Coupled dynamics of fast spins and slow interactions in neural networks and spin systems

The authors examine an Ising spin system in which both spins and the interactions between them may evolve in time, although on disparate timescales, such that the couplings change adiabatically. In thermal equilibrium they find a novel application of the replica method, but for finite replica number, representing the ratio of the temperatures of the spin and interaction systems. Regimes where the motion of the couplings has non-trivial effects are found in addition to those where solely the stochasticity of these interaction weights in significant, and this issue is closely related to the orders of the transitions between the various phases observed. Simulation results lend support to the analysis.