The Kuramoto model: A simple paradigm for synchronization phenomena

Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.

Synchronization reveals topological scales in complex networks

We study the relationship between topological scales and dynamic time scales in complex networks. The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In the synchronization process, modular structures corresponding to well-defined communities of nodes emerge in different time scales, ordered in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology, and spectral graph analysis.

Remote Synchronization Reveals Network Symmetries and Functional Modules

We study a Kuramoto model in which the oscillators are associated with the nodes of a complex network and the interactions include a phase frustration, thus preventing full synchronization. The system organizes into a regime of remote synchronization where pairs of nodes with the same network symmetry are fully synchronized, despite their distance on the graph. We provide analytical arguments to explain this result, and we show how the frustration parameter affects the distribution of phases. An application to brain networks suggests that anatomical symmetry plays a role in neural synchronization by determining correlated functional modules across distant locations.

Thermal resonance in signal transmission

We use temperature tuning to control signal propagation in simple one-dimensional arrays of masses connected by hard anharmonic springs and with no local potentials. In our numerical model a sustained signal is applied at one site of a chain immersed in a thermal environment and the signal-to-noise ratio is measured at each oscillator. We show that raising the temperature can lead to enhanced signal propagation along the chain, resulting in thermal resonance effects akin to the resonance observed in arrays of bistable systems.

The mammalian circadian timing system: from gene expression to physiology.

Many physiological processes in organisms from bacteria to man are rhythmic, and some of these are controlled by self-sustained oscillators that persist in the absence of external time cues. Circadian clocks are perhaps the best characterized biological oscillators and they exist in virtually all light-sensitive organisms. In mammals, they influence nearly all aspects of physiology and behavior, including sleep-wake cycles, cardiovascular activity, endocrinology, body temperature, renal activity, physiology of the gastro-intestinal tract, and hepatic metabolism. The master pacemaker is located in the suprachiasmatic nuclei, two small groups of neurons in the ventral part of the hypothalamus. However, most peripheral body cells contain self-sustained circadian oscillators with a molecular makeup similar to that of SCN (suprachiasmatic nucleus) neurons. This organization implies that the SCN must synchronize countless subsidiary oscillators in peripheral tissues, in order to coordinate cyclic physiology. In this review, we will discuss some recent studies on the structure and putative functions of the mammalian circadian timing system, but we will also point out some apparent inconsistencies in the currently publicized model for rhythm generation.

Increased eeg synchronization in schizophrenia patients by the glutathione precursor, n-acetyl-cysteine

Background: Glutathione (GSH) dysregulation at the gene, protein and functional levels observed in schizophrenia patients, and schizophrenia-like anomalies in GSH deficit experimental models, suggest that genetic glutathione synthesis impairments represent one major risk factor for the disease (Do et al., 2009). In a randomized, double blind, placebo controlled, add-on clinical trial of 140 patients, the GSH precursor N-Acetyl-Cysteine (NAC, 2 g/day, 6 months) significantly improved the negative symptoms and reduced side-effects due to antipsychotics (Berk et al., 2008). In a subset of patients (n=7), NAC (2 g/day, 2 months, cross-over design) also improved auditory evoked potentials, the NMDAdependent mismatch negativity (Lavoie et al, 2008). Methods: To determine whether increased GSH levels would modulate the topography of functional brain connectivity, we applied a multivariate phase synchronization (MPS) estimator (Knyazeva et al, 2008) to dense-array EEGs recorded during rest with eyes closed at the protocol onset, the point of crossover, and at its end. Phase synchronization phenomena are appealing because they can be associated to synchronized phases while the amplitudes stay uncorrelated. MPS measures the degree of interactions among the recorded neuronal oscillators by quantifiying to what extent they behave like a macro-oscillator (i.e. the oscillators are phase synchronous). To assess the whole-head synchronization topography, we computed the MPS sensor-wise over the cluster of locations defined by the sensor itself and he surrounding ones belonging to its second-order neighborhood (Carmeli et al, 2005). Such a cluster spans about 12 cm on average. Abstracts 245 Results: The whole-head imaging revealed a specific synchronization landscape in NAC compared to placebo condition. In particular, NAC increased MPS over frontal and left temporal regions in a frequency-specific manner. Importantly, the topography and direction of MPS changes were similar and robust in all 7 patients. Moreover, these changes correlated with the changes in the Liddle's score of disorganization (Liddle, 1987) thus linking EEG synchronization to the improvement of clinical picture. Discussion: The data suggest an important pathway towards new therapeutic strategies that target GSH dysregulation in schizophrenia. They also show the utility of MPS mapping as a marker of treatment efficacy.

