Wave propagation in discrete media

We have considered infinite systems of nonlinear ODEs on the one-dimensional integer lattice which describes the activity in an excitatorily coupled network of excitable cells. For an ideal nonlinearity, we calculated the speed of propagation of an activity and derived the condition for its existence. We also studied the existence and stability of the traveling wave solution and gave, in the simplest case, its explicit expression. We established that some unstable traveling waves lead to propagation with an enlarging profile defined by a front velocity and a wake velocity. We generalized some results to inhomogeneous medium and network with long range connections.