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Traveling waves, standing waves, and spiral waves occur spontaneously in the brain neural network in some brain states. The occurrence of these ordered spatiotemporal patterns is often related to some neurological diseases. However, the mechanisms behind the generation of the ordered pattern are not fully understood. How to quantitatively describe the nature of these spatiotemporal patterns still needs further exploring. In order to solve these problems, the Hindmarsh-Rose neuron model is used to study the dynamic behavior of the two-dimensional (2D) neuronal network with double-coupling layer, which is composed of nearest-neighbor excitatory coupling and long-range repulsive coupling layers and evolves from an initial state with a random phase distribution. An improved cluster entropy is proposed to describe the spatiotemporal pattern of the neuronal network. The numerical simulation results show that the repulsive coupling can either promote the formation of ordered patterns or suppress the formation of ordered patterns. When the repulsive coupling strength and excitatory coupling strength are appropriately selected, the chaotic network can spontaneously generate single spiral wave, multiple spiral wave, traveling wave, the coexistence of spiral wave and others wave state, the coexistence of target wave and others wave state, the coexistence of traveling wave and standing wave, etc. The probability with which spiral wave and traveling wave occur reach 0.4555 and 0.1667 respectively. The probability with which target wave and other states co-occur, and the probability with which the traveling wave and the standing wave co-occur, are 0.0389 and 0.1056, respectively. These ordered wave patterns and chaotic states can be distinguished by using the proposed cluster entropy. When the repulsive coupling strength is large enough, the neuronal network is generally in chaotic state. It is found by calculating cluster entropy that a large cluster can appear in the neuronal network when the excitatory coupling strength and repulsive coupling strength are both weak. These results can conduce to understanding the self-organization phenomena occurring in the experiments and also to treating various neurological diseases.
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Keywords:
- spiral wave /
- standing wave /
- Hindmarsh-Rose neuron /
- repulsive coupling
[1] Larionova Y, Egorov O, Cabrera-Granado E, Esteban-Martin A 2005 Phys. Rev. A 72 033825
Google Scholar
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Google Scholar
[3] Bär M, Gottschalk N, Eiswirth M, Ertl G 1994 J. Phys. Chem. B 100 1202
Google Scholar
[4] Müller S C, Plesser T, Hess B 1985 Science 230 661
Google Scholar
[5] Vanag V K, Epstein I R 2001 Science 294 835
Google Scholar
[6] Davidenko J M, Pertsov A V, Salomonsz R, Baxter W, Jalife J 1992 Nature 355 349
Google Scholar
[7] Huang X Y, Xu W F, Liang J M, Takagaki K, Gao X, Wu J Y 2010 Neuron 68 978
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[8] Huang X Y, Troy W C, Yang Q, Ma H T, Laing C R, Schiff S J, Wu J Y 2004 J. Neurosci. 24 9897
