Velocity of pulses in discrete excitable systems

• Published in: Nonlinear analysis: real world applications 2012-12, Vol.13 (6), p.2794-2803
• Main Authors: Arana, J.I., Bonilla, L.L.
• Format: Article
• Language: English
• Subjects: Approximation; Asymptotic expansions; Dimensional measurements; Discrete FitzHugh–Nagumo system; Excitable systems; Excitation; Matched asymptotic expansions; Mathematical analysis; Mathematical models; Nonlinear pulses and wave fronts; Nonlinearity; Wave fronts
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2012.03.016

The pulse solution of the spatially discrete excitable FitzHugh–Nagumo (FHN) system is approximately constructed using matched asymptotic expansions in the limit of large time scale separation (as measured by a small dimensionless parameter ϵ). The pulse profile typically consists of slowly varying regions of the excitatory variable separated by sharp wave fronts. In the FHN system, the velocity of a pulse is decided by the interaction between its leading and trailing fronts, but the leading order approximation gives only a fair result when compared with direct numerical solutions. A higher order approximation to the wave fronts comprising the FHN pulse is found. Our approximation provides an ϵ-dependent pulse velocity that approximates much better the velocity obtained from numerical solutions. As a result, the reconstruction of the FHN pulse using the improved wave fronts is much closer to the numerically obtained pulse.