Qualitative similarities between the behavior of coupled oscillators and circadian rhythms

This paper summarizes in a nonmathematical way the major properties of coupled oscillators which relate to circadian rhythms. For certain values of the coupling strength it is far easier to maintain synchrony than to achieve it among the various interacting units. This property not only simulates the free run period lability but also the effects of critical pulses.     

Further evidence that the circadian clock in Drosophila is a population of coupled ultradian oscillators

We hypothesize that ultradian oscillators are coupled to yield a composite circadian clock in Drosophila. In such a system, period would be a function of the tightness of coupling of these oscillators, increasing as coupling loosens. Ultradian oscillations would become apparent under weak coupling or in the absence of coupling. A new technique for calculating signal-to-noise ratio (SNR) for biological rhythms to characterize their precision has yielded support for this hypothesis. SNR of rhythms of the allelic series of mutations at the period (per) locus of Drosophila melanogaster were compared. Per(o) was the noisiest, grading through perL, per+, and pers, the least noisy. SNR decreases significantly with increasing period in pers, per+, and perL; per(o) typically has multiple ultradian oscillations and the lowest SNR. At least 70% of perL individuals also exhibit ultradian periodicities.     

Two coupled oscillators: simulations of the circadian pacemaker in mammalian activity rhythms

In the activity rhythms of captive small mammals a variety of features, most notably “splitting”, suggest that two coupled oscillators may constitute the pacemaker system which underlies the rhythms. A phenomenological model proposed by Pittendrigh is developed and expanded here using an explicit quantitative structure. It is found that such a system can simulate several qualitative features in the experimental data: the inter-dependence of free-running period (gt) and activity time (α) with changing light intensity described in Aschoff's rule, after-effects on τ and α of prior conditions, and the occasional existence of two stable phase relationships, with different τ values for a given light intensity, as observed in “splitting”. It is hoped that the model will suggest experiments aimed at the elucidation of the physiological basis of these phenomena.     

Human sleep and circadian rhythms: a simple model based on two coupled oscillators

We propose a model of the human circadian system. The sleep-wake and body temperature rhythms are assumed to be driven by a pair of coupled nonlinear oscillators described by phase variables alone. The novel aspect of the model is that its equations may be solved analytically. Computer simulations are used to test the model against sleep-wake data pooled from 15 studies of subjects living for weeks in unscheduled, time-free environments. On these tests the model performs about as well as the existing models, although its mathematical structure is far simpler.Supported by NIGMS Grant No. 5-R01-GM-30719-03     

Spontaneous synchronization of coupled circadian oscillators

In mammals, the circadian pacemaker, which controls daily rhythms, is located in the suprachiasmatic nucleus (SCN). Circadian oscillations are generated in individual SCN neurons by a molecular regulatory network. Cells oscillate with periods ranging from 20 to 28 h, but at the tissue level, SCN neurons display significant synchrony, suggesting a robust intercellular coupling in which neurotransmitters are assumed to play a crucial role. We present a dynamical model for the coupling of a population of circadian oscillators in the SCN. The cellular oscillator, a three-variable model, describes the core negative feedback loop of the circadian clock. The coupling mechanism is incorporated through the global level of neurotransmitter concentration. Global coupling is efficient to synchronize a population of 10,000 cells. Synchronized cells can be entrained by a 24-h light-dark cycle. Simulations of the interaction between two populations representing two regions of the SCN show that the driven population can be phase-leading. Experimentally testable predictions are: 1), phases of individual cells are governed by their intrinsic periods; and 2), efficient synchronization is achieved when the average neurotransmitter concentration would dampen individual oscillators. However, due to the global neurotransmitter oscillation, cells are effectively synchronized.     

Pregnancy-induced changes in ultradian rhythms persist in circadian arrhythmic Siberian hamsters

The impact of pregnancy and lactation on ultradian rhythms (URs) and circadian rhythms (CRs) of locomotor activity was assessed in circadian rhythmic and arrhythmic Siberian hamsters maintained in a long-day photoperiod (16 h light/day). Progressive decrements in CR robustness and amplitude over the course of gestation were accompanied by enhanced URs. Dark-phase UR period and amplitude increased during early gestation and complexity and robustness increased during late gestation. The persistence of pregnancy-associated enhancements of URs in circadian arrhythmic (ARR) hamsters suggests that reproductive modulation of the UR waveform is not dependent on coherent circadian organization. The increased incidence of dark-phase URs appeared more rapidly in ARR dams than entrained (ENTR) dams. Throughout gestation, the percentage of dams with dark-phase URs was significantly greater in the ARR group. Gestational increases in UR complexity and robustness emerged earlier and were greater in ARR than ENTR dams. The attenuation of CRs during lactation is correlated with increased expression of URs. Relaxation of circadian control of the dam's behavior may increase fitness by permitting more efficient interactions with circadian arrhythmic pups.     

