We provide a refined hierarchical classification of first-order recurrent neural networks made up of McCulloch and Pitts cells. The classification is achieved by first proving the equivalence between the expressive powers of such neural networks and Muller automata, and then translating the Wadge classification theory from the automata-theoretic to the neural network context. The obtained hierarchical classification of neural networks consists of a decidable pre-well ordering of width 2 and height omega(omega), and a decidability procedure of this hierarchy is provided. Notably, tins classification is shown to be intimately related to the attractive properties of the networks, and hence provides a new refined measurement of the computational power of these networks in terms of their attractive behaviours.