  We use a generic model for type-I excitability (known as the SNIPER or SNIC
model) to describe the local dynamics of nodes within a network in the presence
of non-zero coupling delays. Utilising the method of the Master Stability
Function, we investigate the stability of the zero-lag synchronised dynamics of
the network nodes and its dependence on the two coupling parameters, namely the
coupling strength and delay time. Unlike in the FitzHugh-Nagumo model (a model
for type-II excitability), there are parameter ranges where the stability of
synchronisation depends on the coupling strength and delay time. One important
implication of these results is that there exist complex networks for which the
adding of inhibitory links in a small-world fashion may not only lead to a loss
of stable synchronisation, but may also restabilise synchronisation or
introduce multiple transitions between synchronisation and desynchronisation.
To underline the scope of our results, we show using the Stuart-Landau model
that such multiple transitions do not only occur in excitable systems, but also
in oscillatory ones.
