September  2013, 18(7): 1735-1754. doi: 10.3934/dcdsb.2013.18.1735

Interaction of waves in a one dimensional stochastic PDE model of excitable media

1. 

Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Saudi Arabia

2. 

Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

Received  June 2012 Revised  February 2013 Published  May 2013

We study the Barkley PDE model of excitable media in a one dimensional periodic domain with additive space time noise. Regions of excitation and kinks (i.e., boundaries between regions of excitation) form due to the additive noise and propagate due to the underlying dynamics of the excitable media. We study the resulting distribution of excitation and kinks by developing a reduced model.
Citation: Hasan Alzubaidi, Tony Shardlow. Interaction of waves in a one dimensional stochastic PDE model of excitable media. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1735-1754. doi: 10.3934/dcdsb.2013.18.1735
References:
[1]

D. Barkley, A model for fast computer simulation of waves in excitable media,, Physica D: Nonlinear Phenomena, 49 (1991), 61.  doi: 10.1016/0167-2789(91)90194-E.  Google Scholar

[2]

R. Bürger, R. Ruiz-Baier and K. Schneider, Adaptive multiresolution methods for the simulation of waves in excitable media,, Journal of Scientific Computing, 43 (2010), 261.  doi: 10.1007/s10915-010-9356-3.  Google Scholar

[3]

M. Büttiker and T. Christen, Nucleation of weakly driven kinks,, Physical Review Letters, 75 (1995), 1895.   Google Scholar

[4]

B. Costa, Spectral methods for partial differential equations,, CUBO, 6 (2004), 1.   Google Scholar

[5]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[6]

J. García-Ojalvo, F. Sagués, L. Schimansky-Geier and J. M. Sancho, Noise-enhanced excitability in bistable activator-inhibitor media,, Physical Review E, 65 (2002).  doi: 10.1103/PhysRevE.65.011105.  Google Scholar

[7]

J. García-Ojalvo and J. M. Sancho, "Noise in Spatially Extended Systems,", Springer Verlag, (1999).  doi: 10.1007/978-1-4612-1536-3.  Google Scholar

[8]

J. García-Ojalvo and L. Schimansky-Geier, Noise-induced spiral dynamics in excitable media,, Europhysics Letters, 47 (1999), 298.  doi: 10.1209/epl/i1999-00388-9.  Google Scholar

[9]

S. Habib and G. Lythe, Dynamics of kinks: Nucleation, diffusion and annihilation,, Physical Review Letters, 84 (2000), 1070.  doi: 10.1103/PhysRevLett.84.1070.  Google Scholar

[10]

P. Jung and G. Mayer-Kress, Noise controlled spiral growth in excitable media,, Chaos, 5 (1995), 458.  doi: 10.1063/1.166117.  Google Scholar

[11]

S. Kádár, J. Wang and K. Showalter, Noise-supported travelling waves in sub-excitable media,, Nature, 391 (1998), 770.   Google Scholar

[12]

P. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,", Springer-Verlag, (1992).   Google Scholar

[13]

B. Lindner, J. García-Ojalvo, A. Neimand and L. Schimansky-Geier, Effects of noise in excitable systems,, Physics Reports, 392 (2004), 321.  doi: 10.1016/j.physrep.2003.10.015.  Google Scholar

[14]

G. J. Lord and T. Shardlow, Postprocessing for stochastic parabolic partial differential equations,, SIAM J. Numer. Anal., 45 (2007), 870.  doi: 10.1137/050640138.  Google Scholar

[15]

G. Lythe and S. Habib, Kink stochastics,, Computing in Science and Engineering, 8 (2006), 10.  doi: 10.1109/MCSE.2006.43.  Google Scholar

[16]

R. McLachlan, G. Quispel and P. Tse, Linearization-preserving self-adjoint and symplectic integrators,, BIT Numerical Mathematics, 49 (2009), 177.  doi: 10.1007/s10543-009-0214-3.  Google Scholar

[17]

A. S. Mikhailov, "Foundations of Synergetics I,", Springer Verlag, (1994).  doi: 10.1007/978-3-642-78556-6.  Google Scholar

[18]

B. V. Minchev and W. M. Wright, A review of exponential integrators for first order semi-linear problems,, Technical report, (2005).   Google Scholar

[19]

J. D. Murray, "Mathematical Biology II,", Springer Verlag, (1993).  doi: 10.1007/b98869.  Google Scholar

[20]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proceedings of the IRE, 50 (1962), 2061.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[21]

C. E. Rasmussen and C. Williams, "Gaussian Processes for Machine Learning,", The MIT Press, (2006).   Google Scholar

[22]

A. C. Scott, The electrophysics of a nerve fiber,, Reviews of Modern Physics, 47 (1975), 487.   Google Scholar

[23]

T. Shardlow, Nucleation of waves in excitable media by noise,, Multiscale Modeling and Simulation, 3 (2005), 151.  doi: 10.1137/030602149.  Google Scholar

[24]

T. Shardlow, Numerical simulation of stochastic PDEs for excitable media,, Journal of Computational and Applied Mathematics, 175 (2005), 429.  doi: 10.1016/j.cam.2004.06.020.  Google Scholar

[25]

H. C. Tuckwell and F. Y. Wan, Time to first spike in stochastic Hodgkin-Huxley systems,, Physica A: Statistical Mechanics and its Applications, 351 (2005), 427.  doi: 10.1016/j.physa.2004.11.059.  Google Scholar

[26]

A. T. Winfree, Varieties of spiral wave behavior: An experimentalist's approach to the theory of excitable media,, Chaos, 1 (1991), 303.  doi: 10.1063/1.165844.  Google Scholar

