Crossing the quasi-threshold manifold of a noise-driven excitable system

Chen, Zhen; Zhu, Jinjie; Liu, Xianbin

doi:10.1098/rspa.2017.0058

Cited by 11 publications

(8 citation statements)

References 31 publications

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“…For example, the vector field b can be very complicated, as it is in the Sneppen-Aurell genetic switch model in Lambda Phage [1], so that the location of the transition state is hard to determine by setting b to zero, but becomes apparent from the computed quasipotential [13]. An interesting example of a maximum likelihood escape path not associated with any special-trajectory-type transition state such as saddle or unstable limit cycle was found in the FitzHugh-Nagumo system using the quasipotential computed in an entire region [9].…”

Section: Significance Of Computing the Quasipotential In The Entire Rmentioning

confidence: 99%

Computing the quasipotential for nongradient SDEs in 3D

Yang

Potter

Cameron

2019

Journal of Computational Physics

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Nongradient SDEs with small white noise often arise when modeling biological and ecological time-irreversible processes. If the governing SDE were gradient, the maximum likelihood transition paths, transition rates, expected exit times, and the invariant probability distribution would be given in terms of its potential function. The quasipotential plays a similar role for nongradient SDEs. Unfortunately, the quasipotential is the solution of a functional minimization problem that can be obtained analytically only in some special cases. We propose a Dijkstra-like solver for computing the quasipotential on regular rectangular meshes in 3D. This solver results from a promotion and an upgrade of the previously introduced ordered line integral method with the midpoint quadrature rule for 2D SDEs. The key innovations that have allowed us to keep the CPU times reasonable while maintaining good accuracy are (i) a new hierarchical update strategy, (ii) the use of Karush-Kuhn-Tucker theory for rejecting unnecessary simplex updates, and (iii) pruning the number of admissible simplexes and a fast search for them. An extensive numerical study is conducted on a series of linear and nonlinear examples where the quasipotential is analytically available or can be found at transition states by other methods. In particular, the proposed solver is applied to Tao's examples where the transition states are hyperbolic periodic orbits, and to a genetic switch model by Lv et al. (2014). The C source code implementing the proposed algorithm is available at M. Cameron's web page.

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Section: Significance Of Computing the Quasipotential In The Entire Rmentioning

confidence: 99%

Computing the quasipotential for nongradient SDEs in 3D

Yang

Potter

Cameron

2019

Journal of Computational Physics

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“…Third, the maximum likelihood exit location from B(E) might be unknown in advance. For example, the exit point for the escape to a large loop in the FitzHugh-Nagumo system lying on the separatrix trajectory was discovered by means of computing the quasi-potential in the whole space [12].…”

Section: Significance Of Computing the Quasi-potential On A Meshmentioning

confidence: 99%

“…Nonetheless, there are plenty of interesting and important low-dimensional biological and ecological models (see e.g. [26,12]) where the quasi-potential analysis is instrumental. Furthermore, in some systems, the effective dynamics is effectively restricted to a low-dimensional manifold, and hence the quasi-potential can be computed on it.…”

Section: Significance Of Computing the Quasi-potential On A Meshmentioning

confidence: 99%

An Ordered Line Integral Method for computing the quasi-potential in the case of variable anisotropic diffusion

Dahiya

Cameron

2018

Physica D: Nonlinear Phenomena

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Nongradient stochastic differential equations (SDEs) with position-dependent and anisotropic diffusion are often used in biological modeling. The quasi-potential is a crucial function in the Large Deviation Theory that allows one to estimate transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths. Unfortunately, the quasi-potential can rarely be found analytically. It is defined as the solution to a certain action minimization problem. In this work, the recently introduced Ordered Line Integral Method (OLIM) is extended for computing the quasi-potential for 2D SDEs with anisotropic and position-dependent diffusion scaled by a small parameter on a regular rectangular mesh. The presented solver employs the dynamical programming principle. At each step, a local action minimization problem is solved using straight line path segments and the midpoint quadrature rule. The solver is tested on two examples where analytic formulas for the quasi-potential are available. The dependence of the computational error on the mesh size, the update factor K (a key parameter of OLIMs), as well as the degree and the orientation of anisotropy is established. The effect of anisotropy on the quasi-potential and the maximum likelihood paths is demonstrated on the Maier-Stein model. The proposed solver is applied to find the quasi-potential and the maximum likelihood transition paths in a model of the genetic switch in Lambda Phage between the lysogenic state where the phage reproduces inside the infected cell without killing it, and the lytic state where the phage destroys the infected cell.

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“…Using a nontrivial adjustment of the Ordered Upwind Method, Cameron computed the quasipotential on a mesh to find the most probable paths by numerical integration [2]. Chen, Zhu and Liu [4] considered the noiseinduced escapes in an excitable system possessing a quasi-threshold manifold, along which there exists a certain point of minimal quasipotential. Dahiya and Cameron [5] performed numerical computation of the quasipotential for SDEs with multiplicative noise on a mesh, and discussed an application to the Maier-Stein model with anisotropic diffusion.…”

Section: Introductionmentioning

confidence: 99%

Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning

Yuan

She

2022

Preprint

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The mean exit time escaping basin of attraction in the presence of white noise is of practical importance in various scientific fields. In this work, we propose a strategy to control mean exit time of general stochastic dynamical systems to achieve a desired value based on the quasipotential concept and machine learning. Specifically, we develop a neural network architecture to compute the global quasipotential function. Then we design a systematic iterated numerical algorithm to calculate the controller for a given mean exit time. Moreover, we identify the most probable path between metastable attractors with help of the effective Hamilton-Jacobi scheme and the trained neural network. Numercal experiments demonstrate that our control strategy is effective and sufficiently accurate.

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Crossing the quasi-threshold manifold of a noise-driven excitable system

Cited by 11 publications

References 31 publications

Computing the quasipotential for nongradient SDEs in 3D

Computing the quasipotential for nongradient SDEs in 3D

An Ordered Line Integral Method for computing the quasi-potential in the case of variable anisotropic diffusion

Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning

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