Evolving recurrent neural networks represent a natural model of computation beyond the Turing limits. Here, we consider evolving recurrent neural networks working on infinite input streams. The expressive power of these networks is related to their attractor dynamics and is measured by the topological complexity of their underlying neural ω-languages. In this context, the deterministic and non-deterministic evolving neural networks recognize the (boldface) topological classes of BC(Π^0_2) and Σ^1_1 ω-languages, respectively. These results can actually be significantly refined: the deterministic and nondeterministic evolving networks which employ α ∈ 2^ω as sole binary evolving weight recognize the (lightface) relativized topological classes of BC(Π^0_2)(α) and Σ^1_1 (α) ω-languages, respectively. As a consequence, a proper hierarchy of classes of evolving neural nets, based on the complexity of their underlying evolving weights, can be obtained. The hierarchy contains chains of length ω_1 as well as uncountable antichains.