Delay Fokker-Planck equations, Novikov's theorem, and Boltzmann distributions as small delay approximations

Stochastic switching of delayed feedback suppresses oscillations in genetic regulatory systems

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.

Directional switches in network-organized swarming systems with delay

Coordinated directional switches can emerge between members of moving biological groups. Previous studies have shown that the self-propelled particles model can well reproduce directional switching behaviors, but it neglects the impact of social interactions. Thus, we focus on the influence of social interactions on the ordered directional switching motion of swarming systems, in which homogeneous Erdös-Rényi networks, heterogeneous scale-free networks, networks with community structures, and real-world animal social networks have been considered. The theoretical estimation of mean switching time is obtained, and the results show that the interplay between social and delayed interactions plays an important role in regulating directional switching behavior. To be specific, for homogeneous Erdös-Rényi networks, the increase in mean degree may suppress the directional switching behaviors if the delay is sufficiently small. However, when the delay is large, the large mean degree may promote the directional switching behavior. For heterogeneous scale-free networks, the increase of degree heterogeneity can reduce the mean switching time if the delay is sufficiently small, while the increasing degree heterogeneity may suppress the ordered directional switches if the delay is large. For networks with community structures, higher communities can promote directional switches for small delays, while for large delays, it may inhibit directional switching behavior. For dolphin social networks, delay can promote the directional switching behavior. Our results bring to light the role of social and delayed interactions in the ordered directional switching motion.

Coloured noise induces phenotypic diversity with energy dissipation

Genes integrate many different sources of noise to adapt their survival strategy with energy costs, but how this noise impacts gene phenotype switching is not fully understood. Here, we refine a mechanistic model with multiplicative and additive coloured noise and analyse the influence of noise strength (NS) and autocorrelation time (AT) on gene phenotypic diversity. Different from white noise, we found that in the autocorrelation time-scale plane, increasing the multiplicative noise will broaden the bimodal region of the gene product, and additive noise will induce bimodal region drift from the lower level to the higher level, while the AT will promote this transition. Specifically, the effect of AT on gene expression is similar to a feedback loop; that is, the AT of multiplicative noise will elongate the mean first passage time (MFPT) from the low stable state to the high stable state, but it will reduce the MFPT from the high stable state to the low stable state, and the opposite is true for additive noise. Moreover, these transitions will violate the detailed equilibrium and then consume energy. By effective topology network reconstruction, we found that when the NS is small, the more obvious the bimodality is, the lower the energy dissipation; however, when the NS is large, it will consume more energy with a tendency for bimodality. The overall analysis implies that living organisms will utilize noise strength and its autocorrelation time for better survival in complex and fluctuating environments.

Effect of time delay in a bistable synthetic gene network

The essence of logical stochastic resonance is the dynamic manipulation of potential wells. The effect of time delay on the depth of potential wells and the width of a bistable region can be inferred by logic operations in the bistable system with time delay. In a time-delayed synthetic gene network, time delay in the synthesis process can increase the depth of the potential wells, while that in the degradation process, it can reduce the depth of the potential wells, which will result in a decrease in the width of the bistable region (the reason for time delay to induce logic operations without external driving force) and the instability of the system (oscillation). These two opposite effects imply stretching and folding, leading to complex dynamical behaviors of the system, including period, chaos, bubble, chaotic bubble, forward and reverse period doubling bifurcation, intermittency, and coexisting attractors.

Laminar chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of laminar chaos

Recently, it was shown that certain systems with large time-varying delay exhibit different types of chaos, which are related to two types of time-varying delay: conservative and dissipative delays. The known high-dimensional turbulent chaos is characterized by strong fluctuations. In contrast, the recently discovered low-dimensional laminar chaos is characterized by nearly constant laminar phases with periodic durations and a chaotic variation of the intensity from phase to phase. In this paper we extend our results from our preceding publication [Hart, Roy, Müller-Bender, Otto, and Radons, Phys. Rev. Lett. 123, 154101 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.154101], where it is demonstrated that laminar chaos is a robust phenomenon, which can be observed in experimental systems. We provide a time series analysis toolbox for the detection of robust features of laminar chaos. We benchmark our toolbox by experimental time series and time series of a model system which is described by a nonlinear Langevin equation with time-varying delay. The benchmark is done for different noise strengths for both the experimental system and the model system, where laminar chaos can be detected, even if it is hard to distinguish from turbulent chaos by a visual analysis of the trajectory.

