Propagation and Its Failure in Coupled Systems of Discrete Excitable Cells

We study propagation and its failure in systems of discrete coupled excitable cells. It is shown that propagation fails when coupling is weak, but succeeds if coupling is strong enough. We use a theorem of Moser on maps of the plane to establish the existence of an infinite variety of stable standing solutions when coupling is small. We use upper and lower solution techniques and perturbation analysis to show that propagation is successful when the coupling strength is large enough. These results are applied to the cubic FitzHugh–Nagumo dynamics as well as to the more realistic Beeler–Reuter model of action potentials in myocardial cells.