Noise Effects on Temperature Encoding of Neuronal Spike Trains in a Cold Receptor

We examine how noise interacts with encoding mechanisms of neuronal stimulus in a cold receptor. From ISI series and bifurcation diagrams it is shown that there are considerable differences in interval distributions and impulse patterns caused by purely deterministic simulations and noisy simulations. The ISI-distance can be used as an effective and powerful way to measure the noise effects on spike trains of the cold receptor quantitatively. It is also found that spike trains observed in cold receptors can be more strongly affected by noise for low temperatures than for high temperatures in some aspects; meanwhile, the spike train has greater variability with increasing noise intensity.     

Pattern Synchronization in a Two-Layer Neuronal Network

Pattern synchronization in a two-layer neuronal network is studied. For a single-layer network of Rulkov map neurons, there are three kinds of patterns induced by noise. Additive noise can induce ordered patterns at some intermediate noise intensities in a resonant way; however, for small and large noise intensities there exist excitable patterns and disordered patterns, respectively. For a neuronal network coupled by two single-layer networks with noise intensity differences between layers, we find that the two-layer network can achieve synchrony as the interlayer coupling strength increases. The synchronous states strongly depend on the interlayer coupling strength and the noise intensity difference between layers.     

Synchronization in coupled nonidentical incommensurate fractional-order systems

Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.     

Strange Nonchaotic Attractors in a Time-Delay System

A nonchaotic attractor is observed in an infinite-dimensional system which is related to optical bistability and described by a nonlinear time-delay differential equation. The observed nonchaotic attractor is characterized by the strange trajectory of attractor but with negative value for the largest Lyapunov exponent, as well as the Fourier power spectra.     

Grazing-induced crises in hybrid dynamical systems

In hybrid dynamical systems including both continuous and discrete components, an interplay between a continuous trajectory and a discontinuity boundary can trigger a sudden qualitative change in the system dynamics. Grazing phenomena, which occur when a continuous trajectory hits a boundary tangentially, are well known as a representative of such phenomena. We demonstrate that a grazing phenomenon of a chaotic attractor can result in its sudden disappearance and initiate chaotic transients. The mechanism of this grazing-induced crisis is revealed in an illustrative example. Furthermore, we derive a formula to obtain the critical exponent of the power law on the mean duration of chaotic transients.     

Breakup of Spiral Waves in Coupled Hindmarsh--Rose Neurons

Breakup of spiral wave in the Hindmarsh-Rose neurons with nearest-neighbour couplings is reported. Appropriate initial values and parameter regions are selected to develop a stable spiral wave and then the Gaussian coloured noise with different intensities and correlation times is imposed on all neurons to study the breakup of spiral wave, respectively. Based on the mean field theory, the statistical factor of synchronization is defined to analyse the evolution of spiral wave. It is found that the stable rotating spiral wave encounters breakup with increasing intensity of Gaussian coloured noise or decreasing correlation time to certain threshold.     

Delay-enhanced coherence of spiral waves in noisy Hodgkin-Huxley neuronal networks

We study the spatial dynamics of spiral waves in noisy Hodgkin-Huxley neuronal ensembles evoked by different information transmission delays and network topologies. In classical settings of coherence resonance the intensity of noise is fine-tuned so as to optimize the system's response. Here, we keep the noise intensity constant, and instead, vary the length of information transmission delay amongst coupled neurons. We show that there exists an intermediate transmission delay by which the spiral waves are optimally ordered, hence indicating the existence of delay-enhanced coherence of spatial dynamics in the examined system. Additionally, we examine the robustness of this phenomenon as the diffusive interaction topology changes towards the small-world type, and discover that shortcut links amongst distant neurons hinder the emergence of coherent spiral waves irrespective of transmission delay length. Presented results thus provide insights that could facilitate the understanding of information transmission delay on realistic neuronal networks.     

Effects of time delay on transient behavior of a time-delayed metastable system subjected to cross-correlated noises

The effects of time delay on the transient properties of a time-delayed metastable system subjected to cross-correlated noises are studied by means of a stochastic simulation method. It is found that: (i) Both additive noise and multiplicative noise can produce the noise enhanced stability (NES) effect; (ii) The time delay induces critical behavior on the NES, i.e., there is a critical value of the delay time τ_c1≈2.2, above which the time delay increases the stability of the system enhanced by the additive noise, and below which the NES effect induced by the additive noise disappears; (iii) There exists another critical value of the delay time τ_c2≈3.0, above which the time delay increases the stability of the system enhanced by the multiplicative noise and below which the time delay decreases it.     

Noise-Mediated Generalized Synchronization

We investigate a drive-response system by considering the impacts of noise on generalized synchronization （GS）. It is found that a small amount of noise can turn the system from desynchronization to the GS state in the resonant case no matter how noise is injected into the system. In the non-resonant case, noise with intensity in a certain range is helpful in building GS only when the noise is injected to the driving system. The mechanism behind the observed phenomena is discussed.     

Resonant phenomena in extended chaotic systems subject to external noise: The Lorenz’96 model case

We have investigated the effects of noise on an extended chaotic system. The chosen model is the Lorenz’96, a type of “toy” model used for climate studies. Through the analysis of the system’s time evolution and its time and space correlations, we have obtained numerical evidence for two distinct stochastic resonance-like behaviors. Such behaviors are seen when both the usual and a generalized signal-to-noise ratio functions are depicted as a function of the external noise intensity, or of the system size. The underlying mechanisms seem to be associated with a noise-induced chaos reduction. The possible relevance of these and other findings for an optimal climate prediction are discussed.     

