搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

兴奋性和抑制性自反馈压制靠近Hopf分岔的神经电活动比较

黎丽 赵志国 古华光

引用本文:
Citation:

兴奋性和抑制性自反馈压制靠近Hopf分岔的神经电活动比较

黎丽, 赵志国, 古华光

Suppression effects of excitatory and inhibitory self-feedbacks on neuronal spiking near Hopf bifurcation

Li Li, Zhao Zhi-Guo, Gu Hua-Guang
PDF
HTML
导出引用
  • 突触输入刺激神经元产生的电活动, 在神经编码中发挥着重要作用. 通常认为, 兴奋性输入增强电活动, 抑制性输入压制电活动. 本文选取可调节电流衰减速度的突触模型, 研究了兴奋性自突触在亚临界Hopf分岔附近压制神经元电活动的反常作用, 与抑制性自突触的压制作用进行了比较, 并采用相位响应曲线和相平面分析解释了压制作用的机制. 对于单稳的峰放电, 快速和中速衰减的兴奋性自突触分别可以诱发频率降低的峰放电和混合振荡(峰放电与阈下振荡的交替), 而中速和慢速衰减的抑制性自突触也可以分别诱发频率降低的峰放电和混合振荡. 对于与静息共存的峰放电, 除上述两种行为外, 中速衰减的兴奋性和慢速衰减的抑制性自突触还可以诱发静息. 兴奋性和抑制性自突触电流在不同的衰减速度下, 分别作用在峰放电的不同相位, 才能诱发同类压制行为. 结果丰富了兴奋性突触压制电活动反常作用的实例, 获得了兴奋性和抑制性自突触压制作用机制的不同, 给出了调控神经放电的新手段.
    Neural activities driven by synaptic inputs are important for neural coding. In general, excitatory synaptic inputs facilitate the firing activities of neurons, and inhibitory synaptic inputs suppress the firing activities. In the present work, the autapse model with adjustable current decay speed is considered, and the suppression effects of excitatory autapse on neuronal firing responses are simulated near subcritical Hopf bifurcation in the Morris-Lecar (ML) neuron model, which are compared with the suppression effects of the inhibitory autapse. Furthermore, the dynamical mechanisms of the suppression effects are acquired with the help of the phase response curve and phase trajectories perturbed by excitatory autaptic current. Near the subcritical Hopf bifurcation, the ML neuron exhibits monostable periodic spiking and coexistence of periodic spiking and resting state. For monostable periodic spiking, excitatory autaptic current with fast and middle decay speeds can induce the periodic spiking with reduced firing frequency and the mixed-mode oscillations (MMOs) that are alternations between subthreshold oscillations and a spike, respectively, and inhibitory autaptic current with middle and slow decay speeds can also induce these two behaviors, respectively. For the periodic spiking in the coexistence region, besides the above two behaviors, excitatory autaptic current with middle decay speed and inhibitory autaptic current with slow decay speed can induce the change from spiking to resting state. The results enrich the paradoxical cases that excitatory inputs suppress the neuronal firing responses, and present the different nonlinear mechanisms in the suppression effects of excitatory and inhibitory self-feedbacks on the neuronal spiking, which provide novel measures to modulate neuronal firing activity.
      通信作者: 赵志国, zzg164637758@163.com
    • 基金项目: 国家自然科学基金(批准号: 11802085, 11872276, 12072236)、广州市科技计划项目(批准号: 202102021167)、广东省科学院建设国内一流研究机构行动专项(批准号: 2021GDASYL-20210103088)、河南省科技攻关计划(批准号: 212102310827)和河南省高等学校重点科研项目(批准号: 21A110008)资助的课题
      Corresponding author: Zhao Zhi-Guo, zzg164637758@163.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11802085, 11872276, 12072236), the Science and Technology Project of Guangzhou, China (Grant No. 202102021167), GDAS’ Project of Science and Technology Development, China (Grant No. 2021GDASYL-20210103088), the Science and Technology Development Program of Henan Province, China (Grant No. 212102310827), and the Key Scientific Research Program of Higher Education of Henan Province, China (Grant No. 21A110008).
    [1]

    Izhikevich E M 2000 Int. J. Bifurcat. Chaos 10 1171Google Scholar

    [2]

    Izhikevich E M 2004 IEEE Trans. Neural Networks 15 1063Google Scholar

    [3]

    Bean B P 2007 Nat. Rev. Neurosci. 8 451Google Scholar

    [4]

