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The photoreceptors can receive all kinds of visible light which is translated to the bioelectrical signal for the visual cortex. The function would be simulated by the photoelectric effect. This paper studies the dynamic characteristics of FitzHugh-Nagumo neurons coupled with a phototube. In the parameter space of phototube, the synchronization region of the coupled system in which the neuron mode is in chaos and burst, is discussed in detail; the data show that the forced resonance is prominent in the complete synchronization of the system when the coupling strength is low, while the phase synchronization is observed in numerical experiment when the coupling strength is strong. The active operation of the phototube, as well the inverse cutoff voltage can modulate the synchronization of the system. Our work can be used to understand the mechanism of the retinal diseases, such as macular degeneration.
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Keywords:
- neuron /
- phototube /
- synchronization /
- phase lock
[1] Ma J, Song X, Jin W, Wang C 2015 Chaos, Solitons Fractals 80 31Google Scholar
[2] Iqbal M, Rehan M, Hong K S 2017 Plos One 12 e0176986Google Scholar
[3] Sotero R C, Trujillo-Barreto N J 2008 Neuroimage 39 290Google Scholar
[4] Izhikevich E M 2004 IEEE Trans. Neural Networks 15 1063Google Scholar
[5] Ibarz B, Casado J M, Sanjuán M A F 2011 Phys. Rep. 501 1Google Scholar
[6] Hodgkin A L, Huxley A F 1990 Bull. Math. Biol. 52 25Google Scholar
[7] Fitzhugh R 1961 Biophys. J. 1 445Google Scholar
[8] Shilnikov A 2012 Nonlinear Dyn. 68 305Google Scholar
[9] Miesenbock G, Kevrekidis I G 2005 Annu. Rev. Neurosci. 28 533Google Scholar
[10] Gu H, Pan B 2015 Nonlinear Dyn. 81 2107Google Scholar
[11] Pikovskii A, Rabinovich M 1978 Dokl. Akad. Nauk SSSR 239 301
[12] Lv M, Wang C, Ren G, Ma J, Song X 2016 Nonlinear Dyn. 85 1479Google Scholar
[13] Baines P G 2008 Prog. Phys. Geogr. 32 475Google Scholar
[14] Zhang X, Wang C, Ma J, Ren G 2020 Mod. Phys. Lett. B 2050267Google Scholar
[15] Zhang G, Ma J, Alsaedi A, Ahmad B, Alzahrani F 2018 Appl. Math. Comput. 321 290Google Scholar
[16] Yao Z, Ma J, Yao Y, Wang C 2019 Nonlinear Dyn. 96 205Google Scholar
[17] Xu Y M, Yao Z, Hobiny A, Ma J 2019 Front. Inform. Tech. El. 20 571Google Scholar
[18] Liu Z, Wang C, Jin W, Ma J 2019 Nonlinear Dyn. 97 2661Google Scholar
[19] Tosini G, Doyle S, Geusz M, Menaker M 2000 Proc. Natl. Acad. Sci. 97 11540Google Scholar
[20] Menaker M 1972 Sci. Am. 226 22Google Scholar
[21] Kennedy D 1958 Am. J. Ophthal. 46 19Google Scholar
[22] Martenson M E, Halawa O I, Tonsfeldt K J, et al. 2016 Pain 157 868Google Scholar
[23] Liu Y, Xu W J, Ma J, Alzahrani F, Hobiny A 2020 Front. Inform. Tech. El. 21 1387Google Scholar
[24] Li J R, Wang J P, Jiang L 1994 Biosens. Bioelectron. 9 147Google Scholar
