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

TL;DR

This paper provides an exact path-integral solution for the transient dynamics of an associative memory model with infinite limit cycles, comparing it with signal-to-noise analysis, and explores how storage capacity depends on cycle length.

Contribution

It derives stationary state equations for the model using path-integral analysis without Gaussian assumptions, confirming results with signal-to-noise analysis, and analyzes capacity dependence on cycle length.

Findings

Storage capacity increases with cycle length, approaching 0.269 at l≈10.

Path-integral and signal-to-noise analyses yield identical stationary state results.

Finite-step properties are maintained for small cycle lengths (l=O(1)).

Abstract

It is shown that an exact solution of the transient dynamics of an associative memory model storing an infinite number of limit cycles with l finite steps 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 the completely 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 storage capacity (alpha_c) on the number of patterns per one limit cycle (l). Storage capacity monotonously increases with the number of steps, and converges to alpha_c=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).