ABSTRACT
Spatially extended excitable media are a sub-class within the more general framework of reaction-diffusion systems. As suggested by the name, reaction-diffusion models provide a natural description for the dynamics of a chemical system: the reagents are reacting with each other and the reactants as well as the products being transported through diffusion. Over time, these models have been used to analyze a wide class of spatially extended systems in chemistry, physics, biology, and ecology [Cross and Greenside 2009; Cross and Hohenberg 1993; Murray 2002]. Under coarse-graining, these systems are modeled using partial differential equations (PDEs) having the form:
∂q(x, t)
∂t = D∇2q(x, t) +R(q) (2.1)
where each component of q(x, t) represents one of the several variables describing the state of a system (e.g., concentration of a chemical species in case of chemical reactions), D is the diffusion matrix, and R(q) represents the different (non-linear) reaction terms. Thus, the term on the right-hand side of Eq. (2.1) represents the transport of the different components while the second term contains details of all the local dynamical processes operating on each of the components including production, decay, etc.
