  Neuronal network dynamics depends on network structure. 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{o}s-R\'enyi (ER) networks, regular random (RR) networks, ring
lattices, and all-to-all networks. We solve analytically a neuronal network
model with stochastic binary-state neurons in all the network topologies,
except ring lattices. Given that apart from network structure, all four models
are equivalent, this allows us to understand the role of network structure in
neuronal network dynamics. Whilst ER and RR networks are characterized by
similar phase diagrams, we find strikingly different phase diagrams in the
all-to-all network. Neuronal network dynamics is not only different within
certain parameter ranges, but it also undergoes different bifurcations (with a
richer repertoire of bifurcations in ER and RR compared to all-to-all
networks). This suggests that local heterogeneity in the ratio between
excitation and inhibition plays a crucial role on emergent dynamics.
Furthermore, we also observe one subtle discrepancy between ER and RR networks,
namely ER networks undergo a neuronal activity jump at lower noise levels
compared to RR networks, presumably due to the degree heterogeneity in ER
networks that is absent in RR networks. Finally, a comparison between network
oscillations in RR networks and ring lattices shows the importance of
small-world properties in sustaining stable network oscillations.
