Journal of Biological Rhythms

We examine theoretical models for circadian oscillations based on transcriptional regulation in Drosophila and Neurospora. For Drosophila, the molecular model is based on the negative feedback exerted on the expression of the per and tim genes by the complex formed between the PER and TIM proteins. For Neurospora, similarly, the model relies on the feedback exerted on the expression of the frq gene by its protein product FRQ. In both models, sustained rhythmic variations in protein and mRNA levels occur in continuous darkness, in the form of limit cycle oscillations. The effect of light on circadian rhythms is taken into account in the models by considering that it triggers degradation of the TIM protein in Drosophila, and frq transcription in Neurospora. When incorporating the control exerted by light at the molecular level, we show that the models can account for the entrainment of circadian rhythms by light-dark cycles and for the damping of the oscillations in constant light, though such damping occurs more readily in the Drosophila model. The models account for the phase shifts induced by light pulses and allow the construction of phase response curves. These compare well with experimental results obtained in Drosophila. The model for Drosophila shows that when applied at the appropriate phase, light pulses of appropriate duration and magnitude can permanently or transiently suppress circadian rhythmicity. We investigate the effects of the magnitude of light-induced changes on oscillatory behavior. Finally, we discuss the common and distinctive features of circadian oscillations in the two organisms.

Circadian systems direct many metabolic parameters and, at the same time, they appear to be exquisitely shielded from metabolic variations. Although the recent decade of circadian research has brought insights into how circadian periodicity may be generated at the molecular level, little is known about the relationship between this molecular feedback loop and metabolism both at the cellular and at the organismic level. In this theoretical paper, we conjecture about the interdependence between circadian rhythmicity and metabolism. A mathematical model based on the chemical reactions of photosynthesis demonstrates that metabolism as such may generate rhythmicity in the circadian range. Two additional models look at the possible function of feedback loops outside of the circadian oscillator. These feedback loops contribute to the robustness and sustainability of circadian oscillations and to compensation for long-and short-term metabolic variations. The specific circadian property of temperature compensation is put into the context of metabolism. As such, it represents a general compensatory mechanism that shields the clock from metabolic variations.

Circadian rhythm generation in the suprachiasmatic nucleus was modeled by locally coupled self-sustained oscillators. The model is composed of 10,000 oscillators, arranged in a square array. Coupling between oscillators and standard deviation of (randomly determined) intrinsic oscillator periods were varied. A stable overall rhythm emerged. The model behavior was investigated for phase shifts of a 24-h zeitgeber cycle. Prolongation of either the dark or the light phase resulted in a lengthening of the period, whereas shortening of the dark or the light phase shortened the period. The model's response to shifts in the light-dark cycle was dependent only on the extent of the shift and was insensitive to changes in parameters. Phase response curves (PRC) and amplitude response curves were determined for single and triple 5-h light pulses (1000 lux). Single pulses lead to type 1 PRCs with larger phase shifts for weak coupling. Triple pulses generally evoked type 1 PRCs with the exception of weak coupling, where a type 0 PRC was observed.

This article focuses on the Goodwin oscillator and related minimal models, which describe negative feedback schemes that are of relevance for the circadian rhythms in Neurospora, Drosophila, and probably also in mammals. The temperature behavior of clock mutants in Neurospora crassa and Drosophila melanogaster are well described by the Goodwin model, at least on a semi-quantitative level. A similar semi-quantitative description has been found for Neurospora crassa phase response curves with respect to moderate temperature pulses, heat shock pulses, and pulses of cycloheximide. A characteristic feature in the Goodwin and related models is that degradation of clock-mRNA and clock protein species plays an important role in the control of the oscillator's period. As predicted by this feature, recent experimental results from Neurospora crassa indicate that the clock (FRQ) protein of the long period mutant frq ⁷ is degraded approximately twice as slow as the corresponding wild-type protein. Quantitative RT-PCR indicates that experimental frq ⁷ -mRNA concentrations are significantly higher than wild-type levels. The latter findings cannot be modeled by the Goodwin oscillator.Therefore,athresholdinhibitionmechanismoftranscriptionis proposed.

