Hierarchically coupled ultradian oscillators generating robust circadian rhythms

Dynamics of hierarchical weighted networks of van der Pol oscillators

We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.

The Sudden Infant Death Syndrome (SIDS): A Neuro-Molecular Hypothesis

Most of the children who die before age 1 in developed countries do so for unknown reasons, and these deaths are attributed to the sudden infant death syndrome (SIDS). Prospective cardiorespiratory monitoring of infants has revealed that SIDS victims have subtle differences in breathing and heartbeat patterns compared to controls. Because death must involve cardiorespiratory arrest, a straightforward explanation for SIDS is failure on the part of pacemaker neurons controlling the rhythmical processes of breathing or heartbeat. Genes coding for hyperpolarization-activated pacemaker cation channels have recently been isolated and are expressed in the heart and the brain. The authors propose that mutations in these genes and in other genes required for cardiorespiratory pacemaker activity will predispose an individual to SIDS during a window of vulnerability present in the first year of life. Furthermore, mutations in clock genes can alter a variety of rhythmical processes and may indirectly disturb cardiorespiratory function as well.

Biological Timekeeping: The Business of a Blind Watchmaker

Fundamental understanding of life depends on both structural and functional details at the molecular level. Continually improving means of measurement of spatial and dynamic properties of biochemical constituents and cellular components complement studies of whole organisms. Integration of the interaction of components to provide coherent behaviour depends on highly elaborate orchestration in space and time. Whereas spatial information on a nanometre resolution is available, and fast dynamic analyses provide biochemical reaction rates measured in nanoseconds, functional coordination of the system requires integrated time dependence. While we are well aware of the special complexity of living organisms, appreciation of temporal scales and their organisation in time is still fragmentary. This article summarises current developments in research on biological time on scales from nanoseconds to years, the networks that connect different time domains and the oscillations, rhythms and biological clocks that coordinate and synchronise the complexity of the living state.

“It is the pattern maintained by this homeostasis, which is the touchstone of our personal identity. Our tissues change as we live: the food we eat and the air we breathe become flesh of our flesh, and bone of our bone, and the momentary elements of our flesh and bone pass out of our body every day with our excreta. We are but whirlpools in a river of ever-flowing water. We are not the stuff that abides, but patterns that perpetuate themselves”60.

Wiener, 1954

“What are called structures are slow processes of long duration, functions are quick processes of short duration”61.

Von Bertalanffy, 1952

A mathematical model provides mechanistic links to temporal patterns in Drosophila daily activity

Background

Circadian clocks are endogenous biochemical oscillators that control daily behavioral rhythms in all living organisms. In fruit fly, the circadian rhythms are typically studied using power spectra of multiday behavioral recordings. Despite decades of study, a quantitative understanding of the temporal shape of Drosophila locomotor rhythms is missing. Locomotor recordings have been used mostly to extract the period of the circadian clock, leaving these data-rich time series largely underutilized. The power spectra of Drosophila and mouse locomotion often show multiple peaks in addition to the expected at T ~ 24 h. Several theoretical and experimental studies have previously used these data to examine interactions between the circadian and other endogenous rhythms, in some cases, attributing peaks in the T < 24 h regime to ultradian oscillators. However, the analysis of fly locomotion was typically performed without considering the shape of time series, while the shape of the signal plays important role in its power spectrum. To account for locomotion patterns in circadian studies we construct a mathematical model of fly activity. Our model allows careful analysis of the temporal shape of behavioral recordings and can provide important information about biochemical mechanisms that control fly activity.

Results

Here we propose a mathematical model with four exponential terms and a single period of oscillation that closely reproduces the shape of the locomotor data in both time and frequency domains. Using our model, we reexamine interactions between the circadian and other endogenous rhythms and show that the proposed single-period waveform is sufficient to explain the position and height of >88 % of spectral peaks in the locomotion of wild-type and circadian mutants of Drosophila. In the time domain, we find the timescales of the exponentials in our model to be ~1.5 h⁻¹ on average.

Conclusions

Our results indicate that multiple spectral peaks from fly locomotion are simply harmonics of the circadian period rather than independent ultradian oscillators as previously reported. From timescales of the exponentials we hypothesize that model rates reflect activity of the neuropeptides that likely transduce signals of the circadian clock and the sleep–wake homeostat to shape behavioral outputs.

Electronic supplementary material

The online version of this article (doi:10.1186/s12868-016-0248-9) contains supplementary material, which is available to authorized users.

Ultradian rhythm unmasked in the Pdf clock mutant of Drosophila

A diverse range of organisms shows physiological and behavioural rhythms with various periods. Extensive studies have been performed to elucidate the molecular mechanisms of circadian rhythms with an approximately 24 h period in both Drosophila and mammals, while less attention has been paid to ultradian rhythms with shorter periods. We used a video-tracking method to monitor the movement of single flies, and clear ultradian rhythms were detected in the locomotor behaviour of wild type and clock mutant flies kept under constant dark conditions. In particular, the Pigment-dispersing factor mutant (Pdf 01) demonstrated a precise and robust ultradian rhythmicity, which was not temperature compensated. Our results suggest that Drosophila has an endogenous ultradian oscillator that is masked by circadian rhythmic behaviours.

