Factorized time-dependent distributions for certain multiclass queueing networks and an application to enzymatic processing networks.

Motivated by applications in biological systems, we show for certain multiclass queueing networks that time-dependent distributions for the multiclass queue-lengths can have a factorized form which reduces the problem of computing such distributions to a similar problem for related single-class queueing networks. We give an example of the application of this result to an enzymatic processing network.

Stochastically driven genetic circuits.

Transcriptional regulation in small genetic circuits exhibits large stochastic fluctuations. Recent experiments have shown that a significant fraction of these fluctuations is caused by extrinsic factors. In this paper we review several theoretical and computational approaches to modeling of small genetic circuits driven by extrinsic stochastic processes. We propose a simplified approach to this problem, which can be used in the case when extrinsic fluctuations dominate the stochastic dynamics of the circuit (as appears to be the case in eukaryots). This approach is applied to a model of a single nonregulated gene that is driven by a certain gating process that affects the rate of transcription, and to a simplified version of the galactose utilization circuit in yeast.

Regularization of dynamics in an ensemble of nondiffusively coupled chaotic elements.

We investigate the dynamics in an ensemble of chaotic elements with nondiffusive coupling. First, we analyze the case of global coupling. The type of coupling we consider leads to the suppression of oscillations in the whole ensemble at a high coupling strength. A distinct feature of this transition from high-dimensional chaos at a low coupling strength to the stationary state is that there is no partially ordered phase characterized by a large number of coexisting synchronized clusters. A two-cluster mode emerges abruptly, replacing the asynchronous mode. We focus on the influence of connectivity on the dynamics in the two-cluster modes and their domains of existence. We introduce a parameter that characterizes the connectivity: the range of coupling. Our computational and analytical results indicate that the most significant changes in the dynamics occur in the case of local coupling, when extra connections are added. By contrast, if the range of coupling is high, even substantial changes in this range have a small influence on the dynamics.

Repulsive synchronization in an array of phase oscillators.

We study the dynamics of a repulsively coupled array of phase oscillators. For an array of globally coupled identical oscillators, repulsive coupling results in a family of synchronized regimes characterized by zero mean field. If the number of oscillators is sufficiently large, phase locking among oscillators is destroyed, independently of the coupling strength, when the oscillators' natural frequencies are not the same. In locally coupled networks, however, phase locking occurs even for nonidentical oscillators when the coupling strength is sufficiently strong.

Cooperative dynamics in a network of stochastic elements with delayed feedback.

Networks of globally coupled, noise-activated, bistable elements with connection time delays are considered. The dynamics of these systems is studied numerically using a Langevin description and analytically using (1) a Gaussian approximation as well as (2) a dichotomous model. The system demonstrates ordering phase transitions and multistability. That is, for a strong enough feedback it exhibits nontrivial stationary states and oscillatory states whose frequencies depend only on the mean of the time delay distribution function. Other observed dynamical phenomena include coherence resonance and, in the case of nonuniform coupling strengths, amplitude death and chaos. Furthermore, an increase of the stability of the trivial equilibrium with increasing nonuniformity of the time delays is observed.

Periodic and disordered structures in a modulated gas-driven granular layer.

Experiments with a thin gas-fluidized granular layer revealed a sequence of well-defined transitions as the amplitude and frequency of the gas flow modulation are varied. The observed patterns include subharmonic squares and stripes, quasiperiodic and disordered structures. The wavelength of subharmonic patterns increases with the mean flow rate and decreases with the modulation frequency.

Dynamics of an ensemble of noisy bistable elements with global time delayed coupling.

The dynamics of an ensemble of bistable elements with global time-delayed coupling under the influence of noise is studied analytically and numerically. Depending on the noise level, the system undergoes ordering transitions and demonstrates multistability. That is, for a strong enough positive feedback it exhibits a nonzero stationary mean-field,and a variety of stable oscillatory mean-field states are accessible for positive and negative feedback. The regularity of the oscillatory states is maximal for a certain noise level; i.e., the system demonstrates coherence resonance. While away from the transition points the system dynamics is well described by a Gaussian approximation, near the bifurcation points a description in terms of a dichotomous theory is more adequate.

