  Characterizing metastable neural dynamics in finite-size spiking networks
remains a daunting challenge. We propose to address this challenge in the
recently introduced replica-mean-field (RMF) limit. In this limit, networks are
made of infinitely many replicas of the finite network of interest, but with
randomized interactions across replicas. Such randomization renders certain
excitatory networks fully tractable at the cost of neglecting activity
correlations, but with explicit dependence on the finite size of the neural
constituents. However, metastable dynamics typically unfold in networks with
mixed inhibition and excitation. Here, we extend the RMF computational
framework to point-process-based neural network models with exponential
stochastic intensities, allowing for mixed excitation and inhibition. Within
this setting, we show that metastable finite-size networks admit multistable
RMF limits, which are fully characterized by stationary firing rates.
Technically, these stationary rates are determined as the solutions of a set of
delayed differential equations under certain regularity conditions that any
physical solutions shall satisfy. We solve this original problem by combining
the resolvent formalism and singular-perturbation theory. Importantly, we find
that these rates specify probabilistic pseudo-equilibria which accurately
capture the neural variability observed in the original finite-size network. We
also discuss the emergence of metastability as a stochastic bifurcation, which
can be interpreted as a static phase transition in the RMF limits. In turn, we
expect to leverage the static picture of RMF limits to infer purely dynamical
features of metastable finite-size networks, such as the transition rates
between pseudo-equilibria.
