Grigory Bordyugov

Self-emerging and turbulent chimeras in oscillator chains

We report on a self-emerging chimera state in a homogeneous chain of nonlocally and nonlinearly coupled oscillators. This chimera, i.e., a state with coexisting regions of complete and partial synchrony, emerges via a supercritical bifurcation from a homogeneous state. We develop a theory of chimera based on the Ott-Antonsen equations for the local complex order parameter. Applying a numerical linear stability analysis, we also describe the instability of the chimera and transition to phase turbulence with persistent patches of synchrony.

Timing of circadian genes in mammalian tissues

Circadian clocks are endogenous oscillators driving daily rhythms in physiology. The cell-autonomous clock is governed by an interlocked network of transcriptional feedback loops. Hundreds of clock-controlled genes (CCGs) regulate tissue specific functions. Transcriptome studies reveal that different organs (e.g. liver, heart, adrenal gland) feature substantially varying sets of CCGs with different peak phase distributions. To study the phase variability of CCGs in mammalian peripheral tissues, we develop a core clock model for mouse liver and adrenal gland based on expression profiles and known cis-regulatory sites. ‘Modulation factors’ associated with E-boxes, ROR-elements, and D-boxes can explain variable rhythms of CCGs, which is demonstrated for differential regulation of cytochromes P450 and 12 h harmonics. By varying model parameters we explore how tissue-specific peak phase distributions can be generated. The central role of E-boxes and ROR-elements is confirmed by analysing ChIP-seq data of BMAL1 and REV-ERB transcription factors.

The Interplay of cis-Regulatory Elements Rules Circadian Rhythms in Mouse Liver

The mammalian circadian clock is driven by cell-autonomous transcriptional feedback loops that involve E-boxes, D-boxes, and ROR-elements. In peripheral organs, circadian rhythms are additionally affected by systemic factors. We show that intrinsic combinatorial gene regulation governs the liver clock. With a temporal resolution of 2 h, we measured the expression of 21 clock genes in mouse liver under constant darkness and equinoctial light-dark cycles. Based on these data and known transcription factor binding sites, we develop a six-variable gene regulatory network. The transcriptional feedback loops are represented by equations with time-delayed variables, which substantially simplifies modelling of intermediate protein dynamics. Our model accurately reproduces measured phases, amplitudes, and waveforms of clock genes. Analysis of the network reveals properties of the clock: overcritical delays generate oscillations; synergy of inhibition and activation enhances amplitudes; and combinatorial modulation of transcription controls the phases. The agreement of measurements and simulations suggests that the intrinsic gene regulatory network primarily determines the circadian clock in liver, whereas systemic cues such as light-dark cycles serve to fine-tune the rhythms.

ENTRAINMENT BETWEEN HEART RATE AND WEAK NONINVASIVE FORCING

We demonstrate that the heart rate of a healthy human can be synchronized by means of weak external noninvasive forcing in the form of a sequence of sound and light pulses, being either periodic or aperiodic, the latter forcing given by interbeat intervals of the heart of another subject. The phenomenon of phase locking of n:m type is observed for both situations in about 90% of our experiments. The plot for the ratio of forcing frequency to the average frequency of response versus detuning possesses a plateau and is in agreement with classical synchronization theory.

Computation of the Response Functions of spiral waves in active media

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological natures. A small perturbation causes gradual change in spatial location of spirals rotation center and frequency, i.e., drift. The response functions (RFs) of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues lambda=0,+/-iomega. The RFs describe the spirals sensitivity to small perturbations in the way that a spiral is insensitive to small perturbations where its RFs are close to zero. The velocity of a spirals drift is proportional to the convolution of RFs with the perturbation. Here we develop a regular and generic method of computing the RFs of stationary rotating spirals in reaction-diffusion equations. We demonstrate the method on the FitzHugh-Nagumo system and also show convergence of the method with respect to the computational parameters, i.e., discretization steps and size of the medium. The obtained RFs are localized at the spirals core.