The New Zealand weta, Hemideina thoracica, is a nocturnal orthopteran insect which emerges from holes in trees or from under bark soon after sunset to forage for several hours on plant and animal material before returning to its refuge before dawn. In tests of the internal clock hypothesis it exhibits clear circadian locomotor rhythms in which the period is initially somewhat less than 24 h, but frequently spontaneously increases to over 25 h. The rhythms are entrainable by light and temperature cycles, obey Aschoff's Law and are temperature compensated. A single oscillator feedback model accounts for these basic properties of the weta clock, but does not explain a variety of examples of rhythm lability, such as day skipping, spontaneous change in period, scalloping and desynchrony typically found in the real data. To account for these characteristics the model is expanded into two linked populations of oscillators, which retain the basic properties of the simple model and in addition interact through their coupling to show the various types of free-run lability. To make these control systems models compatible with the molecular interpretation of circadian biology, each of the components in the feedback loop is matched with molecular function and structure.

In 1990, Kronauer proposed a mathematical model of the effects of light on the human circadian pacemaker. This study presents several refinements to Kronauer's original model of the pacemaker that enable it to predict more accurately the experimental results from a number of different studies of the effects of the intensity, timing, and duration of light stimuli on the human circadian pacemaker. These refinements include the following: The van der Pol oscillator from Kronauer’s model has been replaced with a higher order limit cycle oscillator so that the system’s amplitude recovery is slower near the singularity and faster near the limit cycle; the phase and amplitude of the circadian rhythm in sensitivity to light from Kronauer’s model has been refined so that the peak sensitivity to light on the limit cycle now occurs 4 h before the core body temperature minimum (CBT_min) and is three times as great as the minimum sensitivity on the limit cycle; the critical phase (at which type 1 phase response curves [PRCs] can be distinguished from type 0 PRCs) that occurs at CBT_min now corresponds to 0.8 h after the minimum of x (x _min) in this refined model rather than to the exact timing of x _min as in Kronauer’s model; a direct effect of light on circadian period was incorporated into the model such that as light intensity increases, the period decreases, which is in accordance with Aschoff’s rule.

The authors' previous models have been able to describe accurately the effects of extended (5 h) bright-light (>4000 lux) stimuli on the phase and amplitude of the human circadian pacemaker, but they are not sufficient to represent the surprising human sensitivity to both brief pulses of bright light and light of more moderate intensities. Therefore, the authors have devised a new model in which a dynamic stimulus processor (Process L) intervenes between the light stimuli and the traditional representation of the circadian pacemaker as a self-sustaining limit-cycle oscillator (Process P). The overall model incorporating Process L and Process P is intended to allow the prediction of phase shifts to photic stimuli of any temporal pattern (extended and brief light episodes) and any light intensity in the photopic range. Two time constants emerge in the Process Lmodel: the characteristic duration for necessary bright-light pulses to achieve their full effect (5-10 min) and the characteristic stimulus-free (dark) interval that can be tolerated without incurring an excessive penalty in phase shifting (30-80 min). The effect of reducing light intensity is incorporated in Process L as an extension of the time necessary for the light pulse to be fully realized (a power-law relation between time and intensity). This new model generates a number of new testable hypotheses, including the surprising prediction that 24-h cycles consisting of8hof darkness and 16 h of only 3.5 lux would be capable of entraining a large fraction of the adult population (45%). Experimental data on the response of the human circadian system to lower light intensities and briefer stimuli are needed to allow for further refinement and validation of the model proposed here.

A model based on the van der Pol equation has been developed to predict the pattern of adaptation of aircrew and other travellers to rapid time-zone transitions, when the exposure to light cannot be quantified. The parameters of the model include the stiffness ([.mu]) and the intrinsic period (T ₀), which together define the free-running period, and the external force (F). The parameter values were estimated by using a simplex minimization technique to fit the output from the model to body temperature data from 12 individuals before, and over a 12-day period immediately after, a 10-h eastward transition between London and Sydney. Data were collected at three equally spaced points during each sleep period and at the end of four 45-min rest periods during the day. The fitting procedure enabled the parameters of the temperature rhythm to be estimated after correcting for the masking effect of sleep. The average estimates of [.mu] (0.38 h) and T ₀ (24.24 h) were close to earlier estimates based on forced desynchronization experiments, and the mean free-running period, calculated from these, was 24.50 h. The mean value of the external force F (0.54) was surprisingly high, and this may reflect the strong outdoor light levels during the days in Sydney. Estimates of phase, based on the model solutions, suggested that 11 subjects adapted by a phase delay and 1 by a phase advance. However, the amplitude of the rhythms was much reduced at times when the phase was changing rapidly. Simulations using the range of the model parameters for the 12 individuals predicted that adaptation to within 1 h after a 10-h eastward transition would be achieved within between 3 and 11 days. However, since these predictions are dependent on the choice of external force, estimates may need to be more conservative in real-life situations when light exposure cannot be measured.

