title: Correspondence between discrete and continuous models of excitable media: trigger waves.
abstract: We present a theoretical framework for relating continuous partial differential equation (PDE) models of excitable media to discrete cellular automata (CA) models on a randomized lattice. These relations establish a quantitative link between the CA model and the specific physical system under study. We derive expressions for the CA model's plane wave speed, critical curvature, and effective diffusion constant in terms of the model's internal parameters (the interaction radius, excitation threshold, and time step). We then equate these expressions to the corresponding quantities obtained from solution of the PDEs (for a fixed excitability). This yields a set of coupled equations with a unique solution for the required CA parameter values. Here we restrict our analysis to "trigger" wave solutions obtained in the limiting case of a two-dimensional excitable medium with no recovery processes. We tested the correspondence between our CA model and two PDE models (the FitzHugh-Nagumo medium and a medium with a "sawtooth" nonlinear reaction source) and found good agreement with the numerical solutions of the PDEs. Our results suggest that the behavior of trigger waves is actually controlled by a small number of parameters.
authors.affiliation: Division of Health Sciences and Technology, Harvard University-Massachusetts Institute of Technology, Cambridge, USA. yurich@atrium.mit.edu
authors.forename: Y B
authors.initials: YB
authors.lastname: Chernyak
authors.forename: A B
authors.initials: AB
authors.lastname: Feldman
authors.forename: R J
authors.initials: RJ
authors.lastname: Cohen
mesh_terms: D000818:Animals; D001145:Arrhythmias, Cardiac / Q000503:physiopathology; D055592:Biophysical Phenomena; D001703:Biophysics; D002468:Cell Physiological Phenomena; D004594:Electrophysiology; D006331:Heart Diseases / Q000503:physiopathology*; D006801:Humans; D008433:Mathematics; D008564:Membrane Potentials; D008954:Models, Biological; D009206:Myocardium; D009469:Neuromuscular Junction / Q000502:physiology*; D017711:Nonlinear Dynamics; D016276:Ventricular Function