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簇放电是神经系统复杂的、多时间尺度的非线性现象, 具有多样性, 在兴奋性或抑制性作用下实现生理功能. 近期较多研究发现了与通常概念(抑制性作用引起电活动降低、兴奋性作用引起放电增强)相反的现象, 丰富了非线性科学的内涵. 本文关注于抑制性和兴奋性自突触反馈都会诱发的一类复杂的混合簇放电产生的反常现象及其分岔机制. 利用快慢变量分离, 确认了放电的复杂之处: 簇结束于极限环的鞍结分岔之后要先经过去极化阻滞才到休止期. 进一步, 揭示了该鞍结分岔在反常现象的产生中起到了关键作用. 抑制性自反馈引起了该分岔的左移导致簇的参数范围变宽, 引起簇内峰个数增多和平均放电频率增加; 而兴奋性自突触则引起该分岔右移导致电活动降低. 与其他类簇放电只在抑制性自反馈下产生反常现象和慢突触诱发的反常现象不同, 该结果给出了簇放电的反常现象的新示例及调控机制, 展示了反常现象的多样性, 有助于认识脑神经元簇放电和自反馈调控的潜在功能.Bursting is a complex multiple-time-scale nonlinear phenomenon in a nervous system and exhibits diverse patterns, which is modulated by the excitatory or inhibitory effect to achieve the physiological functions. According to the bifurcations of the fast subsystem, bursting is classified as different patterns including the mixed-mode bursting. Recently, many studies have found the paradoxical phenomena contrary to the common concept that the inhibitory effect induces the electrical activity to decrease or the excitatory effect induces the discharge activity to increase, which enriches the connotation of the nonlinear dynamics. To identify more examples of paradoxical phenomena of different bursting patterns and the underlying nonlinear mechanism, in the present study the paradoxical phenomena for the complex mixed-mode oscillations of the bursting pattern induced by the self-feedback mediated by both the inhibitory autapse and excitatory autapse and the bifurcation mechanism are acquired in the modified Morris-Lecar model. By using the fast/slow variable dissection method, the complex dynamics of the bursting is acquired, which is that the depolarization block behavior appears after the burst and before the quiescent state. The burst begins from a saddle-node bifurcation in an invariant cycle (SNIC) and terminates at a fold limit cycle (FLC) bifurcation. Furthermore, the FLC bifurcation is identified to play a key role in generating the paradoxical phenomenon. The inhibitory autapse induces the FLC bifurcation to shift leftward. However, the SNIC point remains unchanged. The change of FLC bifurcation point leads the parameter range of the burst to widen, the number of spikes per burst to become larger, and the average firing frequency to turn higher. Unlike the inhibitory autapse, the excitatory autapse induces the FLC bifurcation to shift rightward, and SNIC to be unchanged, thus reducing the bursting activity. Such results are different from those of the paradoxical phenomenon induced by the inhibitory autapse instead of excitatory autapse for the other bursting pattern and by the slow auatpse, which present a novel example and regulation mechanism of the paradoxical phenomena of the bursting patterns and show the diversity of the paradoxical phenomena, thus helping understand the potential functions of the bursting and self-feedback modulations of the brain neurons.
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Keywords:
- bifurcation /
- bursting /
- self-feedback /
- paradoxical behavior /
- mixed-mode oscillation
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图 1 MML模型未加自突触的混合簇放电的膜电位 (a) 周期3,
$ {V_u}{{ = }}0.02 $ ; (b) 周期4,$ {V_u}{{ = }}0.05 $ ; (c) 周期6,$ {V_u}{{ = }}0.1 $ , 数字1—6表示一个簇内的第1—6个峰; (d) 周期8,$ {V_u}{{ = }}0.12 $ Fig. 1. Membrane potential of mixed oscillations in the MML model without autapse: (a) Period-3 bursting when
$ {V_u}{{ = }}0.02 $ ; (b) Period-4 bursting when$ {V_u}{{ = }}0.05 $ ; (c) Period-6 bursting when$ {V_u}{{ = }}0.1 $ , The numbers 1–6 represent the 1st–6th spikes within a burst; (d) Period-8 bursting when$ {V_u}{{ = }}0.12 $ .
