  In this work we determine a process-level Large Deviation Principle (LDP) for
a model of interacting neurons indexed by a lattice $\mathbb{Z}^d$. The neurons
are subject to noise, which is modelled as a correlated martingale. The
probability law governing the noise is strictly stationary, and we are
therefore able to find a LDP for the probability laws $\Pi^n$ governing the
stationary empirical measure $\hat{\mu}^n$ generated by the neurons in a cube
of length $(2n+1)$. We use this LDP to determine an LDP for the neural network
model. The connection weights between the neurons evolve according to a
learning rule / neuronal plasticity, and these results are adaptable to a large
variety of neural network models. This LDP is of great use in the mathematical
modelling of neural networks, because it allows a quantification of the
likelihood of the system deviating from its limit, and also a determination of
which direction the system is likely to deviate. The work is also of interest
because there are nontrivial correlations between the neurons even in the
asymptotic limit, thereby presenting itself as a generalisation of traditional
mean-field models.
