  This article presents a biological neural network model driven by
inhomogeneous Poisson processes accounting for the intrinsic randomness of
synapses. The main novelty is the introduction of local interactions: each
firing neuron triggers an instantaneous increase in electric potential to a
fixed number of randomly chosen neurons. We prove that, as the number of
neurons approaches infinity, the finite network converges to a nonlinear
meanfield process characterised by a jump-type stochastic differential
equation. We show that this process displays a phase transition: the activity
of a typical neuron in the infinite network either rapidly dies out, or
persists forever, depending on the global parameters describing the intensity
of interconnection. This provides a way to understand the emergence of
persistent activity triggered by weak input signals in large neural networks.
