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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.
We review results on the scaling of the optimal path length in random networks with weighted links or nodes. In strong disorder we find that the length of the optimal path increases dramatically compared to the known small world result for the minimu
We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdos-Renyi networks, based on recursion equations for the shells around a reference node and for the paths originating from it. The resul
The stochastization of the Jacobi second equality of classical mechanics, by Gaussian white noises for the Lagrangian of a particle in an arbitrary field is considered. The quantum mechanical Hamilton operator similar to that in Euclidian quantum the
Recent progress in the development of efficient computational algorithms to price financial derivatives is summarized. A first algorithm is based on a path integral approach to option pricing, while a second algorithm makes use of a neural network pa
Recently it has been suggested that fermions whose hopping amplitude is quenched to extremely low values provide a convenient source of local disorder for lattice bosonic systems realized in current experiment on ultracold atoms. Here we investigate