Geodesic
Calculation of stability domains of discrete models of big size small world networks
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 5 (2016) no. 3, pp. 69-75 Cet article a éte moissonné depuis la source Math-Net.Ru

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The article is devoted to description of discrete models of small world networks with a large number of neurons with a certain parameter $p$ varying from $0$ to $1$. For $p = 0$ have model, regular neural networks, which is a ring network in which each neuron interacts with several neighbors on the ring. In the case $p = 1$ have a model with randomly distributed connections. When the values of $p$ not exceeding $0,1$ have the Watts–Strogatz small world network. Such a neural network can be models of different neural structures in living organisms, for example, the hipocampus of the mammalian brain. This paper examines the dynamics of change areas of stability of such neural networks when $0 \leqslant p\leqslant 0,1$. Numerical experiments show an increase in sustainability in the transition from a regular network to small world.
Keywords: Watts–Strogatz discrete models, small world, stability. 
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S. A. Ivanov. Calculation of stability domains of discrete models of big size small world networks. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 5 (2016) no. 3, pp. 69-75. http://geodesic.mathdoc.fr/item/VYURV_2016_5_3_a4/

[1] D. Watts, S. Strogatz, “Collective Dynamics of «Small-World» Networks”, Nature, 393 (1998), 440–442 | DOI

[2] R.T. Gray, C.K.C. Fung, P.A. Robinson, “Stability of Small-World Networks of Neural Populations”, Neurocomputing, 72(7–9) (2009), 1565–1574 | DOI

[3] S. Sinha, “Complexity vs Stability in Small-World Networks”, Physica A, 346 (2005), 147–153 | DOI

[4] M.G. Hart, R.J.F. Ypma, R. Romero-Garcia, S.J. Price, J. Suckling, “Graph Theory Analysis of Complex Brain Networks: New Concepts in Brain Maping Aplied to Neurosurgery”, Journal of Neurosurgery, 124:6 (2016), 1665–1678 | DOI

[5] T.I. Netoff, R. Clewley, S. Arno, T. Keck, A. John, “White Epilepsy in Small-World Networks”, The Journal of Neuroscience, 24(37) (2004), 8075–8083

[6] M.A. Arbib, P. Érdi, J. Szentágothai, Neural Organization: Structure, Function, and Dynamics, MIT Press, Cambridge, MA, 1998, 420 pp.

[7] M. Arbib, The Handbook of Brain Theory and Neural Networks, MIT Press, Cambridge, MA, 2003, 1308 pp.

[8] S.A. Ivanov, M.M. Kipnis, “Stability Analysis Discrete-Time Neural Networks with Delayed Interactions: Torus, Ring, Grid, Line”, International Journal of Pure and Aplied Math, 78(5) (2012), 691–709

[9] S.A. Ivanov, M.M. Kipnis, R. Medina, “On the Stability of the Cartesian Product of a Neural Ring and an Arbitrary Neural Network”, Advances in Difference Equations, 2014 (2014), 1–7 | DOI

[10] M.M. Kipnis, V.V. Malygina, “The Stability Cone for a Matrix Delay Difference Equation”, International Journal of Mathematics and Mathematical Sciences, 2011 (2011), 1–15

[11] S.A. Ivanov, M.M. Kipnis, V.V. Malygina, “The Stability Cone for a Difference Matrix Equation with Two Delays”, ISRN Aplied Math, 2011 (2011), 1–19 | DOI

[12] T.N. Khokhlova, “The Construction of Regions of Stability of a Circular Neural Networks”, Chronicles of OFERNiO, 1(32) (2012), 4–5

[13] T.N. Khokhlova, M.M. Kipnis, “The Breaking of a Delayed Ring Neural Network Contributes to Stability: The Rule and Exceptions”, Neural Networks, 48 (2013), 148–152 | DOI