Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks

Bahruz Gadjiev; Tatiana Progulova

doi:doi:10.11648/j.ajpa.20190703.12

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Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks

Bahruz Gadjiev, Tatiana Progulova

Published in American Journal of Physics and Applications (Volume 7, Issue 3)

Received: 20 April 2019 Accepted: 11 June 2019 Published: 25 June 2019

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Abstract

We study the dynamics of the processes in the small-world networks with a power-law degree distribution where every node is considered to be in one of the two available statuses. We present an algorithm for generation of such network and determine analytically a temporal dependence of the network nodes degrees and using the maximum entropy principle we define a degree distribution of the network. We discuss the results of the Ising discrete model for small-world networks and in the framework of the continuous approach using the principle of least action, we derive an equation of motion for the order parameter in these networks in the form of a fractional differential equation. The obtained equation enables the description of the problem of a spontaneous symmetry breaking in the system and determination of the spatio-temporal dependencies of the order parameter in varies stable phases of the system. In the cases of one and two component order parameters with taken into account major and secondary order parameters we obtain analytical solutions of the equation of motion for the order parameters and determine solutions for various regimes of the system functioning. We apply the obtained results to the description of the processes in the brain and discuss the problems of emergence of mind.

Published in	American Journal of Physics and Applications (Volume 7, Issue 3)
DOI	10.11648/j.ajpa.20190703.12
Page(s)	68-72
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Phase Transition, Small-World Networks, Order Parameter, Brain Dynamics, Fractional Differential Equation

References

[1]	H. Haken, Principles of Brain Functioning: Synergetic Approach to Brain Activity, Behavior, and Cognition. Berlin: Springer, 1996.
[2]	H. Haken, Brain Dynamics: An Introduction to Models and Simulations. Berlin: Springer, 2008.
[3]	R. Penrose, A. Shimony, N. Cartwright, and S. Hawking, The Large, the Small and the Human Mind. Cambridge: Cambridge University Press, 1997.
[4]	V. M. Eguiluz, D. R. Chialvo, G. A. Cecchi, M. Baliki, and A. V. Apkarian, “Scale-free brain functional networks”, Physical Review Letters, vol. 94, 018102, 2005.
[5]	E. Bullmore and O. Sporns, “Complex brain networks: graph theoretical analysis of structural and functional systems”, Nature Reviews Neuroscience, vol. 10, 2009, pp. 186–198.
[6]	O. Sporns and R. F. Betzel, “Modular Brain Networks”, Annual Review of Psychology, vol. 67, 2016, pp. 613–640.
[7]	D. J. Watts and S. H. Strogatz, “Collective dynamics of “small-world” networks”, Nature, vol. 393, 1998, pp. 440–442.
[8]	B. Gadjiev and T. Progulova, “Origin of generalized entropies and generalized statistical mechanics for superstatistical multifractal systems”, AIP Conference Proceedings, vol. 1641, 2015, pp. 595–602.
[9]	S. N. Dorogovtsev, Lectures on Complex Networks. Oxford: Oxford University Press, 2010.
[10]	B. R. Gadjiev and T. B. Progulova, “Phase transitions in small-world systems: application to functional brain networks”, Journal of Physics, vol. 597, 2015, 012038.
[11]	M. Gitterman and V. H. Halpern, Phase Transitions: A Brief Account With Modern Applications. World Scientific Publishing, 2004.
[12]	Vik S. Dotsenko, “Critical phenomena and quenched disorder”, Phys. Usp. vol. 38 (5), 1995, pp. 457–496.
[13]	T. Abdeljawad, “On Conformable Fractional Calculus”. Journal of Computational and Applied Mathematics, vol. 279, 2015, pp. 57–66.
[14]	K. G. Wilson and J. Kogut, “The renormalization group and the ԑ-expansion”, Phys. Reports, vol. 12, 1974, pp. 75–200.
[15]	H. Haken, Synergetic. An Introduction. Berlin‐Heidelberg‐New York: Springer-Verlag, 1978.
[16]	A. A. Abrikosov, “Type-II superconductors and the vortex lattice”, Reviews of Modern Physics, vol. 76, 2004, pp. 975-979.

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Bahruz Gadjiev, Tatiana Progulova. (2019). Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks. American Journal of Physics and Applications, 7(3), 68-72. https://doi.org/10.11648/j.ajpa.20190703.12

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ACS Style

Bahruz Gadjiev; Tatiana Progulova. Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks. Am. J. Phys. Appl. 2019, 7(3), 68-72. doi: 10.11648/j.ajpa.20190703.12

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AMA Style

Bahruz Gadjiev, Tatiana Progulova. Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks. Am J Phys Appl. 2019;7(3):68-72. doi: 10.11648/j.ajpa.20190703.12