Respiratory rhythms in stingless bee workers: circadian and ultradian components throughout adult development

The workers of the stingless bee, Melipona quadrifasciata, assume different tasks during their adult life. Newly emerged individuals remain inside the nest, without contact with the external environment. Maturing workers go to more peripheral regions and only the oldest, the foragers, leave the nest. As this diversity of activities implies different metabolic patterns, oxygen consumption has been measured in workers of three different ages: 24–48 h (nurses), 10–15 days (builders), and older than 25 days (foragers). Oxygen consumption of individually isolated workers was determined by intermittent respirometry, under constant darkness and temperature of 25 ± 1°C. Sets of 24-h measurements were obtained from individuals belonging to each of the three worker groups. Rhythmicity has been assessed in the daily (24 h) and ultradian (5–14 h) domains. This experimental design allowed detection of endogenous rhythms without the influence of the social group and without inflicting stress on the individuals, as would be caused by their longer isolation from the colony. Significant 24-h rhythms in oxygen consumption were present in nurses, builders and foragers; therefore, workers are rhythmic from the age of 24–48 h. However, the amplitude of the circadian rhythm changed according to age: nurses showed the lowest values, while foragers consistently presented the largest ones, about ten times larger than the amplitude of nurses’ respiratory rhythm. Ultradian frequencies were detected for all worker groups, the power and frequencies of which varied little with age. This means that the ultradian strength was relatively larger in nurses and apparently maintains some relationship with the queen’s oviposition episodes.     

Enhancement and suppression of ultradian and circadian rhythms across the female hamster reproductive cycle

The impact of ovarian hormones on hamster ultradian rhythms (URs) is unknown. We concurrently monitored URs and circadian rhythms (CRs) of home cage locomotor activity during the estrous cycle, pregnancy, and lactation of Syrian hamsters. URs with a mean period of 4-5 h were evident during the dark phase in more than 90% of females on days 1 and 2 of the estrous cycle but were significantly less prevalent on cycle days 3 and 4. The period of the UR did not vary as a function of estrous cycle stage, but at all stages, the UR period was longer in the dark than the light phase. The UR acrophase occurred significantly earlier on cycle day 4 than on days 1 and 2, and UR robustness and amplitude were reduced on days 3 and 4. Robustness, mesor, and amplitude of CRs were greater during cycle days 3 and 4; timing of the CR acrophase was delayed on day 4 relative to all other cycle days. Effects of the estrous cycle on URs were evident only during the dark phase. The proportion of hamsters displaying dark phase URs increased significantly during early and late gestation and decreased during lactation. Pregnancy significantly increased UR complexity, robustness, and amplitude. The emergence of URs over gestation was paralleled by decrements in the robustness and amplitude of CRs, which also were absent in a significant proportion of dams during lactation but re-emerged at weaning of litters. The changing endocrine profile of the estrous cycle, hormonal dynamics of pregnancy and lactation, and nursing demands placed on dams are each associated with alterations in the expression of ultradian and circadian locomotor rhythms. Diminution of CRs and augmentation of URs may afford greater behavioral flexibility during life stages when interactions with mates and offspring are less predictable.     

A case for multiple oscillators controlling different circadian rhythms in Drosophila melanogaster

A population of the fruit fly Drosophila melanogaster was raised in periodic light/dark (LD) cycles of 12:12 h for about 35 generations. Eclosion, locomotor activity, and oviposition were found to be rhythmic in these flies, when assayed in constant laboratory conditions where the light intensity, temperature, humidity and other factors which could possibly act as time cue for these flies, were kept constant. These rhythms also entrained to a LD cycle of 12:12 h in the laboratory with each of them adopting a different temporal niche. The free-running periods (tau) of the eclosion, locomotor activity and oviposition rhythms were significantly different from each other. The peak of eclosion and the onset of locomotor activity occurred during the light phase of the LD cycle, whereas the peak of oviposition was found to occur during the dark phase of the LD cycle. Based on these results, we conclude that different circadian oscillators control the eclosion, locomotor activity and oviposition rhythms in the fruit fly D. melanogaster.     

