  We examine traveling-wave solutions on a regular ring network with one
additional long-range link that spans a distance d. The nodes obey the
FitzHugh-Nagumo kinetics in the excitable regime. The additional shortcut
induces a plethora of spatio-temporal behavior that is not present without it.
We describe the underlying mechanisms for different types of patterns:
propagation failure, period decreasing, bistability, shortcut blocking and
period multiplication. For this purpose, we investigate the dependence on d,
the network size, the coupling range in the original ring and the global
coupling strength and present a phase diagram summarizing the different
scenarios. Furthermore, we discuss the scaling behavior of the critical
distance by analytical means and address the connection to spatially continuous
excitable media.
