Wolfram's celebrated three-input Cellular Automata is further developed and extended from the perspective of neural networks. A single explicit formula involving two nested absolute-value functions and eight adjustable parameters called synaptic weights, is presented. Such a neuron is proved to be universal by specifying the synaptic weights of all 256 local rules. Applying the nonlinear dynamics concepts developed from Part I of this multipart series of papers, we present the rational for partitioning the entire set of 256 local rules into 16 distinct gene families, each composed of 16 gene siblings. Such a partitioning allows us to explain, if not predict, the pattern features generated from each local rule. Finally, these 16 gene families of Cellular Automata rules are encoded onto a new compact and insightful representation called the "double-helix torus."
