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PubMed2004

Global spatiotemporal order and induced stochastic resonance due to a locally applied signal.

We study the phenomenon of spatiotemporal stochastic resonance (STSR) in a chain of diffusively coupled bistable oscillators. In particular, we examine the situation in which the global STSR response is controlled by a locally applied signal and reveal a wave-front propagation. In order to deepen the understanding of the system dynamics, we introduce, on the time scale of STSR, the study of the effective statistical renormalization of a generic lattice system. Using this technique we provide a criterion for STSR, and predict and observe numerically a bifurcationlike behavior that reflects the difference between the most probable value of the local quasiequilibrium density and its mean value. Our results, tested with a chain of nonlinear oscillators, appear to possess some universal qualities and may stimulate a deeper search for more generic phenomena.

Journal Article↗
PubMed2004

Rate description of Fokker-Planck processes with time-dependent parameters.

The reduction of a continuous Markov process with multiple metastable states to a discrete rate process is investigated in the presence of slow time-dependent parameters such as periodic external forces or slowly fluctuating barrier heights. A quantitative criterion is provided under which condition a kinetic description with time-dependent frozen rates applies and nonadiabatic corrections to the frozen rates are obtained. Finally it is shown how the long-time behavior of the underlying continuous process can be retrieved from the knowledge of the discrete process by means of an appropriate random decoration of the discrete states. As a particular example of the presented theory an overdamped bistable Brownian oscillator with periodic driving is discussed.

Journal Article↗
PubMed2004

Hopf bifurcation cascade in small-alpha laser diodes subject to optical feedback.

We analyze theoretically the dynamics of a semiconductor laser subject to optical feedback, on the basis of the well-known Lang-Kobayashi equations. Previous investigations on this laser system suggest that a small linewidth enhancement factor (alpha factor) stabilizes the laser dynamics. By contrast, we unveil here optical feedback induced instabilities which are present for a small value of alpha but which disappear when alpha increases above alpha approximately 1. By combining numerical simulations and modern continuation methods for delay-differential equations, we unveil cascades of subcritical and supercritical Hopf bifurcations on the first external-cavity mode (ECM). We unveil for the first time, to our knowledge, the occurrence of subcritical Hopf bifurcation points for intermediate values of the EC length, i.e. close to the boundary between the short and the long EC regimes. They lead to severe laser instabilities such as large intensity and possibly chaotic pulsations. Moreover, these Hopf bifurcation cascades for small values of alpha are shown to be responsible for different bifurcation scenarios leading to restabilization of the first ECM and to ECM bistability.

Journal Article↗
PubMed2004

Macroscopic limit cycle via pure noise-induced phase transitions.

Bistability generated via a pure noise-induced phase transition is reexamined from the view of bifurcations in macroscopic cumulant dynamics. It allows an analytical study of the phase diagram in more general cases than previous methods. In addition, using this approach we investigate spatially extended systems with two degrees of freedom per site. For this system, the analytic solution of the stationary Fokker-Planck equation is not available and a standard mean field approach cannot be used to find noise-induced phase transitions. A different approach based on cumulant dynamics predicts a noise-induced phase transition through a Hopf bifurcation leading to a macroscopic limit cycle motion, which is confirmed by numerical simulation.

Journal Article↗
PubMed2004

Transmission of stimulus-locked responses in two coupled phase oscillators.

A model of two n:m coupled phase oscillators is studied, where both oscillators are subject to random forces, but only one oscillator is repetitively stimulated with a pulsatile stimulus. The focus of the paper is on transmission of transient responses as well as transient synchronization and desynchronization, which are stimulus locked, i.e., tightly time locked to the stimulus. A bistability or multistability of stable synchronized states of the two-phase oscillators (modulo 2pi ) occurs due to the n:m coupling. Accordingly, after stimulation the two oscillators may tend to qualitatively different stable states, which leads to a cross-trial (CT) response clustering (i.e., a switching between qualitatively different poststimulus responses across trials) of either one of the oscillators or both. A stochastic CT phase resetting analysis allows one to detect such transient responses and provides a reliable estimation of the transmission time. In contrast, CT averaging (averaging over an ensemble of responses), CT standard deviation, and CT cross correlation fail in studying the transmission of such stimulus-locked responses, even in the simpler case of 1:1 coupling. In particular, even though being used as golden standard for the analysis of evoked responses in medicine and neuroscience, CT averaging typically causes severe artifacts and misinterpretations.

Animals↗
PubMed2004

Stochastic resonance in biological nonlinear evolution models.

