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Purpose: This study has been carried out to examine the stochastic optical soliton solutions of the nonlinear Schrödinger’s equation (NLSE) with Kerr law nonlinearity by multiplicative noise in Itô sense and the behavioral changes on soliton dynamics. Methodology: In order to see the noise effect better, bright and dark soliton shapes belonging to the NLSE have been obtained using the subversion of the new extended auxiliary equation method (SAEM246) and verified with the computer algorithm that developed for this study. Findings: For the NLSE, the distortion of the soliton solutions obtained by the SAEM246 method under different noise effects is clearly shown and simulated. Originality: This method has never been applied before to a NLSE containing a stochastic function to analyze the noise effect. The novelty of the problem and applied method has resulted in many new and original soliton solutions and their behaviors under the noise effects.

The weak-value (WV) measurement proposed by Aharonov, Albert, and Vaidman (AAV) has attracted a great deal of interest in connection with quantum metrology. In this paper, we extend the analysis beyond the AAV limit and obtain a few main results. (i) We obtain a nonperturbative result for the signal-to-noise ratio (SNR). In contrast to the AAV prediction, we find that the SNR asymptotically gets worse when the AAV WV ${A}_{w}$ becomes large, i.e., in the case $g|{A}_{w}{|}^{2}\ensuremath{\gg}1$, where $g$ is the measurement strength. (ii) With the increase of $g$ (but also small), we find that the SNR is comparable to the result under the AAV limit, while both can reach---actually the former can slightly exceed---the SNR of the standard measurement. However, with a further increase of $g$, the WV technique will become less efficient than the standard measurement, despite the fact that the postselection probability is increased. (iii) We find that the Fisher information can characterize the estimate precision qualitatively well as the SNR, yet the difference will become more prominent with the increase of $g$. (iv) We carry out analytic expressions of the SNR in the presence of technical noises and illustrate the particular advantage of the imaginary WV measurement. The nonperturbative result of the SNR manifests a favorable range of the noise strength and allows an optimal determination.

This study aims to provide a theoretical analysis for investigating the consensus behavior of opinion dynamics in noisy environments. We present how random noises significantly help “synchronize” opinions in the Hegselmann–Krause (HK) model. We prove that when noise strength is below the critical value, the opinions of the noisy HK dynamics tend to achieve “consensus” (referred to as quasi-consensus due to noise) in finite time. Fragmentation of HK dynamics tends to eventually vanish when persistent noise is present. Meanwhile, we verify that the opinions almost surely divide infinite times when noise strength exceeds the critical value.

Multistable non-equilibrium systems are abundant outcomes of nonlinear dynamics with feedback but still relatively little is known about what determines the stability of the steady states and their switching rates in terms of entropy and entropy production. Here, we will link fluctuation theorems for the entropy production along trajectories with the action obtainable from the Freidlin--Wentzell theorem to elucidate the thermodynamics of switching between states in the large volume limit of multistable systems. We find that the entropy production at steady state plays no role, but the entropy production during switching is key. Steady-state entropy and diffusive noise strength can be neglected in this limit. The relevance to biology, ecological, and climate models is apparent., 8 pages, 3 figures

We investigate a modified spatial stochastic Lotka-Volterra formulation of the rock-paper-scissors model using off-lattice stochastic simulations. In this model one of the species moves preferentially in a specific direction -- the level of preference being controlled by a noise strength parameter $\eta \in [0, 1]$ ($\eta = 0$ and $\eta = 1$ corresponding to total preference and no preference, respectively) -- while the other two species have no referred direction of motion. We study the behaviour of the system starting from random initial conditions, showing that the species with asymmetric mobility has always an advantage over its predator. We also determine the optimal value of the noise strength parameter which gives the maximum advantage to that species. Finally, we find that the critical number of individuals, below which the probability of extinction becomes significant, decreases as the noise level increases, thus showing that the addition of a preferred mobility direction studied in the present paper does not favour coexistence., Comment: 5 pages, 7 figures