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@article{10.11648/j.ajpa.20190703.12,
  author = {Bahruz Gadjiev and Tatiana Progulova},
  title = {Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks},
  journal = {American Journal of Physics and Applications},
  volume = {7},
  number = {3},
  pages = {68-72},
  doi = {10.11648/j.ajpa.20190703.12},
  url = {https://doi.org/10.11648/j.ajpa.20190703.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20190703.12},
  abstract = {We study the dynamics of the processes in the small-world networks with a power-law degree distribution where every node is considered to be in one of the two available statuses. We present an algorithm for generation of such network and determine analytically a temporal dependence of the network nodes degrees and using the maximum entropy principle we define a degree distribution of the network. We discuss the results of the Ising discrete model for small-world networks and in the framework of the continuous approach using the principle of least action, we derive an equation of motion for the order parameter in these networks in the form of a fractional differential equation. The obtained equation enables the description of the problem of a spontaneous symmetry breaking in the system and determination of the spatio-temporal dependencies of the order parameter in varies stable phases of the system. In the cases of one and two component order parameters with taken into account major and secondary order parameters we obtain analytical solutions of the equation of motion for the order parameters and determine solutions for various regimes of the system functioning. We apply the obtained results to the description of the processes in the brain and discuss the problems of emergence of mind.},
 year = {2019}
}

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TY  - JOUR
T1  - Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks
AU  - Bahruz Gadjiev
AU  - Tatiana Progulova
Y1  - 2019/06/25
PY  - 2019
N1  - https://doi.org/10.11648/j.ajpa.20190703.12
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T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
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PB  - Science Publishing Group
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UR  - https://doi.org/10.11648/j.ajpa.20190703.12
AB  - We study the dynamics of the processes in the small-world networks with a power-law degree distribution where every node is considered to be in one of the two available statuses. We present an algorithm for generation of such network and determine analytically a temporal dependence of the network nodes degrees and using the maximum entropy principle we define a degree distribution of the network. We discuss the results of the Ising discrete model for small-world networks and in the framework of the continuous approach using the principle of least action, we derive an equation of motion for the order parameter in these networks in the form of a fractional differential equation. The obtained equation enables the description of the problem of a spontaneous symmetry breaking in the system and determination of the spatio-temporal dependencies of the order parameter in varies stable phases of the system. In the cases of one and two component order parameters with taken into account major and secondary order parameters we obtain analytical solutions of the equation of motion for the order parameters and determine solutions for various regimes of the system functioning. We apply the obtained results to the description of the processes in the brain and discuss the problems of emergence of mind.
VL  - 7
IS  - 3
ER  -

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Author Information

Bahruz Gadjiev

Institute of System Analysis and Management, Dubna State University, Dubna, Russia
Tatiana Progulova

Institute of System Analysis and Management, Dubna State University, Dubna, Russia

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APA Style

Bahruz Gadjiev, Tatiana Progulova. (2019). Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks. American Journal of Physics and Applications, 7(3), 68-72. https://doi.org/10.11648/j.ajpa.20190703.12

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ACS Style

Bahruz Gadjiev; Tatiana Progulova. Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks. Am. J. Phys. Appl. 2019, 7(3), 68-72. doi: 10.11648/j.ajpa.20190703.12

Copy | Download

AMA Style

Bahruz Gadjiev, Tatiana Progulova. Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks. Am J Phys Appl. 2019;7(3):68-72. doi: 10.11648/j.ajpa.20190703.12

Copy | Download

@article{10.11648/j.ajpa.20190703.12,
  author = {Bahruz Gadjiev and Tatiana Progulova},
  title = {Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks},
  journal = {American Journal of Physics and Applications},
  volume = {7},
  number = {3},
  pages = {68-72},
  doi = {10.11648/j.ajpa.20190703.12},
  url = {https://doi.org/10.11648/j.ajpa.20190703.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20190703.12},
  abstract = {We study the dynamics of the processes in the small-world networks with a power-law degree distribution where every node is considered to be in one of the two available statuses. We present an algorithm for generation of such network and determine analytically a temporal dependence of the network nodes degrees and using the maximum entropy principle we define a degree distribution of the network. We discuss the results of the Ising discrete model for small-world networks and in the framework of the continuous approach using the principle of least action, we derive an equation of motion for the order parameter in these networks in the form of a fractional differential equation. The obtained equation enables the description of the problem of a spontaneous symmetry breaking in the system and determination of the spatio-temporal dependencies of the order parameter in varies stable phases of the system. In the cases of one and two component order parameters with taken into account major and secondary order parameters we obtain analytical solutions of the equation of motion for the order parameters and determine solutions for various regimes of the system functioning. We apply the obtained results to the description of the processes in the brain and discuss the problems of emergence of mind.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks
AU  - Bahruz Gadjiev
AU  - Tatiana Progulova
Y1  - 2019/06/25
PY  - 2019
N1  - https://doi.org/10.11648/j.ajpa.20190703.12
DO  - 10.11648/j.ajpa.20190703.12
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 68
EP  - 72
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20190703.12
AB  - We study the dynamics of the processes in the small-world networks with a power-law degree distribution where every node is considered to be in one of the two available statuses. We present an algorithm for generation of such network and determine analytically a temporal dependence of the network nodes degrees and using the maximum entropy principle we define a degree distribution of the network. We discuss the results of the Ising discrete model for small-world networks and in the framework of the continuous approach using the principle of least action, we derive an equation of motion for the order parameter in these networks in the form of a fractional differential equation. The obtained equation enables the description of the problem of a spontaneous symmetry breaking in the system and determination of the spatio-temporal dependencies of the order parameter in varies stable phases of the system. In the cases of one and two component order parameters with taken into account major and secondary order parameters we obtain analytical solutions of the equation of motion for the order parameters and determine solutions for various regimes of the system functioning. We apply the obtained results to the description of the processes in the brain and discuss the problems of emergence of mind.
VL  - 7
IS  - 3
ER  -

Copy | Download