Semianalytical approach to criteria for ignition of excitation waves
Simitev, R. D.
Biktashev, V. N.
Physical Review E
American Physical Society
© 2015 American Physical Society. The version of record is available from the publisher via: doi: 10.1103/PhysRevE.92.042917
We consider the problem of ignition of propagating waves in one-dimensional bistable or excitable systems by an instantaneous spatially extended stimulus. Earlier we proposed a method [I. Idris and V. N. Biktashev, Phys. Rev. Lett. 101, 244101 (2008)PRLTAO0031-900710.1103/PhysRevLett.101.244101] for analytical description of the threshold conditions based on an approximation of the (center-)stable manifold of a certain critical solution. Here we generalize this method to address a wider class of excitable systems, such as multicomponent reaction-diffusion systems and systems with non-self-adjoint linearized operators, including systems with moving critical fronts and pulses. We also explore an extension of this method from a linear to a quadratic approximation of the (center-)stable manifold, resulting in some cases in a significant increase in accuracy. The applicability of the approach is demonstrated on five test problems ranging from archetypal examples such as the Zeldovich-Frank-Kamenetsky equation to near realistic examples such as the Beeler-Reuter model of cardiac excitation. While the method is analytical in nature, it is recognized that essential ingredients of the theory can be calculated explicitly only in exceptional cases, so we also describe methods suitable for calculating these ingredients numerically.
Engineering & Physical Sciences Research Council (EPSRC)
London Mathematical Society
McArthur Foundation (Nigeria)
Vol. 92, No 4, article 042917
Place of publication