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[9] Viventi J, Kim D H, Vigeland L, Frechette E S, Blanco J A, Kim Y S, Avrin A E, et al. 2011 Nat. Neurosci. 14 1599
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Google Scholar
[11] Qin H X, Ma J, Wang C N, Chu R T 2014 Sci. China: Phys. Mech. Astron. 57 1918
Google Scholar
[12] 汪芃, 李倩昀, 黄志精, 唐国宁 2018 物理学报 67 170501
Google Scholar
Wang P, Li Q Y, Huang Z J, Tang G N 2018 Acta Phys. Sin. 67 170501
Google Scholar
[13] Lip G Y H, Fauchier L, Freedman S B, Gelder V I, Natale A, Gianni C, et al. 2016 Nat. Rev. Dis. Primers 2 16016
Google Scholar
[14] Yengi D, Tinsley M R, Showalter K 2018 Chaos 28 045114
Google Scholar
[15] Wang Q Y, Chen G R, Perc M 2011 PloS One 6 e15851
Google Scholar
[16] Vreeswijk C V, Abbott L F, Ermentrout G B 1994 J. Comput. Neurosci. 1 313
Google Scholar
[17] Leyva I, Sendiña-Nadal I, Almendral J A, Sanjuán M A F 2006 Phys. Rev. E 74 056112
Google Scholar
[18] Scholkmann F 2015 J. Integr. Neurosci. 14 135
Google Scholar
[19] Veeraraghavan R, Lin J, Hoeker G S, Keener J P, Robert G. Gourdie R G, Poelzing S 2015 Pflug. Arch Eur. J. Physiol. 467 2093
Google Scholar
[20] Copene E D, Keener J P 2008 J. Math. Biol. 57 265
Google Scholar
[21] Shen J, Zhang J H, Xiao H, Wu J M, He K M, Lv Z Z, Li Z J, Xu M, Zhang Y Y 2018 Cell Death and Dis. 9 81
Google Scholar
[22] Ma J, Tang J 2017 Nonlinear Dyn. 89 1569
Google Scholar
[23] Weinberg S H 2017 Chaos 27 093908
Google Scholar
[24] Shlens J, Field G D, Gauthier J L, Grivich M I, Petrusca D, Sher A, Litke A M, Chichilnisky E J 2006 J. Neurosci. 26 8254
Google Scholar
[25] Marre O, Boustani S E, Frégnac Y, Destexhe A 2009 Phys. Rev. Lett. 102 138101
Google Scholar
[26] Nghiem T A, Telenczuk B, Marre O, Destexhe A, Ferrari U 2018 Phys. Rev. E 98 012402
Google Scholar
[27] Jung P, Wang J, Wackerbauer R, Showalter K 2000 Phys. Rev. E 61 2095
Google Scholar
[28] Hindmarsh J L, Rose R M 1984 Pro. R. Soc. Lond. B 221 87
Google Scholar
[29] Adhikari B M, Prasad A, Dhamala M 2011 Chaos 21 023116
Google Scholar
[30] Nunez P L, Srinivasan R 2006 Clin. Neurophysiol. 117 2424
Google Scholar
[31] Müller M F, Rummel C, Goodfellow M, Schindler K 2014 Brain Connectivity 4 131
Google Scholar
[32] Schneidman E, Berry M J, Segev R, Bialek W 2006 Nature 440 1007
Google Scholar
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图 1 不同耦合强度下的神经元
${x_{i,j}}$ 变量斑图, 每一个图下方为相应斑图的熵值 (a)$w = 0,\;g = {\rm{0}}$ ; (b)$w = 0.{\rm{17}},\;g = {\rm{0}}.{\rm{3}}$ ; (c)$w = 0.05,\;g = 1.2$ ; (d)$w = 0.05,\;g = 1.1 $ ; (e)$w = 0.07,\;g = 1.3$ ; (f)$w = 0.0{\rm{9}},\;g = 1.{\rm{9}}$ ; (g) w = 0.05,$g = 1.0$ ; (h)$w = 0.01,\;g = 0.1$ ; (i)$w = 0,\;g = {\rm{0}}.{\rm{1}}$ ; (j)$w = 0.03,\;g = 1.0$ ; (k)$w = 0.07,\;g = 0.4$ ; (l) w = 0.05,$g = 0.4$ Figure 1. Pattern of the variable
${x_{i,j}}$ for different values of coupling strength. Entropy of the corresponding pattern is given underneath each panel: (a)$w = 0,\;g = {\rm{0}}$ ; (b)$w = 0.{\rm{17}},\;g = {\rm{0}}.{\rm{3}}$ ; (c)$w = 0.05,\;g = 1.2$ ; (d)$w = 0.05,\;g = 1.1 $ ; (e)$w = 0.07,\;g = 1.3$ ; (f)$w = 0.0{\rm{9}},\;g = 1.{\rm{9}}$ ; (g)$w = 0.05,\;g = 1.0$ ; (h)$w = 0.01,\;g = 0.1$ ; (i)$w = 0,\;g = {\rm{0}}.{\rm{1}}$ ; (j)$w = 0.03,\;g = 1.0$ ; (k) w$ = 0.07,\;g = 0.4$ ; (l)$w = 0.05,\;g = 0.4$ .
