In this work we study of the dynamics of large size random neural networks. Different methods have been developed to analyse their behavior, most of them rely on heuristic methods based on Gaussian assumptions regarding the fluctuations in the limit of infinite sizes. These approaches, however, do not justify the underlying assumptions systematically. Furthermore, they are incapable of deriving in general the stability of the derived mean field equations, and they are not amenable to analysis of finite size corrections. Here we present a systematic method based on Path Integrals which overcomes these limitations. We apply the method to a large non-linear rate based neural network with random asymmetric connectivity matrix. We derive the Dynamic Mean Field (DMF) equations for the system, and derive the Lyapunov exponent of the system. Although the main results are well known, here for the first time, we calculate the spectrum of fluctuations around the mean field equations from which we derive the general stability conditions for the DMF states. The methods presented here, can be applied to neural networks with more complex dynamics and architectures. In addition, the theory can be used to compute systematic finite size corrections to the mean field equations.
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