Four Models of Hopfield-Type Octonion Neural Networks and Their Existing Conditions of Energy Functions

Recently, models of neural networks in the real domain have been extended into the high dimensional domain such as the complex number 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 Hopfield-type neural networks into the octonion domain and discuss their dynamics. The octonions represent a particular extension of the quaternions which also represent a particular extension of the complex numbers and have 7 imaginary parts. They are non-commutative and non-associative on multiplication and do not belong to Clifford algebra due to the latter fact. With this in mind we propose four models of octonion Hopfield-type neural networks. We derive existence conditions of an energy function and construct energy function for each model.