How relevant is connectivity patterns for the mean-field dynamics of the balanced state?

Abstract: We develop a unified theory that encompasses the macroscopic mean-field dynamics of recurrent interactions of binary units within arbitrary network architectures in the balanced state. Using the martingale theory, we show the deterministic mean-field limit stays invariant if and only if in-degrees and out-degrees of the network connectivity matrix are statically equivalent (i.e., homogeneous networks). Additionally, we demonstrate a dynamic state in which a recurrent network of binary units with statistically inhomogeneous interactions, along with an asynchronous behavior, also exhibits collective nontrivial stochastic fluctuations in the thermodynamical limit.