Models of clifford recurrent neural networks and their dynamics

Recently, models of neural networks in the real domain have been extended into the high dimensional domain such as the complex and quaternion domain, and several high-dimensional models have been proposed. These extensions are generalized by introducing Clifford algebra (geometric algebra). In this paper we extend conventional real-valued models of recurrent neural networks into the domain defined by Clifford algebra and discuss their dynamics. Since geometric product is non-commutative, some different models can be considered. We propose three models of fully connected recurrent neural networks, which are extensions of the real-valued Hopfield type neural networks to the domain defined by Clifford algebra. We also study dynamics of the proposed models from the point view of existence conditions of an energy function. We discuss existence conditions of an energy function for two classes of the Hopfield type Clifford neural networks.