Temporal pattern of LH secretion: regulation by multiple ultradian oscillators versus a single circadian oscillator

The possibility that the 24h rhythm output is the composite expression of ultradian oscillators of varying periodicities was examined by assessing the effect of external continuously or pulsed (20-minute) Gonadotropinreleasing hormone (GnRH) infusions on in vitro luteinizing hormone (LH) release patterns from female mouse pituitaries during 38h study spans. Applying stepwise analyses (spectral, cosine fit, best-fit curve, and peak detection analyses) revealed the waveform shape of LH release output patterns over time is composed of several ultradian oscillations of different periods. The results further substantiated previous observations indicating the pituitary functions as an autonomous clock. The GnRH oscillator functions as a pulse generator and amplitude regulator, but it is not the oscillator that drives the ultradian LH release rhythms. At different stages of the estrus cycle, the effect of GnRH on the expression of ultradian periodicities varies, resulting in the modification of their amplitudes but not their periods. The functional output from the system of ultradian oscillators may superimpose a “circadian or infradian phenotype” on the observed secretion pattern. An “amplitude control” hypothesis is proposed: The temporal pattern of LH release is governed by several oscillators that function in conjunction with one another and are regulated by an amplitude-controlled mechanism. Simulated models show that such a mechanism results in better adaptive response to environmental requirements than does a single circadian oscillator. (Chronobiology International, 18(3), 399-412, 2001)     

Two coupled neural oscillators as a model of the circadian pacemaker

Pittendrigh first found that the circadian rhythm of locomotor activity in nocturnal rodents split into two components. Hoffman then reported that the splitting phenomenon was even more reproducible in the small diurnal primate Tupaia. These “splitting” experiments and many other experiments suggest that two coupled oscillators may constitute the circadian pacemaker system. Pittendrigh proposed a phenomenological two-oscillator model. Daan and Berde developed a quantitative model assuming that the interaction between the two constituent oscillators is by instantaneous resets. Their model system can simulate several qualitative features in the experimental data. As the assumption of instantaneous resets seems to be unnatural, we study two limit cycle oscillators, which are coupled continuously to each other, as a model of the circadian pacemaker. We assume the following points, (i) One oscillator in a resting state does not affect another oscillator, (ii) Two oscillators are identical, (iii) The coupling is symmetrical. By the theory of Hopf bifurcation it is found that the general two-oscillator system has two stable periodic solutions. One is the in-phase solution where the two constituent oscillators oscillate in phase synchrony. Another is the anti-phase solution where the two oscillators oscillate 180 ° out of phase. The former corresponds to a single pattern of locomotor activity and the latter corresponds to a splitting pattern. Furthermore, we study specific two-neural oscillators, which are linearly coupled to each other. By the method of secondary bifurcation we find that the model shows simultaneous stability of the two alternative phase relationships and the hysteresis phenomena found in Tupaia. A natural period of the uncoupled constituent oscillator is longer than that of the in-phase solution but it is shorter than that of the anti-phase solution. This is in agreement with the data of Tupaia.     

Effects of suprachiasmatic nuclei lesions on circadian and ultradian rhythms in body temperature in ocular enucleated rats

Abstract

To test the hypothesis that an oscillator located outside the suprachiasmatic nuclei (SCN) controls the circadian rhythm of body temperature, we conducted a study with 14 blinded rats, 10 of which receiving a SCN lesion. Body temperature was automatically and continuously recorded for about one month by intraperitoneal radio transmitters. Food intake, drinking and locomotor activity were also recorded. Periodograms revealed that 3 rats with histologically verified total bilateral SCN lesions did not exhibit any circadian rhythmicity. The 7 other rats appeared to have partial lesions. They showed shortening of period and severe amplitude reduction in all functions. Thus, no support was found for the hypothesis of a separate circadian ‘temperature oscillator’ located outside the SCN. Nevertheless, after large partial lesions body temperature showed more persistency than some of the other behavioral rhythms.

Ultradian rhythms in temperature persisted after partial and total lesions. Other functions showed parallel ultradian rhythms. In intact rats the ultradian peaks were restricted predominantly to the subjective night. After total lesions these peaks became more or less homogeneously distributed in time but more heterogeneously after partial lesions. So the SCN plays a role in the temporal structure of ultradian rhythms but does not generate them. Non‐24‐hour actograms showed instabilities of period and phase of ultradian rhythms. Intact and lesioned rats were similar with respect to the mean (about 3.5 hrs) and standard deviation (about 1.5 hrs) of ultradian periods in temperature. These features indicate that a mechanism outside the SCN is underlying ultradian rhythmicity, capable of generating short‐term oscillations. Two approaches, homeostatic sleep‐wake relaxation oscillations and multiple circadian oscillators, are discussed.     

