From simple to complex oscillatory behavior in metabolic and genetic control networks

We present an overview of mechanisms responsible for simple or complex oscillatory behavior in metabolic and genetic control networks. Besides simple periodic behavior corresponding to the evolution toward a limit cycle we consider complex modes of oscillatory behavior such as complex periodic oscillations of the bursting type and chaos. Multiple attractors are also discussed, e.g., the coexistence between a stable steady state and a stable limit cycle (hard excitation), or the coexistence between two simultaneously stable limit cycles (birhythmicity). We discuss mechanisms responsible for the transition from simple to complex oscillatory behavior by means of a number of models serving as selected examples. The models were originally proposed to account for simple periodic oscillations observed experimentally at the cellular level in a variety of biological systems. In a second stage, these models were modified to allow for complex oscillatory phenomena such as bursting, birhythmicity, or chaos. We consider successively (1) models based on enzyme regulation, proposed for glycolytic oscillations and for the control of successive phases of the cell cycle, respectively; (2) a model for intracellular Ca(2+) oscillations based on transport regulation; (3) a model for oscillations of cyclic AMP based on receptor desensitization in Dictyostelium cells; and (4) a model based on genetic regulation for circadian rhythms in Drosophila. Two main classes of mechanism leading from simple to complex oscillatory behavior are identified, namely (i) the interplay between two endogenous oscillatory mechanisms, which can take multiple forms, overt or more subtle, depending on whether the two oscillators each involve their own regulatory feedback loop or share a common feedback loop while differing by some related process, and (ii) self-modulation of the oscillator through feedback from the system's output on one of the parameters controlling oscillatory behavior. However, the latter mechanism may also be viewed as involving the interplay between two feedback processes, each of which might be capable of producing oscillations. Although our discussion primarily focuses on the case of autonomous oscillatory behavior, we also consider the case of nonautonomous complex oscillations in a model for circadian oscillations subjected to periodic forcing by a light-dark cycle and show that the occurrence of entrainment versus chaos in these conditions markedly depends on the wave form of periodic forcing. (c) 2001 American Institute of Physics.     

Periodic oscillatory solution of bidirectional associative memory networks with delays

Periodic oscillatory phenomena of bidirectional associative memory (BAM) networks with axonal signal transmission delays are investigated by constructing suitable Lyapunov functionals and some analysis techniques. Some simple sufficient conditions are derived ensuring the existence and uniqueness of periodic oscillatory solutions of the BAM with delays, and all other solutions of the BAM converge exponentially to a periodic oscillatory solution. These conditions are presented in terms of system parameters, and have an important leading significance in the design and applications of periodic oscillatory neural circuits for the BAM with delays.     

Stability of oscillatory flows past compliant surfaces

The stability of oscillatory flows over compliant surfaces is studied analytically and numerically. The types of compliant surfaces studied are the spring backed wall model, which permits tangential motion of the surface, and the incompressible viscoelastic gel model. The stability is determined using the Floquet analysis, where amplitude of perturbations at time intervals separated by one time period is examined to determine whether perturbations grow or decay. The oscillatory flows past both the spring backed wall model and the viscoelastic gel model exhibit an instability in the limit of zero Reynolds number, and the transition amplitude of the oscillatory velocity increases with the frequency of oscillations. The transition amplitude has a minimum at zero wave number for the spring backed plate model, whereas the minimum occurs at finite wavenumber for the viscoelastic gel model. For the spring backed plate model, it is shown that the instability due to steady mean flow and the purely oscillatory instability reinforce each other, and the regions of instability are mapped in the ( ) plane, where is the steady strain rate and A is the oscillatory strain rate. For the viscoelastic gel model, the instability is found to depend strongly on the gel viscosity , and the effect of oscillations on the continuation of viscous modes at intermediate Reynolds number shows a complicated dependence on the oscillation frequency.Received: 17 March 2004, Published online: 30 September 2004PACS: 47.20.Ft Instability of shear flows - 83.50.-v Deformation and flow - 87.19.Tt Rheology of body fluids     

