Ouyang Qi

Stochastic Simulation of Turing Patterns

We investigate the effects of intrinsic noise on Turing pattern formation near the onset of bifurcation from the homogeneous state to Turing pattern in the reaction-diffusion Brusselator. By performing stochastic simulations of the master equation and using Gillespies algorithm, we check the spatiotemporal behaviour influenced by internal noises. We demonstrate that the patterns of occurrence frequency for the reaction and diffusion processes are also spatially ordered and temporally stable. Turing patterns are found to be robust against intrinsic fluctuations. Stochastic simulations also reveal that under the influence of intrinsic noises, the onset of Turing instability is advanced in comparison to that predicted deterministically.

Distance Distribution and Reliability of Small-World Networks

We have studied the distribution of distances in small-world networks by computer simulation. We found that in a small-world network, the longest distance between two vertices is just slightly longer than the average distance, indicating that the efficiency of the network is absolute rather than in average. We also investigated the robustness of a small-world network by randomly cutting some long paths in the network. The results show that the network is reliable against random cutting.

Pattern Formation in the Turing–Hopf Codimension-2 Phase Space in a Reaction–Diffusion System

We systematically investigate the behaviour of pattern formation in a reaction–diffusion system when the system is located near the Turing–Hopf codimension-2 point in phase space. The chloride–iodide–malonic acid (CIMA) reaction is used in this study. A phase diagram is obtained using the concentration of polyvinyl alcohol (PVA) and malonic acid (MA) as control parameters. It is found that the Turing–Hopf mixed state appears only in a small vicinity near the codimension-2 point, and has the form of hexagonal pattern overlapped with anti-target wave; the boundary line separating the Turing state and the wave state is independent of the concentration of MA, only relies on the concentration of PVA. The corresponding numerical simulation using the Lengyel–Epstein (LE) model gives a similar phase diagram as the experiment; it reproduces most patterns observed in the experiment. However, the mixed state we obtain in simulation only appears in the anti-wave tip area, implying that the 3-D effect in the experiments may change the pattern forming behaviour in the codimension-2 regime.

Responses of a Noisy Excitable System to External Signals with Different Periods

The behaviour of an excitable system under Gaussian white noise and external periodic forcing is systematically studied. In a large range of noise intensity, the n:1 phase locking patterns are obtained for certain ranges of the input periods, where n input periods give one spike. In the phase locking regimes, the system presents low noise-to-signal ratios and shows better regularities. Out of the regimes the system behaves less regularly and the relations between the noise-to-signal ratio and the noise intensity exhibit typical stochastic resonance phenomena. At a higher noise level, the system shows the characteristic behaviour of the noise.

Deterministic Characterization of Intrinsic Noise in Chemical Reactions

The association between intrinsic noises and deterministic descriptions/properties of the rate equations for chemical reactions is analyzed using the linear noise approximation of the master equation. We illustrate that the effect of intrinsic noise is determined in combination by three components: the system size, the matrix associated with reaction kinetics, and the eigenvalues associated with the systems dissipation. Generally, a more attractive dynamics tends to attenuate the internal fluctuations more significantly because intrinsic noises are inversely proportional to the absolute value of the real part of the eigenvalues. In addition, a higher reaction rate and larger stoichiometry coefficients will give rise to stronger intrinsic noise.

Phase Propagations in a Coupled Oscillator?Excitor System of FitzHugh?Nagumo Models

A one-dimensional array of 2N+1 automata with FitzHugh?Nagumo dynamics, in which one is set to be oscillatory and the others are excitable, is investigated with bi-directional interactions. We find that 1:1 rhythm propagation in the array depends on the appropriate couple strength and the excitability of the system. On the two sides of the 1:1 rhythm area in parameter space, two different kinds of dynamical behaviour of the pacemaker, i.e. phase-locking phenomena and canard-like phenomena, are shown. The latter is found in company with chaotic pattern and period doubling bifurcation. When the coupling strength is larger than a critical value, the whole system ends to a steady state.

Additive Temporal Coloured Noise Induced Eckhaus Instability in Complex Ginzburg–Landau Equation System

The effect of additive coloured noises, which are correlated in time, on one-dimensional travelling waves in the complex Ginzburg–Landau equation is studied by numerical simulations. We found that a small coloured noise with temporal correlation could considerably influence the stability of one-dimensional wave trains. There exists an optimal temporal correlation of noise where travelling waves are the most vulnerable. To elucidate the phenomena, we statistically calculated the convective velocities Vg of the wave packets, and found that the coloured noise with an appropriate temporal correlation can decrease Vg, making the system convectively more unstable.

Exploring the Critical Sensitivity in Small-World Networks

Catastrophic phenomena such as earthquakes, avalanches, and critical points in the stock market are hard to predict. Recently, Xia et al. [Pure Appl. Geophys. 159 (2002) 2491] showed that a geometrical system near catastrophic rupture presents a general critical sensitivity: the system becomes significantly sensitive near the catastrophe transition. Here we report that the phenomenon of critical sensitivity also exists in small-world network systems. With the increase of the small-world rewiring probability p, from the regular network p = 0 to the random one p = 1, the system performs more sensitively before the critical point while remaining in better organization through the evolutional progress, and the prediction threshold Ps performs more in advance. The concept of critical sensitivity can be applied to other complex network systems.

Frequency sensitivity of sub-excitable systems coupled with different topology

We have investigated numerically the behaviours of identical FitzHugh–Nagumo systems, which are coupled into topologies of regular one-dimensional lattice, small-world network and scale-free network and driven by white noise and an external signal. We found that when a number of uniform systems are coupled into a network, the systems signal-to-noise ratio remains at a high level in a wider frequency range than in the case of a single oscillator. Different architectures manifest different impact, with the scale-free network being the most remarkable. Results presented here suggest a possible approach to improve the sensitivity of a system to external signals and are helpful for designing communication equipments.

A population-level model from the microscopic dynamics in Escherichia coli chemotaxis via Langevin approximation

Recent extensive studies of Escherichia coli (E. coli) chemotaxis have achieved a deep understanding of its microscopic control dynamics. As a result, various quantitatively predictive models have been developed to describe the chemotactic behavior of E. coli motion. However, a population-level partial differential equation (PDE) that rationally incorporates such microscopic dynamics is still insufficient. Apart from the traditional Keller?Segel (K?S) equation, many existing population-level models developed from the microscopic dynamics are integro-PDEs. The difficulty comes mainly from cell tumbles which yield a velocity jumping process. Here, we propose a Langevin approximation method that avoids such a difficulty without appreciable loss of precision. The resulting model not only quantitatively reproduces the results of pathway-based single-cell simulators, but also provides new inside information on the mechanism of E. coli chemotaxis. Our study demonstrates a possible alternative in establishing a simple population-level model that allows for the complex microscopic mechanisms in bacterial chemotaxis.