Energy relaxation in nonlinear one-dimensional lattices

We study energy relaxation in thermalized one-dimensional nonlinear arrays of the Fermi-Pasta-Ulam type. The ends of the thermalized systems are placed in contact with a zero-temperature reservoir via damping forces. Harmonic arrays relax by sequential phonon decay into the cold reservoir, the lower-frequency modes relaxing first. The relaxation pathway for purely anharmonic arrays involves the degradation of higher-energy nonlinear modes into lower-energy ones. The lowest-energy modes are absorbed by the cold reservoir, but a small amount of energy is persistently left behind in the array in the form of almost stationary low-frequency localized modes. Arrays with interactions that contain both a harmonic and an anharmonic contribution exhibit behavior that involves the interplay of phonon modes and breather modes. At long times relaxation is extremely slow due to the spontaneous appearance and persistence of energetic high-frequency stationary breathers. Breather behavior is further ascertained by explicitly injecting a localized excitation into the thermalized arrays and observing the relaxation behavior.

Genome-wide RNA polymerase II profiles and RNA accumulation reveal kinetics of transcription and associated epigenetic changes during diurnal cycles.

Interactions of cell-autonomous circadian oscillators with diurnal cycles govern the temporal compartmentalization of cell physiology in mammals. To understand the transcriptional and epigenetic basis of diurnal rhythms in mouse liver genome-wide, we generated temporal DNA occupancy profiles by RNA polymerase II (Pol II) as well as profiles of the histone modifications H3K4me3 and H3K36me3. We used these data to quantify the relationships of phases and amplitudes between different marks. We found that rhythmic Pol II recruitment at promoters rather than rhythmic transition from paused to productive elongation underlies diurnal gene transcription, a conclusion further supported by modeling. Moreover, Pol II occupancy preceded mRNA accumulation by 3 hours, consistent with mRNA half-lives. Both methylation marks showed that the epigenetic landscape is highly dynamic and globally remodeled during the 24-hour cycle. While promoters of transcribed genes had tri-methylated H3K4 even at their trough activity times, tri-methylation levels reached their peak, on average, 1 hour after Pol II. Meanwhile, rhythms in tri-methylation of H3K36 lagged transcription by 3 hours. Finally, modeling profiles of Pol II occupancy and mRNA accumulation identified three classes of genes: one showing rhythmicity both in transcriptional and mRNA accumulation, a second class with rhythmic transcription but flat mRNA levels, and a third with constant transcription but rhythmic mRNAs. The latter class emphasizes widespread temporally gated posttranscriptional regulation in the mouse liver.

Rhythmic growth explained by coincidence between internal and external cues.

Most organisms use circadian oscillators to coordinate physiological and developmental processes such as growth with predictable daily environmental changes like sunrise and sunset. The importance of such coordination is highlighted by studies showing that circadian dysfunction causes reduced fitness in bacteria and plants, as well as sleep and psychological disorders in humans. Plant cell growth requires energy and water-factors that oscillate owing to diurnal environmental changes. Indeed, two important factors controlling stem growth are the internal circadian oscillator and external light levels. However, most circadian studies have been performed in constant conditions, precluding mechanistic study of interactions between the clock and diurnal variation in the environment. Studies of stem elongation in diurnal conditions have revealed complex growth patterns, but no mechanism has been described. Here we show that the growth phase of Arabidopsis seedlings in diurnal light conditions is shifted 8-12 h relative to plants in continuous light, and we describe a mechanism underlying this environmental response. We find that the clock regulates transcript levels of two basic helix-loop-helix genes, phytochrome-interacting factor 4 (PIF4) and PIF5, whereas light regulates their protein abundance. These genes function as positive growth regulators; the coincidence of high transcript levels (by the clock) and protein accumulation (in the dark) allows them to promote plant growth at the end of the night. Thus, these two genes integrate clock and light signalling, and their coordinated regulation explains the observed diurnal growth rhythms. This interaction may serve as a paradigm for understanding how endogenous and environmental signals cooperate to control other processes.

Blood-Borne Circadian Signal Stimulates Daily Oscillations in Actin Dynamics and SRF Activity.

In peripheral tissues circadian gene expression can be driven either by local oscillators or by cyclic systemic cues controlled by the master clock in the brain's suprachiasmatic nucleus. In the latter case, systemic signals can activate immediate early transcription factors (IETFs) and thereby control rhythmic transcription. In order to identify IETFs induced by diurnal blood-borne signals, we developed an unbiased experimental strategy, dubbed Synthetic TAndem Repeat PROMoter (STAR-PROM) screening. This technique relies on the observation that most transcription factor binding sites exist at a relatively high frequency in random DNA sequences. Using STAR-PROM we identified serum response factor (SRF) as an IETF responding to oscillating signaling proteins present in human and rodent sera. Our data suggest that in mouse liver SRF is regulated via dramatic diurnal changes of actin dynamics, leading to the rhythmic translocation of the SRF coactivator Myocardin-related transcription factor-B (MRTF-B) into the nucleus.