Modeling the behavior of coupled cellular circadian oscillators in the suprachiasmatic nucleus

The suprachiasmatic nucleus (SCN) in the hypothalamus is the site of the master circadian clock in mammals, a complex tissue composed of multiple, coupled, single-cell circadian oscillators. Mathematical modeling is now providing insights on how individual SCN cells might interact and assemble to create an integrated pacemaker that governs the circadian behavior of whole animals. In this article, we will discuss the neurobiological constraints for modeling SCN behavior, system precision, implications of cellular heterogeneity, and analysis of heterogeneously coupled oscillator networks. Mathematical approaches will be critical for better understanding intercellular interactions within the SCN.     

Socially synchronized circadian oscillators

Daily rhythms of physiology and behaviour are governed by an endogenous timekeeping mechanism (a circadian ‘clock’). The alternation of environmental light and darkness synchronizes (entrains) these rhythms to the natural day–night cycle, and underlying mechanisms have been investigated using singly housed animals in the laboratory. But, most species ordinarily would not live out their lives in such seclusion; in their natural habitats, they interact with other individuals, and some live in colonies with highly developed social structures requiring temporal synchronization. Social cues may thus be critical to the adaptive function of the circadian system, but elucidating their role and the responsible mechanisms has proven elusive. Here, we highlight three model systems that are now being applied to understanding the biology of socially synchronized circadian oscillators: the fruitfly, with its powerful array of molecular genetic tools; the honeybee, with its complex natural society and clear division of labour; and, at a different level of biological organization, the rodent suprachiasmatic nucleus, site of the brain''s circadian clock, with its network of mutually coupled single-cell oscillators. Analyses at the ‘group’ level of circadian organization will likely generate a more complex, but ultimately more comprehensive, view of clocks and rhythms and their contribution to fitness in nature.     

Robust synchronization of coupled circadian and cell cycle oscillators in single mammalian cells

Circadian cycles and cell cycles are two fundamental periodic processes with a period in the range of 1 day. Consequently, coupling between such cycles can lead to synchronization. Here, we estimated the mutual interactions between the two oscillators by time‐lapse imaging of single mammalian NIH3T3 fibroblasts during several days. The analysis of thousands of circadian cycles in dividing cells clearly indicated that both oscillators tick in a 1:1 mode‐locked state, with cell divisions occurring tightly 5 h before the peak in circadian Rev‐Erbα‐YFP reporter expression. In principle, such synchrony may be caused by either unidirectional or bidirectional coupling. While gating of cell division by the circadian cycle has been most studied, our data combined with stochastic modeling unambiguously show that the reverse coupling is predominant in NIH3T3 cells. Moreover, temperature, genetic, and pharmacological perturbations showed that the two interacting cellular oscillators adopt a synchronized state that is highly robust over a wide range of parameters. These findings have implications for circadian function in proliferative tissues, including epidermis, immune cells, and cancer.     

Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators

In mammals, circadian rhythms are driven by a pacemaker located in the suprachiasmatic nuclei (SCN) of the anterior hypothalamus. The firing rate of neurons within the SCN exhibits a circadian rhythm. There is evidence that individual neurons within the SCN act as circadian oscillators. Rhythm generation in the SCN was therefore modeled by a system of self-sustained oscillators. The model is composed of up to 10000 oscillatory elements arranged in a square array. Each oscillator has its own (randomly determined) intrinsic period reflecting the widely dispersed periods observed in the SCN. The model behavior was investigated mainly in the absence of synchronizing zeitgebers. Due to local coupling the oscillators synchronized and an overall rhythm emerged. This indicates that a locally coupled system is capable of integrating the output of individual clock cells with widely dispersed periods. The period of the global output (average of all oscillators) corresponded to the average of the intrinsic periods and was stable even for small amplitudes and during transients. Noise, reflecting biological fluctuations at the cellular level, distorted the global rhythm in small arrays. The period of the rhythm could be stabilized by increasing the array size, which thus increased the robustness against noise. Since different regions of the SCN have separate output pathways, the array of oscillators was subdivided into four quadrants. Sudden deviations of periodicity sometimes appeared in one quadrant, while the periods of the other quadrants were largely unaffected. This result could represent a model for splitting, which has been observed in animal experiments. In summary, the multi-oscillator model of the SCN showed a broad repertoire of dynamic patterns, revealed a stable period (even during transients) with robustness against noise, and was able to account for such a complex physiological behavior as splitting.