The dynamics of non-equilibrium spatially extended systems are often dominated by fluctuations, due to e.g.\ deterministic chaos or to intrinsic stochasticity. This reflects into generic scale invariant or kinetic roughening behavior that can be classified into universality classes defined by critical exponent values and by the probability distribution function (PDF) of field fluctuations. Suitable geometrical constraints are known to change secondary features of the PDF while keeping the values of the exponents unchanged, inducing universality subclasses. Working on the Kuramoto-Sivashinsky equation as a paradigm of spatiotemporal chaos, we show that the physical nature of the prevailing fluctuations (chaotic or stochastic) can also change the universality class while respecting the exponent values, as the PDF is substantially altered. This transition takes place at a non-zero value of the stochastic noise amplitude and may be suitable for experimental verification., Comment: 12 pages, 9 figures, submitted

In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full "$L^1$ regime" $\sigma< \sigma_{L^1}=2\sqrt\pi$ where $\sigma$ is the noise strength. We also describe the main necessary modifications needed for the SRF on general compact surfaces, and list some open questions., Comment: 42 pages

In this paper, the asymptotic behavior of a stochastic Cucker–Smale model with multiplicative excitation is investigated. Firstly, it is proven that stochastic flocking occurs for the communication weight originally proposed by Cucker and Smale if the noise strength is sufficiently small. Then, a non-flocking result under the effect of large noise is provided. In addition, the kinematics of the macroscopic variables is addressed which can be regarded as an effective supplement of consensus dynamics.

We present a new kinetic Cucker–Smale–Fokker–Planck (CS-FP) type equation with a degenerate diffusion, which describes the dynamics for an ensemble of infinitely many Cucker–Smale particles in a random environment. The asymptotic dynamics of the CS-FP equation exhibits a threshold-like phenomenon depending on the relative strength between the coupling strength and the noise strength. In the small coupling regime, the noise effect becomes dominant, which induces the velocity variance to increase to infinity exponentially fast. In contrast, the velocity alignment effect is strong in the large coupling regime, and the velocity variance tends to zero exponentially fast. We present the global existence of classical solutions to the CS-FP equation for a sufficiently smooth initial datum without smallness in its size. For the kinetic CS-FP equation with a metric dependent communication weight, we provide a uniform-in-time mean-field limit from the stochastic CS-model to the kinetic CS-FP equation without convergence rate.

Charge noise can be detrimental to the operation of quantum dot (QD) based semiconductor qubits. We study the low-frequency charge noise by charge offset drift measurements for Si-MOS devices with intentionally implanted donors near the QDs. We show that the MOS system exhibits non-equilibrium drift characteristics, in the form of transients and discrete jumps, that are not dependent on the properties of the donor implants. The equilibrium charge noise indicates a 1/f noise dependence, and a noise strength as low as $$1\,\mu {\rm{eV}}/\sqrt{{\rm{Hz}}}$$ 1 μ eV / Hz , comparable to that reported in more model GaAs and Si/SiGe systems (which have also not been implanted). We demonstrate that implanted qubits, therefore, can be fabricated without detrimental effects on long-term drift or 1/f noise for devices with less than 50 implanted donors near the qubit.

Implementation of logical entangling gates is an important step towards realizing a quantum computer. We use a gradient-based optimization approach to find single-qubit rotations which can be interleaved between applications of a noisy nonlocal gate to dramatically suppress arbitrary logical errors, while steering the evolution operator towards the perfectly entangling subset of SU(4) gates. The modularity of the approach allows for application to any two-qubit system, regardless of the Hamiltonian or details of the experimental implementation. This approach is effective for both quasi-static and time-dependent $1/f^{\alpha}$ noise. We also show how the fidelity of the final operation depends on both the fidelity of the local rotations and the noise strength., Comment: 8 pages, 5 figures, data available at https://gist.github.com/goerz/3e04cca5a5dc1e9bcabd247065c24ebe

