Interplay of noise induced stability and stochastic resetting.

Stochastic resetting and noise-enhanced stability are two phenomena that can affect the lifetime and relaxation of nonequilibrium states. They can be considered measures of controlling the efficiency of the completion process when a stochastic system has to reach the desired state. Here, we study the interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability that increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In this paper, we present a framework to analyze compromises between the two contrasting phenomena in noise-driven kinetics subject to random restarts.

Observing changes in human functioning during induced sleep deficiency and recovery periods.

Prolonged periods of sleep restriction seem to be common in the contemporary world. Sleep loss causes perturbations of circadian rhythmicity and degradation of waking alertness as reflected in attention, cognitive efficiency and memory. Understanding whether and how the human brain recovers from chronic sleep loss is important not only from a scientific but also from a public health perspective. In this work we report on behavioral, motor, and neurophysiological correlates of sleep loss in healthy adults in an unprecedented study conducted in natural conditions and comprising 21 consecutive days divided into periods of 4 days of regular life (a baseline), 10 days of chronic partial sleep restriction (30% reduction relative to individual sleep need) and 7 days of recovery. Throughout the whole experiment we continuously measured the spontaneous locomotor activity by means of actigraphy with 1-minute resolution. On a daily basis the subjects were undergoing EEG measurements (64-electrodes with 500 Hz sampling frequency): resting state with eyes open and closed (8 minutes long each) followed by Stroop task lasting 22 minutes. Altogether we analyzed actigraphy (distributions of rest and activity durations), behavioral measures (reaction times and accuracy from Stroop task) and EEG (amplitudes, latencies and scalp maps of event-related potentials from Stroop task and power spectra from resting states). We observed unanimous deterioration in all the measures during sleep restriction. Further results indicate that a week of recovery subsequent to prolonged periods of sleep restriction is insufficient to recover fully. Only one measure (mean reaction time in Stroop task) reverted to baseline values, while the others did not.

Dichotomous flow with thermal diffusion and stochastic resetting.

We consider properties of one-dimensional diffusive dichotomous flow and discuss effects of stochastic resonant activation (SRA) in the presence of a statistically independent random resetting mechanism. Resonant activation and stochastic resetting are two similar effects, as both of them can optimize the noise-induced escape. Our studies show completely different origins of optimization in adapted setups. Efficiency of stochastic resetting relies on elimination of suboptimal trajectories, while SRA is associated with matching of time scales in the dynamic environment. Consequently, both effects can be easily tracked by studying their asymptotic properties. Finally, we show that stochastic resetting cannot be easily used to further optimize the SRA in symmetric setups.

Nonlinear friction in underdamped anharmonic stochastic oscillators.

Non-equilibrium stationary states of overdamped anharmonic stochastic oscillators driven by Lévy noise are typically multimodal. The very same situation is recorded for an underdamped Lévy noise-driven motion in single-well potentials with linear friction. Within the current article, we relax the assumption that the friction experienced by a particle is linear. Using computer simulations, we study underdamped motions in single-well potentials in the regime of nonlinear friction. We demonstrate that it is relatively easy to observe multimodality in the velocity distribution as it is determined by the friction itself and it is the same as the multimodality in the overdamped case with the analogous deterministic force. Contrary to the velocity marginal density, it is more difficult to induce multimodality in the position. Nevertheless, for a fine-tuned nonlinear friction, the spatial multimodality can be recorded.

Erratum: Subdiffusive continuous-time random walks with stochastic resetting [Phys. Rev. E 99, 052116 (2019)].

This corrects the article DOI: 10.1103/PhysRevE.99.052116.

Delineating elastic properties of kinesin linker and their sensitivity to point mutations.

