COMPUTATIONAL MODELS OF THE SLEEP WAKE CYCLE

R. E. Kronauer
Division of Engineering and Applied Sciences. Harvard University
Cambridge, Massachusetts, USA

INTRODUCTION

Physiological  mathematical models fall into two broad categories. One type seeks to represent physiological details (e.g. neuronal firing rates, biochemical concentrations, reaction rates). The other seeks to provide a mathematical framework on which to organize data for practical appli-cations (e.g. how to use light to overcome jet-lag). For both types the literature shows a progression from elementary concepts to ever more complex ones as more refined data are developed. There must be a constant effort to keep models pruned to essentials lest understanding be compromised.

THE CIRCADIAN PACEMAKER

 A fundamental influence on the human sleep/wake (S/W) process lies in some 10,000 hypothalamic neurons that comprise the circadian “ pacemaker”. These are located just over the optic chiasm and are understandably called the suprachiasmatic nuclei (SCN) [ 1 ]. The SCN are sym-metrically bilateral and they contain two principal cell groups. The SCN neurons receive direct inputs from specialized retinal ganglion cells that contain their own photopigments [ 2 ]. Consequently photic drive to the SCN cells can be maintained in certain blind subjects who have lost all conventional visual perception [ 3 ]. Individual SCN neurons are biochemical circadian oscillators that retain rhythmicity even when electrical activity is suppressed by tetrodotoxin [ 4 ]. Intercellular coupling may be chemical (local) or in the form of directed electrical stimulation between cell groups. Outputs from the SCN are in large part neuronal, but the importance of diffusible low-molecular-weight substances has been demonstrated [ 5 ].

Circadian oscillators in single cells are ubiquitous in plants and animals. The oscillations involve mRNA  production in the cell nucleus, subsequent protein production in the cytoplasm and inhibitory feedback onto the mRNA production via nuclear entry of protein dimers.[ 6 ]. The time delays associated with protein production and dimerization establish the cellular circadian period. This basic process is homologus across a wide range of organisms but the specific genes are particular to the organism. Drosophila and mouse are the most thoroughly studied thus far.

The human circadian pacemaker period within a single individual appears to be very stable, at least over experiments in which relevant data could be obtained for several months   [ 7 ]. (Period variations appear to be no more than ~0.2%.)  It is thought that such stability could be the consequence of averaging over thousands of SCN cells.

The dynamic biochemical processes in a circadian pacemaker cell are complex and the differential equations used to model them are multidimensional. Typically 10 or more state variables are used [8, 9]. The time courses of the variables in these complex models do, under appropriate parameter values, correspond to stable periodic solutions. In some cases the waveforms of these variables are quite non-sinusoidal. However, studies of the way in which the human pacemaker responds to various temporal patterns of light stimulation suggest that functionally the pacemaker rhythm may be close to sinusoidal [ 10 ]. The “smoothness” of the model solution waveforms may ultimately provide a criterion for distinguishing between alternative model constructs.

SLEEP AND WAKE AS A RHYTHM

The human sleep/wake rhythm reflects a process that is distinct from the pacemaker. One form of evidence for this is found when a subject is permitted to “free-run”. That is, the subject is allowed to select his/her own sleep and wake episodes in an environment shielded from time cues and in moderate room light when awake. In most cases the S/W rhythm shows a cycle period of ~25 hours [11] compared to 24.2 hours that is the average for the isolated human pacemaker [12]. That the S/W process is actually a self-sustained oscillator was first shown by Aschoff [13] who identified the simultaneous existence of both pacemaker and S/W rhythms in a phenomenon known as spontaneous internal desynchrony.