Oscillations and chaos in CO+O(2) combustion

The gas-phase reaction between carbon monoxide and oxygen (in the presence of small amounts of hydrogen) shows bistability and oscillatory behavior. Typically, the oscillatory ignition has a period-1 relaxation waveform. The limit cycle is born at a saddle-node loop and terminates via a supercritical Hopf bifurcation. For a mean residence time of 8 s there is a period-doubling to a period-2 solution followed by period-halving to quasisinusoidal period-1 oscillations. At longer residence times, more period-doublings forming a full cascade to chaos with subsequent periodic windows are observed. The chaotic attractor has an underlying single-humped next maximum map.     

Capacity of oscillatory associative-memory networks with error-free retrieval

Networks of coupled periodic oscillators (similar to the Kuramoto model) have been proposed as models of associative memory. However, error-free retrieval states of such oscillatory networks are typically unstable, resulting in a near zero capacity. This puts the networks at disadvantage as compared with the classical Hopfield network. Here we propose a simple remedy for this undesirable property and show rigorously that the error-free capacity of our oscillatory, associative-memory networks can be made as high as that of the Hopfield network. They can thus not only provide insights into the origin of biological memory, but can also be potentially useful for applications in information science and engineering.     

Synchronization and oscillator death in oscillatory media with stirring

The effect of stirring in an inhomogeneous oscillatory medium is investigated. We show that the stirring rate can control the macroscopic behavior of the system producing collective oscillations (synchronization) or complete quenching of the oscillations (oscillator death). We interpret the homogenization rate due to mixing as a measure of global coupling and compare the phase diagrams of stirred oscillatory media and of populations of globally coupled oscillators.     

Noise-induced synchronous stochastic oscillations in small scale cultured heart-cell networks

This paper reports that the synchronous integer multiple oscillations of heart-cell networks or clusters are observed in the biology experiment.The behaviour of the integer multiple rhythm is a transition between super-and subthreshold oscillations,the stochastic mechanism of the transition is identified.The similar synchronized oscillations are theoretically reproduced in the stochastic network composed of heterogeneous cells whose behaviours are chosen as excitable or oscillatory states near a Hopf bifurcation point.The parameter regions of coupling strength and noise density that the complex oscillatory rhythms can be simulated are identified.The results show that the rhythm results from a simple stochastic alternating process between super-and sub-threshold oscillations.Studies on single heart cells forming these clusters reveal excitable or oscillatory state nearby a Hopf bifurcation point underpinning the stochastic alternation.In discussion,the results are related to some abnormal heartbeat rhythms such as the sinus arrest.     

Red blood cells and other nonspherical capsules in shear flow: oscillatory dynamics and the tank-treading-to-tumbling transition

We consider the motion of red blood cells and other nonspherical microcapsules dilutely suspended in a simple shear flow. Our analysis indicates that depending on the viscosity, membrane elasticity, geometry, and shear rate, the particle exhibits either tumbling, tank-treading of the membrane about the viscous interior with periodic oscillations of the orientation angle, or intermittent behavior in which the two modes occur alternately. For red blood cells, we compute the complete phase diagram and identify a novel tank-treading-to-tumbling transition as the shear rate decreases. Observations of such motions coupled with our theoretical framework may provide a sensitive means of assessing capsule properties.     

Improved functional-weight approach to oscillatory patterns in excitable networks

Studies of sustained oscillations on complex networks with excitable node dynamics received much interest in recent years. Although an individual unit is non-oscillatory, they may organize to form various collective oscillatory patterns through networked connections. An excitable network usually possesses a number of oscillatory modes dominated by different Winfree loops and numerous spatiotemporal patterns organized by different propagation path distributions. The traditional approach of the so-called dominant phase-advanced drive method has been well applied to the study of stationary oscillation patterns on a network. In this paper, we develop the functional-weight approach that has been successfully used in studies of sustained oscillations in gene-regulated networks by an extension to the high-dimensional node dynamics. This approach can be well applied to the study of sustained oscillations in coupled excitable units. We tested this scheme for different networks, such as homogeneous random networks, small-world networks, and scale-free networks and found it can accurately dig out the oscillation source and the propagation path. The present approach is believed to have the potential in studies competitive non-stationary dynamics.     