We analyze free energy estimators from simulation trials mimicking single-molecule pulling experiments on a neck linker of a kinesin motor. For that purpose, we have performed a version of steered molecular dynamics (SMD) calculations. The sample trajectories have been analyzed to derive distribution of work done on the system. In order to induce stretching of the linker, we have applied a constant pulling force to the molecule and allowed for a subsequent relaxation of its structure. The use of fluctuation relations (FR) relevant to non-equilibrium systems subject to thermal fluctuations allows us to assess the difference in free energy between stretched and relaxed conformations. To further understand effects of potential mutations on elastic properties of the linker, we have performed similar in silico studies on a structure formed of a polyalanine sequence (Ala-only) and on three other structures, created by substituting selected types of amino acid residues in the linker's sequence with alanine (Ala) ones. The results of SMD simulations indicate a crucial role played by the Asparagine (Asn) and Lysine (Lys) residues in controlling stretching and relaxation properties of the linker domain of the motor.

Peculiarities of escape kinetics in the presence of athermal noises.

Stochastic evolution of various dynamic systems and reaction networks is commonly described in terms of noise assisted escape of an overdamped particle from a potential well, as devised by the paradigmatic Langevin equation in which additive Gaussian stochastic force reproduces effects of thermal fluctuations from the reservoir. When implemented for systems close to equilibrium, the approach correctly explains the emergence of the Boltzmann distribution for the ensemble of trajectories generated by the Langevin equation and relates the intensity of the noise strength to the mobility. This scenario can be further generalized to include effects of non-Gaussian, burstlike forcing modeled by Lévy noise. In this case, however, the pulsatile additive noise cannot be treated as the internal (thermal) since the relation between the strength of the friction and variance of the noise is violated. Heavy tails of Lévy noise distributions not only facilitate escape kinetics, but also, more importantly, change the escape protocol by altering the final stationary state to a non-Boltzmann, nonequilibrium form. As a result, contrary to the kinetics induced by a Gaussian white noise, escape rates in environments with Lévy noise are determined not by the barrier height, but instead by the barrier width. We further discuss consequences of simultaneous action of thermal and Lévy noises on statistics of passage times and population of reactants in double-well potentials.

Subdiffusive continuous-time random walks with stochastic resetting.

We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stationary distributions, and first hitting time statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.

Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces.

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of "white noise" to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α < 2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation-dissipation theorem derived for weak external forcing.

Lévy flights versus Lévy walks in bounded domains.

Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities is the discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. As a consequence, a well-developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flight and Lévy walk models on bounded domains, examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time, and stationary probability density functions. It is demonstrated that the similarity of the models is affected by the type of boundary conditions and the value of the stability index defining the asymptotics of the jump length distribution.

Underdamped stochastic harmonic oscillator driven by Lévy noise.

We investigate the distribution of potential and kinetic energy in stationary states of the linearly damped stochastic oscillator driven by Lévy noises. In the long time limit distributions of kinetic and potential energies of the oscillator follow the power-law asymptotics and do not fulfill the equipartition theorem. The partition of the mechanical energy is controlled by the damping coefficient. In the limit of vanishing damping a stochastic analog of the equipartition theorem can be proposed, namely, the statistical properties of potential and kinetic energies attain distributions characterized by the same widths. For larger damping coefficient the larger fraction of energy is stored in its potential form. In the limit of very strong damping the contribution of kinetic energy becomes negligible. Finally, we demonstrate that the ratio of instantaneous kinetic and potential energies, which signifies departure from the mechanical energy equipartition, follows universal power-law asymptotics, regardless of the symmetric α-stable noise parameters. Altogether our investigations clearly indicate strongly nonequilibrium character of Lévy-stable fluctuations with the stability index α<2.

Phase transitions and entropies for synchronizing oscillators.

