Delayed feedback control of noise-induced patterns in excitable media

Role of Noise-Modulated Self-Propulsion in Driving Spatiotemporal Orders in Active Systems

Fluctuations play a pivotal role in driving spatiotemporal order in active matter systems. In this study, we employ a novel analytical framework to investigate the impact of dichotomous noise on the self-propelling velocity of active particle systems such as polymerizing actin filaments or reproducing elongated bacteria. By incorporating dichotomous fluctuations with Ornstein-Zernike correlations into a continuum-based model, we derive a bifurcation condition in the noise parameter space, revealing a noise-induced instability that drives the emergence of traveling waves. This approach demonstrates how specific noise strengths and correlation times expand the instability region by introducing effective new degrees of freedom that alter the system's stability matrix. Advance numerical simulations, meticulously designed to handle the properties of dichotomous noise, validate these theoretical predictions and reveal excellent agreement. A key finding is the observation of wave-reversal behavior, driven by the sign alternation of the noise-modulated advection term and modulated by the relaxation time. Remarkably, we identify a finite parameter range where this reversal is suppressed, offering new insights into noise-induced bifurcations and spatiotemporal dynamics. Our combined analytical and numerical approach provides a deeper understanding of the role of noise in shaping self-organization and pattern formation in biological and synthetic active systems.

Noisy-flow-induced instability in a reaction-diffusion system

We consider a generic reaction-diffusion-advection system where the flow velocity of the advection term is subjected to dichotomous noise with zero mean and Ornstein-Zernike correlation. A general condition for noisy-flow-induced instability is derived in the flow velocity-correlation rate parameter plane. Full numerical simulations on Gierer-Meinhardt model with activator-inhibitor kinetics have been performed to show how noisy differential flow can lead to symmetry breaking of a homogeneous stable state in the presence of noise resulting in traveling waves.

Transient behavior in systems with time-delayed feedback

We investigate the transient times for the onset of control of steady states by time-delayed feedback. The optimization of control by minimizing the transient time before control becomes effective is discussed analytically and numerically, and the competing influences of local and global features are elaborated. We derive an algebraic scaling of the transient time and confirm our findings by numerical simulations in dependence on feedback gain and time delay.

Chimera states induced by spatially modulated delayed feedback

Recently, we have presented spatially modulated delayed feedback as a novel mechanism, which generically generates chimera states, remarkable spatiotemporal patterns in which coherence coexists with incoherence [O. E. Omel'chenko, Phys. Rev. Lett. 100, 044105 (2008)]. Remarkably, such chimera states serve as a natural link between completely coherent states and completely incoherent states. So far, we have studied this mechanism with a self-consistency-based numerical analysis only. In contrast, in this paper we perform a thorough dynamical description and, in particular, a stability analysis of the emerging chimera states. For this, we apply a recently developed reduction procedure [A. Pikovsky and M. Rosenblum, Phys. Rev. Lett. 101, 264103 (2008)]. By combining analytical and numerical approaches, we systematically describe the relationship between the parameters of the delayed feedback on one hand and the properties of the chimera states on the other hand. We provide the general rules for an effective control and manipulation of the chimera states.

Pattern formation controlled by time-delayed feedback in bistable media

Effects of time-delayed feedback on pattern formation are studied both numerically and theoretically in a bistable reaction-diffusion model. The time-delayed feedback applied to the activator and/or the inhibitor alters the behavior of the nonequilibrium Ising-Bloch (NIB) bifurcation. If the intensities of the feedbacks applied to the two species are identical, only the velocities of Bloch fronts are changed. If the intensities are different, both the critical point of the NIB bifurcation and the threshold of stability of front to transverse perturbations are changed. The effect of time-delayed feedback on the activator opposes the effect of time-delayed feedback on the inhibitor. When the time-delayed feedback is applied individually to one of the species, positive and negative feedbacks make the bifurcation point shift to different directions. The time-delayed feedback provides a flexible way to control the NIB bifurcation and the pattern formation.