The systemic control of circadian gene expression.

The mammalian circadian timing system consists of a central pacemaker in the brain's suprachiasmatic nucleus (SCN) and subsidiary oscillators in nearly all body cells. The SCN clock, which is adjusted to geophysical time by the photoperiod, synchronizes peripheral clocks through a wide variety of systemic cues. The latter include signals depending on feeding cycles, glucocorticoid hormones, rhythmic blood-borne signals eliciting daily changes in actin dynamics and serum response factor (SRF) activity, and sensors of body temperature rhythms, such as heat shock transcription factors and the cold-inducible RNA-binding protein CIRP. To study these systemic signalling pathways, we designed and engineered a novel, highly photosensitive apparatus, dubbed RT-Biolumicorder. This device enables us to record circadian luciferase reporter gene expression in the liver and other organs of freely moving mice over months in real time. Owing to the multitude of systemic signalling pathway involved in the phase resetting of peripheral clocks the disruption of any particular one has only minor effects on the steady state phase of circadian gene expression in organs such as the liver. Nonetheless, the implication of specific pathways in the synchronization of clock gene expression can readily be assessed by monitoring the phase-shifting kinetics using the RT-Biolumicorder.

Desenvolvimento de um detector piezelétrico para cromatógrafo a gás interfaceado a microcomputador

A piezoelectric detector based on a commercial quartz crystal of 3.57MHz is shown. A PTFE cell attached to the GC-column outlet holds the polymer-coated crystal, whose terminals were removed. Two TTL oscillators (reference and sensor) are used and the beating of their frequencies is monitored through a home-made counter card inserted in a microcomputer. The original thermal conductivity detector was also interfaced, but through a 12-bit analog-to-digital converter. In spite of the well-known fact that QCM has low sensitivity at high temperatures, results show significant peak deformation for low temperatures. In addition, the phenomenon is also dependent on the chemical composition of the absorbent film besides instantaneous partial pressure and chemical composition of the analyte.

Um modelo para a superfície líquida no estudo da dinâmica do espalhamento de Xe e Ne pelo esqualano

In this work we present a theoretical model to investigate the scattering of Xe and Ne by a liquid squalane surface. The liquid surface is modeled as a grid of harmonic oscillators with frequencies adjusted to experimental vibration as frequencies of the liquid squalane and the atom-surface interaction potential is modeled by a Lennard-Jones function. The three dimensional description of the dynamics of the process which occurs at the gas-liquid interface is obtained by the classical trajectory method. The general characteristics of the dynamics of the scattering process are in good agreement with experimental data.

Dinâmica complexa no sistema bromato/hipofosfito/acetona/manganês e ferroína

New chemical systems have been recently designed for the study of complex phenomena such as oscillatory dynamics in the temporal domain and spatiotemporal pattern formation. Systems derived from oscillators based on the chemistry of bromate are the most extensively studied, with the celebrated Belousov-Zhabotinsky (BZ) reaction being the most popular example. Problems such as the formation of bubbles (CO2) and solid precipitate in the course of the reaction and the occurrence of simply short-lived oscillations under batch conditions are very common and, in some cases, compromise the use of some of these systems. It is investigated in this paper the dynamic behavior of the bromate/hypophosphite/acetone/dual catalyst system, which has been sugested as an interesting alternative to circumvent those inconvenients. In this work, manganese and ferroin are employed as catalysts and the complete system (BrO3-/H2PO2-/acetone/Mn(II)-ferroin) is studied under batch conditions. Temporal symmetry breaking was studied in a reactor under agitation by means of simultaneous records of the potential changes of platinum and Ag/AgBr electrodes, both measured versus a reversible hydrogen electrode. Additionally, spatio-temporal formation of target patterns and spiral waves were obtained when the oscillating mixture was placed in a quasi two-dimensional reactor.

Integração numérica de leis de velocidade diferenciais com o uso do SCILAB

In this work, we applied the free open source SCILAB software for the numerical integration of differential rate law equations to obtain the concentration profiles of chemical species involved in the kinetics of some complex reactions. An automated method was applied to construct the system of ordinary differential equations (ODE) from the postulated chemical models. The solutions of the ODEs were obtained numerically by standard SCILAB functions. We successfully simulated even complex chemical systems such as pH oscillators. This communication opens up the possibility of using SCILAB in simulations and modeling by our chemistry undergraduate students.