Global dynamics on one-dimensional excitable media

The FitzHugh--Nagumo system has been studied extensively for several decades. It has been shown numerically that pulses are generated to propagate and then some of the pulses are annihilated after collision. For the mathematical understanding of these complicated dynamics, we investigate the global dynamics of a one-dimensional free boundary problem in the singular limit of a FitzHugh--Nagumo type reaction-diffusion system. By introducing the notion of symbolic dynamics, we show that the asymptotic behaviors of solutions are classified into three categories: (i) the solution converges uniformly to the resting state; (ii) the solution converges to a series of traveling pulses propagating in either the same direction or both directions; and (iii) the solution converges to a propagating wave consisting of multiple traveling pulses and two traveling fronts propagating in both directions.