  We investigate temporal coherence and spatial synchronization on small-world
networks consisting of noisy Terman-Wang (TW) excitable neurons in dependence
on two types of time-delayed coupling: $\{x_j(t-\tau)-x_i (t)\}$ and
$\{x_j(t-\tau)-x_i(t-\tau)\}$. For the former case, we show that time delay in
the coupling can dramatically enhance temporal coherence and spatial synchrony
of the noise-induced spike trains. In addition, if the delay time $\tau$ is
tuned to nearly match the intrinsic spike period of the neuronal network, the
system dynamics reaches a most ordered state, which is both periodic in time
and nearly synchronized in space, demonstrating an interesting resonance
phenomenon with delay. For the latter case, however, we can not achieve a
similar spatiotemporal ordered state, but the neuronal dynamics exhibits
interesting synchronization transition with time delay from zigzag fronts of
excitations to dynamic clustering anti-phase synchronization (APS), and further
to clustered chimera states which have spatially distributed anti-phase
coherence separated by incoherence. Furthermore, we also show how these
findings are influenced by the change of the noise intensity and the rewiring
probability. Finally, qualitative analysis is given to illustrate the numerical
results.
