Research paperQuasi-threshold phenomenon in noise-driven Higgins model
Introduction
Environmental noise is an inevitable attribute of any physical system, which has a profound impact on the dynamical behaviors. Due to the presence of random forces, a variety of unexpected phenomena which cannot be observed in noise-free cases are emerged, such as noise-induced escape [1], [2], [3], stochastic excitability [4], [5], [6], [7], noise-induced chaos [8,9], coherence resonance [10] and control problem [11], [12], [13]. In recent years these phenomena have attracted the universal attention of researchers.
For an excitable system, even weak noise may essentially deform the dynamical behaviors due to the non-uniformity of the vector field. Generally, the event of noise-induced excitability is associated with the multi-stability of the corresponding deterministic dynamical system. For the system driven by random forces, a new regime of large excursion will be observed after traversing through the boundary between basins of the coexisting attractors [5]. Therefrom a spike can be divided into two phases by the threshold: a long time initial phase and a transient excitation phase. The former can be regarded as noise-induced escaping phenomenon from the rest state to the threshold, while within the latter phase, the system almost follows the deterministic flow returning to the stable state. Thus the whole process will be clear if the phenomenon of noise-induced escape is uncovered. Actually, even in the cases without well-defined thresholds due to the absence of saddle points, they also share many features in common with those multi-stable systems through defining a quasi-threshold [5,14].
In order to investigate the escaping phenomenon, Freidlin-Wentzell large deviation theory [15] have proposed a concept of action functional to estimate how impossible for a stochastic path to realize. The theory asserts that the escaping from a domain follows the optimal path with overwhelming probability, along which the action functional attaches its minimum. Although the system is stochastic, the optimal path itself is not. The last several decades have witnessed an increasing number of applications for large deviation theory to a variety of dynamical systems with various structures, such as equilibriums [16,17], limit cycles [18] and chaotic attractors [19,20]. Furthermore, this elegant theory has also been verified by both the numerical simulations proposed by Dykman [21] and the analogue electronic experiments designed by Luchinsky [22].
MFPT is another crucial notion which is exponentially dominated by quasi-potential, i.e., the constrained minimization of action functional [15]. It is of great relevance to the reliability of engineering structures of first passage failure [23] and to the stochastic stability [24]. Utilizing the singular perturbation method and the method of characteristics, Schuss and his co-workers deduced the approximate expression of MFPT between fixed points for the systems with characteristic and uncharacteristic boundary [25], [26], [27]. Then stochastic averaged method was introduced by Roy into this theory to transform the exit problem between cycles into the one between fixed points [28]. Kong and Liu have also used these methods to derive MFPT and taken it as a measure to describe noise-induced chaos quantitatively in a piecewise linear system [29].
By using stochastic sensitivity function and constructing the confidence ellipses, Ryashko [30] uncovered the switching behavior from small amplitude oscillation regime to the bursting one with the noise intensity increasing for Higgins model. However, the second-order approximation of the quasi-potential within the region far from the fixed point is inaccurate. Therefore, we tried to compute quasi-potential precisely and used its results to calculate theoretically the optimal path, MFPT and exit location distribution which provide the information of how, when and where to exit. Our major contributions are: (i) to select the most sensitive trajectory as the specific quasi-threshold; (ii) to demonstrate the global quasi-potential landscape and the momentum to analyze the behavior of the optimal path before and after the quasi-threshold.; (iii) to reveal the divergence between the practical exit point and the position given by the minimal action on the threshold and to explain it by the exit location distribution; and (iv) to compute the more accurate expression of MFPT theoretically than the pure exponential estimate.
The article is organized as follows. In Section 2, the dynamical behaviors of Higgins model and its quasi-threshold phenomenon are described. Then the application of WKB approximation to this model is performed in Section 3, through which a set of ordinary differential equations (ODEs) for the computation of the quasi-potential and the exit paths is formulated. In Section 4, the optimal path is integrated and verified by numerical simulations. Besides, the exit location distribution and MFPT are evaluated in Section 5 by both theoretical prediction and Monte Carlo simulations, and they both agree well. Conclusions are then present in Section 6.
Section snippets
Formulation
Glycolysis is a classic example of an oscillatory biochemical reaction. In a glycolytic process, glucose and other sugars decompose, and the compounds, containing six molecules of carbon, turn into tricarbon acids that include three carbon molecules. Due to the excess of free energy in the glycolytic process, two ATP molecules form per molecule of the six-carbon sugar. The main role in the generation of observed concentration oscillations of the reaction components fructose-6-phosphate,
WKB approximation
As mentioned above, an excitable event can be divided into a long time initial phase and a transient large amplitude motion with a threshold defined as the specific separatrix in the previous section. In fact, the initial phase can be regarded as the rare event of noise-induced escape from the fixed point to the threshold. Generally, WKB approximation is an effective and common method to deal with it.
Under random disturbances, the stationary probability distribution ps(x) of stochastic states
Quasi-potential and the optimal path
Based on the quasi-threshold defined in Section 2 and the theoretical results of WKB approximation in Section 3, we now investigate the escape problem starting from the fixed point and crossing the threshold. The method of action plot [35] is employed to provide the actions of all the extreme paths. The so-called action is the value of action functional on the extreme paths. Thus the quasi-potential is given by the minimal action of all the extreme paths arriving at this point. Eighty thousand
Exit location distribution and mean first passage time
Actually, through comparing the escaping trajectory found by numerical simulations with the theoretical optimal path, one can find a discrepancy that the statistical path crosses the separatrix transversely before reaching S rather than running along it. Such a behavior has been observed earlier [38,39], which is referred to as saddle point avoidance and can be characterized by exit location distribution. This phenomenon essentially stems from the fact that the only reservation of lowest-order
Conclusion
In conclusion, we have investigated the quasi-threshold phenomenon in Higgins model driven by additive Gaussian white noise. For the convenience of quantitative description, a specific phase trajectory is defined as a quasi-threshold which is generally occupied by the stable manifold of a saddle point. This quasi-threshold is selected as the location that is most sensitive about the initial condition. Under weak random perturbations, the particle spends a long time wandering around the stable
Data availability statement
The data that support the findings of this study are openly available in GitHub [40].
CRediT authorship contribution statement
Yang Li: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft. Jianlong Wang: Software, Resources, Writing - review & editing. Xianbin Liu: Formal analysis, Writing - review & editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant Nos. 11472126 and 11232007) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
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