The neuroidal tabula rasa (NTR) as a hypothetical device that is capable of performing tasks related to cognitive processes in the brain was introduced by L. G. Valiant in 1994. Neuroidal nets represent a computational model of the NTR. Their basic computational element is a kind of a programmable neuron called neuroid. Essentially, it is a combination of a standard threshold element with a mechanism that allows modification of the neuroid's computational behaviour. This is done by changing its state and the settings of its weights and of threshold in the course of computation. The computational power of an NTR crucially depends both on the functional properties of the underlying update mechanism that allows changing of neuroidal parameters and on the universe of allowable weights. We will define instances of neuroids for which the computational power of the respective finite-size NTR ranges from that of finite automata, through Turing machines, up to that of a certain restricted type of BSS machines that possess super-Turing computational power. The latter two results are surprising since similar results were known to hold only for certain kinds of analog neural networks.