Cluster synchronization and spatio-temporal dynamics in networks of oscillatory and excitable Luo-Rudy cells

We study collective phenomena in nonhomogeneous cardiac cell culture models, including one- and two-dimensional lattices of oscillatory cells and mixtures of oscillatory and excitable cells. Individual cell dynamics is described by a modified Luo-Rudy model with depolarizing current. We focus on the transition from incoherent behavior to global synchronization via cluster synchronization regimes as coupling strength is increased. These regimes are characterized qualitatively by space-time plots and quantitatively by profiles of local frequencies and distributions of cluster sizes in dependence upon coupling strength. We describe spatio-temporal patterns arising during this transition, including pacemakers, spiral waves, and complicated irregular activity.     

A bifurcation analysis of two coupled calcium oscillators

In many cell types, asynchronous or synchronous oscillations in the concentration of intracellular free calcium occur in adjacent cells that are coupled by gap junctions. Such oscillations are believed to underlie oscillatory intercellular calcium waves in some cell types, and thus it is important to understand how they occur and are modified by intercellular coupling. Using a previous model of intracellular calcium oscillations in pancreatic acinar cells, this article explores the effects of coupling two cells with a simple linear diffusion term. Depending on the concentration of a signal molecule, inositol (1,4,5)-trisphosphate, coupling two identical cells by diffusion can give rise to synchronized in-phase oscillations, as well as different-amplitude in-phase oscillations and same-amplitude antiphase oscillations. Coupling two nonidentical cells leads to more complex behaviors such as cascades of period doubling and multiply periodic solutions. This study is a first step towards understanding the role and significance of the diffusion of calcium through gap junctions in the coordination of oscillatory calcium waves in a variety of cell types. (c) 2001 American Institute of Physics.     

Phase transitions in small systems: Microcanonical vs. canonical ensembles

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g., temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.     

Noise-induced excitability in oscillatory media

A noise-induced phase transition to excitability is reported in oscillatory media with FitzHugh-Nagumo dynamics. This transition takes place via a noise-induced stabilization of a deterministically unstable fixed point of the local dynamics, while the overall phase-space structure of the system is maintained. Spatial coupling is required to prevent oscillations through suppression of fluctuations (via clustering in the case of local coupling). Thus, the joint action of coupling and noise leads to a different type of phase transition and results in a stabilization of the system. The resulting regime is shown to display characteristic traits of excitable media, such as stochastic resonance and wave propagation. This effect thus allows the transmission of signals through an otherwise globally oscillating medium.     

Oscillating viscosity in a lyotropic lamellar phase under shear flow

We report some time-dependent behavior of lyotropic lamellar phase under shear flow. At fixed stress, near a layering instability, the system presents an oscillating shear rate. We build up a new stress versus shear rate diagram that includes temporal behavior. This diagram is made of two distinct branches of stationary states which correspond, respectively, to disordered and ordered multilamellar vesicle phases. When increasing the shear stress, prior to the transition to the ordered structural state, sustained oscillations of the viscosity are recorded. They correspond to periodic structural change of the entire sample between a disordered and a ordered state of multilamellar vesicles.     