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.

Control of noise-induced coherent behaviors in an array of excitable elements by time-delayed feedback

We investigate feedback-controlled coherent dynamics in a two-dimensional array of excitable elements. We demonstrate that one can freely enhance or reduce the spatiotemporal coherence of noise-induced oscillation, such as coherence resonance and phase synchronization, by controlling both the delay time and the feedback gain. Furthermore, we find that noise-induced oscillations are entrained by the feedback force in a certain range of the delay time. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator model.

Interplay of time-delayed feedback control and temporally correlated noise in excitable systems

The interplay of time-delayed feedback and temporally correlated coloured noise in a single and two coupled excitable systems is studied in the framework of the FitzHugh-Nagumo (FHN) model. By using coloured noise instead of white noise, the noise correlation time is introduced as an additional time scale. We show that in a single FHN system the major time scale of oscillations is strongly influenced by the noise correlation time, which in turn affects the maxima of coherence with respect to the delay time. In two coupled FHN systems, coloured noise input to one subsystem influences coherence resonance and stochastic synchronization of both subsystems. Application of delayed feedback control to the coloured noise-driven subsystem is shown to change coherence and time scales of noise-induced oscillations in both systems, and to enhance or suppress stochastic synchronization under certain conditions.

Feedback-controlled dynamics in a two-dimensional array of active elements

We investigate collective behaviors in a two-dimensional array of active elements controlled by time-delayed feedback, where elements are prepared by localizing the Belousov-Zhabotinsky reaction in a gel matrix. We demonstrate that both the spatial and temporal coherence can be effectively controlled by varying feedback parameters, such as the time delay and the gain. For a sufficiently high feedback gain, the fully synchronized state with low temporal coherence appears, which might be the state induced only by the delay feedback. Experimental results are approximately reproduced in a numerical simulation with a forced Oregonator reaction-diffusion model.

Time-delayed feedback in neurosystems

The influence of time delay in systems of two coupled excitable neurons is studied in the framework of the FitzHugh-Nagumo model. A time delay can occur in the coupling between neurons or in a self-feedback loop. The stochastic synchronization of instantaneously coupled neurons under the influence of white noise can be deliberately controlled by local time-delayed feedback. By appropriate choice of the delay time, synchronization can be either enhanced or suppressed. In delay-coupled neurons, antiphase oscillations can be induced for sufficiently large delay and coupling strength. The additional application of time-delayed self-feedback leads to complex scenarios of synchronized in-phase or antiphase oscillations, bursting patterns or amplitude death.

Robust entrainment phenomena of oscillations by delay time in the photosensitive Oregonator model

The influences of delayed feedback on the oscillating behaviors are numerically investigated by using the photosensitive Oregonator model with a Hopf point. We find that the time delay in the robust entrainment phenomena determines the time scale of the system, that is, T_{m}=(tau+delta)N (N=1,2,cdots, three dots, centered) , where T_{m} is the mean period of the oscillation and delta is a small constant compared with the delay time tau . Further, our numerical simulation shows that, when the system has a characteristic period T0 under the feedback with time delay, there exists an asymptotical line delta=delta_{0}T_{0} ( delta_{0} independent of any parameters) in the entrainment region with increasing strength of the feedback control c ; when the system has no characteristic period, the above linear relation is also kept, and delta decreases with increasing c .

Resonant control of stochastic spatiotemporal dynamics in a tunnel diode by multiple time-delayed feedback

We study the control of noise-induced spatiotemporal current density patterns in a semiconductor nanostructure (double-barrier resonant tunneling diode) by multiple time-delayed feedback. We find much more pronounced resonant features of noise-induced oscillations compared to single time feedback, rendering the system more sensitive to variations in the delay time tau . The coherence of noise-induced oscillations measured by the correlation time exhibits sharp resonances as a function of tau , and can be strongly increased by optimal choices of tau . Similarly, the peaks in the power spectral density are sharpened. We provide analytical insight into the control mechanism by relating the correlation times and mean frequencies of noise-induced breathing oscillations to the stability properties of the deterministic stationary current density filaments under the influence of the control loop. Moreover, we demonstrate that the use of multiple time delays enlarges the regime in which the deterministic dynamical properties of the system are not changed by delay-induced bifurcations.