Mitigation of tipping point transitions by time-delay feedback control

In stochastic multistable systems driven by the gradient of a potential, transitions between equilibria are possible because of noise. We study the ability of linear delay feedback control to mitigate these transitions, ensuring that the system stays near a desirable equilibrium. For small delays, we show that the control term has two effects: (i) a stabilizing effect by deepening the potential well around the desirable equilibrium and (ii) a destabilizing effect by intensifying the noise by a factor of (1-τα)^-1/2, where τ and α denote the delay and the control gain, respectively. As a result, successful mitigation depends on the competition between these two factors. We also derive analytical results that elucidate the choice of the appropriate control gain and delay that ensure successful mitigations. These results eliminate the need for any Monte Carlo simulations of the stochastic differential equations and, therefore, significantly reduce the computational cost of determining the suitable control parameters. We demonstrate the application of our results on two examples.

Fokker-Planck representations of non-Markov Langevin equations: application to delayed systems

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker-Planck equations for the n-time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker-Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Traveling band formation in feedback-driven colloids

Using simulation and theory we study the dynamics of a colloidal suspension in two dimensions subject to a time-delayed repulsive feedback that depends on the positions of the colloidal particles. The colloidal particles experience an additional potential that is a superposition of repulsive potential energies centered around the positions of all the particles a delay time ago. Here we show that such a feedback leads to self-organization of the particles into traveling bands. The width of the bands and their propagation speed can be tuned by the delay time and the range of the imposed repulsive potential. The emerging traveling band behavior is observed in Brownian dynamics computer simulations as well as microscopic dynamic density functional theory. Traveling band formation also persists in systems of finite size leading to rotating traveling waves in the case of circularly confined systems.

Uncertainty relations for time-delayed Langevin systems

The thermodynamic uncertainty relation, which establishes a universal trade-off between nonequilibrium current fluctuations and dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian systems; therefore, we investigate the thermodynamic uncertainty relation for time-delayed Langevin systems. We prove that the fluctuation of arbitrary dynamical observables is constrained by the Kullback-Leibler divergence between the distributions of the forward path and its reversed counterpart. Specifically, for observables that are antisymmetric under time reversal, the fluctuation is bounded from below by a function of a quantity that can be identified as a generalization of the total entropy production in Markovian systems. We also provide a lower bound for arbitrary observables that are odd under position reversal. The term in this bound reflects the extent to which the position symmetry has been broken in the system and can be positive even in equilibrium. Our results hold for finite observation times and a large class of time-delayed systems because detailed underlying dynamics are not required for the derivation. We numerically verify the derived uncertainty relations using two single time-delay systems and one distributed time-delay system.

Heat flow due to time-delayed feedback

Many stochastic systems in biology, physics and technology involve discrete time delays in the underlying equations of motion, stemming, e. g., from finite signal transmission times, or a time lag between signal detection and adaption of an apparatus. From a mathematical perspective, delayed systems represent a special class of non-Markovian processes with delta-peaked memory kernels. It is well established that delays can induce intriguing behaviour, such as spontaneous oscillations, or resonance phenomena resulting from the interplay between delay and noise. However, the thermodynamics of delayed stochastic systems is still widely unexplored. This is especially true for continuous systems governed by nonlinear forces, which are omnipresent in realistic situations. We here present an analytical approach for the net steady-state heat rate in classical overdamped systems subject to time-delayed feedback. We show that the feedback inevitably leads to a finite heat flow even for vanishingly small delay times, and detect the nontrivial interplay of noise and delay as the underlying reason. To illustrate this point, and to provide an understanding of the heat flow at small delay times below the velocity-relaxation timescale, we compare with the case of underdamped motion where the phenomenon of "entropy pumping" has already been established. Application to an exemplary (overdamped) bistable system reveals that the feedback induces heating as well as cooling regimes and leads to a maximum of the medium entropy production at coherence resonance conditions. These observations are, in principle, measurable in experiments involving colloidal suspensions.

Feeble object detection of underwater images through LSR with delay loop

Feeble object detection is a long-standing problem in vision based underwater exploration work. However, because of the complicated light propagation situation and high background noise, underwater images are highly degraded. Noise is not always detrimental. Logical stochastic resonance (LSR) can be a useful tool for amplifying feeble signals by utilizing the constructive interplay of noise and a nonlinear system. In the present study, an appropriate LSR structure with a delay loop is proposed to process a low-quality underwater image for enhancing the vision detection accuracy of underwater feeble objects. Ocean experiments are conducted to demonstrate the effectiveness of the proposed structure. We also give explicit numerical results to illustrate the relationship between the structure of LSR and the correct detection probability. Methods presented in this paper are quite general and can thus be potentially extended to other applications for obtaining better performance.

