Neural Processing Letters

, Volume 47, Issue 3, pp 935–948 | Cite as

Gauge Neural Network with Z(2) Synaptic Variables: Phase Structure and Simulation of Learning and Recalling Patterns

  • Atsutomo Murai
  • Tetsuo Matsui


We study the Z(2) gauge-invariant neural network which is defined on a partially connected random network and involves Z(2) neuron variables \(S_i\) (\(=\pm \)1) and Z(2) synaptic connection (gauge) variables \(J_{ij}\) (\(=\pm \)1). Its energy consists of the Hopfield term \(-c_1S_iJ_{ij}S_j\), double Hopfield term \(-c_2 S_iJ_{ij}J_{jk} S_k\), and the reverberation (triple Hopfield) term \(-c_3 J_{ij}J_{jk}J_{ki}\) of synaptic self interactions. For the case \(c_2=0\), its phase diagram in the \(c_3-c_1\) plane has been studied both for the symmetric couplings \(J_{ij}=J_{ji}\) and asymmetric couplings (\(J_{ij}\) and \(J_{ji}\) are independent); it consists of the Higgs, Coulomb and confinement phases, each of which is characterized by the ability of learning and/or recalling patterns. In this paper, we consider the phase diagram for the case of nonvanishing \(c_2\), and examine its effect. We find that the \(c_2\) term enlarges the region of Higgs phase and generates a new second-order transition. We also simulate the dynamical process of learning patterns of \(S_i\) and recalling them and measure the performance directly by overlaps of \(S_i\). We discuss the difference in performance for the cases of Z(2) variables and real variables for synaptic connections.


Hopfield model Gauge neural network Phase diagram Higgs phase Confinement phase Simulation of learning and recalling process 



The authors would like to thank Dr. Yuki Nakano and Mr. Munetada Juta for discussion.


  1. 1.
    Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79:2554–2558MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Haykin S (1994) Neural networks; a comprehensive foundation. Prentice Hall, New JerseyzbMATHGoogle Scholar
  3. 3.
    Matsui T (2001) Gauge symmetry and neural networks. In: Janke W et al (eds) Fluctuating paths and fields. World Scientific, Singapore, pp 271–280 (cond-mat/0112463)CrossRefGoogle Scholar
  4. 4.
    Kemuriyama M, Matsui T, Sakakibara K (2005) Gauged neural network: phase structure, learning, and associative memory. Physica A 356:525–553 (cond-mat/0203136)CrossRefGoogle Scholar
  5. 5.
    Yang C-N, Mills RL (1954) Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. 96:191–195MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Wegner FJ (1971) Duality in generalized ising models and phase transitions without local order parameters. J. Math. Phys. 12:2259MathSciNetCrossRefGoogle Scholar
  7. 7.
    Wilson KG (1974) Confinement of quarks. Phys. Rev. D 10:2445–2458CrossRefGoogle Scholar
  8. 8.
    Takafuji Y, Nakano Y, Matsui T (2012) Z(2) gauge neural network and its phase structure. Physica A 391:5258–5304 (arXiv:1206.1110)CrossRefGoogle Scholar
  9. 9.
    Fujita Y, Matsui T (2002) Quantum gauged neural network: U(1) gauge theory. In: Wang L et al (ed) Proceedings of 9th international conference on neural information processing, pp 1360–1367 (arXiv:cond-mat/0207023)
  10. 10.
    Fujita Y, Hiramatsu T, Matsui T (2005) Quantum gauged neural networks: learning and recalling. In: Proceedings of international joint conference on neural networks. Montreal, Canada, pp 1108–1113Google Scholar
  11. 11.
    Sakane S, Hiramatsu T and Matsui T (2016) Neural network for quantum brain dynamics: 4D CP1+U(1) gauge theory on lattice and its phase structure. In: Proceedings pati IV of neural information processing 23rd international conference, ICONIP 2016, Kyoto, Japan, October 16–21, pp 522–530. (arXiv:1610.05433)
  12. 12.
    Ichinose I, Matsui T (2014) Lattice gauge theory for condensed matter physics: ferromagnetic superconductivity as its example. Mod. Phys. Lett. B28(1-33):1430012. (arXiv:1408.0089)
  13. 13.
    Drouffe JM (1980) Large dimensional lattice gauge theories: (I). Z2 pure gauge system. Nucl. Phys. B 170:211–227MathSciNetCrossRefGoogle Scholar
  14. 14.
    Elitzur S (1975) Impossibility of spontaneously breaking local symmetries. Phys. Rev. D 12:3978–3982CrossRefGoogle Scholar
  15. 15.
    Hebb DO (1949) The organization of behavior: a neuropsychological theory. Wiley, New YorkGoogle Scholar
  16. 16.
    Murai A, Matsui T (2016) Asymmetric synaptic connections in Z(2) gauge neural network. In: Proceedings pati IV of neural information processing 23rd international conference, ICONIP 2016, Kyoto, Japan, October 16–21, 522–530Google Scholar
  17. 17.
    Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AM, Teller E (1953) Equation of state calculations by fast computing machines. J. Chem. Phys. 21:1087–1092CrossRefGoogle Scholar
  18. 18.
    Gell-Mann M, Hartle JB (1993) Classical equations for quantum systems. Phys. Rev. D 47:3345–3382MathSciNetCrossRefGoogle Scholar
  19. 19.
    van Kampen NG (1981) Stochastic processes in physics and chemistry. Elsevier, AmsterdamzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of PhysicsKindai UniversityHigashi-OsakaJapan

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