This research contribution instantiates a framework of a hybrid cascade neural network based on the application of a specific sort of neo-fuzzy elements and a new peculiar adaptive training rule. The main trait of the offered system is its competence to continue intensifying its cascades until the required accuracy is gained. A distinctive rapid training procedure is also covered for this case that offers the possibility to operate with non-stationary data streams in an attempt to provide online training of multiple parametric variables. A new training criterion is examined for handling non-stationary objects. Additionally, there is always an occasion to set up (increase) the inference order and the number of membership relations inside the extended neo-fuzzy neuron.