Spike firing pattern of output neurons of the Limulus circadian clock

The lateral eyes of the horseshoe crab (Limulus polyphemus) show a daily rhythm in visual sensitivity that is mediated by efferent nerve signals from a circadian clock in the crab's brain. How these signals communicate circadian messages is not known for this or other animals. Here the authors describe in quantitative detail the spike firing pattern of clock output neurons in living horseshoe crabs and discuss its possible significance to clock organization and function. Efferent fiber spike trains were recorded extracellularly for several hours to days, and in some cases, the electroretinogram was simultaneously acquired to monitor eye sensitivity. Statistical features of single- and multifiber recordings were characterized via interval distribution, serial correlation, and power spectral analysis. The authors report that efferent feedback to the eyes has several scales of temporal structure, consisting of multicellular bursts of spikes that group into clusters and packets of clusters that repeat throughout the night and disappear during the day. Except near dusk and dawn, the bursts occur every 1 to 2 sec in clusters of 10 to 30 bursts separated by a minute or two of silence. Within a burst, each output neuron typically fires a single spike with a preferred order, and intervals between bursts and clusters are positively correlated in length. The authors also report that efferent activity is strongly modulated by light at night and that just a brief flash has lasting impact on clock output. The multilayered firing pattern is likely important for driving circadian rhythms in the eye and other target organs.

Modeling circadian clocks: Roles, advantages, and limitations

Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

A model for generating circadian rhythm by coupling ultradian oscillators

Background

Organisms ranging from humans to cyanobacteria undergo circadian rhythm, that is, variations in behavior that cycle over a period about 24 hours in length. A fundamental property of circadian rhythm is that it is free-running, and continues with a period close to 24 hours in the absence of light cycles or other external cues. Regulatory networks involving feedback inhibition and feedforward stimulation of mRNA transcription and translation are thought to be critical for many circadian mechanisms, and genes coding for essential components of circadian rhythm have been identified in several organisms. However, it is not clear how such components are organized to generate a circadian oscillation.

Results

We propose a model in which two independent transcriptional-translational oscillators with periods much shorter than 24 hours are coupled to drive a forced oscillator that has a circadian period, using mechanisms and parameters of conventional molecular biology. Furthermore, the resulting circadian oscillator can be entrained by an external light-dark cycle through known mechanisms. We rationalize the mathematical basis for the observed behavior of the model, and show that the behavior is not dependent on the details of the component ultradian oscillators but occurs even if quite generalized basic oscillators are used.

Conclusion

We conclude that coupled, independent, transcriptional-translational oscillators with relatively short periods can be the basis for circadian oscillators. The resulting circadian oscillator can be entrained by 24-hour light-dark cycles, and the model suggests a mechanism for its evolution.

The association of hypertension and secondary cardiovascular disease with sleep-disordered breathing

We present a review of the English-language literature from 1972 through 2000 pertaining to systemic high BP in patients with sleep-disordered breathing (SDB). We reviewed studies assessing the relationship between obstructive sleep apnea, central sleep apnea or periodic breathing, and systemic high BP, and present an approach to the management of these patients. Complications of obesity and the role of the sympathetic nervous system are reviewed as well. It is the aim of these reviews to draw qualified conclusions, based on the current literature, with regard to SDB as a causative or contributory factor in systemic hypertension.

[Chrono-endocrinology--quo vadis?]

The present review deals with important new chronobiological results especially in the field of chronoendocrinology, shedding new light on the circadian organisation of mammals including man. In vitro studies have shown that the concept of the existence of a single circadian oscillator located in the suprachiasmatic nucleus has to be extended. Circadian oscillators have also been found to exist in the retina, islets of Langerhans, liver, lung, and fibroblasts. Another major result is the detection of a new photopigment, melanopsin, present in a subpopulation of retinal ganglion cells which are lightsensitive and project to the suprachiasmatic nucleus, acting as zeitgeber for the photic entrainment of the circadian rhythm. We are only beginning to understand how the circadian oscillator transmits the circadian message to the endocrine system. The generation of circadian and seasonal rhythms of hormone synthesis is best understood in the pineal gland and its hormone melatonin. Seasonal changes of melatonin synthesis are transduced in the pars tuberalis of the adenohypophysis which is now entering the limelight of chronoendocrinological research. Currently, the elucidation of the genetic basis and the molecular organisation of the circadian oscillator within individual cells is a major thrust in chronobiological research.

Link between truncated fractals and coupled oscillators in biological systems

This article aims at providing a new theoretical insight into the fundamental question of the origin of truncated fractals in biological systems. It is well known that fractal geometry is one of the characteristics of living organisms. However, contrary to mathematical fractals which are self-similar at all scales, the biological fractals are truncated, i.e. their self-similarity extends at most over a few orders of magnitude of separation. We show that nonlinear coupled oscillators, modeling one of the basic features of biological systems, may generate truncated fractals: a truncated fractal pattern for basin boundaries appears in a simple mathematical model of two coupled nonlinear oscillators with weak dissipation. This fractal pattern can be considered as a particular hidden fractal property. At the level of sufficiently fine precision technique the truncated fractality acts as a simple structure, leading to predictability, but at a lower level of precision it is effectively fractal, limiting the predictability of the long-term behavior of biological systems. We point out to the generic nature of our result.