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 and 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), 61-70. doi: 10.1016/0167-2789(91)90194-E.

[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), 261-290. doi: 10.1007/s10915-010-9356-3.

[3]

M. Büttiker and T. Christen, Nucleation of weakly driven kinks, Physical Review Letters, 75 (1995), 1895-1898.

[4]

B. Costa, Spectral methods for partial differential equations, CUBO, A Mathematical Journal, 6 (2004), 1-32.

[5]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.

[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), 011105. doi: 10.1103/PhysRevE.65.011105.

[7]

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

[8]

J. García-Ojalvo and L. Schimansky-Geier, Noise-induced spiral dynamics in excitable media, Europhysics Letters, 47 (1999), 298-303. doi: 10.1209/epl/i1999-00388-9.

[9]

S. Habib and G. Lythe, Dynamics of kinks: Nucleation, diffusion and annihilation, Physical Review Letters, 84 (2000), 1070-1073. doi: 10.1103/PhysRevLett.84.1070.

[10]

P. Jung and G. Mayer-Kress, Noise controlled spiral growth in excitable media, Chaos, 5 (1995), 458-462. doi: 10.1063/1.166117.

[11]

S. Kádár, J. Wang and K. Showalter, Noise-supported travelling waves in sub-excitable media, Nature, 391 (1998), 770-772.

[12]

P. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, Berlin, 1992.

[13]

B. Lindner, J. García-Ojalvo, A. Neimand and L. Schimansky-Geier, Effects of noise in excitable systems, Physics Reports, 392 (2004), 321-424. doi: 10.1016/j.physrep.2003.10.015.

[14]

G. J. Lord and T. Shardlow, Postprocessing for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 45 (2007), 870-889. doi: 10.1137/050640138.

[15]

G. Lythe and S. Habib, Kink stochastics, Computing in Science and Engineering, 8 (2006), 10-15. doi: 10.1109/MCSE.2006.43.

[16]

R. McLachlan, G. Quispel and P. Tse, Linearization-preserving self-adjoint and symplectic integrators, BIT Numerical Mathematics, 49 (2009), 177-197. doi: 10.1007/s10543-009-0214-3.

[17]

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

[18]

B. V. Minchev and W. M. Wright, A review of exponential integrators for first order semi-linear problems, Technical report, The Norwegian University of Science and Technology, Norway, 2005.

[19]

J. D. Murray, "Mathematical Biology II," Springer Verlag, Berlin, 1993. doi: 10.1007/b98869.

[20]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.

[21]

C. E. Rasmussen and C. Williams, "Gaussian Processes for Machine Learning," The MIT Press,Cambridge, UK, 2006.

[22]

A. C. Scott, The electrophysics of a nerve fiber, Reviews of Modern Physics, 47 (1975), 487-533.

[23]

T. Shardlow, Nucleation of waves in excitable media by noise, Multiscale Modeling and Simulation, 3 (2005), 151-167. doi: 10.1137/030602149.

[24]

T. Shardlow, Numerical simulation of stochastic PDEs for excitable media, Journal of Computational and Applied Mathematics, 175 (2005), 429-446. doi: 10.1016/j.cam.2004.06.020.

[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), 427-438. doi: 10.1016/j.physa.2004.11.059.

[26]

A. T. Winfree, Varieties of spiral wave behavior: An experimentalist's approach to the theory of excitable media, Chaos, 1 (1991), 303-334. doi: 10.1063/1.165844.

[27]

A. M. Zhabotinsky, A history of chemical oscillations and waves, Chaos, 1 (1991), 379-386. doi: 10.1063/1.165848.

[28]

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

show all references

References:
[1]

D. Barkley, A model for fast computer simulation of waves in excitable media, Physica D: Nonlinear Phenomena, 49 (1991), 61-70. doi: 10.1016/0167-2789(91)90194-E.

[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), 261-290. doi: 10.1007/s10915-010-9356-3.

[3]

M. Büttiker and T. Christen, Nucleation of weakly driven kinks, Physical Review Letters, 75 (1995), 1895-1898.

[4]

B. Costa, Spectral methods for partial differential equations, CUBO, A Mathematical Journal, 6 (2004), 1-32.

[5]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.

[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), 011105. doi: 10.1103/PhysRevE.65.011105.

[7]

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

[8]

J. García-Ojalvo and L. Schimansky-Geier, Noise-induced spiral dynamics in excitable media, Europhysics Letters, 47 (1999), 298-303. doi: 10.1209/epl/i1999-00388-9.

[9]

S. Habib and G. Lythe, Dynamics of kinks: Nucleation, diffusion and annihilation, Physical Review Letters, 84 (2000), 1070-1073. doi: 10.1103/PhysRevLett.84.1070.

[10]

P. Jung and G. Mayer-Kress, Noise controlled spiral growth in excitable media, Chaos, 5 (1995), 458-462. doi: 10.1063/1.166117.

[11]

S. Kádár, J. Wang and K. Showalter, Noise-supported travelling waves in sub-excitable media, Nature, 391 (1998), 770-772.

[12]

P. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, Berlin, 1992.

[13]

B. Lindner, J. García-Ojalvo, A. Neimand and L. Schimansky-Geier, Effects of noise in excitable systems, Physics Reports, 392 (2004), 321-424. doi: 10.1016/j.physrep.2003.10.015.

[14]

G. J. Lord and T. Shardlow, Postprocessing for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 45 (2007), 870-889. doi: 10.1137/050640138.

[15]

G. Lythe and S. Habib, Kink stochastics, Computing in Science and Engineering, 8 (2006), 10-15. doi: 10.1109/MCSE.2006.43.

[16]

R. McLachlan, G. Quispel and P. Tse, Linearization-preserving self-adjoint and symplectic integrators, BIT Numerical Mathematics, 49 (2009), 177-197. doi: 10.1007/s10543-009-0214-3.

[17]

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

[18]

B. V. Minchev and W. M. Wright, A review of exponential integrators for first order semi-linear problems, Technical report, The Norwegian University of Science and Technology, Norway, 2005.

[19]

J. D. Murray, "Mathematical Biology II," Springer Verlag, Berlin, 1993. doi: 10.1007/b98869.

[20]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.

[21]

C. E. Rasmussen and C. Williams, "Gaussian Processes for Machine Learning," The MIT Press,Cambridge, UK, 2006.

[22]

A. C. Scott, The electrophysics of a nerve fiber, Reviews of Modern Physics, 47 (1975), 487-533.

[23]

T. Shardlow, Nucleation of waves in excitable media by noise, Multiscale Modeling and Simulation, 3 (2005), 151-167. doi: 10.1137/030602149.

[24]

T. Shardlow, Numerical simulation of stochastic PDEs for excitable media, Journal of Computational and Applied Mathematics, 175 (2005), 429-446. doi: 10.1016/j.cam.2004.06.020.

[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), 427-438. doi: 10.1016/j.physa.2004.11.059.

[26]

A. T. Winfree, Varieties of spiral wave behavior: An experimentalist's approach to the theory of excitable media, Chaos, 1 (1991), 303-334. doi: 10.1063/1.165844.

[27]

A. M. Zhabotinsky, A history of chemical oscillations and waves, Chaos, 1 (1991), 379-386. doi: 10.1063/1.165848.

[28]

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

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