A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

Dangerfield, Ciara; Kay, David; MacNamara, Shev; Burrage, Kevin

doi:10.1007/s10543-011-0351-3

Cited by 29 publications

(38 citation statements)

References 27 publications

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“…The SDE describing the dynamics of the sodium and potassium channels was first derived in [26] and later implemented in [29,30]. We use the representation of the noise term in the SDE described in [49], which removes the need to find the square root of a matrix and so the sodium and potassium channel dynamics are given by the following SDEs [32,49]:…”

Section: Conductance Model With Bounded Stochastic Ion Channel Dymentioning

confidence: 99%

“…Alterations to the numerical scheme can be made to force the solution to remain positive, for example the Wiener increment can be continually resampled [5]. However, such alterations can bias the results [31][32][33]. Another approach is to replace the variable (i.e., the state of the system) in the noise term with its equilibrium value [24,29] so that the square root term is independent of the state of the system.…”

Section: Introductionmentioning

confidence: 99%

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Modeling ion channel dynamics through reflected stochastic differential equations

2012

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Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks.

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Section: Conductance Model With Bounded Stochastic Ion Channel Dymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Modeling ion channel dynamics through reflected stochastic differential equations

2012

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“…The extension of the Wright-Fisher model to the multidimensional case has been proposed in [9] and [10]. Here, we will treat a three-state system as in [6,Sec. 6] given by the following system of SDEs…”

Section: Introductionmentioning

confidence: 99%

A boundary preserving numerical scheme for the Wright–Fisher model

Stamatiou¹

2018

Journal of Computational and Applied Mathematics

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Abstract. We are interested in the numerical approximation of non-linear stochastic differential equations (SDEs) with solution in a certain domain. Our goal is to construct explicit numerical schemes that preserve that structure. We generalize the semi-discrete method Halidias N. and Stamatiou I.S. (2016), On the numerical solution of some non-linear stochastic differential equations using the Semi-Discrete method, Computational Methods in Applied Mathematics,16(1) and propose a numerical scheme, for which we prove a strong convergence result, to a class of SDEs that appears in population dynamics and ion channel dynamics within cardiac and neuronal cells. We furthermore extend our scheme to a multidimensional case.

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“…We know that E(|∆W n |) = 2h π , also s n and s n−m are F tn −measurable, so with substituting inequalities (18), (19) and (20) in (17), we obtain…”

Section: Bim and Boundedness Of The P-moments In Continuous Casementioning

confidence: 99%

Convergence and non-negativity preserving of the solution of balanced method for the delay CIR model with jump

Aghda

Hosseini

Tahmasebi

2018

Journal of Computational and Applied Mathematics

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In this work, we propose the balanced implicit method (BIM) to approximate the solution of the delay Cox-Ingersoll-Ross (CIR) model with jump which often gives rise to model an asset price and stochastic volatility dependent on past data. We show that this method preserves non-negativity property of the solution of this model with appropriate control functions. We prove the strong convergence and investigate the pth moment boundedness of the solution of BIM. Finally, we illustrate those results in the last section.

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A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

Cited by 29 publications

References 27 publications

Modeling ion channel dynamics through reflected stochastic differential equations

Modeling ion channel dynamics through reflected stochastic differential equations

A boundary preserving numerical scheme for the Wright–Fisher model

Convergence and non-negativity preserving of the solution of balanced method for the delay CIR model with jump

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