    Bartos M, Vida I, Jonas P 2007 Nat. Rev. Neurosci. 8 45Google Scholar

    [5]

    Tateno T, Pakdaman K 2004 Chaos 14 511Google Scholar

    [6]

    Lee S G, Neiman A, Kim S 1998 Phys. Rev. E 57 3292Google Scholar

    [7]

    Prescott S A, De Koninck Y, Sejnowski T J 2008 PloS Comput. Biol. 4 e1000198Google Scholar

    [8]

    Ermentrout B 1996 Neural Comput. 8 979Google Scholar

    [9]

    Smeal R M, Ermentrout G B, White J A 2010 Philos. Trans. R. Soc. London, Ser. B 365 2407Google Scholar

    [10]

    Goldwyn J H, Slabe B R, Travers J B, Terman D 2018 PloS Comput. Biol. 14 e1006292Google Scholar

    [11]

    Guo D Q, Wang Q Y, Perc M 2012 Phys. Rev. E 85 061905Google Scholar

    [12]

    Silver R A 2010 Nat. Rev. Neurosci. 11 474Google Scholar

    [13]

    Ferrante M, Shay C F, Tsuno Y, Chapman G W, Hasselmo M E 2017 Cereb. Cortex 27 2111

    [14]

    Zhao Z G, Li L, Gu H G, Gao Y 2020 Nonlinear Dyn. 99 1129Google Scholar

    [15]

    Dodla R, Rinzel J 2006 Phys. Rev. E 73 010903Google Scholar

    [16]

    Dodla R, Svirskis G, Rinzel J 2006 J. Neurophysiol. 95 2664Google Scholar

    [17]

    Van Vreeswijk C, Abbott L F, Bard Ermentrout G 1994 J. Comput. Neurosci. 1 313Google Scholar

    [18]

    Wang X J, Buzsaki G 1996 J. Neurosci. 16 6402Google Scholar

    [19]

    Vida I, Bartos M, Jonas P 2006 Neuron 49 107Google Scholar

    [20]

    Li L, Zhao Z G, Gu H G 2019 Int. J. Bifurcat. Chaos 29 1950147Google Scholar

    [21]

    Uzuntarla M, Torres J J, Calim A, Barreto E 2019 Neural Networks 110 131Google Scholar

    [22]

    Van Der Loos H, Glaser E M 1972 Brain Res. 48 355Google Scholar

    [23]

    Bacci A, Huguenard J R 2006 Neuron 49 119Google Scholar

    [24]

    Cobb S R, Halasy K, Vida I, Nyiri G, Tamas G, Buhl E H, Somogyi P 1997 Neuroscience 79 629Google Scholar

    [25]

    Yin L P, Zheng R, Ke W, He Q S, Zhang Y, Li J L, Wang B, Mi Z, Long Y S, Rasch M J, Li T F, Luan G M, Shu Y S 2018 Nat. Commun. 9 4890Google Scholar

    [26]

    Saada R, Miller N, Hurwitz I, Susswein A J 2009 Curr. Biol. 19 479Google Scholar

    [27]

    Bacci A, Huguenard J R, Prince D A 2003 J. Neurosci. 23 859Google Scholar

    [28]

    Wang H T, Chen Y 2015 Chin. Phys. B 24 128709Google Scholar

    [29]

    Yilmaz E, Ozer M, Baysal V, Perc M 2016 Sci. Rep. 6 30914Google Scholar

    [30]

    Yang X L, Yu Y H, Sun Z K 2017 Chaos 27 083117Google Scholar

    [31]

    Qin H X, Wu Y, Wang C N, Ma J 2015 Commun. Nonlinear Sci. Numer. Simul. 23 164Google Scholar

    [32]

    Yilmaz E, Baysal V, Perc M, Ozer M 2016 Sci. China Technol. Sci. 59 364Google Scholar

    [33]

    Qin H X, Ma J, Wang C N, Chu R T 2014 Sci. China Phys. Mech. Astron. 57 1918Google Scholar

    [34]

    Ma J, Song X L, Tang J, Wang C N 2015 Neurocomputing 167 378Google Scholar

    [35]

    Yilmaz E, Baysal V, Ozer M, Perc M 2016 Physica A 444 538Google Scholar

    [36]

    Wang H T, Wang L F, Chen Y L, Chen Y 2014 Chaos 24 043113Google Scholar

    [37]

    Li Y Y, Schmid G, Hanggi P, Schimansky-Geier L 2010 Phys. Rev. E 82 061907Google Scholar