Cell population represents an intrinsically heterogeneous and stochastic system, in which individual cells often behave very differently in molecular contents, functions and even genotypes from the population average in response to uniform physiological stimuli. The traditional bulk cellular analysis often overlooks cellular heterogeneity and does not provide information on cell-cell variations. Single-cell measurements can reveal information obscured in population averages, and enable us to determine distributions rather than averaged properties within a cell population. The level of complexity, with numerous variables acting at the same time, requires multiparametric and dynamic investigation of a large number of single cells. Multiplexed study can provide quantitative correlations or inter-relationships among multiple cellular components and molecular markers within a protein network or family in biological processes. In this paper, we applied multiple fluorophore-conjugated primary antibodies to detect multiple proteins expressed on the same singe cells from a clonal population. To reveal cell-cell heterogeneity, we quantified the histograms of six proteins within a cell population as functions of TNF-α stimulation time. Then, we quantified noise and noise strength of these protein histograms as functions of TNF-α stimulation time. Thirdly, we quantified correlation coefficients of multiple proteins expressed on same single-cells as functions of TNF-α stimulation time. Above parameters demonstrated nonlinear relationships with TNF-α stimulation. Quantification of above parameters on independent cell subpopulations further reveals the cell-cell heterogeneity when exposed to identical environmental conditions. Such cellular heterogeneity will be useful to characterize the disease progression and disease diagnoses.

We analyze \emph{RXTE} and \emph{Swift} observations of SWIFT J1729.9$-$3437 after its outburst from 2010 July 20 to 2010 August 12. We calculate a spin frequency and spin frequency derivative of $1.8734(8) \times 10^{-3}$ Hz and $6.42(6) \times 10^{-12}$ Hz/s respectively from the quadratic fit of pulse arrival times. The quadratic fit residuals fit well to a circular orbital model with a period of $15.3(2)$ days and a mass function of about $1.3M_{\odot}$, but they can also be explained by a torque noise strength of $6.8 \times 10^{-18}$ Hz sec$^{-2}$. Pulse profiles switches from double-peaked to single-peaked as the source flux continues to decrease. We find that the pulse shape generally shows no strong energy dependence. The hardness ratios reveal that the source becomes softer with decreasing flux. We construct a single spectrum from all the available RXTE and Swift observations. We find that adding an \emph{Fe} line complex feature around 6.51 keV slightly improves the spectral fit and that this feature is more likely to originate from the source rather than the Galactic ridge. From the pulse phase spectral analysis, it is shown that that photon index and folding energy of the high energy cut-off vary with varying pulse phase., 15 pages, 9 figures, accepted for publication in MNRAS

For stimuli just beyond threshold and certain noise strength, the Hodgkin–Huxley HH neuron model shows sensitivity to noise with respect to latency to first spike after stimulus onset. This effect has been called “noise delayed decay” NDD . Here we investigate how the NDD is affected by the variation in temperature using a HH model that includes stochastic ion channels. We show that the NDD effect increases and emerges for smaller noise strengths or larger cell sizes as the temperature increases. We also show that the potassium channel noise plays a key role on the appearance of the NDD effect.

We consider large systems of stochastic interacting particles through discontinuous kernels which has vision geometrical constrains. We rigorously derive a Vlasov–Fokker–Planck type of kinetic mean-field equation from the corresponding stochastic integral inclusion system. More specifically, we construct a global-in-time weak solution to the stochastic integral inclusion system and derive the kinetic equation with the discontinuous kernels and the inhomogeneous noise strength by employing the 1-Wasserstein distance.

Bistable switches are frequently encountered in biological systems. Typically, a bistable switch models a binary decision where each decision corresponds to one of the two stable equilibria. Recently, we showed that the global decision-making process in bistable switches strongly depends on a particular equilibrium point of these systems, their saddle point. In particular, we showed that a saddle point with a time-scale separation between its attractive and repulsive directions can delay the decision-making process. In this paper, we study the effects of white Gaussian noise on this mechanism of delayed decision-making induced by the saddle point. Results show that the mean decision-time strongly depends on the balance between the initial distance to the separatrix and the noise strength.