Tuning the phase of circadian entrainment

The circadian clock coordinates daily physiological, metabolic and behavioural rhythms. These endogenous oscillations are synchronized with external cues (‘zeitgebers’), such as daily light and temperature cycles. When the circadian clock is entrained by a zeitgeber, the phase difference ψ between the phase of a clock-controlled rhythm and the phase of the zeitgeber is of fundamental importance for the fitness of the organism. The phase of entrainment ψ depends on the mismatch between the intrinsic period τ and the zeitgeber period T and on the ratio of the zeitgeber strength to oscillator amplitude. Motivated by the intriguing complexity of empirical data and by our own experiments on temperature entrainment of mouse suprachiasmatic nucleus (SCN) slices, we present a theory on how clock and zeitgeber properties determine the phase of entrainment. The wide applicability of the theory is demonstrated using mathematical models of different complexity as well as by experimental data. Predictions of the theory are confirmed by published data on Neurospora crassa strains for different period mismatches τ − T and varying photoperiods. We apply a novel regression technique to analyse entrainment of SCN slices by temperature cycles. We find that mathematical models can explain not only the stable asymptotic phase of entrainment, but also transient phase dynamics. Our theory provides the potential to explore seasonal variations of circadian rhythms, jet lag and shift work in forthcoming studies.

How Coupling Determines the Entrainment of Circadian Clocks

AbstractAutonomous circadian clocks drive daily rhythms in physiology and behaviour. A network of coupled neurons, the suprachiasmatic nucleus (SCN), serves as a robust self-sustained circadian pacemaker. Synchronization of this timer to the environmental light-dark cycle is crucial for an organism’s fitness. In a recent theoretical and experimental study it was shown that coupling governs the entrainment range of circadian clocks. We apply the theory of coupled oscillators to analyse how diffusive and mean-field coupling affects the entrainment range of interacting cells. Mean-field coupling leads to amplitude expansion of weak oscillators and, as a result, reduces the entrainment range. We also show that coupling determines the rigidity of the synchronized SCN network, i.e. the relaxation rates upon perturbation. Our simulations and analytical calculations using generic oscillator models help to elucidate how coupling determines the entrainment of the SCN. Our theoretical framework helps to interpret experimental data.

A Theoretical Study on Seasonality

In addition to being endogenous, a circadian system must be able to communicate with the outside world and align its rhythmicity to the environment. As a result of such alignment, external Zeitgebers can entrain the circadian system. Entrainment expresses itself in coinciding periods of the circadian oscillator and the Zeitgeber and a stationary phase difference between them. The range of period mismatches between the circadian system and the Zeitgeber that Zeitgeber can overcome to entrain the oscillator is called an entrainment range. The width of the entrainment range usually increases with increasing Zeitgeber strength, resulting in a wedge-like Arnold tongue. This classical view of entrainment does not account for the effects of photoperiod on entrainment. Zeitgebers with extremely small or large photoperiods are intuitively closer to constant environments than equinoctial Zeitgebers and hence are expected to produce a narrower entrainment range. In this paper, we present theoretical results on entrainment under different photoperiods. We find that in the photoperiod-detuning parameter plane, the entrainment zone is shaped in the form of a skewed onion. The bottom and upper points of the onion are given by the free-running periods in DD and LL, respectively. The widest entrainment range is found near photoperiods of 50%. Within the onion, we calculated the entrainment phase that varies over a range of 12 h. The results of our theoretical study explain the experimentally observed behavior of the entrainment phase in dependence on the photoperiod.

Mathematical Modeling in Chronobiology

Circadian clocks are autonomous oscillators entrained by external Zeitgebers such as light-dark and temperature cycles. On the cellular level, rhythms are generated by negative transcriptional feedback loops. In mammals, the suprachiasmatic nucleus (SCN) in the anterior part of the hypothalamus plays the role of the central circadian pacemaker. Coupling between individual neurons in the SCN leads to precise self-sustained oscillations even in the absence of external signals. These neuronal rhythms orchestrate the phasing of circadian oscillations in peripheral organs. Altogether, the mammalian circadian system can be regarded as a network of coupled oscillators. In order to understand the dynamic complexity of these rhythms, mathematical models successfully complement experimental investigations. Here we discuss basic ideas of modeling on three different levels (1) rhythm generation in single cells by delayed negative feedbacks, (2) synchronization of cells via external stimuli or cell-cell coupling, and (3) optimization of chronotherapy.

Anomalous pulse interaction in dissipative media.

We review a number of phenomena occurring in one-dimensional excitable media due to modified decay behind propagating pulses. Those phenomena can be grouped in two categories depending on whether the wake of a solitary pulse is oscillatory or not. Oscillatory decay leads to nonannihilative head-on collision of pulses and oscillatory dispersion relation of periodic pulse trains. Stronger wake oscillations can even result in a bistable dispersion relation. Those effects are illustrated with the help of the Oregonator and FitzHugh-Nagumo models for excitable media. For a monotonic wake, we show that it is possible to induce bound states of solitary pulses and anomalous dispersion of periodic pulse trains by introducing nonlocal spatial coupling to the excitable medium.