[25] Zou W, Senthilkumar D V, Zhan M, Kurths J 2013 Phys. Rev. Lett. 111 014101Google Scholar
[26] Wu Y, Xiao J, Hu G, Zhan M 2012 EPL 97 40005Google Scholar
[27] Perc M 2009 Biophys. Chem. 141 175Google Scholar
[28] Lin W, Wang Y, Ying H, Lai Y C, Wang X 2015 Phys. Rev. E 92 012912Google Scholar
[29] 张平伟, 唐国宁, 罗晓曙 2005 物理学报 54 3497Google Scholar
Zhang P W, Tang G N, Luo X S 2005 Acta Phys. Sin. 54 3497Google Scholar
[30] Wouapi K M, Fotsin B H, Louodop F P, Feudjio K F, Njitacke Z T, Djeudjo T H 2020 Cogn. Neurodyn. 14 375Google Scholar
[31] Shafiei M, Jafari S, Parastesh F, Ozer M, Kapitaniak T, Perc M 2020 Commun. Nonlinear Sci. Numer. Simul. 84 105175Google Scholar
[32] Phan C, You Y 2020 Nonlinear. Anal.-Real 55 103139Google Scholar
[33] Moayeri M M, Rad J A, Parand K 2020 Comput. Math. Appl. 80 1887Google Scholar
[34] Makovkin S Y, Shkerin I V, Gordleeva S Y, Ivanchenko M V 2020 Chaos, Solitons Fractals 138 109951Google Scholar
[35] Zou Y L, Zhu J, Chen G, Luo X S 2005 Chaos, Solitons Fractals 25 1245Google Scholar
[36] Zhou S, Hong Y, Yang Y, Lü L, Li C 2020 Pramana J. Phys. 94 34Google Scholar
[37] Venkatesh P, Venkatesan A, Lakshmanan M 2016 Pramana J. Phys. 86 1195Google Scholar
[38] Sivaganesh G, Sweetlin M D, Arulgnanam A 2016 J. Korean Phys. Soc. 69 124Google Scholar
[39] Binczak S, Jacquir S, Bilbault J M, Kazantsev V B, Nekorkin V I 2006 Neural Networks 19 684Google Scholar
[40] Wade J J, Mcdaid L J, Harkin J, Crunelli V, Kelso J S 2011 PloS One 6 e29445Google Scholar
[41] Sambas A, WS M S, Mamat M 2015 J. Eng. Sci. Tech. Rev. 8 89Google Scholar
[42] Daoudal G, Hanada Y, Debanne D 2002 PNAS 99 14512Google Scholar
[43] Chorev E, Brecht M 2012 J. Neurophysiol. 108 1584Google Scholar
[44] 杨永霞, 李玉叶, 古光华 2020 物理学报 69 040501Google Scholar
Yhang Y X, Li Y Y, Gu G H 2020 Acta Phys. Sin. 69 040501Google Scholar
[45] 汪芃, 李倩昀, 唐国宁 2018 物理学报 67 030502Google Scholar
Wang P, Li Q Y, Tang G N 2018 Acta Phys. Sin. 67 030502Google Scholar
[46] FitzHugh R 1955 Bull. Math. Biophys. 17 257Google Scholar
[47] Nagumo J, Arimoto S, Yoshizawa S 1962 Proc. IRE 50 2061Google Scholar
[48] Kawato M, Suzuki R 1980 J. Theor. Biol. 86 547Google Scholar
[49] Okuda M 1981 Prog. Theor. Phys. 66 90Google Scholar
[50] Treutlein H, Schulten K 1985 Ber. Bunse. Ges. Phys. Chem. 89 710Google Scholar
[51] Rajasekar S, Lakshmanan M 1988 J. Theor. Biol. 133 473Google Scholar
[52] Einstein A 1905 Ann. Physik. 17 132
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图 5 耦合系统中神经元的ISI和放电序列(f = 0.16) (a) ua = 0.1; (b) I0 = 1.5, ua = 0.1; (c) I0 = 2.5, ua = 0.1; (d) I0 = 0.3; (e) ua = 1.5, I0 = 0.3; (f) ua = 2.3, I0 = 0.3
Fig. 5. ISI and the firing sequence of neuron in the coupled system (f = 0.16): (a) ua = 0.1; (b) I0 = 1.5, ua = 0.1; (c) I0 = 2.5, ua = 0.1; (d) I0 = 0.3; (e) ua = 1.5, I0 = 0.3; (f) ua = 2.5, I0 = 0.3.