We show that a thermally isolated system driven across a quantum phase transition by a noisy control field exhibits anti-Kibble-Zurek behavior, whereby slower driving results in higher excitations. We characterize the density of excitations as a function of the ramping rate and the noise strength. The optimal driving time to minimize excitations is shown to scale as a universal power law of the noise strength. Our findings reveal the limitations of adiabatic protocols such as quantum annealing and demonstrate the universality of the optimal ramping rate., 5 pages, 3 figures, 2 pages of supplemental material

In this work we studied the diffusive behavior of active brownian particles under lateral parabolic confinement. The results showed that we go from subdiffusion to ballistic motion as we vary the angular noise strength and confinement intensity. We argued that the subdiffusion regimes appear as consequence of the restricted space available for diffusion (achieved either through large confinement and/or large noise); we saw that when there are large confinement and noise intensity, a similar configuration to single file diffusion appears; on the other hand, normal and superdiffusive regimes may occur due to low noise (longer persistent motion), either through exploring a wider region around the potential minimum in the transverse direction (low confinement), or by forming independent clusters (high confinement)., 6 pages, 6 figures, accepted for publication in Physica A

Absolute quantification of macromolecules in single cells is critical for understanding and modeling biological systems that feature cellular heterogeneity. Here we show extremely sensitive and absolute quantification of both proteins and mRNA in single mammalian cells by a very practical workflow that combines proximity ligation assay (PLA) and digital PCR. This digital PLA method has femtomolar sensitivity, which enables the quantification of very small protein concentration changes over its entire 3-log dynamic range, a quality necessary for accounting for single-cell heterogeneity. We counted both endogenous (CD147) and exogenously expressed (GFP-p65) proteins from hundreds of single cells and determined the correlation between CD147 mRNA and the protein it encodes. Using our data, a stochastic two-state model of the central dogma was constructed and verified using joint mRNA/protein distributions, allowing us to estimate transcription burst sizes and extrinsic noise strength and calculate the transcription and translation rate constants in single mammalian cells.

The stability of a class of delayed cellular neural networks (DCNN) with or without noise perturbation is studied. After presenting a simple and easily checkable condition for the global exponential stability of a deterministic system, we further investigate the case with noise perturbation. When DCNN is perturbed by external noise, the system is globally stable. An important fact is that, when the system is perturbed by internal noise, it is globally exponentially stable only if the total noise strength is within a certain bound. This is significant since the stochastic resonance phenomena have been found to exist in many nonlinear systems.

We study the overdamped dynamics of a Brownian particle in the double-well potential under the influence of an external periodic (AC) force with zero mean. We obtain a dependence of the jump rate on the frequency of the external force. The dependence shows a maximum at a certain driving frequency. We explain the phenomenon as a switching between different time scales of the system: interwell relaxation time (the mean residence time) and the intrawell relaxation time. Dependence of the resonant peak on the system parameters, namely the amplitude of the driving force A and the noise strength (temperature) D has been explored. We observe that the effect is well pronounced when A/D > 1 and if A/D 1 the enhancement of the jump rate can be of the order of magnitude with respect to the Kramers rate., Published in J. Phys. A: Math. Gen. 37 (2004) 6043-6051; 6 figures

Recent advances in the periodic orbit theory of stochastically perturbed systems have permitted a calculation of the escape rate of a noisy chaotic map to order 64 in the noise strength. Comparison with the usual asymptotic expansions obtained from integrals and with a previous calculation of the electrostatic potential of exactly selfsimilar fractal charge distributions, suggests a remarkably accurate form for the late terms in the expansion, with parameters determined independently from the fractal repeller and the critical point of the map. Two methods give a precise meaning to the asymptotic expansion, Borel summation and Shafer approximants. These can then be compared with the escape rate as computed by alternative methods., Comment: 15 pages, 5 postscript figures incorporated into the text; v2: Quadratic Pade (Shafer) method added, also a few references

Phase turbulence is suppressed by applying common noise additively to the Kuramoto-Sivashinsky type equation, and the noise-induced phase synchronization is realized. The noise strength necessary for the suppression of phase turbulence is evaluated theoretically., 7 pages, 4 figures

A deterministic computer model of the signal transduction pathway mediating bacterial chemotaxis was used to examine the variation in both unstimulated swimming behaviour and adaptation time to stimuli in clonal populations of cells. Copy numbers of proteins in the pathway were computed from a simplified model of transcription and translation that predicts greater-than-Poissonian statistics. Simulated and experimental individuality data could be brought into good agreement on varying the noise strength of the protein copy number distributions. In the simulations, all the proteins in the pathway are involved to a significant degree in the appearance of phenotypic diversity, although there is a modest decrease in influence with increasing copy number.