The patterns of light intensity to which humans expose their circadian pacemakers in daily life are very irregular and vary greatly from day to day. The circadian pacemaker can adjust to such irregular exposure patterns by daily phase shifts, such as summarized in a phase response curve. It is demonstrated in this paper on the basis of computer simulations applying actually recorded human light exposure patterns that the pacemaker can substantially improve its accuracy by an additional response to light: For that purpose, it should additionally change its angular velocity (and consequently its period [.tau]) in response to light. Reductions of [.tau] in response to light in the morning and increases of [.tau] in response to light in the evening can lead to an increase in entrained pacemaker accuracy with about 25%. Circadian pacemakers have evolved as accurate internal representations of external time, and investigated diurnal mammals all seem to respond to light by changing the period of their circadian pacemaker (in addition to shifting phase). The authors suggest that also human circadian systems take advantage of this possibility and that their pacemakers respond to light by shifting phase and changing period. As a consequence of this postulated mechanism, the simulations demonstrate that the period of the pacemaker under normally entrained conditions is 24 h. The maximum accuracy corresponds to a day-to-day standard deviation of the time of phase 0 of circa 15 min. This is considerably more accurate than the light signal humans usually perceive.

Numerous studies have used the classic van der Pol oscillator, which contains a cubic nonlinearity, to model the effect of light on the human circadian pacemaker. Jewett and Kronauer demonstrated that Aschoff's rule could be incorporated into van der Pol type models and used a van der Pol type oscillator with higher order nonlinearities. Kronauer, Forger, and Jewett have proposed a model for light preprocessing, Process L, representing a biochemical process that converts a light signal into an effective drive on the circadian pacemaker. In the paper presented here, the authors use the classic van der Pol oscillator with Process L and Jewett and Kronauer's model of Aschoff's rule to model the human circadian pacemaker. This simpler cubic model predicts the results of a three-pulse human phase response curve experiment and a two-pulse amplitude reduction study with as much, or more, accuracy as the models of Jewett and Kronauer and Kronauer, Forger, and Jewett, which both employ a nonlinearity of degree 7. This suggests that this simpler cubic model should be considered as a potential alternative to other models of the human circadian system currently available.

Thermoregulatory mechanisms were hypothesized to provide primary control of non-rapid-eye-movement sleep (NREM). On the basis of this hypothesis, we incorporated the thermoregulatory feedback loops mediated by the “heat memory,” heat load, and loss processes associated with sleep-wake cycles, which were modulated by two circadian oscillators. In addition, hypnogenic warm-sensitive neurons (HWSNs) were assumed to integrate thermoregulation and NREM control. The heat memory described above could be mediated by some sleep-promoting substances. In this paper, considering the possible carrier of the heat memory, its losing process is newly included in the model. The newly developed model can generate the appropriate features of human sleepwake patterns. One of the special features of the model is to generate the bimodal distribution of the sleepiness. This bimodality becomes distinct, as the losing rate of the heat memory decreases or the amplitude of the Y oscillator increases. The theoretical analysis shows the losing rate of the heat memory control's rapidity of model response to a thermal perturbation, which is confirmed by simulating the responses with various losing rates to transient heat loads (“heat load pulse”). The sleepiness exhibits large responses to the heat load pulses applied in the early and late phases of wake period, while the response is significantly reduced to the pulse applied in the supposed wake-maintenance zone. This bimodality of the response appears to reflect the sensitivity of the HWSNs. In addition, the early pulse raises the immediate sleepiness rather than the nocturnal sleepiness, while the heat load pulse applied in the later phase of waking period significantly raises the sleepiness during a nocturnal sleep. In simulations of sleep deprivation, the discontinuous relationship between recovery sleep length and deprivation time is reproduced, where the critical sleep deprivation time at which the recovery sleep length jumps is extended as the losing rate increases. This is possibly due to the dissipation of the heat memory accumulated by the sleep deprivation. The simulation results here qualitatively reproduce the experimental observations or predict the intriguing phenomena of human circadian rhythms. Therefore, our model could provide a novel framework for investigating the relationship between thermoregulation and sleep control processes.

