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Henning D. ., Sailer F. X., Schimansky-Geier L. . Patterns in excitable dynamics driven by additive dichotomic noise. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 4, pp. 49-63. DOI: 10.18500/0869-6632-2009-17-4-49-63

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Patterns in excitable dynamics driven by additive dichotomic noise

Henning Dirk , University of Portsmouth
Sailer Franz Xaver Xaver, University of Portsmouth
Schimansky-Geier Lutz , University of Portsmouth

Pattern formation due the presence of additive dichotomous fluctuations is studied an extended system with diffusive coupling and a bistable FitzHugh–Nagumo kinetics. The fluctuations vary in space and/or time being noise or disorder, respectively. Without perturbations the dynamics does not support pattern formation. With proper dichotomous fluctuations, however, the homogeneous steady state is destabilized either via a Turing instability or the fluctuations create spatial nuclei of an inhomogeneous states. Finally, for purely static dichotomous disorder we find destabilization of homogeneous steady states for finite nonzero correlation length of the disorder resulting again in spatial patterns.

  1. Koch AJ, Meinhardt H. Biological pattern formation: from basic mechanisms to complex strucctures. Rev. of Mod. Phys. 1994;66(4):1481–1507.
  2. Mikhailov AS. Foundations of Synergetics I. 2nd Ed. Berlin, Heidelberg, New York: Springer; 1994. 213 p.
  3. Garcia-Ojalvo J, Sancho JM. Noise in Spatially Extended Systems. New York: Springer- Verlag; 1999. 307 p.
  4. Anishchenko V, Neiman A, Astakhov A, Vadivasova T, Schimansky-Geier L. Chaotic and Stochastic Processes in Dynamic Systems. Berlin-Heidelberg-New York: Springer; 2002.
  5. Lindner B, Garcia-Ojalvo J, Neiman A, Schimansky-Geier L. Effects of noise in excitable systems. Phys. Rep. 2004;392(6):321–424. DOI: 10.1016/J.PHYSREP.2003.10.015.
  6. Sagues F, Sancho J, Garcia-Ojalvo J. Spatiotemporal order out of noise. Rev Mod Phys. 2007;79:829–82. DOI: 10.1103/RevModPhys.79.829.
  7. Mikhailov AS. Effects of diffusion in fluctuating media: A noise-induced phase transition. Z. Phys. B. 1981;41(3):277–282. DOI: 10.1007/BF01294434.
  8. Garcia-Ojalvo J, Sancho JM, Ramirez-Piscina L. A nonequilibrium phase transition with colored noise. Phys. Lett. A. 1992;168(1):35–39.
  9. Parrondo JMR, C. van den Broeck, Buceta J, FJ. de la Rubia. Noise-Induced Spatial Patterns. Physica A. 1996;224(1-2):153–161. DOI:10.1016/0378-4371(95)00350-9.
  10. Zaikin AA, Schimansky-Geier L. Spatial patterns induced by additive noise. Phys. Rev. E. 1998;58(4):4355–4360.
  11. Kawai R, Sailer X, Schimansky-Geier L, Van den Broeck C. Macroscopic limit cycle via pure noise-induced phase transitions. Phys. Rev. E. 2004;69(5):051104. DOI: 10.1103/PhysRevE.69.051104.
  12. Buceta J, Ibanes M, Sancho JM, Lindenberg K. Noise-driven mechanism for pattern formation. Phys. Rev. E. 2003;67(2):021113. DOI: 10.1103/PhysRevE.67.021113.
  13. Buceta J, Lindenberg K, Parrondo JMR. Stationary and oscillatory spatial patterns induced by global periodic switching. Phys. Rev. Lett. 2002;88(2):024103. DOI: 10.1103/PhysRevLett.88.024103.
  14. Buceta J, Lindenberg K, Parrondo JMR. Pattern formation induced by nonequilibrium global alternation of dynamics. Phys. Rev. E. 2002;66(3):036216. DOI: 10.1103/PHYSREVE.66.036216.
  15. Buceta J, Lindenberg K. Switching-induced Turing instability. Phys. Rev. E. 2002;66():046202. DOI: 10.1103/PHYSREVE.66.046202.
  16. Buceta J, Lindenberg K. Spatial patterns induced purely by dichotomous disorder. Phys Rev E Stat Nonlin Soft Matter Phys. 2003;68(1):011103. DOI: 10.1103/PhysRevE.68.011103.
  17. Fitzhugh R. Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophys J. 1961;1(6):445-466. DOI: 10.1016/s0006-3495(61)86902-6.
  18. Nagumo J, Arimoto S, Yoshitzawa S. An active pulse transmission line simulating nerve axon. Proc. IRE. 1962;50:2061–2070. DOI: 10.1109/JRPROC.1962.288235.
  19. Vasilev VA, Romanovski YuM, Yakhno VG. Autowave processes in distributed kinetic systems. Phys. Usp. 1979;22(8):615–639. DOI: 10.3367/UFNr.0128.197908c.0625.
  20. Elphick C, Hagberg A, Meron E. Dynamic front transitions and spiral-vortex nucleation. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1995;51(4):3052–3058. DOI: 10.1103/physreve.51.3052.
  21. Martinez K, Lin AL, Kharrazian R, Sailer X, Swinney HL. Resonance in periodically inhibited reaction-diffusion systems. Physica D. 2002;168:1–9. DOI: 10.1016/S0167-2789(02)00490-6.
  22. Sailer X, Hennig D, Beato V, Engel H, Schimansky-Geier L. Regular patterns in dichotomically driven activator-inhibitor dynamics. Phys Rev E Stat Nonlin Soft Matter Phys. 2006;73(5):056209. DOI: 10.1103/PhysRevE.73.056209.
  23. Koga S, Kuramoto Y. Localized patterns in reaction-diffusin systems. Prog. of Theor. Phys. 1980;63:106–121. DOI: 10.1143/PTP.63.106.
  24. Rinzel J, Keller JB. Traveling wave solutions of a nerve conduction equation. Biophys J. 1973;13(12):1313-1337. DOI: 10.1016/S0006-3495(73)86065-5.
  25. Ohta T, Mimura M, Kobayashi R. Higher-dimensional localized patterns in excitable media. Physica D. 1989;34(1-2):115–144.
  26. Ohta T, Ito A, Tetsuka A. Self-organization in an excitable reaction-diffusion system: Synchronization of oscillatory domains in one dimension. Phys. Rev. A. 1990;42(6):3225–3232. DOI: 10.1103/physreva.42.3225.
  27. Harmer GP, Abott D. Losing strategies can win by Parrondo’s paradox. Nature (London). 1999;199(6764):864–864.
  28. Schimansky-Geier L, Hempel H, Bartussek R, Zulicke C. Analysis of domain-solutions in reaction-diffusion systems. Z. Physik B. 1995;96:417–427. DOI: 10.1007/BF01313065.
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