Delayed self-synchronization in homoclinic chaos.

The chaotic spike train of a homoclinic dynamical system is self-synchronized by applying a time-delayed correction proportional to the laser output intensity. Due to the sensitive nature of the homoclinic chaos to external perturbations, stabilization of very long-periodic orbits is possible. On these orbits, the dynamics appears chaotic over a finite time, but then it repeats with a recurrence time that is slightly longer than the delay time. The effect, called delayed self-synchronization, displays analogies with neurodynamic events that occur in the buildup of long-term memories.

Noise-induced dynamics in bistable systems with delay.

Noise-induced dynamics of a prototypical bistable system with delayed feedback is studied theoretically and numerically. For small noise and magnitude of the feedback, the problem is reduced to the analysis of the two-state model with transition rates depending on the earlier state of the system. Analytical solutions for the autocorrelation function and the power spectrum have been found. The power spectrum has a peak at the frequency corresponding to the inverse delay time, whose amplitude has a maximum at a certain noise level, thus demonstrating coherence resonance. The linear response to the external periodic force also has maxima at the frequencies corresponding to the inverse delay time and its harmonics.

Swarming ring patterns in bacterial colonies exposed to ultraviolet radiation.

We report a novel morphological transition in a Bacillus subtilis colony initially growing under ambient conditions, after ultraviolet radiation exposure. The bacteria in the central regions of the colonies are observed to migrate towards the colony edge forming a ring during uniform spatial exposure. When the radiation is switched off, the colonies were observed to grow both inward into the evacuated regions as well as outward indicating that the pattern is not formed due to depletion of nutrients at the center of the colony. We also propose a reaction-diffusion model in which waste-limited chemotaxis initiated by the UV radiation leads to the observed phenomenology.

Continuum description of avalanches in granular media.

We develop a continuum description of partially fluidized granular flows. Our theory is based on the hydrodynamic equation for the flow coupled with the order parameter equation, which describes the transition between flowing and static components of the granular system. This theory captures important phenomenology recently observed in experiments with granular flows on rough inclined planes [A. Daerr and S. Douady, Nature (London) 399, 241 (1999)]: layer bistability, and transition from triangular avalanches propagating downhill at small inclination angles to balloon-shaped avalanches also propagating uphill for larger angles.

Discrete stochastic modeling of calcium channel dynamics.

We propose a discrete stochastic model for calcium dynamics in living cells. A set of probabilities for the opening/closing of calcium channels is assumed to depend on the calcium concentration. We study this model in one dimension, analytically in the limit of a large number of channels per site N, and numerically for small N. As the number of channels per site is increased, the transition from a nonpropagating region of activity to a propagating one changes from one described by directed percolation to that of deterministic depinning in a spatially discrete system. Also, for a small number of channels a propagating calcium wave can leave behind a novel fluctuation-driven state.

Continuum theory of axial segregation in a long rotating drum.

We develop a continuum description for the axial segregation of granular materials in a long rotating drum based on the dynamics of the thin near-surface granular flow coupled to bulk flow. The equations of motion are reduced to the one-dimensional system for two local variables only, the concentration difference and the dynamic angle of repose, or the average slope of the free surface. The parameters of the system are established from comparison with experimental data. The resulting system describes both initial transient traveling wave dynamics and the formation of quasi-stationary bands of segregated materials. A long-term evolution proceeds through slow logarithmic coarsening of the band structure which is analogous to the spinoidal decomposition described by the Cahn-Hilliard equation.

Dynamics of a shallow fluidized bed.

The results of the experimental study of the dynamics of a shallow fluidized bed are reported. The behavior of granular material is controlled by the interplay of two factors--levitation due to the upward airflow, and sliding back due to gravity. Near the threshold of instability, the system shows critical behavior with remarkably long transient dynamics. The experimental observations are compared with a simple cellular automata model.