Commutative quaternion and multistate Hopfield neural networks

This paper explores two types of multistate Hopfield neural networks, based on commutative quaternions that are similar to Hamilton's quaternions but with commutative multiplication. In one type of the networks, the state of a neuron is represented by two kinds of phases and one real number. The other type of the networks adopts the decomposed form of commutative quaternion, i.e., the state of a neuron consists of a combination of two complex values. We have investigated the stabilities of these networks, i.e., the energies monotonically decreases with respect to the changes of the network states.