We investigate stochastic resonance in the nonlinear, one-dimensional Fisher-Eigen model (FEM), which represents an archetypal model for biological evolution based on a global coupling scheme. In doing so we consider different periodically driven fitness functions which govern the evolution of a biological phenotype population. For the case of a simple harmonic fitness function we are able to derive the exact analytic solution for the asymptotic probability density. A distinct feature of this solution is a phase lag between the driving signal and the linear response of the system. Furthermore, for more complex systems a general perturbation theory (linear response approximation) is put forward. Using the latter approach, we investigate stochastic resonance in terms of the spectral amplification measure for a quadratic, a quartic single-peaked, and for a bistable fitness function. Our analytical results are also compared with those of detailed numerical simulations. Our findings vindicate that stochastic resonance does occur in these nonlinear, globally coupled biological systems.

Journal Article↗
PubMed2004

Coexistence of stable particle and hole solutions for fixed parameter values in a simple reaction diffusion system.

We present a simple autocatalytic reaction-diffusion model for two variables, which shows for fixed parameter values the simultaneous stable coexistence of particle solutions as well of two types of hole solutions. The associated spatially homogeneous system is characterized by the coexistence of one stable fixed point and a stable limit cycle solution. We compare our results to other dissipative systems which have for fixed parameters either stable particle or stable hole solutions including the quintic complex Ginzburg-Landau equation and the envelope equation for optical bistability as well as other reaction-diffusion models.

Journal Article↗
PubMed2004

Optical patterns with different wavelengths.

The semiconductor resonator is an example of an optical system where two modulational instabilities with different wave numbers coexist. In the limit of nascent bistability, the dynamics is generically described by a nonvariational real order parameter equation, of which we give a detailed derivation. This considerably simplifies the linear and weakly nonlinear stability analyses. When the two instabilities are close together, we derive normal form equations and put special emphasis on "envelope" branches of solutions. These particular solutions may connect the two instability points or form an isola. On the basis of these rigorous results, we finally discuss the case of distant modulational instabilities, in both one and two transverse dimensions.

Journal Article↗
PubMed2004

Nonlinear stochastic resonance with subthreshold rectangular pulses.

We analyze the phenomenon of nonlinear stochastic resonance (SR) in noisy bistable systems driven by pulsed time periodic forces. The driving force contains, within each period, two pulses of equal constant amplitude and duration but opposite signs. Each pulse starts every half period and its duration is varied. For subthreshold amplitudes, we study the dependence of the output signal-to-noise ratio and the SR gain on the noise strength and the relative duration of the pulses. We find that the SR gains can reach values larger than unity, with maximum values showing a nonmonotonic dependence on the duration of the pulses.

Journal Article↗
PubMed2004

Coupled oscillator systems of cultured cardiac myocytes: fluctuation and scaling properties.

Isolated and cultured neonatal cardiac myocytes exhibit autonomous rhythmic contraction, and their dynamics vary dramatically depending on the extent of mutual coupling among individual myocytes. We study the temporal changes of interbeat interval series in aggregated systems of spontaneously beating cultured neonatal rat cardiac myocytes and observe a rich variety of complex, nonlinear features such as frequent alternations, bistability, and periodic spikes. Fluctuation analysis of the interval series reveals that there occurs a transition in scaling behavior from persistent correlations to antipersistent correlations as the coupling develops with incubation time. Additionally, we perform computer simulations using interacting Bonhoeffer-van der Pol oscillators to understand the effects of coupling on the fluctuation dynamics of each constituent oscillator. We find that the formation of strong and heterogeneous coupling among the oscillators is a key factor to yield the complexity in the interval series as well as in the scaling behavior.

Animals↗
PubMed2004

Bifurcation study of regular pulse packages in laser diodes subject to optical feedback.

We study the influence of delayed optical feedback from a short external cavity on the emission dynamics of semiconductor lasers using the Lang and Kobayashi rate equation model. A combination of numerical integration and continuation techniques allows us to bring new light into the bifurcation scenario leading to the regular pulse packages (RPP) regime. We give examples of bistability between RPP and time-periodic or steady state solutions. Our bifurcation study of RPP regime is complemented by an analysis of the dependency of the RPP period on the laser and feedback parameters. We qualitatively study this new dynamical regime by plotting a two-dimensional map in the feedback parameters space. The occurrence of RPP is furthermore associated with a topological change in the bifurcation diagram and accompanied by the creation of a new type of bifurcation bridge between a mode and an antimode.

Journal Article↗
PubMed2004

Synchronization of two interacting populations of oscillators.

We analyze synchronization between two interacting populations of different phase oscillators. For the important case of asymmetric coupling functions, we find a much richer dynamical behavior compared to that of symmetrically coupled populations of identical oscillators. It includes three types of bistabilities, higher order entrainment, and the existence of states with unusual stability properties. All possible routes to synchronization of the populations are presented and some stability boundaries are obtained analytically. The impact of these findings for neuroscience is discussed.

