Stationary pattern formation in a discrete excitable system with strong inhibitory coupling.

We study a discrete model described by coupled excitable elements following the monostable FitzHugh-Nagumo equations. Our model has a weakly coupled activator and a strongly coupled inhibitor. For two-coupled excitable elements, we show that the trivial state always exists stably, while nontrivial stable states appear depending on the coupling strengths. In a one-dimensional array, only the elements near the initial condition step remain at nontrivial states. We discuss stationary pattern formation in a one-dimensional array and a two-dimensional lattice using the analytical results of a two-coupled system.

[1]  Thomas Erneux,et al.  Mechanisms for propagation failure in discrete reaction-diffusion systems , 1992 .

[2]  Hidetoshi Miike,et al.  Realizing visual functions with the reaction-diffusion mechanism , 2003 .

[3]  Daishin Ueyama,et al.  Chaotic Pulses for Discrete Reaction Diffusion Systems , 2005, SIAM J. Appl. Dyn. Syst..

[4]  L. Bonilla,et al.  Dislocations in cubic crystals described by discrete models , 2007, 0805.1221.

[5]  Luis L. Bonilla,et al.  Pulse Propagation in Discrete Systems of Coupled Excitable Cells , 2003, SIAM J. Appl. Math..

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

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

[8]  Etsuro Yokoyama,et al.  Numerical experiments on the Turing instability in the Oregonator model , 1997 .

[9]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[10]  J. E. Pearson.  Complex Patterns in a Simple System , 1993, Science.

[11]  S Kinoshita,et al.  Method for determining a coupling function in coupled oscillators with application to Belousov-Zhabotinsky oscillators. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Irving R Epstein,et al.  Design and control of patterns in reaction-diffusion systems. , 2008, Chaos.

[13]  James P. Keener,et al.  Propagation and its failure in coupled systems of discrete excitable cells , 1987 .

[14]  A Carpio,et al.  Effects of disorder on the wave front depinning transition in spatially discrete systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  H. Swinney,et al.  Transition from a uniform state to hexagonal and striped Turing patterns , 1991, Nature.

[16]  L. Bonilla,et al.  Edge dislocations in crystal structures considered as traveling waves in discrete models. , 2003, Physical review letters.