图 2 周期6簇放电的快慢变量分离 (a) 快子系统的平衡点及分岔. 上支黑色实线表示稳定的焦点, 黑色虚线表示不稳定的焦点, 中支黑色虚线表示鞍点, 下支黑色实线表示稳定的结点, H点为sub-Hopf分岔点, SNIC为不变环上的鞍结分岔点.
$ {V_{\max }} $ 和$ {V_{\min }} $ 表示稳定极限环的极大和极小值. (b) 快子系统的极限环分岔(蓝色实线表示稳定的极限环, 蓝色虚线表示不稳定的极限环)与图(a)的叠加. (c)周期6簇放电的相轨线(红)与图(b)的叠加. (d) 图(c)的局部放大, 数字1—6表示一个簇内的第1—6个峰Fig. 2. Fast/slow variable dissection of the period-6 bursting: (a) Bifurcation diagram of the equilibria of the fast subsystem. Upper black solid line, upper black dotted line, middle black dotted line, and lower black solid line is composed of the stable focus, unstable focus, saddle, and stable node, respectively. H and SNIC represents the subcritical Hopf bifurcation and saddle-node bifurcation on an invariant circle.
$ {V_{\max }} $ and$ {V_{\min }} $ represent the maximal and minimal value of the stable limit cycle, respectively. (b) Bifurcation diagram of the limit cycles (solid and dashed bule lines represent the stable and unstable limit cycle) of the fast subsystem and Fig. (a) plotted in one figure. (c) The trajectory of the period-6 bursting (red) superimposes Fig. (b). (d) Partial enlargement of Fig. (c). The numbers 1–6 represents the 1st–6th spikes within a burst.
图 3 自突触电流(红)影响下的膜电位(黑) (a1)—(a3) 抑制性自突触 (a1) 周期8簇放电,
$ g = 0.01 $ ; (a2) 周期10簇放电,$ g = 0.015 $ ; (a3) 周期19簇,$ g = 0.02 $ . (b1)—(b3) 兴奋性自突触 (b1) 周期3簇放电,$ g = 0.02 $ ; (b2) 周期2簇放电,$ g = 0.03 $ ; (b3) 周期1簇放电,$ g = 0.04 $ Fig. 3. Membrane potential (black) under the influence of autaptic current (red). (a1)–(a3) Inhibitory autapse: (a1) Period-8 bursting when
$ g = 0.01 $ ; (a2) period-10 bursting when$ g = 0.015 $ ; (a3) period-19 bursting when$ g = 0.02 $ . (b1)–(b3) Excitatory autupse: (b1) Period-3 bursting when$ g = 0.02 $ ; (b2) period-2 bursting when$ g = 0.03 $ ; (b3) Period-1 bursting when$ g = 0.04 $ .
图 4 不同自突触电导下的簇放电的峰峰间期ISI随序号的变化 (a) 抑制性自突触,
$ g = 0 $ (蓝),$ g = 0.01 $ (绿),$ g = 0.015 $ (红)和$ g = 0.02 $ (黑); (b) 兴奋性自突触,$ g = 0 $ (蓝),$ g = 0.02 $ (粉)和$ g = 0.03 $ (青)Fig. 4. Change of ISI respect to the sequential number at different levels of autaptic conductance: (a) Inhibitory autaptic current, blue, green, red, and black lines represent
$ g = 0 $ ,$ g = 0.01 $ ,$ g = 0.015 $ , and$ g = 0.02 $ , respectively; (b) excitatory autaptic current, blue, pink, and cyan-blue lines represents$ g = 0 $ ,$ g = 0.02 $ , and$ g = 0.03 $ , respectively.
图 5 平均放电频率(上, 绿)与自突触电流的平均值(下, 蓝)随自突触电导g的变化 (a) 抑制性自突触; (b) 兴奋性自突触
Fig. 5. Change of the mean firing frequency (green, upper) and the mean value of autaptic current (blue, lower) with respect to g: (a) Inhibitory autapse; (b) excitatory autapse.
