Phase diagram and storage capacity of sequence processing neural networks

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, namely the sequence overlap and correlation and response functions, in the thermodynamic limit. 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 alpha(c) similar to 0.269, compared with alpha(c) similar to 0.139 for Hopfield networks storing static patterns. Our results are tested against extensive computer simulations and excellent agreement is found.