Abstract
We consider a system of two-level quantum quasi-spins and gauge bosons put on a 3+1D lattice. As a model of neural network of the brain functions, these spins describe neurons quantum-mechanically, and the gauge bosons describes weights of synaptic connections. It is a generalization of the Hopfield model to a quantum network with dynamical synaptic weights. At the microscopic level, this system becomes a model of quantum brain dynamics proposed by Umezawa et al., where spins and gauge field describe water molecules and photons, respectively. We calculate the phase diagram of this system under quantum and thermal fluctuations, and find that there are three phases; confinement, Coulomb, and Higgs phases. Each phase is classified according to the ability to learn patterns and recall them. By comparing the phase diagram with that of classical networks, we discuss the effect of quantum fluctuations and thermal fluctuations (noises in signal propagations) on the brain functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Hopfield, J.J.: Proc. Nat. Acd. Sci. USA 79, 2554 (1982)
Hebb, D.O.: The Organization of Behavior: A Neuropsychological Theory. Wiley, New York (1949)
Haykin, S.: Neural Networks; A Comprehensive Foundation. Macmillan Pub. Co., New York (1994)
Matsui, T.: Fluctuating Paths and Fields, Janke, W., et al. (eds.). World Scientific (2001) 271 (arXiv:cond-mat/0112463); Kemuriyama, M., Matsui, T., Sakakibara, K.: Physica A 356, 525 (2005) (arXiv:cond-mat/0203136); Takafuji, Y., Nakano, Y., Matsui, T.: Physica A 391, 5258 (2012)
Fujita, Y., Hiramatsu, T., Matsui, T.: Proceedings of International Joint Conference on Neural Networks (Montreal, Canada, 2005) 1108. There is a similar model describing each neuron state by U(1) variables instead of CP1 variable; Fujita, Y., Matsui, T.: Proceedings of 9th International Conference on Neural Information Processing, Wang, L., et al. (eds.) (2002) 1360 (arXiv:cond-mat/0207023)
Stuart, C., Takahashi, Y., Umezawa, H.: J. Theor. Biol. 71, 605 (1978); Found. Phys. 9, 301 (1979); Ricciardi, L.M., Umezawa, H.: Kybernetik 4, 44 (1967)
Jibu, M., Yasue, K.: Informatica 21, 471 (1997); Quantum Brain Dynamics. An Introduction. John Benjamins, Amsterdam (1995); See also Vitiello, G.: Int. J. Mod. Phys. B 9, 973 (1995)
Wilson, K.: Phys. Rev. D 10, 2445 (1974)
Rothe, H.J.: Lattice Gauge Theories: An Introduction. World Scientific, Singapore (2005)
DeGrand, T.A., Toussaint, D.: Phys. Rev. D 22, 2478 (1980)
Takashima, S., Ichinose, I., Matsui, T.: Phys. Rev. B 72, 075112 (2005)
Acknowledgment
The authors thank Prof. K. Sakakibara and Dr. Y. Nakano for discussion. This work was supported by JSPS KAKENHI Grant No. 26400412.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Sakane, S., Hiramatsu, T., Matsui, T. (2016). Neural Network for Quantum Brain Dynamics: 4D CP\(^1\)+U(1) Gauge Theory on Lattice and Its Phase Structure. In: Hirose, A., Ozawa, S., Doya, K., Ikeda, K., Lee, M., Liu, D. (eds) Neural Information Processing. ICONIP 2016. Lecture Notes in Computer Science(), vol 9947. Springer, Cham. https://doi.org/10.1007/978-3-319-46687-3_36
Download citation
DOI: https://doi.org/10.1007/978-3-319-46687-3_36
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46686-6
Online ISBN: 978-3-319-46687-3
eBook Packages: Computer ScienceComputer Science (R0)