图 6 不同自突触电导强度下快子系统的分岔 (a1)—(a4) 抑制性自突触: (a1)平衡点分岔; (a2) 图(a1)的局部放大; (a3) 极限环分岔与图(a1)的叠加; (a4) 图(a3)的局部放大. 电导强度的取值为
$ g = 0 $ (蓝),$ g = 0.01 $ (绿),$ g = 0.015 $ (红)和$ g = 0.02 $ (黑). (b1)—(b4)兴奋性自突触: (b1) 平衡点分岔; (b2) 图(b1)的局部放大; (b3) 极限环分岔与图(b1)的叠加; (b4) 图(b3)的局部放大. 电导强度的取值为$ g = 0 $ (蓝),$ g = 0.02 $ (粉),$ g = 0.03 $ (青)和$ g = 0.04 $ (棕)Fig. 6. Bifurcations of the fast subsystem at different values of the autapse conductance. (a1)–(a4) Inhibitory autapse: (a1) Bifurcations of the equilibria; (a2) partial enlargement of Fig. (a1); (a3) bifurcations of the limit cycles superimposes Fig. (a1); (a4) partial enlargement of Fig. (a3). Blue, green, red, and black lines represent
$ g = 0 $ ,$ g = 0.01 $ ,$ g = 0.015 $ , and$ g = 0.02 $ , respectively. (b1)–(b4) Excitatory autapse: (b1) Bifurcations of the equilibria; (b2) partial enlargement of Fig. (b1); (b3) bifurcation of the limit cycles superimposes Fig. (b1); (b4) partial enlargement of Fig. (b3). Blue, pink, cyan-blue, and brown curves represents$ g = 0 $ ,$ g = 0.02 $ ,$ g = 0.03 $ and$ g = 0.04 $ , respectively.
图 7 不同抑制性自突触电导强度下的簇放电的快慢变量分离 (a1)
$ g = 0 $ ; (a2) 图(a1)的放大; (b1)$ g = 0.01 $ , 数字1—6代表簇内第1—6个放电峰; (b2) 图(b1)的放大; (c1)$ g = 0.015 $ ; (c2) 图(c1)的放大; (d1)$ g = 0.02 $ ; (d2) 图(d1)的放大Fig. 7. Fast/slow variable dissection of bursting at different values of the inhibitory autapse conductance: (a1)
$ g = 0 $ , and the numbers 1–6 represent the 1st–6th spikes within a burst; (a2) enlargement of Fig. (a1); (b1)$ g = 0.01 $ ; (b2) enlargement of Fig. (b1); (c1)$ g = 0.015 $ ; (c2) enlargement of Fig. (c1); (d1)$ g = 0.02 $ ; (d2) enlargement of Fig. (d1).
图 8 不同兴奋性自突触电导强度下簇放电的快慢变量分离 (a)
$ g = 0 $ ; (b)$ g = 0.02 $ ; (c)$ g = 0.03 $ ; (d)$ g = 0.04 $ Fig. 8. Fast/slow variable dissection of the bursting at different values of the excitatory autapse conductance: (a)
$ g = 0 $ ; (b)$ g = 0.02 $ ; (c)$ g = 0.03 $ ; (d)$ g = 0.04 $ .
图 9 快子系统的极限环的周期随分岔参数u的变化 (a) 抑制性自突触, 蓝, 绿, 红和黑色曲线分别对应
$ g = 0 $ ,$ g = 0.01 $ ,$ g = 0.015 $ 和$ g = 0.02 $ ; (b) 兴奋性自突触, 蓝、粉、青和棕色曲线分别对应g = 0, g = 0.02, g = 0.03和g = 0.04Fig. 9. Period of the limit cycle of the fast system changes with respect to the bifurcation parameter u: (a) Inhibitory autupse. The blue, green, red, and black curves correspond to
$ g = 0 $ ,$ g = 0.01 $ ,$ g = 0.015 $ , and$ g = 0.02 $ , respectively. (b) Excitatory autupse. The blue, pink, cyan-blue, and brown curves correspond to g = 0, g = 0.02, g = 0.03, and g = 0.04, respectively.
