Brain waves are one of the most important features of brain dynamics. Waves at a large scale are a direct consequence of the microscopic synchronization of neuronal populations. For this reason, simple models of coupled oscillators are widely used tools to analyse whole-brain dynamics, both in computational and data-driven studies. Experimental evidence suggests that, at this description level, large neuronal regions work close to the edge-of-synchronization, which according to theoretical models would confer optimal capabilities for information processing and computation. However, how this critical state emerges is yet unclear. Different hypotheses have pointed out the hierarchical-modular structural organization of the brain as a relevant piece of the puzzle: on one hand, this structure can produce Griffiths' phases, i.e., large regions in parameter space that present critical-like properties. On the other hand, a core-periphery hierarchical structure has been identified as an optimal structure to balance segregation and integration of information, which could enhance the network computational capabilities. The topological structure is deeply entwined with oscillatory behaviour: in low dimensions, oscillator models display topological defects which manifest as time-dependent, rich activity patterns. In this context, we address two different issues that fill a gap in the literature: first, we study the behaviour of non-linear excitable oscillators. Neuronal tissue is known to be excitable, an ingredient that strikingly changes the macroscopic collective behaviour of the system, leading to complex dynamics in realistic connectivities. Second, we provide insights linking oscillator frustration and the segregation-and-integration balance view, by studying synchronization models with core-periphery synthetic networks for the first time. We present a unifying view, comparing the differences among different dynamical models and structural topologies, thus paving the way for future theoretical and data-driven studies based on oscillator models.