Volume 7, Issue 3, May 2019, Page: 68-72
Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks
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
Received: Apr. 20, 2019;       Accepted: Jun. 11, 2019;       Published: Jun. 25, 2019
DOI: 10.11648/j.ajpa.20190703.12      View  493      Downloads  72
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.
Phase Transition, Small-World Networks, Order Parameter, Brain Dynamics, Fractional Differential Equation
To cite this article
Bahruz Gadjiev, Tatiana Progulova, Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks, American Journal of Physics and Applications. Vol. 7, No. 3, 2019, pp. 68-72. doi: 10.11648/j.ajpa.20190703.12
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
H. Haken, Principles of Brain Functioning: Synergetic Approach to Brain Activity, Behavior, and Cognition. Berlin: Springer, 1996.
H. Haken, Brain Dynamics: An Introduction to Models and Simulations. Berlin: Springer, 2008.
R. Penrose, A. Shimony, N. Cartwright, and S. Hawking, The Large, the Small and the Human Mind. Cambridge: Cambridge University Press, 1997.
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.
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.
O. Sporns and R. F. Betzel, “Modular Brain Networks”, Annual Review of Psychology, vol. 67, 2016, pp. 613–640.
D. J. Watts and S. H. Strogatz, “Collective dynamics of “small-world” networks”, Nature, vol. 393, 1998, pp. 440–442.
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.
S. N. Dorogovtsev, Lectures on Complex Networks. Oxford: Oxford University Press, 2010.
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.
M. Gitterman and V. H. Halpern, Phase Transitions: A Brief Account With Modern Applications. World Scientific Publishing, 2004.
Vik S. Dotsenko, “Critical phenomena and quenched disorder”, Phys. Usp. vol. 38 (5), 1995, pp. 457–496.
T. Abdeljawad, “On Conformable Fractional Calculus”. Journal of Computational and Applied Mathematics, vol. 279, 2015, pp. 57–66.
K. G. Wilson and J. Kogut, “The renormalization group and the ԑ-expansion”, Phys. Reports, vol. 12, 1974, pp. 75–200.
H. Haken, Synergetic. An Introduction. Berlin‐Heidelberg‐New York: Springer-Verlag, 1978.
A. A. Abrikosov, “Type-II superconductors and the vortex lattice”, Reviews of Modern Physics, vol. 76, 2004, pp. 975-979.
Browse journals by subject