According to the two-process model of sleep regulation, the timing and structure of sleep are determined by the interaction of a homeostatic and a circadian process. The original qualitative model was elaborated to quantitative versions that included the ultradian dynamics of sleep in relation to the non-REM-REM sleep cycle. The time course of EEG slow-wave activity, the major marker of non-REM sleep homeostasis, as well as daytime alertness were simulated successfully for a considerable number of experimental protocols. They include sleep after partial sleep deprivation and daytime napping, sleep in habitual short and long sleepers, and alertness in a forced desynchrony protocol or during an extended photoperiod. Simulations revealed that internal desynchronization can be obtained for different shapes of the thresholds. New developments include the analysis of the waking EEG to delineate homeostatic and circadian processes, studies of REM sleep homeostasis, and recent evidence for local, use-dependent sleep processes. Moreover, nonlinear interactions between homeostatic and circadian processes were identified. In the past two decades, models have contributed considerably to conceptualizing and analyzing the major processes underlying sleep regulation, and they are likely to play an important role in future advances in the field.

Quantitative models have been developed to describe salient aspects of human sleep regulation. The two-process model of sleep regulation and the thermoregulatory model of sleep control highlight the interaction between sleep homeostasis and circadian rhythmicity and the association between sleep and temperature regulation, respectively. These models have been successful and inspiring, but continuing progress remains dependent on rigorous testing of some of their basic assumptions. Whereas it has been established that EEG slowwave activity is a marker of sleep homeostasis, its causal role in regulating the timing of sleep and wakefulness remains to be demonstrated conclusively. Likewise, the causal role of the temperature regulatory system in sleep timing requires further investigation. In both models, many parameters have yet to be associated with specific physiologic processes. This makes it challenging, at least within the framework of these models, to account for interindividual differences or age-related changes in such features as sleep duration and sleep timing, as well as changes in the phase angle between the sleep-wake cycle and accepted markers of the circadian pacemaker, such as the body temperature or melatonin rhythm. Although the models may describe adequately global sleep patterns and their circadian modulation, detailed modeling of the frequent short awakenings from, and the subsequent transitions back to, sleep, as well as the variation of the propensity to awaken across the ultradian non-REM-REM cycle, is not addressed. Incoporation of these aspects of sleep in mathematical models of sleep regulation may further our understanding of a key aspect of sleep regulation, that is, its timing.

This paper starts by summarizing the development and refinement of the additive three-process model of alertness first published by Folkard and Åkerstedt in 1987. It reviews some of the successes that have been achieved by the model in not only predicting variations in subjective alertness on abnormal sleep-wake schedules but also in accounting for objective measures of sleep latency and duration. Nevertheless, predictions derived from the model concerning alertness on different shifts, and over successive night shifts, are difficult to reconcile with published data on accident risk. In light of this, we have examined two large sets of alertness ratings with a view to further refining the model and identifying additional factors that may influence alertness at any given point in time. Our results indicate that, at least for the range of sleep durations and wake-up times commonly found on rotating shift systems, we may assume the phase of the endogenous circadian component of alertness (process C) to be “set” by the time of waking. Such an assumption considerably enhanced the predictive power of the model and yielded remarkably similar phase estimates to those obtained by maximizing the post-hoc fit of the model. We then examined the manner in which obtained ratings differed from predicted values over a complete 8-day cycle of two, 12-h shift systems. This revealed a pronounced “first night compensation effect” that resulted in shift workers rating themselves as progressively more alert than would be predicted over the course of the first night shift. However, this appeared to be achieved only at the cost of lowered ratings on the second night shift. Finally, we were able to identify a “time on shift” effect whereby, with the exception of the first night shift, alertness ratings decreased over the course of each shift before showing a modest “end effect.” We conclude that the identification of these additional components offers the possibility that in the future we may be able to predict trends in accident risk on abnormal sleep-wake schedules.

