Characterizing metastable neural dynamics in finite-size spiking networks remains a daunting challenge. We have recently proposed to address this challenge in the simplifying 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 replica. 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. Unfortunately, realistic metastable dynamics typically unfold in networks with mixed inhibition and excitation. To address this point , we extend the RMF computational framework to neural networks modeled by point processes with exponential stochastic intensities  . Within this framework , 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 solutions to a set of delayed differential equations under certain regularity conditions that any physical solutions shall satisfy. We solve this original singular-perturbation problem by adapting the resolvent formalism to our delayed setting . 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 perform the analysis of the stochastic bifurcationof a bistable neural network in terms of static phase transition of its RMF limits. In turn, we expect to leverage the static picture of RMF limit to infer purely dynamical features of metastable , finite-size networks, such as transition rates .
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 replica. 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 solutions to 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 also 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 .
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