the clock genes responsible for these differences cause changes in sleep pressure which accounts for variations in sleep timing and duration.

Last, the authors discover that clock genes also play a role in metabolism. Mice that lacked clock genes were typically obese and unable to adapt to changes in food schedule, to the point of starvation. These results can play a huge role in future research on obesity. Furthermore, since there is a known link between sleep and mood, studies on mood disorders will be able to benefit from the information on clock genes.

External Links

For more information regarding sleep pressure and sleep homeostasis, see Sleep Homeostasis.

An interesting study was conducted to examine the difference in sleep pressure and the circadian rhythm between "night owls" and "early birds". The paper is published by the European Sleep Research Society and can be found at: "The circadian and homeostatic modulation of sleep pressure during wakefulness differs between morning and evening chronotypes".

Another interesting study was conducted to examine the pathological effects of a non-regular circadian sleep schedule. The paper is written by Christopher L. Drake of the Sleep Disorders and Research Center, at Henry Ford Hospital. It is published in the Journal of Family Practice. Drake's focuses included:

  • The social and economic effect of shift-work disorder.
  • The characterization and treatment of pathological sleep schedules.
  • Recognition of shift-work disorder.
  • Managing treatment of a patient with shift-work disorder.

Drake cites the Borbely paper[1]
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