Pacemaker-guided noise-induced spatial periodicity in excitable media

We study the impact of subthreshold periodic pacemaker activity and internal noise on the spatial dynamics of excitable media. For this purpose, we examine two systems that both consist of diffusively coupled units. In the first case, the local dynamics of the units is driven by a simple one-dimensional model of excitability with a piece-wise linear potential. In the second case, a more realistic biological system is studied, and the local dynamics is driven by a model for calcium oscillations. Internal noise is introduced via the τ-leap stochastic integration procedure and its intensity is determined by the finite size of each constitutive system unit. We show that there exists an intermediate level of internal stochasticity for which the localized pacemaker activity maps best into coherent periodic waves, whose spatial frequency is uniquely determined by the local subthreshold forcing. Via an analytical treatment of the simple minimal model for the excitable spatially extended system, we explicitly link the pacemaker activity with the spatial dynamics and determine necessary conditions that warrant the observation of the phenomenon in excitable media. Our results could prove useful for the understanding of interplay between local and global agonists affecting the functioning of tissue and organs.     

Deterministic escape of a dimer over an anharmonic potential barrier

In the present paper we consider the deterministic escape dynamics of a dimer from a metastable state over an anharmonic potential barrier. The underlying dynamics is conservative and noiseless and thus, the allocated energy has to suffice for barrier crossing. The two particles comprising the dimer are coupled through a spring. Their motion takes place in a two-dimensional plane. Each of the two constituents for itself is unable to escape, but as the outcome of strongly chaotic coupled dynamics the two particles exchange energy in such a way that eventually exit from the domain of attraction may be promoted. We calculate the corresponding critical dimer configuration as the transition state and its associated activation energy vital for barrier crossing. It is found that there exists a bounded region in the parameter space where a fast escape entailed by chaotic dynamics is observed. Interestingly, outside this region the system can show Fermi resonance which, however turns out to impede fast escape.     

Modification of instability processes by multiplicative noises

We study two dynamical systems submitted to white and Gaussian random noise acting multiplicatively. The first system is an imperfect pitchfork bifurcation with a noisy departure from onset. The second system is a pitchfork bifurcation in which the noise acts multiplicatively on the non-linear term of lowest order. In both cases noise suppresses some solutions that exist in the deterministic regime. Besides, for the first system, the imperfectness of the bifurcation reduces the regime of on-off intermittency. For the second system, the unstable mode can achieve a jump of finite amplitude at instability but without hysteresis. We finally identify a generic property that is verified by the stationary probability density function of the dynamical variable when a control parameter is varied.     

Torus destruction via global bifurcations in a piecewise-smooth, continuous map with square-root nonlinearity

It has been shown recently that torus formation in piecewise-smooth maps can occur through a special type of border collision bifurcation in which a pair of complex conjugate Floquet multipliers “jump” from the inside to the outside of the unit circle. It has also been shown that a large class of impacting mechanical systems yield piecewise-smooth maps with square-root singularity. In this Letter we investigate the dynamics of a two-dimensional piecewise-smooth map with square-root type nonlinearity, and describe two new routes to chaos through the destruction of two-frequency torus. In the first scenario, we identify the transition to chaos through the destruction of a loop torus via homoclinic bifurcation. In the other scenario, a change of structure in the torus occurs via heteroclinic saddle connections. Further parameter changes lead to a homoclinic bifurcation resulting in the creation of a chaotic attractor. However, this scenario is much more complex, with the appearance of a sequence of heteroclinic and homoclinic bifurcations.     

A fuzzy crisis in a Duffing-van der Pol system

A crisis in a Duffing--van del Pol system with fuzzy uncertainties is studied by means of the fuzzy generalised cell mapping (FGCM) method. A crisis happens when two fuzzy attractors collide simultaneously with a fuzzy saddle on the basin boundary as the intensity of fuzzy noise reaches a critical point. The two fuzzy attractors merge discontinuously to form one large fuzzy attractor after a crisis. A fuzzy attractor is characterized by its global topology and membership function. A fuzzy saddle with a complicated pattern of several disjoint segments is observed in phase space. It leads to a discontinuous merging crisis of fuzzy attractors. We illustrate this crisis event by considering a fixed point under additive and multiplicative fuzzy noise. Such a crisis is fuzzy noise-induced effects which cannot be seen in deterministic systems.     

A Hyperchaotic Attractor with Multiple Positive Lyapunov Exponents

There are many hyperchaotic systems, but few systems can generate hyperchaotic attractors with more than three PLEs （positive Lyapunov exponents）. A new hyperchaotic system, constructed by adding an approximate time-delay state feedback to a five-dimensional hyperchaotic system, is presented. With the increasing number of phase-shift units used in this system, the number of PLEs also steadily increases. Hyperchaotic attractors with 25 PLEs can be generated by this system with 32 phase-shift units. The sum of the PLEs will reach the maximum value when 23 phase-shift units are used. A simple electronic circuit, consisting of 16 operational amplifiers and two analogy multipliers, is presented for confirming hyperchaos of order 5, i.e., with 5 PLEs.     

Experimental Confirmation of a Modified Lorenz System

We experimentally demonstrate the butterfly-shaped chaotic attractor we have proposed before lint. J. Nonlin. Sci. Numerical Simulation 7 （2006） 187]. Some basic dynamical properties and chaotic behaviour of this new butterfly attractor are studied and they are in agreement with the results of our theoretical analysis. Moreover, the proposed system is experimental demonstrated.     

Synchronization in Coupled Oscillators with Two Coexisting Attractors

Dynamics in coupled Dufling oscillators with two coexisting symmetrical attractors is investigated. For a pair of Dufl~ng oscillators coupled linearly, the transition to the synchronization generally consists of two steps： Firstly, the two oscillators have to jump onto a same attractor, then they reach synchronization similarly to coupled monostable oscillators. The transition scenarios to the synchronization observed are strongly dependent on initial conditions.