Control of coherence resonance in semiconductor superlattices

We study the effect of time-delayed feedback control and Gaussian white noise on the spatiotemporal charge dynamics in a semiconductor superlattice. The system is prepared in a regime where the deterministic dynamics is close to a global bifurcation, namely, a saddle-node bifurcation on a limit cycle. In the absence of control, noise can induce electron charge front motion through the entire device, and coherence resonance is observed. We show that with appropriate selection of the time-delayed feedback parameters the effect of coherence resonance can be either enhanced or destroyed, and the coherence of stochastic domain motion at low noise intensity is dramatically increased. Additionally, the purely delay-induced dynamics in the system is investigated, and a homoclinic bifurcation of a limit cycle is found.

Limit-cycle oscillators subject to a delayed feedback

The coexistence of two stable limit cycles exhibiting different periods is examined for a nonlinear oscillator subject to a delayed feedback. For the case of a weakly nonlinear oscillator, we discuss the validity of a previously determined phase equation. For the case of a strongly nonlinear oscillator, we derive a phase equation and analyze its bifurcation diagram. Our analysis is motivated by previous experimental studies of chemical oscillators controlled by a delayed feedback.

Failure of feedback as a putative common mechanism of spreading depolarizations in migraine and stroke

The stability of cortical function depends critically on proper regulation. Under conditions of migraine and stroke a breakdown of transmembrane chemical gradients can spread through cortical tissue. A concomitant component of this emergent spatio-temporal pattern is a depolarization of cells detected as slow voltage variations. The propagation velocity of approximately 3 mm/min indicates a contribution of diffusion. We propose a mechanism for spreading depolarizations (SD) that rests upon a nonlocal or noninstantaneous feedback in a reaction-diffusion system. Depending upon the characteristic space and time scales of the feedback, the propagation of cortical SD can be suppressed by shifting the bifurcation line, which separates the parameter regime of pulse propagation from the regime where a local disturbance dies out. The optimization of this feedback is elaborated for different control schemes and ranges of control parameters.

Chimera states: the natural link between coherence and incoherence

Chimera states are remarkable spatiotemporal patterns in which coherence coexists with incoherence. As yet, chimera states have been considered as nongeneric, since they emerge only for particular initial conditions. In contrast, we show here that in a network of globally coupled oscillators delayed feedback stimulation with realistic (i.e., spatially decaying) stimulation profile generically induces chimera states. Intriguingly, a bifurcation analysis reveals that these chimera states are the natural link between the coherent and the incoherent states.

Suppressing noise-induced intensity pulsations in semiconductor lasers by means of time-delayed feedback

We investigate the possibility to suppress noise-induced intensity pulsations (relaxation oscillations) in semiconductor lasers by means of a time-delayed feedback control scheme. This idea is first studied in a generic normal-form model, where we derive an analytic expression for the mean amplitude of the oscillations and demonstrate that it can be strongly modulated by varying the delay time. We then investigate the control scheme analytically and numerically in a laser model of Lang-Kobayashi type and show that relaxation oscillations excited by noise can be very efficiently suppressed via feedback from a Fabry-Perot resonator.

Long-term correlations in stochastic systems with extended time-delayed feedback

The effects of a feedback with multiple time delays on noise-induced dynamics are studied in a nonlinear system close to the Hopf instability. We show analytically and numerically that such a feedback creates two distinct time scales, which can be tuned independently by the feedback parameters. In this way, the coherence of noise-induced oscillations can be drastically improved, and an arbitrarily large correlation of oscillations can be achieved without inducing a bifurcation. This opens up new perspectives for control of stochastic dynamical systems.