We provide a decidable hierarchical classification of first-order recurrent neural networks made up of McCulloch and Pitts cells. This classification is achieved by proving an equivalence result between such neural networks and deterministic Büuchi automata, and then translating the Wadge classification theory from the abstract machine to the neural network context. The obtained hierarchy of neural networks is proved to have width 2 and height ω+1, and a decidability procedure of this hierarchy is provided. Notably, this classification is shown to be intimately related to the attractive properties of the considered networks.