  Neural computations emerge from myriads of neuronal interactions occurring in
intricate spiking networks. Due to the inherent complexity of neural models,
relating the spiking activity of a network to its structure requires
simplifying assumptions, such as considering models in the thermodynamic
mean-field limit. In the thermodynamic mean-field limit, an infinite number of
neurons interact via vanishingly small interactions, thereby erasing the finite
size of interactions. To better capture the finite-size effects of
interactions, we propose to analyze the activity of neural networks in the
replica-mean-field limit. Replica-mean-field models are made of infinitely many
replicas which interact according to the same basic structure as that of the
finite network of interest. Here, we analytically characterize the stationary
dynamics of an intensity-based neural network with spiking reset and
heterogeneous excitatory synapses in the replica-mean-field limit.
Specifically, we functionally characterize the stationary dynamics of these
limit networks via ordinary differential equations derived from the Poisson
Hypothesis of queuing theory. We then reduce this functional characterization
to a system of self-consistency equations specifying the stationary neuronal
firing rates. Of general applicability, our approach combines rate-conservation
principles from point-process theory and analytical considerations from
generating-function methods. We validate our approach by demonstrating
numerically that replica-mean-field models better capture the dynamics of
neural networks with large, sparse connections than their thermodynamic
counterparts. Finally, we explain that improved performance by analyzing the
neuronal rate-transfer functions, which saturate due to finite-size effects in
the replica-mean-field limit.
