Fixed points incompatibility in neural networks with local interactions

Two conditions of incompatibility between fixed points are proved, which hold for a wide class of discrete-time, discrete-state, nonlinear neural networks with local interactions and no self-feedback. These conditions, which can be checked easily by inspection, can be stated briefly as follows: Let the network be defined on a two-dimensional array. A pair of states cannot both be fixed points of the network dynamics if: (1) in the neighborhood of a different component, there is no other different component; and (2) in the neighborhood of an equal component, there is no other equal component