Force-linearization closure for non-Markovian Langevin systems with time delay

This paper is concerned with the Fokker-Planck (FP) description of classical stochastic systems with discrete time delay. The non-Markovian character of the corresponding Langevin dynamics naturally leads to a coupled infinite hierarchy of FP equations for the various n-time joint distribution functions. Here, we present an approach to close the hierarchy at the one-time level based on a linearization of the deterministic forces in all members of the hierarchy starting from the second one. This leads to a closed equation for the one-time probability density in the steady state. Considering two generic nonlinear systems, a colloidal particle in a sinusoidal or bistable potential supplemented by a linear delay force, we demonstrate that our approach yields a very accurate representation of the density as compared to quasiexact numerical results from direct solution of the Langevin equation. Moreover, the results are significantly improved against those from a small-delay approximation and a perturbation-theoretical approach. We also discuss the possibility of accessing transport-related quantities, such as escape times, based on an additional Kramers approximation. Our approach applies to a wide class of models with nonlinear deterministic forces.

Diffusion of active chiral particles

The diffusion of chiral active Brownian particles in three-dimensional space is studied analytically, by consideration of the corresponding Fokker-Planck equation for the probability density of finding a particle at position x and moving along the direction v[over ̂] at time t, and numerically, by the use of Langevin dynamics simulations. The analysis is focused on the marginal probability density of finding a particle at a given location and at a given time (independently of its direction of motion), which is found from an infinite hierarchy of differential-recurrence relations for the coefficients that appear in the multipole expansion of the probability distribution, which contains the whole kinematic information. This approach allows the explicit calculation of the time dependence of the mean-squared displacement and the time dependence of the kurtosis of the marginal probability distribution, quantities from which the effective diffusion coefficient and the "shape" of the positions distribution are examined. Oscillations between two characteristic values were found in the time evolution of the kurtosis, namely, between the value that corresponds to a Gaussian and the one that corresponds to a distribution of spherical shell shape. In the case of an ensemble of particles, each one rotating around a uniformly distributed random axis, evidence is found of the so-called effect "anomalous, yet Brownian, diffusion," for which particles follow a non-Gaussian distribution for the positions yet the mean-squared displacement is a linear function of time.

TUMOR CELL GROWTH SUBJECTED TO CORRELATED NOISES AND TIME DELAY

The tumor cell growth with time-delayed feedback driven by correlated noises under the immune surveillance are investigated within an anti-tumor model. The effects of the noise correlation strength and time delay on the stationary probability distribution, the average tumor cell population and the mean first passage time (MFPT) are analyzed in detail based on the delay Fokker–Planck equation. The effects of the correlation strength and time delay could play the same role in the average tumor cell population, but play opposite role in the MFPT. In addition, the role of the correlation strength and time delay for different activation thresholds of immune system is explored.

Delay-induced stochastic bifurcations in a bistable system under white noise

In this paper, the effects of noise and time delay on stochastic bifurcations are investigated theoretically and numerically in a time-delayed Duffing-Van der Pol oscillator subjected to white noise. Due to the time delay, the random response is not Markovian. Thereby, approximate methods have been adopted to obtain the Fokker-Planck-Kolmogorov equation and the stationary probability density function for amplitude of the response. Based on the knowledge that stochastic bifurcation is characterized by the qualitative properties of the steady-state probability distribution, it is found that time delay and feedback intensity as well as noise intensity will induce the appearance of stochastic P-bifurcation. Besides, results demonstrated that the effects of the strength of the delayed displacement feedback on stochastic bifurcation are accompanied by the sensitive dependence on time delay. Furthermore, the results from numerical simulations best confirm the effectiveness of the theoretical analyses.

Smoluchowski diffusion equation for active Brownian swimmers

We study the free diffusion in two dimensions of active Brownian swimmers subject to passive fluctuations on the translational motion and to active fluctuations on the rotational one. The Smoluchowski equation is derived from a Langevin-like model of active swimmers and analytically solved in the long-time regime for arbitrary values of the Péclet number; this allows us to analyze the out-of-equilibrium evolution of the positions distribution of active particles at all time regimes. Explicit expressions for the mean-square displacement and for the kurtosis of the probability distribution function are presented and the effects of persistence discussed. We show through Brownian dynamics simulations that our prescription for the mean-square displacement gives the exact time dependence at all times. The departure of the probability distribution from a Gaussian, measured by the kurtosis, is also analyzed both analytically and computationally. We find that for the inverse of Péclet numbers ≲0.1, the distance from Gaussian increases as ∼t(-2) at short times, while it diminishes as ∼t(-1) in the asymptotic limit.

Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities

Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the non-Markovian nature of the dynamics requires a modification of the basic relation linking dissipation to the breaking of time-reversal symmetry at the level of trajectories. The modified relation includes a contribution arising from the acausal character of the reverse process. This, in turn, leads to another second-law-like inequality. We illustrate the general formalism with a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.

Time delay can facilitate coherence in self-driven interacting-particle systems

Directional switching in a self-propelled particle model with delayed interactions is investigated. It is shown that the average switching time is an increasing function of time delay. The presented results are applied to studying collective animal behavior. It is argued that self-propelled particle models with time delays can explain the state-dependent diffusion coefficient measured in experiments with locust groups. The theory is further generalized to heterogeneous groups where each individual can respond to its environment with a different time delay.

Modulating resonance behaviors by noise recycling in bistable systems with time delay

In this paper, the impact of noise recycling on resonance behaviors is studied theoretically and numerically in a prototypical bistable system with delayed feedback. According to the interior cooperating and interacting activity of noise recycling, a theory has been proposed by reducing the non-Markovian problem into a two-state model, wherein both the master equation and the transition rates depend on not only the current state but also the earlier two states due to the recycling lag and the feedback delay. By virtue of this theory, the formulae of the power spectrum density and the linear response function have been found analytically. And the theoretical results are well verified by numerical simulations. It has been demonstrated that both the recycling lag and the feedback delay play a crucial role in the resonance behaviors. In addition, the results also suggest an alternative scheme to modulate or control the coherence or stochastic resonance in bistable systems with time delay.

Neuronal avalanches, epileptic quakes and other transient forms of neurodynamics

Power-law behaviors in brain activity in healthy animals, in the form of neuronal avalanches, potentially benefit the computational activities of the brain, including information storage, transmission and processing. In contrast, power-law behaviors associated with seizures, in the form of epileptic quakes, potentially interfere with the brain's computational activities. This review draws attention to the potential roles played by homeostatic mechanisms and multistable time-delayed recurrent inhibitory loops in the generation of power-law phenomena. Moreover, it is suggested that distinctions between health and disease are scale-dependent. In other words, what is abnormal and defines disease it is not the propagation of neural activity but the propagation of activity in a neural population that is large enough to interfere with the normal activities of the brain. From this point of view, epilepsy is a disease that results from a failure of mechanisms, possibly located in part in the cortex itself or in the deep brain nuclei and brainstem, which truncate or otherwise confine the spatiotemporal scales of these power-law phenomena.

Spectral solution of delayed random walks

We develop a spectral method for computing the probability density function for delayed random walks; for such problems, the method is exact to machine precision and faster than existing approaches. In conjunction with a step function approximation and the weak Euler-Maruyama discretization, the spectral method can be applied to nonlinear stochastic delay differential equations (SDDE). In essence, this means approximating the SDDE by a delayed random walk, which is then solved using the spectral method. We carry out tests for a particular nonlinear SDDE that show that this method captures the solution without the need for Monte Carlo sampling.

The delayed and noisy nervous system: implications for neural control

Recent advances in the study of delay differential equations draw attention to the potential benefits of the interplay between random perturbations ('noise') and delay in neural control. The phenomena include transient stabilizations of unstable steady states by noise, control of fast movements using time-delayed feedback and the occurrence of long-lived delay-induced transients. In particular, this research suggests that the interplay between noise and delay necessitates the use of intermittent, discontinuous control strategies in which corrective movements are made only when controlled variables cross certain thresholds. A potential benefit of such strategies is that they may be optimal for minimizing energy expenditures associated with control. In this paper, the concepts are made accessible by introducing them through simple illustrative examples that can be readily reproduced using software packages, such as XPPAUT.

