Xianbin Liu

Effects of distance-dependent delay on small-world neuronal networks.

We study firing behaviors and the transitions among them in small-world noisy neuronal networks with electrical synapses and information transmission delay. Each neuron is modeled by a two-dimensional Rulkov map neuron. The distance between neurons, which is a main source of the time delay, is taken into consideration. Through spatiotemporal patterns and interspike intervals as well as the interburst intervals, the collective behaviors are revealed. It is found that the networks switch from resting state into intermittent firing state under Gaussian noise excitation. Initially, noise-induced firing behaviors are disturbed by small time delays. Periodic firing behaviors with irregular zigzag patterns emerge with an increase of the delay and become progressively regular after a critical value is exceeded. More interestingly, in accordance with regular patterns, the spiking frequency doubles compared with the former stage for the spiking neuronal network. A growth of frequency persists for a larger delay and a transition to antiphase synchronization is observed. Furthermore, it is proved that these transitions are generic also for the bursting neuronal network and the FitzHugh-Nagumo neuronal network. We show these transitions due to the increase of time delay are robust to the noise strength, coupling strength, network size, and rewiring probability.

Noise induced escape from a nonhyperbolic chaotic attractor of a periodically driven nonlinear oscillator

Noise induced escape from the domain of attraction of a nonhyperbolic chaotic attractor in a periodically excited nonlinear oscillator is investigated. The general mechanism of the escape in the weak noise limit is studied in the continuous case, and the fluctuational path is obtained by statistical analysis. Selecting the primary homoclinic tangency as the initial condition, the action plot is presented by parametrizing the set of escape trajectories and the global minimum gives rise to the optimal path. Results of both methods show good agreements. The entire process of escape is discussed in detail step by step using the fluctuational force. A structure of hierarchical heteroclinic crossings of stable and unstable manifolds of saddle cycles is found, and the escape is observed to take place through successive jumps through this deterministic hierarchical structure.

Vibrational resonance in the FitzHugh-Nagumo system with time-varying delay feedback

In the present paper, the phenomenon of vibrational resonance in a time-varying delayed FitzHugh-Nagumo system that is driven by two-frequency periodic signals is reported. Via a numerical simulation, the periodic vibrational resonances are found to be induced by the time-varying delay feedback under the condition that the delayed feedback strength is small, and then along with the increase of the delayed feedback strength K within the slow period (i.e., the period of low-frequency signal), the single resonance turns into multiple resonances. However, if the delayed feedback strength K is big enough, the resonance no longer occurs. More interestingly, the multiple resonances can also turn into a single resonance in a cycle by modulating the amplitude F of high-frequency signal. Furthermore, both the resonance region and the resonance amplitude are found to be able to be controlled by the time-varying delay. Finally, it is found that the regular motion of the system can be enhanced by the time-varying delay feedback and then more regular motion will present if the resonance does not occur.

Moment Lyapunov Exponent for a Three Dimensional Stochastic System

In the present paper, for an arbitrary finite real number p, the pth moment Lyapunov exponent for a codimension two bifurcation system that is on a three-dimensional center manifold and is subjected to a parametric excitation by a small intensity white noise is investigated. Via a perturbation method and a linear stochastic transformation introduced by Wedig, an eigenvalue problem associated with the moment Lyapunov exponent is obtained. The eigenvalue problem is then solved approximately via a Fourier cosine series, and for whom the convergence rate is illustrated numerically. Furthermore, the stability regions of pth moment are also obtained.

Crossing the quasi-threshold manifold of a noise-driven excitable system

We consider the noise-induced escapes in an excitable system possessing a quasi-threshold manifold, along which there exists a certain point of minimal quasi-potential. In the weak noise limit, the optimal escaping path turns out to approach this particular point asymptotically, making it analogous to an ordinary saddle. Numerical simulations are performed and an elaboration on the effect of small but finite noise is given, which shows that the ridges where the prehistory probability distribution peaks are located mainly within the region where the quasi-potential increases gently. The cases allowing anisotropic noise are discussed and we found that varying the noise term in the slow variable would dramatically raise the whole level of quasi-potentials, leading to significant changes in both patterns of optimal paths and exit locations.

Subtle escaping modes and subset of patterns from a nonhyperbolic chaotic attractor

Noise-induced escape from the domain of attraction of a nonhyperbolic chaotic attractor in a periodically excited nonlinear oscillator is further investigated. Deviations are found to be amplified at the primary homoclinic tangency from which the optimal force begins to fluctuate dramatically. Escaping trajectories turn out to possess several modes to pass through the saddle cycle on the basin boundary, and each mode corresponds to a certain type of value of the action plot, respectively. A subset of the pattern of fluctuational paths from the chaotic attractor is obtained, showing the existence of complicated singularities.

Analysis of a quintic system with fractional damping in the presence of vibrational resonance

Abstract In the present paper, the phenomenon of the vibrational resonance in a quantic oscillator that possesses a fractional order damping and is driven by both the low and the high frequency periodic signals is investigated, and the approximate theoretical expression of the response amplitude at the low-frequency is obtained by utilizing the method of direct partition of motions. Based on the definition of the Caputo fractional derivative, an algorithm for simulating the system is introduced, and the new method is shown to have higher precision and better feasibility than the method based on the Grunwald –Letnikov expansion. Due to the order of the fractional derivative, various new resonance phenomena are found for the system with single-well, double-well, and triple-well potential, respectively. Moreover, the value of fractional order can be treated as a bifurcation parameter, through which, it is found that the slowly-varying system can be transmitted from a bistability system to a monostabillity one, or from tristability to bistability, and finally to monostabillity. Unlike the cases of the integer-order system, the critical resonance amplitude of the high-frequency signal in the fractional system does depend on the damping strength and can be significantly affected by the fractional-order damping. The numerical results given by the new method is found to be in good agreement with the analytical predictions.

Delay-induced locking in bursting neuronal networks

In this paper, the collective behaviors for ring structured bursting neuronal networks with electrical couplings and distance-dependent delays are studied. Each neuron is modeled by the Hindmarsh-Rose neuron. Through changing time delays between connected neurons, different spatiotemporal patterns are obtained. These patterns can be explained by calculating the ratios between the bursting period and the delay which exhibit clear locking relations. The holding and the failure of the lockings are investigated via bifurcation analysis. In particular, the bursting death phenomenon is observed for large coupling strengths and small time delays which is in fact the result of the partial amplitude death in the fast subsystem. These results indicate that the collective behaviors of bursting neurons critically depend on the bifurcation structure of individual ones and thus the variety of bifurcation types for bursting neurons may create diverse behaviors in similar neuronal ensembles.