This monograph addresses the intertwined mathematical, neurological, and cognitive mysteries of the brain. It first evaluates the mathematical performance limits of simple spiking neuron models that both learn and later recognize complex spike excitation patterns in less than one second without using training signals unique to each pattern. Simulations validate these models, while theoretical expressions validate their simpler performance parameters. These single-neuron models are then qualitatively related to the training and performance of multi-layer neural networks that may have...