图 10 快子系统在
$ (u, {\text{g}}) $ 平面的双参数分岔 (a1) 抑制性自突触; (a2) 图(a1)的局部放大; (b1) 兴奋性自突触; (b2) 图(b1)的局部放大Fig. 10. Double-parameter bifurcation in
$ (u, {\text{g}}) $ plane of the fast subsystem: (a1) Inhibitory autapse; (a2) enlargement of Fig (a1); (b1) excitatory autapse; (b2) enlargement of Fig (b1). -
[1] Yang Y, Cui Y H, Sang K N, Dong Y Y, NI Z Y, Ma S S, Hu H L 2018 Nature 554 317
Google Scholar
[2] Wang X J 2010 Phys. Rev. 90 1195
Google Scholar
[3] Mondal A, Upadhyay R K, Ma J, Yadav B K, Sharma S K, Mondal A 2019 Cogn. Neurodyn. 13 393
Google Scholar
[4] Glass L 2001 Nature 410 277
Google Scholar
[5] Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171
Google Scholar
[6] Duan L X, Liang W J, Ji W C, Xi H G 2020 Int. J. Bifurcation Chaos 30 2050192
Google Scholar
[7] Desroches M, Guckenheimer J, Krauskopf B, Kuehn C, Osinga H M, Wechselberger M 2012 SIAM Rev. 54 211
Google Scholar
[8] Lu B, Liu S, Jiang X, Jing W, Wang X 2017 Discrete Cont. Dyn. -S 10 487
Google Scholar
[9] Zhan F B, Liu S Q, Zhang X H, Wang J, Lu B 2018 Nonlinear Dyn. 94 807
Google Scholar
[10] Lisman J E 1997 Trends Neurosci. 20 38
Google Scholar
[11] Braun H A, Wissing H, Schäfer K, Hirsch M C 1994 Nature 367 270
Google Scholar
[12] Gu H G, Pan B B 2015 Front. Comput. Neurosci. 9 108
Google Scholar
[13] Gu H G, Pan B B, Chen G R, Duan L X 2014 Nonlinear Dyn. 78 391
Google Scholar
[14] Yin L P, Zheng R, Ke W, He Q, Zhang Y, Li J, Wang B, Mi Z, Long Y S, Rasch M J, Li T, Luan G, Shu Y 2018 Nat. Commun. 9 4890
Google Scholar
[15] Costa K P, Ulrich E, Michael S, Petra R, Dan R, Peter F, Nikolaos K, Lana K I, Joseph K 2021 Cereb. Cortex 31 2013
[16] Satterlie R A 1985 Science 229 402
Google Scholar
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Google Scholar
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Google Scholar
[19] 华洪涛, 陆博, 古华光 2020 物理学报 69 090502
Google Scholar
Hua H T, Lu B, Gu H G 2020 Acta Phys. Sin. 69 090502
Google Scholar
[20] 杨永霞, 李玉叶, 古华光 2020 物理学报 69 040501
Google Scholar
Yang Y X, Li Y Y, Gu H G 2020 Acta Phys. Sin. 69 040501
Google Scholar
[21] Wu F Q, Gu H G 2020 Int. J. Bifurcation Chaos 30 2030009
Google Scholar
[22] Zhao Z G, Li L, Gu H G 2018 Front. Cell. Neurosci. 12 62
Google Scholar
[23] Dodla R, Rinzel J 2006 Phys. Rev. E 73 010903
Google Scholar
[24] Uzuntarla M, Torres J J, Calim A, Barreto E 2019 Neural Networks 110 131
Google Scholar
[25] Elson R C, Selverston A I, Abarbanel H D I, Rabinovich M I 2002 J. Neurophysiol. 88 1166
Google Scholar
[26] Jia B, Wu Y, He D, Guo B H, Xue L 2018 Nonlinear Dyn. 93 1599
Google Scholar
[27] Yao C G, He Z W, Nakano T, Qian Y, Shuai J W 2019 Nonlinear Dyn. 97 1425
Google Scholar
[28] Jia Y B, Gu H G, Li Y Y, Ding X L 2021 Commun. Nonlinear Sci. Numer. Simul. 95 105643
Google Scholar
[29] Van Der Loos H, Glaser E M 1972 Brain Res. 48 355
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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