Metastable spiking networks in the replica-mean-field limit

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.Author summary: Electrophysiological recordings show that neural circuits process information by dynamically switching between quasi-stationary states, whereby neurons exhibit sustained, stereotypic activity. Such alternations of stable and unstable bouts of activity is referred to as neural metastability. The observation of metastability supports the view that neural computations are implemented by sequences of input-dependent transitions between quasi-stationary states. Therefore, understanding neural computation conceptually hinges on characterizing metastable dynamics in biophysically relevant network models. Modeling-wise, metastable dynamics can only emerge in finite-size neural networks, for which irreducible neural variability controls the rate of transition between various quasi-stationary states. Unfortunately, the quantitative analysis of neural networks typically requires simplifying assumptions that effectively erase finite-size induced metastability. To remedy this point, we apply a “multiply-and-conquer” approach that consider neural networks made of infinitely many copies of the original network of interest. This setting allows us to introduce simplifying assumptions that render the dynamics of certain biophysically relevant networks tractable, while remaining predictive of metastability in the finite-size original network.