Before revision:  It is often assumed that neurons are connected at random when their actual connectivity structure is unknown. Such models are then often approximated by replacing the random network by ans-R\'enyi (ER) networks, regular random (RR) networks, ring lattices, and } all-to-all network, where every neuron is connected to all other neurons.
After revision:  In this paper we study how network topology underpins the emergence of different dynamical behaviors in neuronal networks. In particular, we consider neuronal network dynamics on Erd\H{os-R\'enyi (ER) networks, regular random (RR) networks, ring lattices, and } all-to-all network, where every neuron is connected to all other neurons.