Frequency adaptation in controlled stochastic resonance utilizing delayed feedback method: two-pole approximation for response function

Stochastic resonance (SR) enhanced by time-delayed feedback control is studied. The system in the absence of control is described by a Langevin equation for a bistable system, and possesses a usual SR response. The control with the feedback loop, the delay time of which equals to one-half of the period (2π/Ω) of the input signal, gives rise to a noise-induced oscillatory switching cycle between two states in the output time series, while its average frequency is just smaller than Ω in a small noise regime. As the noise intensity D approaches an appropriate level, the noise constructively works to adapt the frequency of the switching cycle to Ω, and this changes the dynamics into a state wherein the phase of the output signal is entrained to that of the input signal from its phase slipped state. The behavior is characterized by power loss of the external signal or response function. This paper deals with the response function based on a dichotomic model. A method of delay-coordinate series expansion, which reduces a non-Markovian transition probability flux to a series of memory fluxes on a discrete delay-coordinate system, is proposed. Its primitive implementation suggests that the method can be a potential tool for a systematic analysis of SR phenomenon with delayed feedback loop. We show that a D-dependent behavior of poles of a finite Laplace transform of the response function qualitatively characterizes the structure of the power loss, and we also show analytical results for the correlation function and the power spectral density.

Correlation times in stochastic equations with delayed feedback and multiplicative noise

We obtain the characteristic correlation time associated with a model stochastic differential equation that includes the normal form of a pitchfork bifurcation and delayed feedback. In particular, the validity of the common assumption of statistical independence between the state at time t and that at t-τ, where τ is the delay time, is examined. We find that the correlation time diverges at the model's bifurcation line, thus signaling a sharp bifurcation threshold, and the failure of statistical independence near threshold. We determine the correlation time both by numerical integration of the governing equation, and analytically in the limit of small τ. The correlation time T diverges as T~a(-1), where a is the control parameter so that a=0 is the bifurcation threshold. The small-τ expansion correctly predicts the location of the bifurcation threshold, but there are systematic deviations in the magnitude of the correlation time.

Analytical determination of the bifurcation thresholds in stochastic differential equations with delayed feedback

Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for a nonlinear stochastic differential delay equation with feedback. Our results assume that the delay time τ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average <x(t)|x(t-τ)>, where x(t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the interaction between fluctuation correlations and delay affect the stability of the solutions of the model equation studied.

Pitchfork and Hopf bifurcation thresholds in stochastic equations with delayed feedback

The bifurcation diagram of a model stochastic differential equation with delayed feedback is presented. We are motivated by recent research on stochastic effects in models of transcriptional gene regulation. We start from the normal form for a pitchfork bifurcation, and add multiplicative or parametric noise and linear delayed feedback. The latter is sufficient to originate a Hopf bifurcation in that region of parameters in which there is a sufficiently strong negative feedback. We find a sharp bifurcation in parameter space, and define the threshold as the point in which the stationary distribution function p(x) changes from a delta function at the trivial state x=0 to p(x) approximately x(alpha) at small x (with alpha=-1 exactly at threshold). We find that the bifurcation threshold is shifted by fluctuations relative to the deterministic limit by an amount that scales linearly with the noise intensity. Analytic calculations of the bifurcation threshold are also presented in the limit of small delay tau-->0 that compare quite favorably with the numerical solutions even for moderate values of tau .

Long delay times in reaction rates increase intrinsic fluctuations

In spatially distributed cellular systems, it is often convenient to represent complicated auxiliary pathways and spatial transport by time-delayed reaction rates. Furthermore, many of the reactants appear in low numbers necessitating a probabilistic description. The coupling of delayed rates with stochastic dynamics leads to a probability conservation equation characterizing a non-Markovian process. A systematic approximation is derived that incorporates the effect of delayed rates on the characterization of molecular noise valid in the limit of long delay time. By way of a simple example, we show that delayed reaction dynamics can only increase intrinsic fluctuations about the steady state. The method is general enough to accommodate nonlinear transition rates allowing characterization of fluctuations around a delay-induced limit cycle.

Balancing with positive feedback: the case for discontinuous control

Experimental observations indicate that positive feedback plays an important role for maintaining human balance in the upright position. This observation is used to motivate an investigation of a simple switch-like controller for postural sway in which corrective movements are made only when the vertical displacement angle exceeds a certain threshold. This mechanism is shown to be consistent with the experimentally observed variations in the two-point correlation for human postural sway. Analysis of first-passage times for this model suggests that this control strategy may slow escape by taking advantage of two intrinsic properties of a stochastic unstable first-order delay differential equation: (i) time delay and (ii) the possibility that the dynamics can be 'temporarily confined' near the origin.

