  We here unify the field theoretical approach to neuronal networks with large
deviations theory. For a prototypical random recurrent network model with
continuous-valued units, we show that the effective action is identical to the
rate function and derive the latter using field theory. This rate function
takes the form of a Kullback-Leibler divergence which enables data-driven
inference of model parameters and calculation of fluctuations beyond mean-field
theory. Lastly, we expose a regime with fluctuation-induced transitions between
mean-field solutions.
