  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, and a bias term which is the interactions of the Hopfield model. We
assume that synaptic interactions change much slower than neurons and study the
stationary states of synaptic interactions by the replica method. We find that
the order of the phase transition changes from the second to the first and that
the existence regions of the Hopfield attractor and mixed states increase as
the coefficient of the learning term increases. We also study the AT 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 and find that the stability of the
attractor which emerges after partial annealing is enhanced and those of the
coexistent attractors are reduced.
