Wave Propagation in Spatially Distributed Excitable Media

Consider wave propagation through regions of no excitability (gaps) in an otherwise excitable medium. Propagation in the gaps takes place via simple diffusion. We extend the geometric method for a one-gap system developed by Lewis and Keener to the case of two and three gaps, and we obtain conditions for successful wave propagation and failure. We show that, like the one-gap system, steady-state multiplicity for the case of two gaps arises via a limit point bifurcation. We also demonstrate that in some cases the presence of a large number of gaps promotes wavefront propagation.

[1]  J. Solà-Morales,et al.  On the number and indices of equilibria in a space-dependent bistable parabolic equation , 2001 .

[2]  Alison L. Kay,et al.  On the persistence of spatiotemporal oscillations generated by invasion , 1999 .

[3]  James P. Keener,et al.  Mathematical physiology , 1998 .

[4]  Raymond Kapral,et al.  Asynchronous algorithm for integration of reaction-diffusion equations for inhomogeneous excitable media. , 2000, Chaos.

[5]  D. Aronson,et al.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation , 1975 .

[6]  J. McLeod,et al.  The approach of solutions of nonlinear diffusion equations to travelling wave solutions , 1975 .

[7]  James Sneyd,et al.  On the Propagation of Calcium Waves in an Inhomogeneous Medium , 1997, SIAM J. Appl. Math..

[8]  James P. Keener,et al.  Wave-Block in Excitable Media Due to Regions of Depressed Excitability , 2000, SIAM J. Appl. Math..

[9]  M. Mimura,et al.  Wave-blocking phenomena in bistable reaction-diffusion systems , 1989 .

[10]  P. Grindrod,et al.  A model of a myelinated nerve axon: threshold behaviour and propagation , 1985, Journal of mathematical biology.

[11]  Autowave tunneling through a non-excitable area of active media. , 1997, General physiology and biophysics.

[12]  J. Pauwelussen.  Nerve impulse propagation in a branching nerve system: A simple model , 1981 .

[13]  Jack Xin,et al.  Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media , 1993 .

[14]  James P. Keener,et al.  Homogenization and propagation in the bistable equation , 2000 .

[15]  P. Grindrod.  A model for a myelinated nerve axon , 1985 .

[16]  Jack Xin,et al.  Quenching and propagation of bistable reaction-diffusion fronts in multidimensional periodic media , 1995 .

[17]  Alison L. Kay,et al.  Spatial Noise Stabilizes Periodic Wave Patterns in Oscillatory Systems on Finite Domains , 2000, SIAM J. Appl. Math..