The presence of noise in non linear dynamical systems can play a constructive role, increasing the degree of order and coherence or evoking improvements in the performance of the system. An example of this positive influence in a biological system is the impulse transmission in neurons and the synchronization of a neural network. Integrating numerically the Fokker-Planck equation we show a self-induced synchronized oscillation. Such an oscillatory state appears in a neural network coupled with a feedback term, when this system is excited by noise and the noise strength is within a certain range., Comment: 12 pages, 18 figures

We analyze the effects of noise on the permutation entropy of dynamical systems. We take as numerical examples the logistic map and the Rossler system. Upon varying the noise strength we find a transition from an almost-deterministic regime, where the permutation entropy grows slower than linearly with the pattern dimension, to a noise-dominated regime, where the permutation entropy grows faster than linearly with the pattern dimension. We perform the same analysis on experimental time-series by considering the stochastic spiking output of a semiconductor laser with optical feedback, and find that the permutation entropy always increases faster than linearly. Nevertheless, the analysis allows to detect regularities of the underlying dynamics and model simulations are in a good agreement with the empirical data. By comparing the results of these three different examples, we discuss the possibility of determining from a time series whether the underlying dynamics is dominated by noise or not.

We study the two time correlation for the noise driven dynamics of the double-well oscillator in the Suzuki regime. It is seen that for very small noise strength the correaltion function shows a lack of translational invariance for very long times characterised by the Suzuki scaling variable. We see that in this strongly out-of-equilibrium situation, the conventional mode-coupling approximation is not a convenient tool., Comment: 8 pages, no figure

We consider the effect of introducing a small number of non-aligning agents in a well-formed flock. To this end, we modify a minimal model of active Brownian particles with purely repulsive (excluded volume) forces to introduce an alignment interaction that will be experienced by all the particles except for a small minority of "dissenters". We find that even a very small fraction of dissenters disrupts the flocking state. Strikingly, these motile dissenters are much more effective than an equal number of static obstacles in breaking up the flock. For the studied system sizes we obtain clear evidence of scale invariance at the flocking-disorder transition point and the system can be effectively described with a finite-size scaling formalism. We develop a continuum model for the system which reveals that dissenters act like annealed noise on aligners, with a noise strength that grows with the persistence of the dissenters' dynamics., Comment: 22 pages, 8 figures. Version accepted for publication in New Journal of Physics

Repeated game theory has been one of the most prevailing tools for understanding the long-run relationships, which are footstones in building human society. Recent works have revealed a new set of "zero-determinant (ZD)" strategies, which is an important advance in repeated games. A ZD strategy player can exert a unilaterally control on two players' payoffs. In particular he can deterministically set the opponent's payoff, or enforce an unfair linear relationship between the players' payoffs, thereby always seizing an advantageous share of payoffs. One of the limitations of the original ZD strategy, however, is that it does not capture the notion of robustness when the game is subjected to stochastic errors. In this paper, we propose a general model of ZD strategies for noisy repeated games, and find that ZD strategies have high robustness against errors. We further derive the pinning strategy under noise, by which the ZD strategy player coercively set the opponent's expected payoff to his desired level, although his payoff control ability declines with the increase of noise strength. Due to the uncertainty caused by noise, the ZD strategy player cannot secure his payoff to be higher than the opponent's, which implies strong extortions do not exist even under low noise. While we show that the ZD strategy player can still establish a novel kind of extortions, named weak extortions, where any increase of his own payoff always exceeds that of the opponent's by a fixed percentage, and the conditions under which the weak extortions can be realized are more stringent as the noise becomes stronger., 8 pages, 4 figures

A random multiplicative process with additive noise is described by a Langevin equation. We show that the fluctuation-dissipation relation is satisfied in the Langevin model, if the noise strength is not so strong., Comment: 11 pages, 6 figures, other comments

