The path-integral analysis of an associative memory model storing an infinite number of finite limit cycles

An exact solution of the transient dynamics of an associative memory model storing an infinite number of limit cycles with l finite steps is shown by means of the path-integral analysis. Assuming the Maxwell construction ansatz, we have succeeded in deriving the stationary state equations of the order parameters from the macroscopic recursive equations with respect to the finite-step sequence processing model which has retarded self-interactions. We have also derived the stationary state equations by means of the signal-to-noise analysis (SCSNA). The signal-to-noise analysis must assume that crosstalk noise of an input to spins obeys a Gaussian distribution. On the other hand, the path-integral method does not require such a Gaussian approximation of crosstalk noise. We have found that both the signal-to-noise analysis and the path-integral analysis give completely the same result with respect to the stationary state in the case where the dynamics is deterministic, when we assume the Maxwell construction ansatz. We have shown the dependence of the storage capacity (αc) on the number of patterns per one limit cycle (l). At l = 1, the storage capacity is αc = 0.138 as in the Hopfield model. The storage capacity monotonically increases with the number of steps, and converges to αc = 0.269 at l ≅ 10. The original properties of the finite-step sequence processing model appear as long as the number of steps of the limit cycle has order l = O(1)