The usual real-valued artificial neural networks have been applied to various fields such as telecommunications, robotics, bioinformatics, image processing and speech recognition, in which complex numbers (two dimensions) are often used with the Fourier transformation. This indicates the usefulness of complex-valued neural networks whose input and output signals and parameters such as weights and thresholds are all complex numbers, which are an extension of the usual real-valued neural networks. In addition, in the human brain, an action potential may have different pulse patterns, and the distance between pulses may be different. This suggests that it is appropriate to introduce complex numbers representing phase and amplitude into neural networks. Aizenberg, Ivaskiv, Pospelov and Hudiakov (1971) (former Soviet Union) proposed a complex-valued neuron model for the first time, and although it was only available in Russian literature, their work can now be read in English (Aizenberg, Aizenberg & Vandewalle, 2000). Prior to that time, most researchers other than Russians had assumed that the first persons to propose a complex-valued neuron were Widrow, McCool and Ball (1975). Interest in the field of neural networks started to grow around 1990, and various types of complex- valued neural network models were subsequently proposed. Since then, their characteristics have been researched, making it possible to solve some problems which could not be solved with the real-valued neuron, and to solve many complicated problems more simply and efficiently.
