Definition

A neural population model of sleep is a mathematical model of neural populations that regulate the timing and expression of sleep/wake patterns. Specific circuits and nuclei have been identified in the mammalian brainstem and hypothalamus that play a key role in modulating the brain’s overall arousal state with a circadian (daily) rhythm. These neural populations have been the focus of most sleep modeling.

Detailed Description

Sleep is an arousal state characterized by physical inactivity, reduced sensitivity to environmental stimuli, and a range of characteristic physiological changes, including changes to the EEG associated with rapid eye movement (REM) and non-REM (NREM) sleep in mammals. Sleep is regulated by a variety of physiological and biochemical processes (Krueger et al. 2008; Saper et al. 2010), including specific neural populations in the brainstem and hypothalamus. Various mathematical models have now been developed, providing a conduit between the underlying physiology and the sleep/wake dynamics observed at the behavioral level.

The Sleep/Wake Switch

The arousal states of sleep and wakefulness typically manifest as global patterns of brain activity; notable exceptions to this rule are unihemispheric sleep (sleeping with one brain hemisphere) and local sleep (emergence of sleep-like activity patterns in overworked neural assemblies while the rest of the brain is awake), which demonstrate that sleep can in principle be regulated on more local scales. Global synchronization of sleep and wake states across brain regions is achieved by control systems in the brainstem and hypothalamus. As shown in Fig. 1, a group of monoaminergic, cholinergic, and orexinergic neural populations projects diffusely to the cortex and thalamus. The coherent activation of these populations modulates corticothalamic activity, so as to promote and sustain wakefulness.

Fig. 1
figure 1

The brain’s overall arousal state is modulated by ascending projections to the cortex and thalamus from nuclei in the brainstem and hypothalamus. These include (i) wake-promoting nuclei that release monoaminergic neurotransmitters (blue), the tuberomammillary nucleus (TMN), the ventral tegmental area (VTA), the dorsal raphe (DR), and the locus coeruleus (LC); (ii) wake-promoting and REM sleep-promoting nuclei that release acetylcholine (green), the laterodorsal tegmentum (LDT) and the pedunculopontine tegmentum (PPT); and (iii) wake-promoting neurons in the lateral hypothalamus (yellow) that release orexin (Orx). These nuclei are all GABAergically inhibited by sleep-promoting neurons (red) in the ventrolateral preoptic area (VLPO) and median preoptic area (MnPO). Excitatory (pointed arrows) and inhibitory (rounded arrows) interactions between nuclei are indicated. Mutual inhibition between sleep-promoting and wake-promoting nuclei forms the basis for the sleep/wake switch

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A population of sleep-promoting neurons in the ventrolateral preoptic area (VLPO) and median preoptic area (MnPO) of the hypothalamus GABAergically inhibits the wake-promoting monoaminergic, cholinergic, and orexinergic neural populations. Interestingly, the wake-promoting monoaminergic populations in turn inhibit the sleep-promoting VLPO/MnPO populations. This mutual inhibition between sleep-promoting and wake-promoting neural populations provides a basis for a switch-like system, with stable sleep and wake states, and relatively rapid transitions between states (Saper et al. 2010). This system is called the sleep/wake switch.

Sleep-Regulatory Processes

Prior to the recent detailed mapping of sleep-regulatory circuits in the brain, mathematical models of sleep were developed. In the absence of physiological data, these models were phenomenological rather than physiological. The most influential of these models was the two-process model of sleep (Daan et al. 1984), which provided the conceptual basis for most subsequent models of sleep and human performance (Van Dongen 2004). The two-process model assumes that sleep can be understood in terms of two key sleep-regulatory processes: the circadian process and the sleep homeostatic process. The circadian process represents an approximately 24-h biological rhythm in sleepiness and alertness, typically described by a periodic function or nonlinear oscillator. The sleep homeostatic process represents the increasing need to sleep the longer one is awake and the dissipation of this need the longer one is asleep, typically described by exponentially saturating functions.