The authors present here mathematical models in which levels of subjective alertness and cognitive throughput are predicted by three components that interact with one another in a nonlinear manner. These components are (1) a homeostatic component (H) that falls in a sigmoidal manner during wake and rises in a saturating exponential manner at a rate that is determined by circadian phase during sleep; (2) a circadian component (C) that is a function of the output of our mathematical model of the effect of light on the circadian pacemaker, with the amplitude further regulated by the level of H; and (3) a sleep inertia component (W) that rises in a saturating exponential manner after waketime. The authors first construct initial models of subjective alertness and cognitive throughput based on the results of sleep inertia studies, sleep deprivation studies initiated across all circadian phases, 28-h forced desynchrony studies, and alertness and performance dose response curves to sleep. These initial models are then refined using data from nearly one hundred fifty 30- to 50-h sleep deprivation studies in which subjects woke at their habitual times. The interactive three-component models presented here are able to predict even the fine details of neurobehavioral data from sleep deprivation studies and, after further validation, may provide a powerful tool for the design of safe shift work and travel schedules, including those in which people are exposed to unusual patterns of light.

Modeling human neurobehavioral functions has the goal of identifying work-rest schedules that are safer and more productive. The models of Folkard et al. and of Jewett and Kronauer illustrate excellent progress toward this goal. Examination of these models reveals four additional areas that need to be addressed to facilitate continued development of accurate models of neurobehavioral functions. (1) The choice of neurobehavioral metrics may have a significant influence on model development. The lack of correlation among different neurobehavioral measures may make comparisons of models difficult. Many neurobehavioral measures are confounded by secondary and random error variance that can lead to model distortion. Although different models may ultimately be required for different neurobehavioral functions, measures that have been extensively validated to be sensitive to circadian variation and sleep loss should take priority in model development. (2) Because error variance in neurobehavioral outcomes can be substantial in uncontrolled environments, model validation should proceed from controlled laboratory protocols to real-world scenarios. Once validated, the ability of a model to predict field data can be tested. (3) While neurobehavioral models have been developed to predict behavior over time (i.e., within-subjects), to be useful in the real world, models will also ultimately have to provide estimates of between-subject variation in vulnerability to neurobehavioral dysfunction during night work or sleep loss (e.g., younger versus older workers). (4) Finally, to be theoretically accurate and practically useful, models of human neurobehavioral functions should be able to predict both cumulative effects (i.e., across days or weeks) and the influence of countermeasures (e.g., light, naps, caffeine).

Mathematical models have played an important role in the analysis of circadian systems. The models include simulation of differential equation systems to assess the dynamic properties of a circadian system and the use of statistical models, primarily harmonic regression methods, to assess the static properties of the system. The dynamical behaviors characterized by the simulation studies are the response of the circadian pacemaker to light, its rate of decay to its limit cycle, and its response to the rest-activity cycle. The static properties are phase, amplitude, and period of the intrinsic oscillator. Formal statistical methods are not routinely employed in simulation studies, and therefore the uncertainty in inferences based on the differential equation models and their sensitivity to model specification and parameter estimation error cannot be evaluated. The harmonic regression models allow formal statistical analysis of static but not dynamical features of the circadian pacemaker. The authors present a paradigm for analyzing circadian data based on the Box iterative scheme for statistical model building. The paradigm unifies the differential equation–based simulations (direct problem) and the model fitting approach using harmonic regression techniques (inverse problem) under a single schema. The framework is illustrated with the analysis of a core-temperature data series collected under a forced desynchrony protocol. The Box iterative paradigm provides a framework for systematically constructing and analyzing models of circadian data.

The classical power spectrum, computed in the frequency domain, outranks traditionally used periodograms derived in the time domain (such as the [.chi]² periodogram) regarding the search for biological rhythms. Unfortunately, classical power spectral analysis is not possible with unequally spaced data (e.g., time series with missing data). The Lomb-Scargle periodogram fixes this shortcoming. However, peak detection in the Lomb-Scargle periodogram of unequally spaced data requires some careful consideration. To guide researchers in the proper evaluation of detected peaks, therefore, a novel procedure and a computer program have recently become available. It is recommended that the Lomb-Scargle periodogram be the default method of periodogram analysis in future biomedical applications of rhythm investigation.