Higher-dimensional localized patterns in excitable media

Abstract The excitable reaction-diffusion equation model of the form eτu t =e 2 ▿ 2 u+ƒ(u)–v , v t =▿ 2 v+u−γv is considered. When ƒ(u) is assumed to be of McKean's piecewise linear type, the interfacial approach can be applied to the stability of various localized patterns in higher-dimensional spaces. It is shown that a band-shaped localized pattern is destabilized into a zig-zag mode or a varicose mode and that a disk-shaped localized pattern is destabilized into static modes and, when τ is small, into an oscillatory mode like a “breather motion”. Numerical simulations are performed to confirm such destabilizations for more general nonlinear functions ƒ(u) .

[1]  S. Amari,et al.  Existence and stability of local excitations in homogeneous neural fields , 1979, Journal of mathematical biology.

[2]  Fife,et al.  Phase-field methods for interfacial boundaries. , 1986, Physical review. B, Condensed matter.

[3]  P. Grindrod,et al.  Three-dimensional waves in excitable reaction-diffusion systems , 1987 .

[4]  A. Winfree.  The geometry of biological time , 1991 .

[5]  H. Meinhardt.  Models of biological pattern formation , 1982 .

[6]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[7]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[8]  M. Mimura,et al.  Pattern formation in interacting and diffusing systems in population biology. , 1982, Advances in biophysics.

[9]  H. McKean.  Nagumo's equation , 1970 .

[10]  G. H. Markstein,et al.  Experimental and Theoretical Studies of Flame-Front Stability , 1951 .

[11]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[12]  J. Keener.  A geometrical theory for spiral waves in excitable media , 1986 .

[13]  Takao Ohta,et al.  Kink dynamics in one-dimensional nonlinear systems , 1982 .

[14]  A. Zippelius,et al.  Disappearance of stable convection between free-slip boundaries , 1982 .

[15]  Shang‐keng Ma.  Modern Theory of Critical Phenomena , 1976 .

[16]  David Terman,et al.  Propagation Phenomena in a Bistable Reaction-Diffusion System , 1982 .

[17]  K. Toko,et al.  Dissipative structure in the Characea: Spatial pattern of proton flux as a dissipative structure in characean cells , 1985 .

[18]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[19]  W. J. Lucas,et al.  The Formation of Alkaline and Acid Regions at the Surface of Chara corallina Cells , 1973 .

[20]  Shinji Koga,et al.  Localized Patterns in Reaction-Diffusion Systems , 1980 .

[21]  J. K. Barr,et al.  Localization of Hydrogen Ion and Chloride Ion Fluxes in Nitella , 1969, The Journal of general physiology.

[22]  J. Keener,et al.  Spiral waves in the Belousov-Zhabotinskii reaction , 1986 .

[23]  J. Murray.  A Pre-pattern formation mechanism for animal coat markings , 1981 .

[24]  N. Rashevsky,et al.  An approach to the mathematical biophysics of biological self-regulation and of cell polarity , 1940 .