We study a generic model of coupled oscillators. In the model there is competition between phase synchronization and diffusive effects. For a model with a finite number of states we derive how a phase transition occurs when the coupling parameter is varied. The phase transition is characterized by a symmetry breaking and a discontinuity in the first derivative of the order parameter. We quantitatively account for how the synchronized pulse is a low-entropy structure that facilitates the production of more entropy by the system as a whole. For a model with many states we apply a continuum approximation and derive a potential Burgers' equation for a propagating pulse. No phase transition occurs in that case. However, positive entropy production by diffusive effects still exceeds negative entropy production by the shock formation.

Optimal first-arrival times in Lévy flights with resetting.

We consider the diffusive motion of a particle performing a random walk with Lévy distributed jump lengths and subject to a resetting mechanism, bringing the walker to an initial position at uniformly distributed times. In the limit of an infinite number of steps and for long times, the process converges to superdiffusive motion with replenishment. We derive a formula for the mean first arrival time (MFAT) to a predefined target position reached by a meandering particle and we analyze the efficiency of the proposed searching strategy by investigating criteria for an optimal (a shortest possible) MFAT.

Stepping molecular motor amid Lévy white noise.

We consider a model of a stepping molecular motor consisting of two connected heads. Directional motion of the stepper takes place along a one-dimensional track. Each head is subject to a periodic potential without spatial reflection symmetry. When the potential for one head is switched on, it is switched off for the other head. Additionally, the system is subject to the influence of symmetric, white Lévy noise that mimics the action of external random forcing. The stepper exhibits motion with a preferred direction which is examined by analyzing the median of the displacement of a midpoint between the positions of the two heads. We study the modified dynamics of the stepper by numerical simulations. We find flux reversals as noise parameters are changed. Speed and direction appear to very sensitively depend on characteristics of the noise.

Taming Lévy flights in confined crowded geometries.

We study two-dimensional diffusive motion of a tracer particle in restricted, crowded anisotropic geometries. The underlying medium is formed from a monolayer of elongated molecules [Ciesla J. Chem. Phys. 140, 044706 (2014)] of known concentration. Within this mesh structure, a tracer molecule is allowed to perform a Cauchy random walk with uncorrelated steps. Our analysis shows that the presence of obstacles significantly influences the motion, which in an obstacle-free space would be of a superdiffusive type. At the same time, the selfdiffusive process reveals different anomalous properties, both at the level of a single trajectory realization and after the ensemble averaging. In particular, due to obstacles, the sample mean squared displacement asymptotically grows sublinearly in time, suggesting a non-Markov character of motion. Closer inspection of survival probabilities indicates, however, that the underlying diffusion is memoryless over long time scales despite a strong inhomogeneity of the motion induced by the orientational ordering.

Scale-free fluctuations in behavioral performance: delineating changes in spontaneous behavior of humans with induced sleep deficiency.

The timing and dynamics of many diverse behaviors of mammals, e.g., patterns of animal foraging or human communication in social networks exhibit complex self-similar properties reproducible over multiple time scales. In this paper, we analyze spontaneous locomotor activity of healthy individuals recorded in two different conditions: during a week of regular sleep and a week of chronic partial sleep deprivation. After separating activity from rest with a pre-defined activity threshold, we have detected distinct statistical features of duration times of these two states. The cumulative distributions of activity periods follow a stretched exponential shape, and remain similar for both control and sleep deprived individuals. In contrast, rest periods, which follow power-law statistics over two orders of magnitude, have significantly distinct distributions for these two groups and the difference emerges already after the first night of shortened sleep. We have found steeper distributions for sleep deprived individuals, which indicates fewer long rest periods and more turbulent behavior. This separation of power-law exponents is the main result of our investigations, and might constitute an objective measure demonstrating the severity of sleep deprivation and the effects of sleep disorders.

Tracer diffusion inside fibrinogen layers.

