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Jinqiao Duan

Publications and source records attributed to Jinqiao Duan.

At least 145 records · Page 8Linked to original sources
arXiv2019

State transitions in the Morris-Lecar model under stable Lévy noise

This paper considers the state transition of the stochastic Morris-Lecar neuronal model driven by symmetric $α$-stable Lévy noise. The considered system is bistable: a stable fixed point (resting state) and a stable limit cycle (oscillating state), and there is an unstable limit cycle (borderline state) between them. Small disturbances may cause a transition between the two stable states, thus a deterministic quantity, namely the maximal likely trajectory, is used to analyze the transition phenomena in non-Gaussian stochastic environment. According to the numerical experiment, we find that smaller jumps of the Lévy motion and smaller noise intensity can promote such transition from the sustained oscillating state to the resting state. It also can be seen that larger jumps of the Lévy motion and higher noise intensity are conducive for the transition from the borderline state to the sustained oscillating state. As a comparison, Brownian motion is also taken into account. The results show that whether it is the oscillating state or the borderline state, the system disturbed by Brownian motion will be transferred to the resting state under the selected noise intensity.

math.DS↗
arXiv2019

Lévy noise induced escape in the Morris-Lecar model

The phenomenon of an excitable system producing a pulse under external or internal stimulation may be interpreted as a stochastic escape problem. This work addresses this issue by examining the Morris-Lecar neural model driven by symmetric α-stable Lévy motion. Two deterministic indices: the first escape probability and the mean first exit time, are adopted to analyse the state transition in this stochastic model. We calculate both indices in order to understand the transition from the escape region to the target region, and the area of higher indices in escape region. Additionally, we consider the special case of (Gaussian) Brownian motion to compare with (non-Gaussian) Lévy motion case. Our main results indicate that higher first escape probability promotes the transition, while the mean first exit time reflects the stability of the rest state with the selected escape region. The higher non-Gaussianity index and relatively small noise intensity are more prone to produce spikes. Moreover, by calculating both deterministic indices as functions of noise intensity ratio and non-Gaussianity index, we find that the effect of ion channel noise is more pronounced on the stochastic Morris-Lecar model than noise in the current. This work provides some mathematical understanding about the impact of non-Gaussian, heavy-tailed, burst-like fluctuations on excitable systems such as the Morris-Lecar system.

math.DS↗
arXiv2019

Hamiltonian Systems with Lévy Noise: Symplecticity, Hamilton's Principle and Averaging Principle

This work focuses on topics related to Hamiltonian stochastic differential equations with Lévy noise. We first show that the phase flow of the stochastic system preserves symplectic structure, and propose a stochastic version of Hamilton's principle by the corresponding formulation of the stochastic action integral and the Euler-Lagrange equation. Based on these properties, we further investigate the effective behaviour of a small transversal perturbation to a completely integrable stochastic Hamiltonian system with Lévy noise. We establish an averaging principle in the sense that the action component of solution converges to the solution of a stochastic differential equation when the scale parameter goes to zero. Furthermore, we obtain the estimation for the rate of this convergence. Finally, we present an example to illustrate these results.

math.DS↗
arXiv2019

Transitions between Metastable States in a Simplified Model for the Thermohaline Circulation under Random Fluctuations

In this work, we study the impact of non-Gaussian alpha-stable Levy motion on transitions between metastable equilibrium states (or attractors) in a stochastic Stommel two-box model for thermohaline circulation (THC). By maximizing the probability density of the solution process associated with a nonlocal Fokker-Planck equation, we compute maximal likely pathways and identify corresponding maximal likely stable equilibrium states. Our numerical results indicate weakened THC may be induced by perturbation with very small noise intensity in a certain range of stability index. Moreover, larger noise intensity and larger stability index induce weakened THC within shorter bifurcation time.

physics.ao-ph↗
arXiv2019

The Onsager-Machlup Function as Lagrangian for the Most Probable Path of a Jump-diffusion Process