Adaptation, Physiological↗
PubMed2004

Discrete kink dynamics in hydrogen-bonded chains: the two-component model.

We study discrete topological solitary waves (kinks and antikinks) in two nonlinear diatomic chain models that describe the collective dynamics of proton transfers in one-dimensional hydrogen-bonded networks. The essential ingredients of the models are (i) a realistic (anharmonic) ion-proton interaction in the hydrogen bond, (ii) a harmonic coupling between the protons in adjacent hydrogen bonds, and (iii) a harmonic coupling between the nearest-neighbor heavy ions (an isolated diatomic chain with the lowest acoustic band) or instead a harmonic on-site potential for the heavy ions (a diatomic chain subject to a substrate with two optical bands), both providing a bistability of the hydrogen-bonded proton. Exact two-component (kink and antikink) discrete solutions for these models are found numerically. We compare the soliton solutions and their properties in both the one- (when the heavy ions are fixed) and two-component models. The effect of stability switchings, discovered previously for a class of one-component kink-bearing models, is shown to exist in these two-component models as well. However, the presence of the second component, i.e., the softness of the heavy-ion sublattice, brings principal differences, like a significant difference in the stability switchings behavior for the kinks and the antikinks. Water-filled carbon nanotubes are briefly discussed as possible realistic systems, where topological discrete (anti)kink states might exist.

Journal Article↗
PubMed2004

Single- and double-vortex vector solitons in self-focusing nonlinear media.

We study two-component spatial optical solitons carrying an angular momentum and propagating in a self-focusing saturable nonlinear medium. When one of the components is small, such vector solitons can be viewed as a self-trapped vortex beam that guides either the fundamental or first-order guided mode, and they are classified as single- and double-vortex vector solitons. For such composite vortex beams, we demonstrate that a large-amplitude guided mode can stabilize the ringlike vortex beam which usually decays due to azimuthal modulational instability. We identify different types of these vector vortex solitons and demonstrate both vortex bistability and mutual stabilization effect.

Journal Article↗
PubMed2005

Critical exponents near a bifurcation point in noise-induced escape over a fluctuating barrier.

If a bistable system is brought closer to the bifurcation point at A= A(c) , the rate of noise-activated escape to the other stable equilibrium increases as ln k proportional, variant- mid R:A- A(c) mid R:(3/2) . Recently, Dykman, Golding, and Ryvkine derived that upon the addition of an oscillating force, the exponent equals 2 in an intermediate-frequency regime [Phys. Rev. Lett. 92, 080602 (2004)]; Phys. Rev. E 69, 061102 (2004)]]. In this paper we study the critical exponents near a bifurcation point for an analytically solvable case; that of a piecewise linear well with an imposed dichotomous fluctuation. It is shown how an exponent 2 emerges when a parameter is changed to move a system from nonadiabaticity to adiabaticity. It is, furthermore, shown that the power law breaks down for low barriers. Finally, it is explained why fundamental differences occur between smooth and piecewise linear setups.

Journal Article↗
PubMed2005

Two-state migration of DNA in a structured microchannel.

DNA migration in topologically structured microchannels with periodic cavities is investigated experimentally and with Brownian dynamics simulations of a simple bead-spring model. The results are in very good agreement with one another. In particular, the experimentally observed migration order of lambda - and T2-DNA molecules is reproduced by the simulations. The simulation data indicate that the mobility may depend on the chain length in a nonmonotonic way at high electric fields. This is found to be the signature of a nonequilibrium bistability between two different migration states, a slow one and a fast one. The latter can also be observed experimentally under appropriate conditions.

Computer Simulation↗
PubMed2005

Nonlinear elasticity of an alpha -helical polypeptide.

We study a minimal extension of the wormlike chain model to describe polypeptides having alpha-helical secondary structure. In this model the presence or absence of secondary structure enters as a scalar variable that controls the local chain bending modulus. Using this model we compute the extensional compliance of an alpha-helix under tensile stress, the bending compliance of the molecule under externally imposed torques, and the nonlinear interaction of such torques and forces on the molecule. We find that, due to coupling of the "internal" secondary structure variables to the conformational degrees of freedom of the polymer, the molecule has a highly nonlinear response to applied stress and force couples. In particular we demonstrate a sharp lengthening transition under applied force and a buckling transition under applied torque. Finally, we speculate that the inherent bistability of the molecule may underlie protein conformational change in vivo.

Computer Simulation↗
PubMed2005

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.

Journal Article↗