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In this paper, by using the Terman-Wang small-world neuronal network with electrical synapse coupling, we investigate the synchronous dynamics of neuronal network system subjected to spatially correlated white noise. First, the dynamical mean-field approximation theory is extended to the small-world network system under spatially correlated white noise, through which the original 2N-dimensional stochastic differential equations of the network system are transformed to 11-dimensional deterministic moment differential equations. Then, based on this set of moment differential equations, the key effects of spatially correlated noise and network structure on the synchronous firing property are discussed in the Terman-Wang neuronal network system. The results show that the synchronization ratio of this considered neuronal network system becomes higher not only as the noise correlation coefficient is increased but also as the coupling strength and the average vertex degree are added. Those results imply that the noise spatial correlation coefficient, the coupling strength, and the average vertex degree can play a positive role in inducing synchronous neuronal behaviors. Furthermore, the synchronous dynamics of the original neuronal network system, obtained by direct numerical simulations, is compared with those obtained by the dynamical mean-field approximation theory, and good consistence between them is revealed.
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Keywords:
- synchronous firing /
- small-world neuronal network /
- spatially correlated noise /
- dynamical mean-field approximation theory
[1] Singer W 1993 Annu. Rev. Physiol. 55 349
[2] Honey C J, Kter R, Breakspear M, Sporns O 2007 Natl. Acad. Sci. 104 10240
[3] van den heuvel M P, Stam C J, Boersma M, Hulshoff Pol H E 2008 NeuroImage 43 528
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[6] Han F, Lu Q S, Marian W, Ji Q B 2009 Chin. Phys. B 18 0482
[7] Gu H G, Jia B, Li Y Y, Chen G R 2013 Physica A 392 1361
[8] Yang X L, Senthilkumar D V, Kurths J 2012 Chaos 22 043150
[9] Lindner B, Garca-Ojalvo J, Neiman A, Schimansky-Geier L 2004 Phys. Rep. 392 321
[10] Wang Q Y, Chen G R, Perc M 2011 PLoS ONE 6 e15851
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[12] Manyakov N V, Van Hulle M M 2008 Chaos 18 037130
[13] Bartsch R, Kantelhardt J W, Penzel T, Havlin S 2007 Phys. Rev. Lett. 98 054102
[14] Yu H T, Wang J, Deng B, Wei X L, Wong Y K, Chan W L, Tsang K M, Yu Z Q 2011 Chaos 21 013127
[15] Wang Q Y, Lu Q S 2005 Chin. Phys. Lett. 22 1329
[16] Shi X, Sun X J, L Y B, Lu Q S, Wang H X 2015 Int. J. Non-Linear Mech. 70 112
[17] Yang X L, Jia Y B, Zhang L 2014 Physica A 393 617
[18] Hasegawa H 2003 Phys. Rev. E 67 041903
[19] Hasegawa H 2004 Phys. Rev. E 70 066107
[20] Hasegawa H 2005 Phys. Rev. E 72 056139
[21] Zhou C S, Kurths J, Hu B 2001 Phys. Rev. Lett. 87 098101
[22] Doiron B, Lindner B, Longtin A, Maler L, Bastian J 2004 Phys. Rev. Lett. 93 048101
[23] Liu F, Hu B, Wang W 2001 Phys. Rev. E 63 031907
[24] Lindner B, Doiron B, Longtin A 2005 Phys. Rev. E 72 061919
[25] Sun X J, Lu Q S, Kurths J 2008 Physica A 387 6679
[26] Shao Y G, Kang Y M 2014 Theoret. Appl. Mech. Lett. 4 013006
[27] Terman D, Wang D L 1995 Physica D 81 148
[28] Watts D J, Strogatz S H 1998 Nature 393 440
[29] Tanabe S, Pakdaman K 2001 Phys. Rev. E 63 031911
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[1] Singer W 1993 Annu. Rev. Physiol. 55 349
[2] Honey C J, Kter R, Breakspear M, Sporns O 2007 Natl. Acad. Sci. 104 10240
[3] van den heuvel M P, Stam C J, Boersma M, Hulshoff Pol H E 2008 NeuroImage 43 528
[4] Wang D G, Liang X M, Wang J, Yang C F, Liu K, L H P 2010 Chin. Phys. B 19 110515
[5] Zhou X R, Luo X S 2008 Acta Phys. Sin. 57 2849 (in Chinese) [周小荣, 罗晓曙 2008 物理学报 57 2849]
[6] Han F, Lu Q S, Marian W, Ji Q B 2009 Chin. Phys. B 18 0482
[7] Gu H G, Jia B, Li Y Y, Chen G R 2013 Physica A 392 1361
[8] Yang X L, Senthilkumar D V, Kurths J 2012 Chaos 22 043150
[9] Lindner B, Garca-Ojalvo J, Neiman A, Schimansky-Geier L 2004 Phys. Rep. 392 321
[10] Wang Q Y, Chen G R, Perc M 2011 PLoS ONE 6 e15851
[11] Elson R C, Selverston A I, Huerta R, Rulkov N F, Rabinovich M I, Abarbanel H D I 1998 Phys. Rev. Lett. 81 5692
[12] Manyakov N V, Van Hulle M M 2008 Chaos 18 037130
[13] Bartsch R, Kantelhardt J W, Penzel T, Havlin S 2007 Phys. Rev. Lett. 98 054102
[14] Yu H T, Wang J, Deng B, Wei X L, Wong Y K, Chan W L, Tsang K M, Yu Z Q 2011 Chaos 21 013127
[15] Wang Q Y, Lu Q S 2005 Chin. Phys. Lett. 22 1329
[16] Shi X, Sun X J, L Y B, Lu Q S, Wang H X 2015 Int. J. Non-Linear Mech. 70 112
[17] Yang X L, Jia Y B, Zhang L 2014 Physica A 393 617
[18] Hasegawa H 2003 Phys. Rev. E 67 041903
[19] Hasegawa H 2004 Phys. Rev. E 70 066107
[20] Hasegawa H 2005 Phys. Rev. E 72 056139
[21] Zhou C S, Kurths J, Hu B 2001 Phys. Rev. Lett. 87 098101
[22] Doiron B, Lindner B, Longtin A, Maler L, Bastian J 2004 Phys. Rev. Lett. 93 048101
[23] Liu F, Hu B, Wang W 2001 Phys. Rev. E 63 031907
[24] Lindner B, Doiron B, Longtin A 2005 Phys. Rev. E 72 061919
[25] Sun X J, Lu Q S, Kurths J 2008 Physica A 387 6679
[26] Shao Y G, Kang Y M 2014 Theoret. Appl. Mech. Lett. 4 013006
[27] Terman D, Wang D L 1995 Physica D 81 148
[28] Watts D J, Strogatz S H 1998 Nature 393 440
[29] Tanabe S, Pakdaman K 2001 Phys. Rev. E 63 031911
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