  An associative memory model and a neural network model with a Mexican-hat
type interaction are the two most typical attractor networks used in the
artificial neural network models. The associative memory model has discretely
distributed fixed-point attractors, and achieves a discrete information
representation. On the other hand, a neural network model with a Mexican-hat
type interaction uses a line attractor to achieves a continuous information
representation, which can be seen in the working memory in the prefrontal
cortex and columnar activity in the visual cortex. In the present study, we
propose a neural network model that achieves discrete and continuous
information representation. We use a statistical-mechanical analysis to find
that a localized retrieval phase exists in the proposed model, where the memory
pattern is retrieved in the localized subpopulation of the network. In the
localized retrieval phase, the discrete and continuous information
representation is achieved by using the orthogonality of the memory patterns
and the neutral stability of fixed points along the positions of the localized
retrieval. The obtained phase diagram suggests that the antiferromagnetic
interaction and the external field are important for generating the localized
retrieval phase.
