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

Absbact.~ We examine an Ising spin system io which both spins and the interactions between them may evolve in time, although on d i s p t e timesales, such thaf the couplings change adiabatically. In thermal &yilibrium we find B novel application of the replica method. but for finite replica number. representing the ratio of the temperatures of the spm and interaction systems. Regimes where the motion of the coupliogs has non-uivial effects are found in addition to those where solely the stochasticiry of these interaction weights is signifianf and this issue is claselv related to the orders of the Innsitions between the various Dhases obsemd. Simulation results lend ruppon to the analpis.