Efficiency of generation of net unidirectional current in an adiabatically driven symmetric periodic potential system is studied. The efficiency shows a maximum, in the case of an inhomogeneous system with spatially varying periodic friction coefficient, as a function of temperature. The ratchet is not most efficient when it gives maximum current. The direction of current may also be reversed as a function of noise strength when, instead, an asymmetric periodic potential is considered., Comment: 14 pages, including 5 figures

Electronic analog experiments on escape over a fluctuating potential barrier are performed for the case when the fluctuations are caused by Ornstein-Uhlenbeck noise (OUN). In its dependence on the relation between the two OUN parameters (the correlation time tau and noise strength Q) the nonmonotonic variation of the mean escape time T as a function of tau can exhibit either a minimum (resonant activation), or a maximum (inhibition of activation), or both these effects. The possible resonant nature of these features is discussed. We claim that T is not a good quantity to describe the resonancelike character of the problem. Independently of the specific relation between the OUN parameters, the resonance manifests itself as a maximal lowering of the potential barrier during the escape event, and it appears for tau of the order of the relaxation time toward the metastable state.

In this paper we present a hydrodynamic approach to describe the motion of migrating bacteria as a special class of self-propelled systems. Analytical and numerical calculations has been performed to study the behavior of our model in the turbulent-like regime and to show that a phase transition occurs as a function of noise strength. Our results can explain previous experimental observations as well as results of numerical simulations., Comment: 8 pages, 10 figures

We study experimentally the dynamics of vertical-cavity surface-emitting lasers (VCSELs) with polarization-rotated (PR) optical feedback, such that the natural lasing polarization of a VCSEL is rotated by 90 deg and then is reinjected into the laser. We observe noisy, square-wave-like polarization switchings with periodicity slightly longer than twice the delay time, which degrade to (or alternate with) bursts of irregular oscillations. We present results of simulations that are in good agreement with the observations. The simulations demonstrate that close to threshold the regular switching is very sensitive to noise, while well above threshold is less affected by the noise strength. The frequency splitting between the two polarizations plays a key role in the switching regularity, and we identify wide parameter regions where deterministic and robust switching can be observed.

The work discovers a stochastic bifurcation in delayed systems in the presence of both delay and additive noise. To understand this phenomenon we present a stochastic center manifold method to compute a non-delayed stochastic order parameter equation for a scalar delayed system driven by additive uncorrelated noise. The derived order parameter equation includes additive and multiplicative white and coloured noise. An illustrative neural system with delayed self-excitation reveals stationary states that are postponed by combined additive noise and delay. A final brief analytical treatment of the derived order parameter equation reveals analytically the shift of the stationary states which depends on the delay and the noise strength. Copyright c EPLA, 2012

We investigate the effect of noise strength on the macroscopic ordering dynamics of systems with symmetric absorbing states. Using an explicit stochastic microscopic model, we present evidence for a phase transition in the coarsening dynamics, from an Ising-like to a voter-like behavior, as the noise strength is increased past a nontrivial critical value. By mapping to a thermal diffusion process, we argue that the transition arises due to locally-absorbing states being entered more readily in the high-noise regime, which in turn prevents surface tension from driving the ordering process., v2 with improved introduction and figures, to appear in PRL. 4 pages, 4 figures

The phenomenon of stochastic resonance (SR) is known to occur mostly in bistable systems. However, the question of the occurrence of SR in periodic potential systems has not been resolved conclusively. Our present numerical work shows that the periodic potential system indeed exhibits SR in the high-frequency regime, where the linear-response theory yields maximum frequency-dependent mobility as a function of noise strength. The existence of two (and only two) distinct dynamical states of trajectories in this moderately feebly damped periodically driven noisy periodic potential system plays an important role in the occurrence of SR.