图 2 不同耦合强度下一行和一列格点的
${x_{i,j}}$ 变量时空斑图 (a)$w = 0.05,\;g = 1.0$ ; (b)$w = 0.05,\;g = 1.0$ ; (c) w =$ 0.01,\;g = 0.1$ ; (d)$w = 0.01,\;g = 0.1$ Figure 2. Spatiotemporal pattern of the variable
${x_{i,j}}$ of a row and a column of grid points for different values of coupling strength: (a)$w = 0.05,\;g = 1.0$ ; (b)$w = 0.05,\;g = 1.0$ ; (c)$w = 0.01,\;g = 0.1$ ; (d)$w = 0.01,\;g = 0.1$ .
图 3 不同耦合强度下网络的集团熵随时间变化 (a)
$w = 0.05,\;g = 1.2$ ; (b)$w = 0.05,\;g = 1.1$ ; (c)$w = 0.07,\;g = 1.3$ ; (d)$w = 0.05,\;g = 1.0$ ; (e)$w = 0.01,\;g = 0.1$ ; (f)$w = 0.05,\;g = 0.4$ .Figure 3. Time evolution of the cluster entropy of the network for different values of coupling strength: (a)
$w = 0.05,\;g = 1.2$ ; (b)$w = 0.05,\;g = 1.1$ ; (c)$w = 0.07,\;g = 1.3$ ; (d)$w = 0.05,\;g = 1.0$ ; (e)$w = 0.01,\;g = 0.1$ ; (f)$w = 0.05,\;g = 0.4$ .
图 4 在g -w参量平面上的相图
Figure 4. Phase diagram in the g -w parameter plane.
图 5 在g -w参量平面上的熵分布
Figure 5. Entropy distribution in the g -w parameter plane.
图 6 在
$g = 1.2$ 和不同耦合强度w下不同时刻的${x_{i,j}}$ 变量斑图 (a)$w = - 0.15$ ,$t = 0$ ; (b)$w = - 0.15$ ,$t = 1000$ ; (c)$w = - 0.15$ ,$t = 2000$ ; (d)$w = - 0.15$ ,$t = 4000$ ; (e)$w = - 0.15$ ,$t = 6000$ ; (f)$w = - 0.15$ ,$t = 8000$ ; (g)$w = - 0.15$ ,$t = 10000$ ; (h)$w = 0.05$ ,$t = 12000$ ; (i)$w = 0.05$ ,$t = 14000$ ; (j)$w = 0.05$ ,$t = 16000$ ; (k)$w = 0.05$ ,$t = 18000$ ; (l)$w = 0.05$ ,$t = 20000$ .Figure 6. Pattern of the
${x_{i,j}}$ variable at different time moments for g = 1.2 and different values of coupling strength w: (a)$w = - 0.15$ ,$t = 0$ ; (b)$w = - 0.15$ ,$t = 1000$ ; (c)$w = - 0.15$ ,$t = 2000$ ; (d)$w = - 0.15$ ,$t = 4000$ ; (e)$w = - 0.15$ ,$t = 6000$ ; (f) w = –0.15,$t = 8000$ ; (g)$w = - 0.15$ ,$t = 10000$ ; (h)$w = 0.05$ ,$t = 12000$ ; (i)$w = 0.05$ ,$t = 14000$ ; (j)$w = 0.05$ ,$t = 16000$ ; (k)$w = 0.05$ ,$t = 18000$ ; (l)$w = 0.05$ ,$t = 20000$ . -
[1] Larionova Y, Egorov O, Cabrera-Granado E, Esteban-Martin A 2005 Phys. Rev. A 72 033825
Google Scholar
[2] Plapp B P, Egolf D A, Bodenschatz E, Pesch W 1998 Phys. Rev. Lett. 81 5334
Google Scholar
[3] Bär M, Gottschalk N, Eiswirth M, Ertl G 1994 J. Phys. Chem. B 100 1202
Google Scholar
[4] Müller S C, Plesser T, Hess B 1985 Science 230 661
Google Scholar
[5] Vanag V K, Epstein I R 2001 Science 294 835
Google Scholar
[6] Davidenko J M, Pertsov A V, Salomonsz R, Baxter W, Jalife J 1992 Nature 355 349
Google Scholar
[7] Huang X Y, Xu W F, Liang J M, Takagaki K, Gao X, Wu J Y 2010 Neuron 68 978
