A simple format is disclosed and referred to as Elementary Network Description (END). The format can fully describe a large-scale neuronal model and embodiments of software or hardware engines to simulate such a model efficiently. The architecture of such neuromorphic engines is optimal for high-performance parallel processing of spiking networks with spike-timing dependent plasticity. Neuronal network and methods for operating neuronal networks comprise a plurality of units, where each unit has a memory and a plurality of doublets, each doublet being connected to a pair of the plurality of units. Execution of unit update rules for the plurality of units is order-independent and execution of doublet event rules for the plurality of doublets is order-independent.