This paper will firstly review a new theoretical basis for modelling neural Boolean networks by non-linear digital equations. With integer numbers, these digital equations are Heaviside Fixed Functions in the framework of the Threshold Logic. These can represent non-linear neurons which can be split very easily into a set of McCulloch and Pitts formal neurons with hidden neurons. It is demonstrated that any Boolean tables can be very easily represented by such neural networks where the weights are always either an activation weight +1 or an inhibition weight −1, with integer threshold. A fundamental problem in neural systems is the design of memory. This paper will present new memory neural systems based on hyperincursive neurons, that is neurons with multiple output states for the same input, instead of synaptic weights. Finally, a differential equation of membrane neural potential is used as a model of a brain, the incursive, that is the implicit recursive, computation of which gives rise to non-locality effects. ©1999 American Institute of Physics.