Complex-valued Hopfield neural netw

Complex-valued Hopfield neural networks (CHNNs) have been applied to various fields, although they tend to suffer from low noise tolerance. Rotational invariance, which is an inherent property of CHNNs, reduces noise tolerance. CHNNs have been used in attempts to extend by Clifford algebra, such as hy- perbolic and quaternion algebra. In this work, the directional multistate activation function is introduced to hyperbolic Hopfield neural networks (HHNNs), and the stability condition is also given. The proposed models do not have rotational invariance, and have fewer pseudomemories than CHNNs. The projection rule is also introduced to the HHNNs. In general, the diagonal elements should be eliminated from the obtained matrix for noise tolerance, although they cannot be eliminated in HHNNs. We investigate the relation between the diagonal elements and noise tolerance by computer simulations.