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Journal of Scientific Computing

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01-10-2015

Simulation of SPDEs for Excitable Media Using Finite Elements

Authors: Muriel Boulakia , Alexandre Genadot, Michèle Thieullen

Published in: Journal of Scientific Computing | Issue 1/2015

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Abstract

In this paper, we address the question of the discretization of stochastic partial differential equations (SPDEs) for excitable media. Working with SPDEs driven by colored noise, we consider a numerical scheme based on finite differences in time (Euler–Maruyama) and finite elements in space. Motivated by biological considerations, we study numerically the emergence of reentrant patterns in excitable systems such as the Barkley or Mitchell–Schaeffer models.

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Allen, E., Novosel, S., Zhang, Z.: Finite element and difference approximation of some linear stochastic partial differential equations. Stoch. Int. J. Probab. Stoch. Proc. 64(1–2), 117–142 (1998)MathSciNetCrossRefMATH

Babuska, I., Szabo, B., Katz, I.: The p-version of the finite element method. SIAM J. Numer. Anal. 18(3), 515–545 (1981)MathSciNetCrossRefMATH

Barkley, D.: A model for fast computer simulation of waves in excitable media. Phys. D Nonlinear Phenom. 49(1–2), 61–70 (1991)CrossRef

Barkley, D.: Linear stability analysis of rotating spiral waves in excitable media. Phys. Rev. Lett. 68(13), 2090–2093 (1992)CrossRef

Barkley, D.: Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett. 72(1), 164–167 (1994)CrossRef

Barkley, D., Kness, M., Tuckerman, L.: Spiral-wave dynamics in a simple model of excitable media: the transition from simple to compound rotation. Phys. Rev. A 42(4), 2489–2492 (1990)MathSciNetCrossRef

Berglund, N., Gentz, B.: Noise-Induced Phenomena in Slow–Fast Dynamical Systems: A Sample-Paths Approach, vol. 246. Springer, Berlin (2006)

Bonaccorsi, S., Mastrogiacomo, E.: Analysis of the stochastic Fitzhugh–Nagumo system. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11(3), 427–446 (2008)MathSciNetCrossRefMATH

Boulakia, M., Cazeau, S., Fernández, M., Gerbeau, J.F., Zemzemi, N.: Mathematical modeling of electrocardiograms: a numerical study. Ann. Biomed. Eng. 38(3), 1071–1097 (2010)CrossRef

10.

Bréhier, C.E.: Strong and weak order in averaging for SPDEs. Stoch. Proc. Appl. 122(7), 2553–2593 (2012)CrossRefMATH

11.

Cao, Y., Yang, H., Yin, H.: Finite element methods for semilinear elliptic stochastic partial differential equations. Numer. Math. 106, 181–198 (2007)MathSciNetCrossRefMATH

12.

Cerrai, S., Freidlin, M.: Averaging principle for a class of stochastic reaction-diffusion equations. Probab. Theory Relat. Fields 144(1–2), 137–177 (2009)MathSciNetCrossRefMATH

13.

Ciarlet, P., Lions, J.: Finite Element Methods, Handbook of Numerical Analysis, vol. 2. Elsevier, Amsterdam (1991)

14.

Da Prato, G.: Kolmogorov Equations for Stochastic PDEs. Birkhäuse,r Basel (2004)

15.

Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)CrossRefMATH

16.

Debussche, A., Printems, J.: Weak order for the discretization of the stochastic heat equation. Math. Comput. 78(266), 845–863 (2009)MathSciNetCrossRefMATH

17.

FitzHugh, R.: Mathematical models of excitation and propagation in nerve. In: Schwan, H.P. (ed.) Biological Engineering, chap. 1, pp. 1–85. McGraw–Hill Book Co., New York (1969)

18.

Goudenège, L., Martin, D., Vial, G.: High order finite element calculations for the Cahn–Hilliard equation. J. Sci. Comput. 52(2), 294–321 (2012)MathSciNetCrossRefMATH

19.