This work is devoted to deriving the Onsager-Machlup function for a class of stochastic dynamical systems under (non-Gaussian) Levy noise as well as (Gaussian) Brownian noise, and examining the corresponding most probable paths. This Onsager-Machlup function is the Lagrangian giving the most probable path connecting metastable states for jump-diffusion processes. This is done by applying the Girsanov transformation for measures induced by jump-diffusion processes. Moreover, we have found this Lagrangian function is consistent with the result in the special case of diffusion processes. Finally, we apply this new Onsager-Machlup function to investigate dynamical behaviors analytically and numerically in several examples. These include the transitions from one metastable state to another metastable state in a double-well system, with numerical experiments illustrating most probable transition paths for various noise parameters.

math-ph↗
arXiv2019

Characterization of the Most Probable Transition Paths of Stochastic Dynamical Systems with Stable Lévy Noise

This work is devoted to the investigation of the most probable transition path for stochastic dynamical systems driven by either symmetric $α$-stable Lévy motion ($0<α<1$) or Brownian motion. For stochastic dynamical systems with Brownian motion, minimizing an action functional is a general method to determine the most probable transition path. We have developed a method based on path integrals to obtain the most probable transition path of stochastic dynamical systems with symmetric $α$-stable Lévy motion or Brownian motion, and the most probable path can be characterized by a deterministic dynamical system.

math.DS↗
arXiv2019

Most Probable Evolution Trajectories in a Genetic Regulatory System Excited by Stable Lévy Noise

We study the most probable trajectories of the concentration evolution for the transcription factor activator in a genetic regulation system, with non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account. We calculate the most probable trajectory by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We especially examine those most probable trajectories from low concentration state to high concentration state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period); (ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including (a) a smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric Lévy noise. This phenomenon does not occur in the case of symmetric Lévy noise; (b) the symmetric Lévy motion always induces transition to high concentration, but certain asymmetric Lévy motions do not trigger the switch to transcription. These findings provide insights for further experimental research, in order to achieve or to avoid specific gene transcriptions, with possible relevance for medical advances.

q-bio.MN↗
arXiv2018

Most probable dynamics of a genetic regulatory network under stable Lévy noise

Numerous studies have demonstrated the important role of noise in the dynamical behaviour of a complex system. The most probable trajectories of nonlinear systems under the influence of Gaussian noise have recently been studied already. However, there has been only a few works that examine how most probable trajectories in the two-dimensional system (MeKS network) are influenced under non-Gaussian stable Lévy noise. Therefore, we discuss the most probable trajectories of a two-dimensional model depicting the competence behaviour in B. subtilis under the influence of stable Lévy noise. On the basis of the Fokker-Planck equation, we describe the noise-induced most probable trajectories of the MeKS network from the low ComK protein concentration (vegetative state) to the high ComK protein concentration (competence state) under stable Lévy noise. We demonstrate choices of the non-Gaussianity index $α$ and the noise intensity $ε$ which generate the ComK protein escape from the low concentration to the high concentration. We also reveal the optimal combination of both parameters $α$ and $ε$ making the tipping time shortest. Moreover, we find that different initial concentrations around the low ComK protein concentration evolve to a metastable state, and provide the optimal $α$ and $ε$ such that the distance between the deterministic competence state and the metastable state is smallest.

math.DS↗
arXiv2018

Mean escape time for randomly switching narrow gates in a cellular flow

The escape of particles through a narrow absorbing gate in confined domains is a abundant phenomenon in various systems in physics, chemistry and molecular biophysics. We consider the narrow escape problem in a cellular flow when the two gates randomly switch between different states with a switching rate k between the two gates. After briefly deriving the coupled partial differential equations for the escape time through two gates, we compute the mean escape time for particles escaping from the gates with different initial states. By numerical simulation under nonuniform boundary conditions, we quantify how narrow escape time is affected by the switching rate k between the two gates, arc length s between two gates, angular velocity w of the cellular flow and diffusion coefficient D. We reveal that the mean escape time decreases with the switching rate k between the two gates, angular velocity w and diffusion coefficient D for fixed arc length, but takes the minimum when the two gates are evenly separated on the boundary for any given switching rate k between the two gates. In particular, we find that when the arc length size ε for the gates is sufficiently small, the average narrow escape time is approximately independent of the gate arc length size. We further indicate combinations of system parameters (regions located in the parameter space) such that the mean escape time is the longest or shortest. Our findings provide mathematical understanding for phenomena such as how ions select ion channels and how chemicals leak in annulus ring containers, when drift vector fields are present.

