Finite-size effects in separable recurrent neural networks

We perform a systematic analytical study of finite-size effects in separable recurrent neural network models with sequential dynamics, away from saturation. We find two types of finite-size effects: thermal fluctuations, and disorder-induced `frozen' corrections to the mean-field laws. The finite-size effects are described by equations that correspond to a time-dependent Ornstein-Uhlenbeck process. We show how the theory can be used to understand and quantify various finite-size phenomena in recurrent neural networks, with and without detailed balance.