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Neuronal network dynamics depends on network structure. In this paper westudy how network topology underpins the emergence of different dynamicalbehaviors in neuronal networks. In particular, we consider neuronal networkdynamics on Erd\H{o}s-R\'enyi (ER) networks, regular random (RR) networks, ringlattices, and all-to-all networks. We solve analytically a neuronal networkmodel with stochastic binary-state neurons in all the network topologies,except ring lattices. Given that apart from network structure, all four modelsare equivalent, this allows us to understand the role of network structure inneuronal network dynamics. Whilst ER and RR networks are characterized bysimilar phase diagrams, we find strikingly different phase diagrams in theall-to-all network. Neuronal network dynamics is not only different withincertain parameter ranges, but it also undergoes different bifurcations (with aricher repertoire of bifurcations in ER and RR compared to all-to-allnetworks). This suggests that local heterogeneity in the ratio betweenexcitation 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 levelscompared to RR networks, presumably due to the degree heterogeneity in ERnetworks that is absent in RR networks. Finally, a comparison between networkoscillations in RR networks and ring lattices shows the importance ofsmall-world properties in sustaining stable network oscillations.

Distinct dynamical behavior in Erd\H{o}s-R\'enyi networks, regular random networks, ring lattices, and all-to-all neuronal networks