图 9 耦合系统中神经元的ISI和放电序列(f = 0.002) (a) ua = 0.01; (b) I0 = 0.5, ua = 0.01; (c) I0 = 1.5, ua = 0.01; (d) I0 = 0.3; (e) ua = 0.5, I0 = 0.3; (f) ua = 1.5, I0 = 0.3
Fig. 9. ISI and the firing sequence of neuron in the coupled system (f = 0.002): (a) ua = 0.01; (b) I0 = 0.5, ua = 0.1; (c) I0 = 1.5, ua = 0.1; (d) I0 = 0.3; (e) ua = 0.5, I0 = 0.3; (f) ua = 1.5, I0 = 0.3.
图 11 最大误差函数和最大相位差随参数的变化(灰色曲线为(6)式中的非线性耦合, 红色曲线为线性耦合) (a), (b) ua = 0.01; (c), (d) I0 = 0.001
Fig. 11. The maximum error function and the maximum phase difference change with the parameters, the grey curve represents the nonlinear coupling in Eq. (6) and the red curve is the linear one: (a), (b) ua = 0.01; (c), (d) I0 = 0.001.
图 12 系统误差、相位差和光电管功率随时间的演化(ua = 0.01) (a), (e), (i) I0 = 0.001; (b), (f), (j) I0 = 0.01; (c), (g), (k) I0 = 0.013; (d), (h), (l) I0 = 0.014
Fig. 12. Evolution of error function, phase error and phototube power (ua = 0.01): (a), (e), (i) I0 = 0.001; (b), (f), (j) I0 = 0.01; (c), (g), (k) I0 = 0.013; (d), (h), (l) I0 = 0.014.
表 1 不同外界刺激频率下的耦合FHN神经元分类
Table 1. Category of the coupled FHN neurons driven by external stimulation with different frequencies.
频率f 0.16 0.002 0.012 0.06 放电状态 混沌放电 簇放电 尖峰放电 周期放电 反向截止电压ua 0.1 0.5 0.01 0.5 0.01 0.5 0.01 0.5 耦合分类 case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 -
[1] Ma J, Song X, Jin W, Wang C 2015 Chaos, Solitons Fractals 80 31Google Scholar
[2] Iqbal M, Rehan M, Hong K S 2017 Plos One 12 e0176986Google Scholar
[3] Sotero R C, Trujillo-Barreto N J 2008 Neuroimage 39 290Google Scholar
[4] Izhikevich E M 2004 IEEE Trans. Neural Networks 15 1063Google Scholar
[5] Ibarz B, Casado J M, Sanjuán M A F 2011 Phys. Rep. 501 1Google Scholar
[6] Hodgkin A L, Huxley A F 1990 Bull. Math. Biol. 52 25Google Scholar
[7] Fitzhugh R 1961 Biophys. J. 1 445Google Scholar
[8] Shilnikov A 2012 Nonlinear Dyn. 68 305Google Scholar
[9] Miesenbock G, Kevrekidis I G 2005 Annu. Rev. Neurosci. 28 533Google Scholar
[10] Gu H, Pan B 2015 Nonlinear Dyn. 81 2107Google Scholar
[11] Pikovskii A, Rabinovich M 1978 Dokl. Akad. Nauk SSSR 239 301
[12] Lv M, Wang C, Ren G, Ma J, Song X 2016 Nonlinear Dyn. 85 1479Google Scholar
[13] Baines P G 2008 Prog. Phys. Geogr. 32 475Google Scholar
[14] Zhang X, Wang C, Ma J, Ren G 2020 Mod. Phys. Lett. B 2050267Google Scholar
[15] Zhang G, Ma J, Alsaedi A, Ahmad B, Alzahrani F 2018 Appl. Math. Comput. 321 290Google Scholar
[16] Yao Z, Ma J, Yao Y, Wang C 2019 Nonlinear Dyn. 96 205Google Scholar
[17] Xu Y M, Yao Z, Hobiny A, Ma J 2019 Front. Inform. Tech. El. 20 571Google Scholar
[18] Liu Z, Wang C, Jin W, Ma J 2019 Nonlinear Dyn. 97 2661Google Scholar
[19] Tosini G, Doyle S, Geusz M, Menaker M 2000 Proc. Natl. Acad. Sci. 97 11540Google Scholar
[20] Menaker M 1972 Sci. Am. 226 22Google Scholar