The physiological basis for the circadian process has now been well elucidated. Circadian rhythms are generated endogenously on a molecular level by virtually all mammalian cells. These cellular rhythms are synchronized by a master circadian clock that resides in the suprachiasmatic nucleus (SCN) of the hypothalamus. SCN neurons act as tightly coupled circadian pacemaker cells, sending a robust circadian signal to many other brain regions. The SCN receives direct input from the retina, allowing it to entrain to the daily light/dark cycle (Foster and Kreitzman 2005).

The physiological basis for the sleep homeostatic process is still not well understood, but evidence points towards the accumulation of certain sleep-promoting factors in the brain during wakefulness. These substances include adenosine, nitric oxide, TNF-α, Interleukin-1, and prostaglandin D2 (Krueger et al. 2008).

Both the circadian and sleep homeostatic processes have been found to act on the neural populations in the sleep/wake switch (Pace-Schott and Hobson 2002). The SCN has multiple relays to both the sleep-promoting and wake-promoting neural populations. The sleep-promoting factor adenosine has also been found to have effects on the VLPO. The sleep/wake switch therefore provides a neural basis for the integration of the circadian and sleep homeostatic processes.

Neural Population Models of Sleep

In order to test the theoretical value of the sleep/wake switch theory of sleep regulation, and to link the dynamics of underlying neural systems to behavioral manifestations, various mathematical models of the neural populations in the sleep/wake switch have recently been developed (Tamakawa et al. 2006; Phillips and Robinson 2007; Behn et al. 2007; Behn and Booth 2010; Rempe et al. 2010; Kumar et al. 2012; Sedigh-Sarvestani et al. 2012). These models have included different levels of detail in terms of the neural populations included, as well as slightly different mathematical formalisms, including neural mass, single neuron, and neural mass/neurotransmitter concentration implementations. Central to all of these models is mutual inhibition between sleep-promoting and wake-promoting neural populations, i.e., the sleep/wake switch.

Here, we present a simplified mathematical description of the core dynamics of these neural population models of the sleep/wake switch. We base our equations on the Phillips–Robinson model, since it is the simplest and most widely used of the models (Robinson et al. 2011).

We model two neural populations: a wake-promoting population (subscript w) and a sleep-promoting population (subscript s). Each population has a mean firing rate, \( {Q}_{{}_j} \), and a mean cell body voltage relative to resting, V j . We assume that the firing rate is a sigmoidal function of voltage, Q j = S(V j ). We also assume that the postsynaptic effects on voltage for each population are proportional to the presynaptic firing rates. The dynamics can then be represented by a pair of coupled first-order differential equations,

$$ {\tau}_wd{V}_w/ dt+{V}_w={\nu}_{ws}{Q}_s+A,\kern0.84em {\tau}_sd{V}_s/ dt+{V}_s={\nu}_{sw}{Q}_w+D(t), $$

where τ j is a time constant for the saturation time constant; ν ws < 0 and ν sw < 0 represent the strengths of the inhibitory synaptic connections between the two populations; A represents any constant net input to the wake-promoting population; and D(t) represents the net drive to the sleep-promoting population from circadian and sleep homeostatic processes. In general, the circadian and sleep homeostatic processes could also have a direct effect on the wake-promoting population. Here, we consider the most simple case for analysis.

The dynamics of the neural populations occur on a timescale of milliseconds to minutes (determined by the τ j parameters), whereas D(t) varies on a daily timescale by definition. A timescale separation can therefore be performed, treating D as approximately constant on timescales relevant to neural dynamics (Fulcher et al. 2008). The system dynamics can then be summarized by the reduced manifold of equilibrium values for various fixed values of D, as shown in Fig. 2.

