Title: Distinct dynamical behavior in random and all-to-all neuronal networks,
Abstract: 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 an 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 network structure. 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.