Studies on self-sustained reaction-rate oscillations: II. The role of carbon and oxides in the oscillatory oxidation of carbon monoxide on platinum

CO oxidation on a platinum foil was studied in a high pressure flow cell (10²−10² Pa) and an UHV chamber (10⁻⁸ −5 × 10⁴ Pa) both interfaced to a surface infrared spectrometer. Real-time surface infrared and calorimetry experiments performed in the cell during oscillatory oxidation indicated a slow periodic variation (∼ 40%) in the number of active sites, the period of which was commensurate with that of the reaction-rate oscillations. Auger spectroscopy performed in the UHV chamber showed that surface carbon quantitatively accounted for the surface deactivation, as evidenced by the inverse correlation of the number of surface sites active towards CO adsorption with the surface carbon concentration and by the demonstration that, at the oscillation temperatures, carbon can diffuse from the bulk to the surface, oxygen can remove surface carbon and adsorbed CO can block carbon diffusion. Although silicon oxide was always detected on the surface with infrared spectroscopy, no periodic variation in it could be observed during the reaction-rate oscillations. Auger studies confirmed that the maximum and the variations in surface concentration of silicon oxide could not account for the variations in the number of active sites. A mechanism is therefore proposed in which carbon is driving the long-period self-sustained oscillations in the rate of CO oxidation on Pt.     

Global organization of dynamics in oscillatory heterogeneous excitable media

An oscillatory heterogeneous excitable medium undergoes a transition from periodic target patterns to a bursting rhythm driven by the spontaneous initiation and termination of spiral waves as coupling or density is reduced. We illustrate these phenomena in monolayers of chick embryonic heart cells using calcium-sensitive fluorescent dyes. These results are modeled in a heterogeneous cellular automaton in which the neighborhood of interaction and cell density is modified. Parameters that give rise to bursting rhythms are organized in distinct zones in parameter space, leading to a global organization that should be applicable to the dynamics in a large class of excitable media.     

Isothermal sustained oscillations in a very simple surface reaction

A simple model is used to illustrate that a bimolecular Langmuir—Hinshelwood surface reaction with two empty sites in its reaction step, non-equilibrium in the adsorption steps, and coverage independent parameters may lead to sustained oscillatory reaction rates. The two empty sites in the reaction step play an essential role in the establishment of these oscillations. Numerical simulation is used to demonstrate the periodic behavior predicted by the model. Several similar surface reaction models with coverage independent parameters can also yield oscillations. A mechanism with one vacant site in the adsorption steps, two vacant sites in the reaction step and only two dimensionless non-zero parameters may lead to sustained oscillations.     

Dynamical quorum sensing and synchronization in collections of excitable and oscillatory catalytic particles

We present experimental studies of interacting excitable and oscillatory catalytic particles in well-stirred and spatially distributed systems. A number of distinct paths to synchronized oscillatory behavior are described. We present an example of a Kuramoto type transition in a well-stirred system with a collective rhythm emerging on increasing the number density of oscillatory particles. Groups of spatially distributed oscillatory particles become entrained to a common frequency by organizing centers. Quorum sensing type transitions are found in populations of globally and locally coupled excitable particles, with a sharp transition from steady state to fully synchronized behavior at a critical density or group size.     

Dynamic changes of nucleolar DNA configuration and distribution during the cell cycle in Allium sativum cells

To investigate the correlation between subnucleolar structure and function, the precise distribution and configuration of nucleolar DNA during the cell cycle of Allium sativum were determined using the NAMA-Ur DNA-specific staining technique. We showed that nucleolar DNA is present in two forms: compacted chromatin clumps and a decondensed DNA cloud. The form of the DNA within the nucleolus varied greatly as the cell cycle progressed. During telophase, chromosomes extended into the prenucleolar body. In early G1 phase, DNA was only located in the fibrillar centers in the form of the condensed chromatin clump, while in mid-G1, S and G2 phases, the two forms of DNA were distributed in the fibrillar centers (FC) and dense fibrillar component (DFC). In prophase of mitosis, nucleolar DNA, along with FC and DFC, was linked into a network structure and condensed into a large chromatin clump. The area of the DNA cloud in the dense fibrillar component changed during different phases of the cell cycle. Our results demonstrated that the configuration of nucleolar DNA undergoes a series of decondensations and condensations during the cell cycle to fulfill the function of the nucleoli during the different phases.