  We study general phase structures of neural-network models that have Z(2)
local gauge symmetry. The Z(2) spin variable Si = \pm1 on the i-th site
describes a neuron state as in the Hopfield model, and the Z(2) gauge variable
Jij = \pm1 describes a state of the synaptic connection between j-th and i-th
neurons. The gauge symmetry allows for a self-coupling energy among Jij's such
as JijJjkJki, which describes reverberation of signals. Explicitly, we consider
the three models; (I) annealed model with full and partial connections of Jij,
(II) quenched model with full connections where Jij is treated as a slow
quenched variable, and (III) quenched three-dimensional lattice model with the
nearest-neighbor connections. By numerical simulations, we examine their phase
structures paying attention to the effect of reverberation term, and compare
them each other and with the annealed 3D lattice model which has been studied
beforehand. By noting the dependence of thermodynamic quantities upon the total
number of sites and the connectivity among sites, we obtain a coherent
interpretation to understand these results. Among other things, we find that
the Higgs phase of the annealed model is separated into two stable spin-glass
phases in the quenched cases (II) and (III).
