  Replica-mean-field models have been proposed to decipher the activity of
neural networks via a multiply-and-conquer approach. In this approach, one
considers limit networks made of infinitely many replicas with the same basic
neural structure as that of the network of interest, but exchanging spikes in a
randomized manner. The key point is that these replica-mean-field networks are
tractable versions that retain important features of the finite structure of
interest. To date, the replica framework has been discussed for first-order
models, whereby elementary replica constituents are single neurons with
independent Poisson inputs. Here, we extend this replica framework to allow
elementary replica constituents to be composite objects, namely, pairs of
neurons. As they include pairwise interactions, these pair-replica models
exhibit non-trivial dependencies in their stationary dynamics, which cannot be
captured by first-order replica models. Our contributions are two-fold: $(i)$
We analytically characterize the stationary dynamics of a pair of
intensity-based neurons with independent Poisson input. This analysis involves
the reduction of a boundary-value problem related to a two-dimensional
transport equation to a system of Fredholm integral equations---a result of
independent interest. $(ii)$ We analyze the set of consistency equations
determining the full network dynamics of certain replica limits. These limits
are those for which replica constituents, be they single neurons or pairs of
neurons, form a partition of the network of interest. Both analyses are
numerically validated by computing input/output transfer functions for neuronal
pairs and by computing the correlation structure of certain pair-dominated
network dynamics.