cond-mat.stat-mech↗
arXiv2018

Effective filtering analysis for non-Gaussian dynamic systems

This work is about a slow-fast data assimilation system under non-Gaussian noisy fluctuations. Firstly, we show the existence of a random invariant manifold for a stochastic dynamical system with non-Gaussian noise and two-time scales. Secondly, we obtain a low dimensional reduction of this system via a random invariant manifold. Thirdly, we prove that the low dimensional filter on the random invariant manifold approximates the original filter, in a probabilistic sense.

math.PR↗
arXiv2018

Geometric Methods for Stochastic Dynamical Systems

Noisy fluctuations are ubiquitous in complex systems. They play a crucial or delicate role in the dynamical evolution of gene regulation, signal transduction, biochemical reactions, among other systems. Therefore, it is essential to consider the effects of noise on dynamical systems. It has been a challenging topic to have better understanding of the impact of the noise on the dynamical behaviors of complex systems.

math.DS↗
arXiv2018

Bifurcation in mean phase portraits for stochastic dynamical systems with multiplicative Gaussian noise

We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker-Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e., saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, and have found some interesting phenomena in contrast to the corresponding deterministic counterparts.

math.DS↗
arXiv2018

Slow manifold and parameter estimation for a nonlocal fast-slow stochastic evolutionary system

We establish a slow manifold for a fast-slow stochastic evolutionary system with anomalous diffusion, where both fast and slow components are influ- enced by white noise. Furthermore, we prove the exponential tracking property for the random slow manifold and this leads to a lower dimensional reduced sys- tem based on the slow manifold. Also we consider parameter estimation for this nonlocal fast-slow stochastic dynamical system, where only the slow component is observable. In quantifying parameters in stochastic evolutionary systems, this offers an advantage of dimension reduction.

math.DS↗
arXiv2018

Target Search of a Protein on DNA in the Presence of Position-dependent Bias

We study the target searching on the DNA for proteins in the presence of non-constant drift and non-Gaussian $α$-stable Lévy fluctuations. The target searching is realized by the facilitated diffusion process. The existing works are about this problem in the case of constant drift. Starting from a non-local Fokker-Planck equation with a "sink" term, we obtain the possibility density function for the protein occurring at position $x$ on time $t$. Based on this, we further compute the survival probability and the first arrival density in order to quantify the searching mechanisms. The numerical results show that in the linear drift case, there is an optimal $α$ index for the search to be most likely successful (searching reliability reaches its maximum). This optimal $α$ index depends on initial position-target separation. It is also found that the diffusion intensity plays a positive role in improving the searching success. The nonlinear double-well drift could drive the protein to reach the target with a larger possibility than the linear drag at initial time period, but viewing at the long time evolution, the linear drift is more beneficial for target searching success. In contrast to the linear drift case, the search reliability and efficiency with nonlinear drift have a monotonic relationship with the $α$ index, that is, the smaller the $α$ index is, the more possibly a protein finds its target.

cond-mat.stat-mech↗
arXiv2018

A parameter estimator based on Smoluchowski-Kramers approximation

We devise a simplified parameter estimator for a second order stochastic differential equation by a first order system based on the Smoluchowski-Kramers approximation. We establish the consistency of the estimator by using Γ-convergence theory. We further illus- trate our estimation method by an experimentally studied movement model of a colloidal particle immersed in water under conservative force and constant diffusion.

math.ST↗
arXiv2018

Maximum principles for nonlocal parabolic Waldenfels operators

As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for `parabolic' equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with $α$-stable Lévy processes are presented to illustrate the maximum principles.

math.AP↗
arXiv2018

A role of random slow manifolds in detecting stochastic bifurcation

We consider the relation for the stochastic equilibrium states between the reduced system on a random slow manifold and the original system. This provides a theoretical basis for the reduction about sophisti- cated detailed models by the random slow manifold without significant damage to the overall qualitative properties. Based on this result, we reveal a role of random slow manifolds in detecting stochastic bifurca- tion by an example. The example exhibits a stochastic bifurcation phenomenon and possesses a random slow manifold that carries the stochastic bifurcation information of the system. Specifically, the lower dimensional reduced system on the random slow manifold retains the stochastic bifurcation phenomenon from the original system.

math.DS↗