Order-parameter flow in the fully connected Hopfield model near saturation

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Abstract
We present an exact dynamical theory, valid on finite time scales, to describe the fully connected Hopfield model near saturation in terms of deterministic flow equations for order parameters. Two transparent assumptions allow us to perform a replica calculation of the distribution of intrinsic-noise components of the alignment fields. Numerical simulations support our assumptions and indicate that our equations describe the shape of the intrinsic-noise distribution and the macroscopic dynamics correctly in the region where replica symmetry is stable. In equilibrium our theory reproduces the saddle-point equations obtained in the thermodynamic analysis by Amit, Gutfreund, and Sompolinsky [Phys. Rev. A 32, 1007 (1985); Phys. Rev. Lett. 55, 1530 (1985)], the only difference being the absence in the present formalism of negative entropies at low temperatures.

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