We compare a hierarchy of three stochastic models in gene regulation. In each case, genes produce mRNA molecules which in turn produce protein. The simplest model, as described by Thattai and Van Oudenaarden (Proc. Nat. Acad. Sci., 2001), assumes that a gene is always active, and uses a first-order chemical kinetics framework in the continuous-time, discrete-space Markov jump (Gillespie) setting. The second model, proposed by Raser and O'Shea (Science, 2004), generalizes the first by allowing the gene to switch randomly between active and inactive states. Our third model accounts for other effects, such as the binding/unbinding of a transcription factor, by using two independent on/off switches operating in AND mode. We focus first on the noise strength, which has been defined in the biological literature as the ratio of the variance to the mean at steady state. We show that the steady state variance in the mRNA and protein for the three models can either increase or decrease when switches are incorporated, depending on the rate constants and initial conditions. Despite this, we also find that the overall noise strength is always greater when switches are added, in the sense that one or two switches are always noisier than none. On the other hand, moving from one to two switches may either increase or decrease the noise strength. Moreover, the steady state values may not reflect the relative noise levels in the transient phase. We then look at a hybrid version of the two-switch model that uses stochastic differential equations to describe the evolution of mRNA and protein. This is a simple example of a multiscale modelling approach that allows for cheaper numerical simulations. Although the underlying chemical kinetics appears to be second order, we show that it is possible to analyse the first and second moments of the mRNA and protein levels by applying a generalized version of Ito's lemma. We find that the hybrid model matches the moments of underlying Markov jump model for all time. By contrast, further simplifying the model by removing the diffusion in order to obtain an ordinary differential equation driven by a switch causes the mRNA and protein variances to be underestimated.

Using a rhythmic task where human subjects bounced a ball with a handheld racket, fine-grained analyses of stability and variability extricated contributions from open-loop control, noise strength, and active error compensation. Based on stability analyses of a stochastic-deterministic model of the task—a surface contacting the ball by periodic movements—open-loop or dynamic stability was assessed by the acceleration of the racket at contact. Autocovariance analyses of model and data were further used to gauge the contributions of open-loop stability and noise strength. Variability and regression analyses estimated active error compensation. Empirical results demonstrated that experienced actors exploited open-loop stability more than novices, had lower noise strength, and applied more active error compensations. By manipulating the model parameter coefficient of restitution, task stability was varied and showed that actors graded these three components as a function of task stability. It is concluded that actors tune into task stability when stability is high but use more active compensation when stability is reduced. Implications for the neural underpinnings for passive stability and active control are discussed. Further, results showed that stability and variability are not simply the inverse of each other but contain more quantitative information when combined with model analyses.

The escape rate of a stochastic dynamical system can be found as an expansion in powers of the noise strength. In previous work the coefficients of such an expansion for a one-dimensional map were fitted to a general form containing a few parameters. These parameters were found to be related to the fractal structure of the repeller of the system. The parameter alpha, the "noise dimension", remains to be interpreted. This report presents new data for alpha showing that the relation to the dimensions is more complicated than predicted in earlier work and oscillates as a function of the map parameter, in contrast to other dimension-like quantities., Comment: 7 pages, 5 figures

Calcium (Ca) sparks are elementary events of biological Ca signaling. A normal Ca spark has a brief duration in the range of 10 to 100 ms, but long-lasting sparks with durations of several hundred milliseconds to seconds are also widely observed. Experiments have shown that the transition from normal to long-lasting sparks can occur when ryanodine receptor (RyR) open probability is either increased or decreased. Here, we demonstrate theoretically and computationally that long-lasting sparks emerge as a collective dynamical behavior of the network of diffusively coupled Ca release units (CRUs). We show that normal sparks occur when the CRU network is monostable and excitable, while long-lasting sparks occur when the network dynamics possesses multiple metastable attractors, each attractor corresponding to a different spatial firing pattern of sparks. We further highlight the mechanisms and conditions that produce long-lasting sparks, demonstrating the existence of an optimal range of RyR open probability favoring long-lasting sparks. We find that when CRU firings are sparse and sarcoplasmic reticulum (SR) Ca load is high, increasing RyR open probability promotes long-lasting sparks by potentiating Ca-induced Ca release (CICR). In contrast, when CICR is already strong enough to produce frequent firings, decreasing RyR open probability counter-intuitively promotes long-lasting sparks by decreasing spark frequency. The decrease in spark frequency promotes intra-SR Ca diffusion from neighboring non-firing CRUs to the firing CRUs, which helps to maintain the local SR Ca concentration of the firing CRUs above a critical level to sustain firing. In this setting, decreasing RyR open probability further suppresses long-lasting sparks by weakening CICR. Since a long-lasting spark terminates via the Kramers’ escape process over a potential barrier, its duration exhibits an exponential distribution determined by the barrier height and noise strength, which is modulated differently by different ways of altering the Ca release flux strength., Author Summary Calcium (Ca) sparks, resulting from Ca-induced Ca release, are elementary events of biological Ca signaling. Sparks are normally brief, but long-lasting sparks have been widely observed experimentally under various conditions. The underlying mechanisms of spark duration or termination and the corresponding determinants remain a topic of debate. In this study, we demonstrate theoretically and computationally that normal brief sparks are excitable transients, while long-lasting sparks are multiple metastable states emerging in the diffusively coupled Ca release unit network, as a result of cooperativity and release competition among the Ca release units. Termination of a long-lasting spark is a Kramers’ escape process over a potential barrier, and the spark duration is the first-passage time, exhibiting an exponential distribution.

