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

TL;DR

This paper analytically compares how different network topologies like ER, RR, ring lattices, and all-to-all influence neuronal dynamics, revealing topology-specific bifurcations, activity jumps, and the role of heterogeneity and small-world properties.

Contribution

It provides an analytical study of neuronal network dynamics across multiple topologies, highlighting the impact of network structure on bifurcations and activity patterns, especially the role of heterogeneity and small-world features.

Findings

ER and RR networks have similar phase diagrams

All-to-all networks show distinct bifurcation patterns

ER networks exhibit activity jumps at lower noise levels

Abstract

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.