[27]

A. M. Zhabotinsky, A history of chemical oscillations and waves,, Chaos, 1 (1991), 379.  doi: 10.1063/1.165848.  Google Scholar

[28]

V. S. Zykov and A. T. Winfree, "Simulation of Wave Processes in Excitable Media,", Manchester University Press, (1987).   Google Scholar

show all references

References:
[1]

D. Barkley, A model for fast computer simulation of waves in excitable media,, Physica D: Nonlinear Phenomena, 49 (1991), 61.  doi: 10.1016/0167-2789(91)90194-E.  Google Scholar

[2]

R. Bürger, R. Ruiz-Baier and K. Schneider, Adaptive multiresolution methods for the simulation of waves in excitable media,, Journal of Scientific Computing, 43 (2010), 261.  doi: 10.1007/s10915-010-9356-3.  Google Scholar

[3]

M. Büttiker and T. Christen, Nucleation of weakly driven kinks,, Physical Review Letters, 75 (1995), 1895.   Google Scholar

[4]

B. Costa, Spectral methods for partial differential equations,, CUBO, 6 (2004), 1.   Google Scholar

[5]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[6]

J. García-Ojalvo, F. Sagués, L. Schimansky-Geier and J. M. Sancho, Noise-enhanced excitability in bistable activator-inhibitor media,, Physical Review E, 65 (2002).  doi: 10.1103/PhysRevE.65.011105.  Google Scholar

[7]

J. García-Ojalvo and J. M. Sancho, "Noise in Spatially Extended Systems,", Springer Verlag, (1999).  doi: 10.1007/978-1-4612-1536-3.  Google Scholar

[8]

J. García-Ojalvo and L. Schimansky-Geier, Noise-induced spiral dynamics in excitable media,, Europhysics Letters, 47 (1999), 298.  doi: 10.1209/epl/i1999-00388-9.  Google Scholar

[9]

S. Habib and G. Lythe, Dynamics of kinks: Nucleation, diffusion and annihilation,, Physical Review Letters, 84 (2000), 1070.  doi: 10.1103/PhysRevLett.84.1070.  Google Scholar

[10]

P. Jung and G. Mayer-Kress, Noise controlled spiral growth in excitable media,, Chaos, 5 (1995), 458.  doi: 10.1063/1.166117.  Google Scholar

[11]

S. Kádár, J. Wang and K. Showalter, Noise-supported travelling waves in sub-excitable media,, Nature, 391 (1998), 770.   Google Scholar

[12]

P. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,", Springer-Verlag, (1992).   Google Scholar

[13]

B. Lindner, J. García-Ojalvo, A. Neimand and L. Schimansky-Geier, Effects of noise in excitable systems,, Physics Reports, 392 (2004), 321.  doi: 10.1016/j.physrep.2003.10.015.  Google Scholar

[14]

G. J. Lord and T. Shardlow, Postprocessing for stochastic parabolic partial differential equations,, SIAM J. Numer. Anal., 45 (2007), 870.  doi: 10.1137/050640138.  Google Scholar

[15]

G. Lythe and S. Habib, Kink stochastics,, Computing in Science and Engineering, 8 (2006), 10.  doi: 10.1109/MCSE.2006.43.  Google Scholar

[16]

R. McLachlan, G. Quispel and P. Tse, Linearization-preserving self-adjoint and symplectic integrators,, BIT Numerical Mathematics, 49 (2009), 177.  doi: 10.1007/s10543-009-0214-3.  Google Scholar

[17]

A. S. Mikhailov, "Foundations of Synergetics I,", Springer Verlag, (1994).  doi: 10.1007/978-3-642-78556-6.  Google Scholar

[18]

B. V. Minchev and W. M. Wright, A review of exponential integrators for first order semi-linear problems,, Technical report, (2005).   Google Scholar

[19]

J. D. Murray, "Mathematical Biology II,", Springer Verlag, (1993).  doi: 10.1007/b98869.  Google Scholar

[20]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proceedings of the IRE, 50 (1962), 2061.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[21]

C. E. Rasmussen and C. Williams, "Gaussian Processes for Machine Learning,", The MIT Press, (2006).   Google Scholar

[22]

A. C. Scott, The electrophysics of a nerve fiber,, Reviews of Modern Physics, 47 (1975), 487.   Google Scholar

[23]

T. Shardlow, Nucleation of waves in excitable media by noise,, Multiscale Modeling and Simulation, 3 (2005), 151.  doi: 10.1137/030602149.  Google Scholar

[24]

T. Shardlow, Numerical simulation of stochastic PDEs for excitable media,, Journal of Computational and Applied Mathematics, 175 (2005), 429.  doi: 10.1016/j.cam.2004.06.020.  Google Scholar

[25]

H. C. Tuckwell and F. Y. Wan, Time to first spike in stochastic Hodgkin-Huxley systems,, Physica A: Statistical Mechanics and its Applications, 351 (2005), 427.  doi: 10.1016/j.physa.2004.11.059.  Google Scholar

[26]

A. T. Winfree, Varieties of spiral wave behavior: An experimentalist's approach to the theory of excitable media,, Chaos, 1 (1991), 303.  doi: 10.1063/1.165844.  Google Scholar

[27]

A. M. Zhabotinsky, A history of chemical oscillations and waves,, Chaos, 1 (1991), 379.  doi: 10.1063/1.165848.  Google Scholar

[28]

V. S. Zykov and A. T. Winfree, "Simulation of Wave Processes in Excitable Media,", Manchester University Press, (1987).   Google Scholar

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