Spatiotemporal dynamics on small-world neuronal networks: The roles of two types of time-delayed coupling

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: {xj(t−τ)−xi(t)} and {xj(t−τ)−xi(t−τ)}. 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 τ 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 cannot achieve a similar spatiotemporal ordered state, but the neuronal dynamics exhibits interesting synchronization transitions 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 of the small-world networks. Finally, qualitative analysis is given to illustrate the numerical results.