  Mathematical models of biological neural networks are associated to a rich
and complex class of stochastic processes. In this paper, we consider a simple
{\em plastic} neural network whose {\em connectivity/synaptic strength}
$(W(t))$ depends on a set of activity-dependent processes to model {\em
synaptic plasticity}, a well-studied mechanism from neuroscience. A general
class of stochastic models has been introduced in
\cite{robert_mathematical_2020} to study the stochastic process $(W(t))$. It
has been observed experimentally that its dynamics occur on much slower
timescale than that of the main cellular processes. The purpose of this paper
is to establish limit theorems for the distribution of $(W(t))$ with respect to
the fast timescale of neuronal processes.
  The central result of the paper is an averaging principle for the stochastic
process $(W(t))$. Mathematically, the key variable is the point process whose
jumps occur at the instants of neuronal spikes. A thorough analysis of several
of its unbounded additive functionals is achieved in the slow-fast limit.
Additionally, technical results on interacting shot-noise processes are
developed and used in the general proof of the averaging principle.