Background The concentration gradient of Bicoid protein which determines the developmental pathways in early Drosophila embryo is the best characterized morphogen gradient at the molecular level. Because different developmental fates can be elicited by different concentrations of Bicoid, it is important to probe the limits of this specification by analyzing intrinsic fluctuations of the Bicoid gradient arising from small molecular number. Stochastic simulations can be applied to further the understanding of the dynamics of Bicoid morphogen gradient formation at the molecular number level, and determine the source of the nucleus-to-nucleus expression variation (noise) observed in the Bicoid gradient. Results We compared quantitative observations of Bicoid levels in immunostained Drosophila embryos with a spatially extended Master Equation model which represents diffusion, decay, and anterior synthesis. We show that the intrinsic noise of an autonomous reaction-diffusion gradient is Poisson distributed. We demonstrate how experimental noise can be identified in the logarithm domain from single embryo analysis, and then separated from intrinsic noise in the normalized variance domain of an ensemble statistical analysis. We show how measurement sensitivity affects our observations, and how small amounts of rescaling noise can perturb the noise strength (Fano factor) observed. We demonstrate that the biological noise level in data can serve as a physical constraint for restricting the model's parameter space, and for predicting the Bicoid molecular number and variation range. An estimate based on a low variance ensemble of embryos suggests that the steady-state Bicoid molecular number in a nucleus should be larger than 300 in the middle of the embryo, and hence the gradient should extend to the posterior end of the embryo, beyond the previously assumed background limit. We exhibit the predicted molecular number gradient together with measurement effects, and make a comparison between conditions of higher and lower variance respectively. Conclusion Quantitative comparison of Master Equation simulations with immunostained data enabled us to determine narrow ranges for key biophysical parameters, which for this system can be independently validated. Intrinsic noise is clearly detectable as well, although the staining process introduces certain limits in resolution.

We make a short review about the synchronization in coupled phase oscillator models. Next, we study the common-noise-induced synchronization among active rotators. At an intermediate noise strength, the noise-induced synchronization takes place most effectively, which is analogous to the stochastic resonance. Finally, we study the synchronization of coupled phase oscillators with nonvariational interaction on scale-free networks. We find a sharp transition and a weak hysteresis in the nonvariational systems. The sharp transition is found also in the mean-field approximation., 8 pages, 5 figures

A trapping mechanism is observed and proposed as the origin of the anomalous behavior recently discovered in transport properties of overdamped ratchets subject to external oscillatory drive in the presence of quenched noise. In particular, this mechanism is shown to appear whenever the quenched disorder strength is greater than a threshold value. The minimum disorder strength required for the existence of traps is determined by studying the trap structure in a disorder configuration space. An approximation to the trapping probability density function in a disordered region of finite length included in an otherwise perfect ratchet lattice is obtained. The mean velocity of the particles and the diffusion coefficient are found to have a non-monotonic dependence on the quenched noise strength due to the presence of the traps., Comment: 21 pages, 6 figures, to appear in PRE