Complex-Valued Neurons with Phase-Dependent Activation Functions

  • Igor Aizenberg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6114)


In this paper, we observe two artificial neurons with complex-valued weights. There are a multi-valued neuron and a universal binary neuron. Both neurons have activation functions depending on the argument (phase) of the weighted sum. A multi-valued neuron may learn multiple-valued threshold functions. A universal binary neuron may learn arbitrary (not only linearly-separable) Boolean functions. It is shown that a multi-valued neuron with a periodic activation function may learn non-threshold functions by their projection to the space corresponding to the larger valued logic. A feedforward neural network with multi-valued neurons and its learning are also considered.


complex-valued neural networks derivative-free learning multi-valued neuron 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hirose, A.: Complex-Valued Neural Networks. Springer, Heidelberg (2006)zbMATHCrossRefGoogle Scholar
  2. 2.
    Aizenberg, I., Moraga, C.: Multilayer Feedforward Neural Network Based on Multi-Valued Neurons (MLMVN) and a Backpropagation Learning Algorithm. Soft Computing 11(2), 169–183 (2007)CrossRefGoogle Scholar
  3. 3.
    Aizenberg, I., Paliy, D., Zurada, J.M., Astola, J.: Blur Identification by Multilayer Neural Network based on Multi-Valued Neurons. IEEE Transactions on Neural Networks 19(5), 883–898 (2008)CrossRefGoogle Scholar
  4. 4.
    Aizenberg, I.: Solving the XOR and Parity n Problems Using a Single Universal Binary Neuron. Soft Computing 12(3), 215–222 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Aizenberg, I.: A Multi-Valued Neuron with a Periodic Activation Function. In: International Joint Conference on Computational Intelligence, Funchal-Madeira, Portugal, October 5-7, pp. 347–354 (2009)Google Scholar
  6. 6.
    Aizenberg, N.N., Ivaskiv, Y.L., Pospelov, D.A.: About one generalization of the threshold function Doklady Akademii Nauk SSSR (The Reports of the Academy of Sciences of the USSR), vol. 196(6), pp. 1287–1290 (1971) (in Russian)Google Scholar
  7. 7.
    Aizenberg, N.N., Aizenberg, I.N.: CNN Based on Multi-Valued Neuron as a Model of Associative Memory for Gray-Scale Images. In: The Second IEEE Int. Workshop on Cellular Neural Networks and their Applications, October 1992, pp. 36–41. Technical University Munich, Germany (1992)CrossRefGoogle Scholar
  8. 8.
    Aizenberg, N.N., Ivaskiv, Y.L.: Multiple-Valued Threshold Logic. Naukova Dumka Publisher House, Kiev (1977) (in Russian)Google Scholar
  9. 9.
    Aizenberg, I., Aizenberg, N., Vandewalle, J.: Multi-valued and universal binary neurons: theory, learning, applications. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  10. 10.
    Aizenberg, I., Moraga, C., Paliy, D.: A Feedforward Neural Network based on Multi-Valued Neurons. In: Reusch, B. (ed.) Computational Intelligence, Theory and Applications. Advances in Soft Computing, vol. XIV, pp. 599–612. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Aizenberg, I.N.: A Universal Logic Element over the Complex Field. Kibernetika Cybernetics and Systems Analysis 27(3), 116–121; English version is available from Springer 27(3), 467-473 (1991) (in Russian)Google Scholar
  12. 12.
    Jankowski, S., Lozowski, A., Zurada, J.M.: Complex-Valued Multistate Neural Associative Memory. IEEE Trans. Neural Networks 7(6), 1491–1496 (1996)CrossRefGoogle Scholar
  13. 13.
    Aoki, H., Kosugi, Y.: An Image Storage System Using Complex-Valued Associative Memory. In: 15th International Conference on Pattern Recognition, Barcelona, vol. 2, pp. 626–629. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  14. 14.
    Muezzinoglu, M.K., Guzelis, C., Zurada, J.M.: A New Design Method for the Complex-Valued Multistate Hopfield Associative Memory. IEEE Trans. Neural Networks 14(4), 891–899 (2003)CrossRefGoogle Scholar
  15. 15.
    Aoki, H., Watanabe, E., Nagata, A., Kosugi, Y.: Rotation-Invariant Image Association for Endoscopic Positional Identification Using Complex-Valued Associative Memories. In: Mira, J., Prieto, A.G. (eds.) IWANN 2001. LNCS, vol. 2085, pp. 369–374. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Lee, D.L.: Improving the capacity of complex-valued neural networks with a modified gradient descent learning rule. IEEE Transactions on Neural Networks 12(2), 439–443 (2001)CrossRefGoogle Scholar
  17. 17.
    Aoki, H.: A complex-valued neuron to transform gray level images to phase information. In: Wang, L., Rajapakse, J.C., Fukushima, K., Lee, S.-Y., Yao, X. (eds.) 9th International Conference on Neural Information Processing (ICONIP 2002), vol. 3, pp. 1084–1088 (2002)Google Scholar
  18. 18.
    Aizenberg, I., Myasnikova, E., Samsonova, M., Reinitz, J.: Temporal Classification of Drosophila Segmentation Gene Expression Patterns by the Multi-Valued Neural Recognition Method. Mathematical Biosciences 176(1), 145–159 (2002)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Igor Aizenberg
    • 1
  1. 1.Department of Computer ScienceTexas A&M University-TexarkanaTexarkanaUSA

Personalised recommendations