Theoretical and computational aspects of quaternionic multivalued Hopfield neural networks

Multivalued quaternionic Hopfield neural networks (MV-QHNN) extend the widely known Hopfield network from {-1, +1} to unit quaternions. The first MV-QHNN model, introduced by Isokawa and collaborators, uses a multivalued signum function based on the phase-angle representation of a quaternion. In this paper, we point out that the quaternionic multivalued signum function proposed initially by Isokawa et al. is numerically unstable. As a consequence, unexpected dynamics may be observed in computer simulations of the MV-QHNN. Also, we investigate a modified MV-QHNN, introduced recently by Minemoto et al., which overcomes the aforementioned limitations. Precisely, we observe that the network of Minemoto et al. is numerically stable. Furthermore, under the usual conditions on the synaptic weight matrix, we remark that it always settles to equilibrium if the phase-angle of a neuron are updated simultaneously.