Fig. 2
figure 2

If the sleep drive, D, is assumed to be slowly varying, the model dynamics can be visually summarized in terms of the equilibrium values for the wake-promoting population firing rate, Q w , and the sleep-promoting population firing rate, Q s . When the drive for sleep is low, the model has a stable wake state, with high Q w and low Q s . When the drive for sleep is high, the model has a stable sleep state with low Q w and high Q s . For an intermediate range of drive values, the model is bistable, with stable sleep and wake states. Saddle node bifurcations occur at critical thresholds (dotted lines). The stable wake and sleep branches are linked by an unstable equilibrium branch within the bistable zone. As D slowly oscillates on a daily timescale, the model therefore undergoes hysteresis as it is dragged across the thresholds where sleep and wake states become unstable

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For reasonable parameter values, the reduced system has two stable states, corresponding to sleep and wake. For a certain intermediate range of values for D, the system is bistable. As D oscillates with a period of ~24 h, the system is dragged back and forth across the bistable region, resulting in daily transitions between sleep and wake. The model exhibits hysteresis, because the thresholds for sleep-to-wake and wake-to-sleep transitions are different, as shown in Fig. 2.

This hysteresis behavior describes the most basic, reduced dynamics of most neural population models of the sleep/wake switch. Interestingly, these physiologically based models recapitulate the basic assumptions of the phenomenological two-process model: two separate thresholds for sleep-to-wake and wake-to-sleep transitions (Phillips and Robinson 2008; Rempe et al. 2010). In other words, these models provide a physiological justification for the two-process model, as well as providing new insights by explicitly modeling the underlying physiological dynamics.

Neural population models of the sleep/wake switch have been highly successful in reproducing and predicting many aspects of human sleep and circadian rhythms (Robinson et al. 2011). In addition, mathematical models have been developed to describe how modulation of the corticothalamic system can give rise to the neural activity patterns associated with sleep and wakefulness, as well as transitions between arousal states (Robinson et al. 2002; Hill and Tononi 2005; Steyn-Ross et al. 2005; Deco et al. 2013). Linking corticothalamic and subcortical models poses a significant theoretical and numerical challenge due to their inherently different spatiotemporal scales, but such models are on the horizon (Robinson et al. 2010).

Recent Findings

Neural population models of the sleep/wake switch have been highly successful in reproducing and predicting many aspects of human sleep and circadian rhythms. Specifically, models of the sleep/wake switch have been used to understand the physiological mechanisms that underlie interindividual (Phillips et al. 2010a; Robinson et al. 2011) and interspecies (Phillips et al. 2010b, 2013) differences in sleep timing and duration. They have been used to reproduce normal human sleep (Phillips and Robinson 2007) and mouse sleep (Behn et al. 2007), as well as the effects of neurotransmitter microinjections (Behn and Booth 2010). They have also been applied to predicting human performance, including the effects of sleep deprivation (Fulcher et al. 2010) and caffeine on subjective fatigue (Puckeridge et al. 2010) and the effects of shiftwork on performance (Postnova et al. 2012).

Given the current uncertainty regarding the physiological mechanisms that underlie the REM/REM sleep cycle, neural population models have also been used to probe the possible dynamics of the system. This has led to concrete hypotheses regarding physiological mechanisms and circuits that could potentially generate the REM/NREM sleep cycle (Rempe et al. 2010; Behn and Booth 2012; Behn et al. 2013). Sleep/wake switch models have also recently been applied to reproducing the desynchronization of sleep/wake cycles from the circadian process that can occur when individuals are completely isolated from environmental time cues and are free to self-select their sleep/wake timings (Phillips et al. 2011; Gleit et al. 2013).

In addition, mathematical models have been developed to describe how modulation of the corticothalamic system can give rise to the neural activity patterns associated with sleep and wakefulness, as well as transitions between arousal states (Robinson et al. 2002; Hill and Tononi 2005; Steyn-Ross et al. 2005; Deco et al. 2013). Linking corticothalamic and subcortical models poses a significant theoretical and numerical challenge due to their inherently different spatiotemporal scales, but such models are on the horizon (Robinson et al. 2010).

Cross-References

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