We study the dynamics of excitable integrate-and-fire neurons in a small-world network. At low densities $p$ of directed random connections, a localized transient stimulus results either in self-sustained persistent activity or in a brief transient followed by failure. Averages over the quenched ensemble reveal that the probability of failure changes from $0$ to $1$ over a narrow range in $p$; this failure transition can be described analytically through an extension of an existing mean-field result. Exceedingly long transients emerge at higher densities $p$; their activity patterns are disordered, in contrast to the mostly periodic persistent patterns observed at low $p$. The times at which such patterns die out follow a stretched-exponential distribution, which depends sensitively on the propagation velocity of the excitation.