Neuronal network dynamics depends on network structure. 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. This mean-field approximation is a common approach in statistical physics. In this paper we show that such approximation can be invalid . We solve analytically a neuronal network model with  binary-state neurons in both random and all-to-all networks . We find strikingly different phase diagrams corresponding to each networkstructure . Neuronal network dynamics is not only different within certain parameter ranges, but it also undergoes different bifurcations . Our results therefore suggest cautiousness when using mean-field models based on all-to-all network topologies to represent random networks .
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{os-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 .
R
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 an
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
R
network, where every neuron is connected to all other neurons. This mean-field approximation is a common approach in statistical physics. In this paper we show that such approximation can be invalid
networks
A
stochastic
R
both random and all-to-all networks . We
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
R
corresponding to each networkstructure
in the all-to-all network
R
. Our results therefore suggest cautiousness when using mean-field models based on
(with a richer repertoire of bifurcations in ER and RR compared to
R
network topologies to represent random networks
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