The S/W oscillator and the pacemaker influence one another. In the free-run protocol with ordinary room light during wake episodes it is principally via the periodic light/dark timing that the S/W oscillator can affect the pacemaker. The nonphotic rest/activity pattern associated with S/W has a weak influence on the pacemaker that is difficult to quantify. Studies in totally blind subjects suggest that this rest/activity influence may be able to shift the pacemaker rhythm by only 0.2 to 0.3 hours per day [13 ]. In contrast, the cyclic timing of darkness and room light is able to shift the pacemaker by 0.7 to 1.3 hours per day. The influence of the pacemaker on the S/W oscillator is much stronger. The pacemaker is able to shift the S/W rhythm by as much as 4 to 5 hours per day [14].

The S/W oscillator is very labile and in laboratory free-run experiments its period has been observed to lengthen by as much as 60 to 80% over several months [ 7 ]. When the periods of the S/W oscillator and the pacemaker differ by ~5h the interaction mechanisms are unable to enforce synchrony. The rhythms become separately apparent – first in a “beat” phenomenon that is known as “phase trapping” and then as overt desynchrony that has a different “beat” pattern known as “relative coordination” [14]

The physiological substrate for the S/W oscillator is still unknown. Since its period is so labile it may very likely not be a localized neuropil in the brain but may exist in the form of a neuronal feedback loop interconnecting the centers that direct the sleep or wake states. The observation of phase trapping in free-run experiments implies that the S/W oscillator is not “stiff” in the mathematical sense. That is, it does not have the characteristics of a relaxation oscillator, despite the fact that it necessarily embodies the representation of only two states: sleep and wake. These apparently disparate aspects can be reconciled if the S/W oscillator is seen as governing the probability of being in one of the two states and that the mechanism interpreting this probability is strongly nonlinear (i.e. approximately of the threshold type).

An alternative model for self-selected S/W timing (free-run) postulates a measure, S , that builds up during wake time and degrades during sleep. Transitions between wake and sleep are triggered at high and low thresholds [15]. This is in fact an elementary form for homeostatic regulation of S about a mean value. The model is sometimes described as the two-process model (i. e. build-up and decay). It has also been proposed that the decay of S during sleep can be measured by the decline of EEG “slow-wave” power (power in the very-low-frequency band, 0.5 to 4 Hz.).

SLEEP UNDER IMPOSED SCHEDULES

Aside from the use of the process S model in explaining spontaneous internal desynchrony, there is a widespread view that in daily life, where clocks and schedules play a dominant role in setting the hours at which sleep is attempted, there should exist some physiological measure of sleep propensity (or sleep drive) that increases during wake and declines when sleep actually occurs. Presumably sleep propensity will be reflected in various characteristics of sleep quality (such as “slow wave” power mentioned above).

A new emphasis in sleep research is on sleep fragment-ation. Primates, and humans in particular, are special among mammals in their ability to consolidate sleep. However, if wake bouts of duration one minute or more are counted, the average young adult will wake approximately 7 times per bed-rest episode while older adults ( > 65 years ) will wake approximately 15 times [16] Most of these wake bouts terminate in a return to sleep within a few minutes, but about 15% are 16 minutes or even much longer. Remarkably, these wake bout statistics show no significant age dependence.

Transitions between the states of sleep and wake are very rapid compared to the duration of the wake or sleep bouts themselves That is, the S/W system has the character of a bistable circuit (flip-flop). New models are being developed in which the transitions between states are supposed to be controlled by random processes. The transition probabilities are conditioned both on phase relative to the circadian pacemaker and on the timing within the bed-rest episode (e.g. the probability of transition from a sleep bout to a wake bout will be greater near the end of bed-rest when sleep drive is lower).

CONCLUSION

It may be some time before the biochemical cycle underlying the circadian rhythm in human SCN neurons is accurately modeled. However the pacemaker function, including its response to light is, for practical applications such as jet-lag or shift-work, reasonably well understood. In comparison, the S/W system is more complex since it involves at least half a dozen brain centers. However, through the mathematical structure of a two-state system in which the transitions are described by conditional probability functions derived from human experiments it may be possible to attain a practical  understanding of the sleep and wake processes.

REFERENCES

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