Abstract
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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