Google Scholar
[8] Huang X Y, Troy W C, Yang Q, Ma H T, Laing C R, Schiff S J, Wu J Y 2004 J. Neurosci. 24 9897
Google Scholar
[9] Viventi J, Kim D H, Vigeland L, Frechette E S, Blanco J A, Kim Y S, Avrin A E, et al. 2011 Nat. Neurosci. 14 1599
[10] Perc M 2007 Chaos, Solitons Fractals 31 280
Google Scholar
[11] Qin H X, Ma J, Wang C N, Chu R T 2014 Sci. China: Phys. Mech. Astron. 57 1918
Google Scholar
[12] 汪芃, 李倩昀, 黄志精, 唐国宁 2018 物理学报 67 170501
Google Scholar
Wang P, Li Q Y, Huang Z J, Tang G N 2018 Acta Phys. Sin. 67 170501
Google Scholar
[13] Lip G Y H, Fauchier L, Freedman S B, Gelder V I, Natale A, Gianni C, et al. 2016 Nat. Rev. Dis. Primers 2 16016
Google Scholar
[14] Yengi D, Tinsley M R, Showalter K 2018 Chaos 28 045114
Google Scholar
[15] Wang Q Y, Chen G R, Perc M 2011 PloS One 6 e15851
Google Scholar
[16] Vreeswijk C V, Abbott L F, Ermentrout G B 1994 J. Comput. Neurosci. 1 313
Google Scholar
[17] Leyva I, Sendiña-Nadal I, Almendral J A, Sanjuán M A F 2006 Phys. Rev. E 74 056112
Google Scholar
[18] Scholkmann F 2015 J. Integr. Neurosci. 14 135
Google Scholar
[19] Veeraraghavan R, Lin J, Hoeker G S, Keener J P, Robert G. Gourdie R G, Poelzing S 2015 Pflug. Arch Eur. J. Physiol. 467 2093
Google Scholar
[20] Copene E D, Keener J P 2008 J. Math. Biol. 57 265
Google Scholar
[21] Shen J, Zhang J H, Xiao H, Wu J M, He K M, Lv Z Z, Li Z J, Xu M, Zhang Y Y 2018 Cell Death and Dis. 9 81
Google Scholar
[22] Ma J, Tang J 2017 Nonlinear Dyn. 89 1569
Google Scholar
[23] Weinberg S H 2017 Chaos 27 093908
Google Scholar
[24] Shlens J, Field G D, Gauthier J L, Grivich M I, Petrusca D, Sher A, Litke A M, Chichilnisky E J 2006 J. Neurosci. 26 8254
Google Scholar
[25] Marre O, Boustani S E, Frégnac Y, Destexhe A 2009 Phys. Rev. Lett. 102 138101
Google Scholar
[26] Nghiem T A, Telenczuk B, Marre O, Destexhe A, Ferrari U 2018 Phys. Rev. E 98 012402
Google Scholar
[27] Jung P, Wang J, Wackerbauer R, Showalter K 2000 Phys. Rev. E 61 2095
Google Scholar
[28] Hindmarsh J L, Rose R M 1984 Pro. R. Soc. Lond. B 221 87
Google Scholar
[29] Adhikari B M, Prasad A, Dhamala M 2011 Chaos 21 023116
Google Scholar
[30] Nunez P L, Srinivasan R 2006 Clin. Neurophysiol. 117 2424
Google Scholar
[31] Müller M F, Rummel C, Goodfellow M, Schindler K 2014 Brain Connectivity 4 131
Google Scholar
[32] Schneidman E, Berry M J, Segev R, Bialek W 2006 Nature 440 1007
Google Scholar
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