Attractor modulation and proliferation in (1+infinity)-dimensional neural networks

We extend a recently introduced class of exactly solvable models for recurr ent neural networks with competition between one-dimensional nearest-neighb our and infinite-range information processing. We increase the potential fo r further frustration and competition in these models, as well as their bio logical relevance, by adding next-nearest-neighbour couplings, and we allow for modulation of the attractors so that we can interpolate continuously b etween situations with different numbers of stored patterns. Our models are solved by combining mean-field acid random-field techniques. They exhibit increasingly complex phase diagrams with novel phases, separated by multipl e first- and second-order transitions (dynamical and thermodynamic ones), a nd, upon modulating the attractor strengths, non-trivial scenarios of phase diagram deformation. Our predictions are in excellent agreement with numer ical simulations.