The Artificial Grammar Learning (AGL) paradigm provides a means to study the nature of syntactic processing and implicit sequence learning. With mere exposure and without performance feedback, human beings implicitly acquire knowledge about the structural regularities implemented by complex rule systems. We investigate to which extent a generic cortical microcircuit model can support formally explicit symbolic computations, instantiated by the same grammars used in the human AGL literature and how a functional network emerges, in a self-organized manner, from exposure to this type of data. We use a concrete implementation of an input-driven recurrent network composed of noisy, spiking neurons, built according to the reservoir computing framework and dynamically shaped by a variety of synaptic and intrinsic plasticity mechanisms operating concomitantly. We show that, when shaped by plasticity, these models are capable of acquiring the structure of a simple grammar. When asked to judge string legality (in a manner similar to human subjects), the networks perform at a qualitatively comparable level.