Phase Synchronization of Nonidentical Light-Sensitive Belousov−Zhabotinsky Systems Induced by Variability in a High−Low Illumination Program

Synchronization transition in the two-dimensional Kuramoto model with dichotomous noise

We numerically study the celebrated Kuramoto model of identical oscillators arranged on the sites of a two-dimensional periodic square lattice and subject to nearest-neighbor interactions and dichotomous noise. In the nonequilibrium stationary state attained after a long time, the model exhibits a Berezinskii-Kosterlitz-Thouless (BKT)-like transition between a phase at a low noise amplitude characterized by quasi long-range order (critically ordered phase) and an algebraic decay of correlations and a phase at a high noise amplitude that is characterized by complete disorder and an exponential decay of correlations. The interplay between the noise amplitude and the noise-correlation time is investigated, and the complete, nonequilibrium stationary-state phase diagram of the model is obtained. We further study the dynamics of a single topological defect for various amplitudes and correlation time of the noise. Our analysis reveals that a finite correlation time promotes vortex excitations, thereby lowering the critical noise amplitude of the transition with an increase in correlation time. In the suitable limit, the resulting phase diagram allows one to estimate the critical temperature of the equilibrium BKT transition, which is consistent with that obtained from the study of the dynamics in the Gaussian white noise limit.

Dichotomous noise models of gene switches

Molecular noise in gene regulatory networks has two intrinsic components, one part being due to fluctuations caused by the birth and death of protein or mRNA molecules which are often present in small numbers and the other part arising from gene state switching, a single molecule event. Stochastic dynamics of gene regulatory circuits appears to be largely responsible for bifurcations into a set of multi-attractor states that encode different cell phenotypes. The interplay of dichotomous single molecule gene noise with the nonlinear architecture of genetic networks generates rich and complex phenomena. In this paper, we elaborate on an approximate framework that leads to simple hybrid multi-scale schemes well suited for the quantitative exploration of the steady state properties of large-scale cellular genetic circuits. Through a path sum based analysis of trajectory statistics, we elucidate the connection of these hybrid schemes to the underlying master equation and provide a rigorous justification for using dichotomous noise based models to study genetic networks. Numerical simulations of circuit models reveal that the contribution of the genetic noise of single molecule origin to the total noise is significant for a wide range of kinetic regimes.

Effective phase dynamics of noise-induced oscillations in excitable systems

We develop an effective description of noise-induced oscillations based on deterministic phase dynamics. The phase equation is constructed to exhibit correct frequency and distribution density of noise-induced oscillations. In the simplest one-dimensional case the effective phase equation is obtained analytically, whereas for more complex situations a simple method of data processing is suggested. As an application an effective coupling function is constructed that quantitatively describes periodically forced noise-induced oscillations.