Coherence versus reliability of stochastic oscillators with delayed feedback

For noisy self-sustained oscillators, both reliability, the stability of a response to a noisy driving, and coherence, understood in the sense of constancy of oscillation frequency, are important characteristics. Although both characteristics and techniques for controlling them have received great attention from researchers, owing to their importance for neurons, lasers, clocks, electric generators, etc., these characteristics were previously considered separately. In this paper, a strong quantitative relation between coherence and reliability is revealed for a limit cycle oscillator subject to a weak noisy driving and a linear delayed feedback, a convection control tool. The analytical findings are verified and enriched with a numerical simulation for the Van der Pol-Duffing oscillator.

Effects of time delay on symmetric two-species competition subject to noise

Noise and time delay act simultaneously on real ecological systems. The Lotka-Volterra model of symmetric two-species competition with noise and time delay was investigated in this paper. By means of stochastic simulation, we find that (i) the time delay induces the densities of the two species to periodically oscillate synchronously; (ii) the stationary probability distribution function of the two-species densities exhibits a transition from multiple to single stability as the delay time increases; (iii) the characteristic correlation time for the sum of the two-species densities squared exhibits a nonmonotonic behavior as a function of delay time. Our results have the implication that the combination of noise and time delay could provide an efficient tool for understanding real ecological systems.

Exact time-average distribution for a stationary non-Markovian massive Brownian particle coupled to two heat baths

Using a time-averaging technique we obtain exactly the probability distribution for position and velocity of a Brownian particle under the influence of two heat baths at different temperatures. These baths are expressed by a white noise term, representing the fast dynamics, and a colored noise term, representing the slow dynamics. Our exact solution scheme accounts for inertial effects that are not present in approaches that assume the Brownian particle in the overdamped limit. We are also able to obtain the contributions associated with the fast noise that are usually neglected by other approaches.

Time-delayed feedback control of a flashing ratchet

Closed-loop or feedback control ratchets use information about the state of the system to operate with the aim of maximizing the performance of the system. In this paper we investigate the effects of a time delay in the feedback for a protocol that performs an instantaneous maximization of the center-of-mass velocity. For the one and the few particle cases the flux decreases with increasing delay, as an effect of the decorrelation of the present state of the system with the information that the controller uses, but the delayed closed-loop protocol succeeds to perform better than its open-loop counterpart provided the delays are smaller than the characteristic times of the Brownian ratchet. For the many particle case, we also show that for small delays the center-of-mass velocity decreases for increasing delays. However, for large delays we find the surprising result that the presence of the delay can improve the performance of the nondelayed feedback ratchet and the flux can attain the maximum value obtained with the optimal periodic protocol. This phenomenon is the result of the emergence of a dynamical regime where the presence of the delayed feedback stabilizes one quasiperiodic solution or several (multistability), which resemble the solutions obtained in the so-called threshold protocol. Our analytical and numerical results point towards the feasibility of an experimental implementation of a feedback controlled ratchet that performs equal or better than its optimal open-loop version.

Stochastic order parameter equation of isometric force production revealed by drift-diffusion estimates

We address two questions that are central to understanding human motor control variability: what kind of dynamical components contribute to motor control variability (i.e., deterministic and/or random ones), and how are those components structured? To this end, we derive a stochastic order parameter equation for isometric force production from experimental data using drift-diffusion estimates. We show that the force variability increases with the required force output because of a decrease of deterministic stability and an accompanying increase of noise intensity. A structural analysis reveals that the deterministic component consists of a linear control loop, while the random component involves a noise source that scales with force output. In addition, we present evidence for the existence of a subject-independent overall noise level of human isometric force production.

Brownian motor with time-delayed feedback

An inertial Brownian motor with time-delayed feedback driven by an unbiased time-periodic force is investigated. It is found that the mean velocity and the rectification efficiency are decreased when the noise intensity is increased. While the shape of the mean velocity and the rectification efficiency can be changed from one peak to two peaks when the time delay is increased, the symmetry in the velocity probability distribution function is broken when the delay time is increased.

Theoretical analysis of destabilization resonances in time-delayed stochastic second-order dynamical systems and some implications for human motor control

A linear stochastic delay differential equation of second order is studied that can be regarded as a Kramers model with time delay. An analytical expression for the stationary probability density is derived in terms of a Gaussian distribution. In particular, the variance as a function of the time delay is computed analytically for several parameter regimes. Strikingly, in the parameter regime close to the parameter regime in which the deterministic system exhibits Hopf bifurcations, we find that the variance as a function of the time delay exhibits a sequence of pronounced peaks. These peaks are interpreted as delay-induced destabilization resonances arising from oscillatory ghost instabilities. On the basis of the obtained theoretical findings, reinterpretations of previous human motor control studies and predictions for future human motor control studies are provided.