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Abstract:
We solve the dynamics of Hopfield-type neural networks which
store sequences of patterns, close to saturation. The asymmetry of
the interaction matrix in such models leads to violation of
detailed balance, ruling out an equilibrium statistical mechanical
analysis. Using generating functional methods we derive exact
closed equations for dynamical order parameters, viz. the sequence
overlap and correlation and response functions, in the limit of an
infinite system size. We calculate the time translation invariant
solutions of these equations, describing stationary limit--cycles,
which leads to a phase diagram. The effective retarded
self-interaction usually appearing in symmetric models is here
found to vanish, which causes a significantly enlarged storage
capacity of alphac=0.269, compared to alphac=0.139 for Hopfield
networks storing static patterns. Our results are tested against
extensive computer simulations and excellent agreement is
found.
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