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Spiral waves are a particular form of propagating waves, which rotate around a center point known as a rotor. Spiral waves have been found to play an important role in cardiac arrhythmia. Using voltage-sensitive dye imaging, one can find that spiral waves and plannar waves can occur in the mammalian cortex in vivo. The electrode array conduces to discovering that the seizures may manifest as recurrent spiral waves which propagate in the neocortex. However, the formation mechanism of the ordered waves and its potential function in the nervous system remain uncertain. In order to understand the formation mechanism of the ordered waves, we construct a double-layer two-dimensional -network of neuron, which is composed of nearest-neighbor excitatory coupling and long-range inhibitory coupling layers. The inhibitory grid points account for 25% of total number of grid points in the network. We propose a modified Hindmarsh-Rose neuron model to study whether differently ordered waves can occur spontaneously in the chaotic neuronal network evolving from the initial state with a random phase distribution. The numerical simulation results show that when the inhibitory coupling strength is small the spontaneous formation of ordered wave does not generally appear in the network. The larger inhibitory coupling strength, the more easily the system generates an ordered wave for sufficiently large strength of excitatory coupling. The appearance of differently ordered waves is closely related to the initial state of the system and coupling strength. As the excitatory and inhibitory coupling strengths are appropriately selected, the system can spontaneously generate the maze pattern, planar wave, single spiral wave, multiple spiral wave, paired spiral waves rotating in the opposite directions, two-arm spiral wave, target wave and inward square wave and so on. The probability for spontaneously forming a single spiral wave is far less than that for forming a small spiral wave. The occurrence probabilities of spiral wave, maze pattern and inward square wave reach 27.5%, 21.5% and 10%, respectively. The maze pattern is composed of many plane waves with different propagation directions. The occurrence probabilities of other ordered waves are quite small. These results conduce to understanding the self-organization phenomena occurring in the cerebral cortex.
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Keywords:
- spiral wave /
- maze pattern /
- Hindmarsh-Rose neuron /
- inhibitory coupling
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[2] 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] Ma J, Huang L, Ying H P, Pu Z S 2012 Chin. Sci. Bull. 57 2094
[9] Gu H G, Jia B, Li Y Y, Chen G R 2013 Physica A 392 1361
[10] Hu B, Ma J, Tang J 2013 PloS One 8 e0069251
[11] Qin H X, Ma J, Wang C N, Chu R T 2014 Sci. China:Phys. Mech. Astron. 57 1918
[12] Ma J, Tang J 2015 Sci. China:Tech. Sci. 58 2038
[13] Yao Y G, Deng H Y, Ma C Z, Yi M, Ma J 2017 PloS One 12 e0171273
[14] Yao Y G, Deng H Y, Ma C Z, Yi M, Ma J 2017 Scientific Reports 7 43151
[15] Jung P, Cornell-Bell A, Madden K S, Moss F 1998 J. Neurophysiol. 79 1098
[16] Ma J, Wu Y, Ying H P, Jia Y 2011 Chin. Sci. Bull. 56 151
[17] Wang C N, Ma J, Hu B L, Jin W Y 2015 Int. J. Mod. Phys. B 29 1550043
[18] Wang P, Li Q Y, Tang G N 2018 Acta Phys. Sin. 67 030502 (in Chinese)[汪芃, 李倩昀, 唐国宁 2018 物理学报 67 030502]
[19] Fohlmeister C, Gerstner W, Ritz R, Hemmen J L 1995 Neural Comput. 7 905
[20] Xiao W W, Gu H G, Liu M R 2016 Sci. China:Tech. Sci. 59 1943
[21] Tao Y, Gu H G 2017 Int. J. Mod. Phys. B 31 1750179
[22] Okun M, Lampl I 2008 Nat. Neurosci. 11 535
[23] Soriano J, Martínez M R, Tlusty T, Moses E 2008 PNAS 105 13758
[24] Hindmarsh J L, Rose R M 1984 Proc. R. Soc. Lond. B 221 87
[25] Adhikari B M, Prasad A, Dhamala M 2011 Chaos 21 023116
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[1] Sato T K, Nauhaus I, Carandini M 2012 Neuron 75 218
[2] Huang X Y, Xu W F, Liang J M, Takagaki K, Gao X, Wu J Y 2010 Neuron 68 978
[3] Huang X Y, William C T, Yang Q, Ma H T, Carlo R L, Steven J S, Wu J Y 2004 J. Neurosci. 24 9897
[4] Viventi J, Kim D H, Vigeland L, Frechette E S, Blanco J A, Kim Y S, Avrin A E, Tiruvadi V R, Hwang S W, Vanleer A C, Wulsin D F, Davis K, Gelber C E, Palmer L, Spiegel J V, Wu J, Xiao J L, Huang Y G, Contreras D, Rogers J A, Litt B 2011 Nat. Neurosci. 14 1599
[5] Davidenko J M, Pertsov A V, Salomonsz, Baxter W, Jalife J 1992 Nature 355 349
[6] Yu Y F, Santos L M, Mattiace L A, Costa M L, Ferreira L C, Benabou K, Kim A H, Abrahams J, Bennett M V L, Rozental R 2012 PNAS 109 2585
[7] Wang Q Y, Perc M, Duan Z S, Chen G R 2008 Phys. Lett. A 372 5681
[8] Ma J, Huang L, Ying H P, Pu Z S 2012 Chin. Sci. Bull. 57 2094
[9] Gu H G, Jia B, Li Y Y, Chen G R 2013 Physica A 392 1361
[10] Hu B, Ma J, Tang J 2013 PloS One 8 e0069251
[11] Qin H X, Ma J, Wang C N, Chu R T 2014 Sci. China:Phys. Mech. Astron. 57 1918
[12] Ma J, Tang J 2015 Sci. China:Tech. Sci. 58 2038
[13] Yao Y G, Deng H Y, Ma C Z, Yi M, Ma J 2017 PloS One 12 e0171273
[14] Yao Y G, Deng H Y, Ma C Z, Yi M, Ma J 2017 Scientific Reports 7 43151
[15] Jung P, Cornell-Bell A, Madden K S, Moss F 1998 J. Neurophysiol. 79 1098
[16] Ma J, Wu Y, Ying H P, Jia Y 2011 Chin. Sci. Bull. 56 151
[17] Wang C N, Ma J, Hu B L, Jin W Y 2015 Int. J. Mod. Phys. B 29 1550043
[18] Wang P, Li Q Y, Tang G N 2018 Acta Phys. Sin. 67 030502 (in Chinese)[汪芃, 李倩昀, 唐国宁 2018 物理学报 67 030502]
[19] Fohlmeister C, Gerstner W, Ritz R, Hemmen J L 1995 Neural Comput. 7 905
[20] Xiao W W, Gu H G, Liu M R 2016 Sci. China:Tech. Sci. 59 1943
[21] Tao Y, Gu H G 2017 Int. J. Mod. Phys. B 31 1750179
[22] Okun M, Lampl I 2008 Nat. Neurosci. 11 535
[23] Soriano J, Martínez M R, Tlusty T, Moses E 2008 PNAS 105 13758
[24] Hindmarsh J L, Rose R M 1984 Proc. R. Soc. Lond. B 221 87
[25] Adhikari B M, Prasad A, Dhamala M 2011 Chaos 21 023116
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