We investigate the obstructed motion of tracer (test) particles in crowded environments by carrying simulations of two-dimensional Gaussian random walk in model fibrinogen monolayers of different orientational ordering. The fibrinogen molecules are significantly anisotropic and therefore they can form structures where orientational ordering, similar to the one observed in nematic liquid crystals, appears. The work focuses on the dependence between level of the orientational order (degree of environmental crowding) of fibrinogen molecules inside a layer and non-Fickian character of the diffusion process of spherical tracer particles moving within the domain. It is shown that in general particles motion is subdiffusive and strongly anisotropic, and its characteristic features significantly change with the orientational order parameter, concentration of fibrinogens, and radius of a diffusing probe.

Efficient control of epidemics spreading on networks: balance between treatment and recovery.

We analyse two models describing disease transmission and control on regular and small-world networks. We use simulations to find a control strategy that minimizes the total cost of an outbreak, thus balancing the costs of disease against that of the preventive treatment. The models are similar in their epidemiological part, but differ in how the removed/recovered individuals are treated. The differences in models affect choice of the strategy only for very cheap treatment and slow spreading disease. However for the combinations of parameters that are important from the epidemiological perspective (high infectiousness and expensive treatment) the models give similar results. Moreover, even where the choice of the strategy is different, the total cost spent on controlling the epidemic is very similar for both models.

Understanding disease control: influence of epidemiological and economic factors.

We present a model of disease transmission on a regular and small world network and compare different control options. Comparison is based on a total cost of epidemic, including cost of palliative treatment of ill individuals and preventive cost aimed at vaccination or culling of susceptible individuals. Disease is characterized by pre-symptomatic phase, which makes detection and control difficult. Three general strategies emerge: global preventive treatment, local treatment within a neighborhood of certain size and only palliative treatment with no prevention. While the choice between the strategies depends on a relative cost of palliative and preventive treatment, the details of the local strategy and, in particular, the size of the optimal treatment neighborhood depend on the epidemiological factors. The required extent of prevention is proportional to the size of the infection neighborhood, but depends on time till detection and time till treatment in a non-nonlinear (power) law. The optimal size of control neighborhood is also highly sensitive to the relative cost, particularly for inefficient detection and control application. These results have important consequences for design of prevention strategies aiming at emerging diseases for which parameters are not nessecerly known in advance.

Searching for the most cost-effective strategy for controlling epidemics spreading on regular and small-world networks.

We present a combined epidemiological and economic model for control of diseases spreading on local and small-world networks. The disease is characterized by a pre-symptomatic infectious stage that makes detection and control of cases more difficult. The effectiveness of local (ring-vaccination or culling) and global control strategies is analysed by comparing the net present values of the combined cost of preventive treatment and illness. The optimal strategy is then selected by minimizing the total cost of the epidemic. We show that three main strategies emerge, with treating a large number of individuals (global strategy, GS), treating a small number of individuals in a well-defined neighbourhood of a detected case (local strategy) and allowing the disease to spread unchecked (null strategy, NS). The choice of the optimal strategy is governed mainly by a relative cost of palliative and preventive treatments. If the disease spreads within the well-defined neighbourhood, the local strategy is optimal unless the cost of a single vaccine is much higher than the cost associated with hospitalization. In the latter case, it is most cost-effective to refrain from prevention. Destruction of local correlations, either by long-range (small-world) links or by inclusion of many initial foci, expands the range of costs for which the NS is most cost-effective. The GS emerges for the case when the cost of prevention is much lower than the cost of treatment and there is a substantial non-local component in the disease spread. We also show that local treatment is only desirable if the disease spreads on a small-world network with sufficiently few long-range links; otherwise it is optimal to treat globally. In the mean-field case, there are only two optimal solutions, to treat all if the cost of the vaccine is low and to treat nobody if it is high. The basic reproduction ratio, R(0), does not depend on the rate of responsive treatment in this case and the disease always invades (but might be stopped afterwards). The details of the local control strategy, and in particular the optimal size of the control neighbourhood, are determined by the epidemiology of the disease. The properties of the pathogen might not be known in advance for emerging diseases, but the broad choice of the strategy can be made based on economic analysis only.