[21] Kennedy D 1958 Am. J. Ophthal. 46 19Google Scholar
[22] Martenson M E, Halawa O I, Tonsfeldt K J, et al. 2016 Pain 157 868Google Scholar
[23] Liu Y, Xu W J, Ma J, Alzahrani F, Hobiny A 2020 Front. Inform. Tech. El. 21 1387Google Scholar
[24] Li J R, Wang J P, Jiang L 1994 Biosens. Bioelectron. 9 147Google Scholar
[25] Zou W, Senthilkumar D V, Zhan M, Kurths J 2013 Phys. Rev. Lett. 111 014101Google Scholar
[26] Wu Y, Xiao J, Hu G, Zhan M 2012 EPL 97 40005Google Scholar
[27] Perc M 2009 Biophys. Chem. 141 175Google Scholar
[28] Lin W, Wang Y, Ying H, Lai Y C, Wang X 2015 Phys. Rev. E 92 012912Google Scholar
[29] 张平伟, 唐国宁, 罗晓曙 2005 物理学报 54 3497Google Scholar
Zhang P W, Tang G N, Luo X S 2005 Acta Phys. Sin. 54 3497Google Scholar
[30] Wouapi K M, Fotsin B H, Louodop F P, Feudjio K F, Njitacke Z T, Djeudjo T H 2020 Cogn. Neurodyn. 14 375Google Scholar
[31] Shafiei M, Jafari S, Parastesh F, Ozer M, Kapitaniak T, Perc M 2020 Commun. Nonlinear Sci. Numer. Simul. 84 105175Google Scholar
[32] Phan C, You Y 2020 Nonlinear. Anal.-Real 55 103139Google Scholar
[33] Moayeri M M, Rad J A, Parand K 2020 Comput. Math. Appl. 80 1887Google Scholar
[34] Makovkin S Y, Shkerin I V, Gordleeva S Y, Ivanchenko M V 2020 Chaos, Solitons Fractals 138 109951Google Scholar
[35] Zou Y L, Zhu J, Chen G, Luo X S 2005 Chaos, Solitons Fractals 25 1245Google Scholar
[36] Zhou S, Hong Y, Yang Y, Lü L, Li C 2020 Pramana J. Phys. 94 34Google Scholar
[37] Venkatesh P, Venkatesan A, Lakshmanan M 2016 Pramana J. Phys. 86 1195Google Scholar
[38] Sivaganesh G, Sweetlin M D, Arulgnanam A 2016 J. Korean Phys. Soc. 69 124Google Scholar
[39] Binczak S, Jacquir S, Bilbault J M, Kazantsev V B, Nekorkin V I 2006 Neural Networks 19 684Google Scholar
[40] Wade J J, Mcdaid L J, Harkin J, Crunelli V, Kelso J S 2011 PloS One 6 e29445Google Scholar
[41] Sambas A, WS M S, Mamat M 2015 J. Eng. Sci. Tech. Rev. 8 89Google Scholar
[42] Daoudal G, Hanada Y, Debanne D 2002 PNAS 99 14512Google Scholar
[43] Chorev E, Brecht M 2012 J. Neurophysiol. 108 1584Google Scholar
[44] 杨永霞, 李玉叶, 古光华 2020 物理学报 69 040501Google Scholar
Yhang Y X, Li Y Y, Gu G H 2020 Acta Phys. Sin. 69 040501Google Scholar
[45] 汪芃, 李倩昀, 唐国宁 2018 物理学报 67 030502Google Scholar
Wang P, Li Q Y, Tang G N 2018 Acta Phys. Sin. 67 030502Google Scholar
[46] FitzHugh R 1955 Bull. Math. Biophys. 17 257Google Scholar
[47] Nagumo J, Arimoto S, Yoshizawa S 1962 Proc. IRE 50 2061Google Scholar
[48] Kawato M, Suzuki R 1980 J. Theor. Biol. 86 547Google Scholar
[49] Okuda M 1981 Prog. Theor. Phys. 66 90Google Scholar
[50] Treutlein H, Schulten K 1985 Ber. Bunse. Ges. Phys. Chem. 89 710Google Scholar
[51] Rajasekar S, Lakshmanan M 1988 J. Theor. Biol. 133 473Google Scholar
[52] Einstein A 1905 Ann. Physik. 17 132
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