Hairer, M., Ryser, M., Weber, H.: Triviality of the 2d stochastic Allen–Cahn equation. Electron. J. Probab. 17(39), 1–14 (2012)MathSciNet

20.

Hecht, F., Le Hyaric, A., Ohtsuka, K., Pironneau, O.: Freefem++, finite elements software

21.

Hinch, R.: An analytical study of the physiology and pathology of the propagation of cardiac action potentials. Prog. Biophys. Mol. Biol. 78(1), 45–81 (2002)CrossRef

22.

Hodgkin, A., Huxley, A.: Propagation of electrical signals along giant nerve fibres. Proc. R. Soc. Lond. 140(899), 177–183 (1952)CrossRef

23.

Jentzen, A.: Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients. Potential Anal. 31(4), 375–404 (2009)MathSciNetCrossRefMATH

24.

Jentzen, A., Röckner, M.: A Milstein scheme for SPDEs. arXiv preprint arXiv:1001.2751 (2012)

25.

Jordan, P., Christini, D.: Cardiac Arrhythmia. Wiley, London (2006)CrossRef

26.

Keener, J.: Waves in excitable media. J. Appl. Math. 39(3), 528–548 (1980)MathSciNetMATH

27.

Kovács, M., Larsson, S., Lindgren, F.: Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise. Numer. Algorithms 53(2–3), 309–320 (2010)MathSciNetCrossRefMATH

28.

Kruse, R.: Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise. IMA J. Numer. Anal. 34(1), 217–251 (2014)MathSciNetCrossRefMATH

29.

Kruse, R., Larsson, S.: Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise. Electron. J. Probab. 17(65), 1–19 (2012)MathSciNet

30.

Lindner, B., Garcia-Ojalvo, J., Neiman, A., Schimansky-Geier, L.: Effects of noise in excitable systems. Phys. Rep. 392(6), 321–424 (2004)CrossRef

31.

Lord, G., Tambue, A.: A modified semi-implict Euler–Maruyama scheme for finite element discretization of SPDEs. arXiv preprint arXiv:1004.1998 (2010)

32.

Lord, G., Tambue, A.: Stochastic exponential integrators for finite element discretization of SPDEs for multiplicative and additive noise. IMA J. Numer. Anal. 33(2), 515–543 (2013)MathSciNetCrossRefMATH

33.

Lord, G., Thümmler, V.: Computing stochastic traveling waves. SIAM J. Sci. Comput. 34(1), 24–43 (2012)CrossRef

34.

Mitchell, C., Schaeffer, D.: A two-current model for the dynamics of cardiac membrane. Bull. Math. Biol. 65(5), 767–793 (2003)CrossRef

35.

Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press, Cambridge (2007)CrossRefMATH

36.

Raviart, P., Thomas, J.: Introduction à l’analyse numérique des équations aux dérivées partielles. Masson (1983)

37.

Shardlow, T.: Numerical simulation of stochastic PDEs for excitable media. J. Comput. Appl. Math. 175(2), 429–446 (2005)MathSciNetCrossRefMATH

38.

Tuckwell, H., Jost, J.: Weak noise in neurons may powerfully inhibit the generation of repetitive spiking but not its propagation. PLoS Comput. Biol. 6(5) 1–13 (2010)

39.

Wagner, D.: Survey of measurable selection theorems: an update. Lecture Notes in Mathematics, vol. 794. Springer, Berlin (1980)

40.

Walsh, J.: Finite element methods for parabolic stochastic PDEs. Potential Anal. 23(1), 1–43 (2005)MathSciNetCrossRefMATH

41.

Wang, W., Roberts, A.: Average and deviation for slow–fast stochastic partial differential equations. J. Differ. Equ. 253(5), 1265–1286 (2012)MathSciNetCrossRefMATH

42.

Yan, Y.: Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numeric. Anal. 43(4), 1363–1384 (2005)CrossRefMATH

Title: Simulation of SPDEs for Excitable Media Using Finite Elements
Authors: Muriel Boulakia
Alexandre Genadot
Michèle Thieullen
Publication date: 01-10-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-014-9960-8

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