Associative memory in quaternionic Hopfield neural network

Associative memory networks based on quaternionic Hopfield neural network are investigated in this paper. These networks are composed of quaternionic neurons, and input, output, threshold, and connection weights are represented in quaternions, which is a class of hypercomplex number systems. The energy function of the network and the Hebbian rule for embedding patterns are introduced. The stable states and their basins are explored for the networks with three neurons and four neurons. It is clarified that there exist at most 16 stable states, called multiplet components, as the degenerated stored patterns, and each of these states has its basin in the quaternionic networks.