Statistical mechanical study of partial annealing of a neural network model

We study a neural network model in which both neurons and synaptic interactions evolve in time simultaneously. The time evolution of synaptic interactions is described by a Langevin equation including a Hebbian learning term with the learning coefficient ε, and a bias term which is the interaction of the Hopfield model. We assume that synaptic interactions change is much slower than neurons and we study the stationary states of synaptic interactions by the replica method. We draw phase diagrams taking into account the stability of solutions, and find that the temperature region in which the Hopfield attractor is stable increases as the learning coefficient increases. Theoretical results are confirmed by the direct numerical integration of the Langevin equation. Further, we study the characteristics of the resultant synaptic interactions by partial annealing in the parameter region where the Hopfield and the mixed states exist. We find two kinds of interactions, one of which has the Hopfield attractor and the other has the mixed state attractor. Each interaction is characterized mainly by the eigenvector belonging to the largest eigenvalue of the interaction as a matrix.