    [38]

    Wang H T, Ma J, Chen Y L, Chen Y 2014 Commun. Nonlinear Sci. Numer. Simul. 19 3242Google Scholar

    [39]

    Guo D Q, Wu S D, Chen M M, Perc M, Zhang Y S, Ma J L, Cui Y, Xu P, Xia Y, Yao D Z 2016 Sci. Rep. 6 26096Google Scholar

    [40]

    Guo D Q, Chen M M, Perc M, Wu S D, Xia C, Zhang Y S, Xu P, Xia Y, Yao D Z 2016 Europhys. Lett. 114 30001Google Scholar

    [41]

    Zhao Z G, Gu H G 2017 Sci. Rep. 7 6760Google Scholar

    [42]

    Song X L, Wang H T, Chen Y 2019 Nonlinear Dyn. 96 2341Google Scholar

    [43]

    Zhao Z G, Li L, Gu H G 2020 Sci. Rep. 10 3646Google Scholar

    [44]

    曹奔, 关利南, 古华光 2018 物理学报 67 240502Google Scholar

    Cao B, Guan L N, Gu H G 2018 Acta Phys. Sin. 67 240502Google Scholar

    [45]

    丁学利, 李玉叶 2016 物理学报 65 210502Google Scholar

    Ding X L, Li Y Y 2016 Acta Phys. Sin. 65 210502Google Scholar

    [46]

    Li Y Y, Gu H G, Jia Bing, Ding X L 2021 Sci. China Technol. Sci. 64 1459Google Scholar

    [47]

    Tikidji-Hamburyan R A, Martinez J J, White J A, Canavier C C 2015 J. Neurosci. 35 15682Google Scholar

    [48]

    谢勇, 程建慧 2017 物理学报 66 090501Google Scholar

    Xie Y, Cheng J H 2017 Acta Phys. Sin. 66 090501Google Scholar

    [49]

    Wilson C J 2017 J. Neurophysiol. 118 855Google Scholar

    [50]

    Canavier C C, Wang S G, Chandrasekaran L 2013 Front. Neural Circuits 7 194

    [51]

    丁学利, 贾冰, 李玉叶 2019 物理学报 68 180502Google Scholar

    Ding X L, Jia B, Li Y Y 2019 Acta Phys. Sin. 68 180502Google Scholar

    [52]

    Zhao Z, Li L, Gu H 2020 Commun. Nonlinear Sci. Numer. Simul. 85 105250Google Scholar

    [53]

    Moss S J, Smart T G 2001 Nat. Rev. Neurosci. 2 240

    [54]

    Prescott S A, Ratte S, De Koninck Y, Sejnowski T J 2008 J. Neurophysiol. 100 3030Google Scholar

    [55]

    Ermentrout B 2002 Simulating, Analyzing, and Animating Dynamical Systems: A guide to XPPAUT for Researchers and Students (Philadelphia: SIAM) pp77–173

    [56]

    Liu C M, Liu X L, Liu S Q 2014 Biol. Cybern. 108 75Google Scholar

    [57]

    Drion G, O'Leary T, Marder E 2015 Proc. Natl. Acad. Sci. U. S. A. 112 E5361Google Scholar

    [58]

    Coggan J S, Ocker G K, Sejnowski T J, Prescott S A 2011 J. Neural Eng. 8 065002Google Scholar

    [59]

    Rho Y A, Prescott S A 2012 PloS Comput. Biol. 8 e1002524Google Scholar

    [60]

    Dipoppa M, Gutkin B S 2013 Proc. Natl. Acad. Sci. U. S. A. 110 12828Google Scholar

  • 图 1  ML神经元模型(方程(1)和方程(2))动力学行为随${I_{{\text{app}}}}$变化 (a) 分岔, 红色(虚)实线为(不)稳定平衡点, 黑色(空)实心圆为(不)稳定极限环的最小值和最大值; (b) 放电频率

    Fig. 1.  Changes of dynamical behavior of the ML neuron model (Equation (1) and (2)) with respect to ${I_{{\text{app}}}}$: (a) Bifurcations, red (dashed) solid curve denotes (unstable) stable equilibria, and black solid (empty) circles denote minima and maxima of (unstable) stable limit cycles; (b) firing frequency.

    图 2  ML神经元模型(方程(1)和方程(2))对不同电流脉冲的相位响应曲线 (a) 兴奋性电流脉冲; (b) 抑制性电流脉冲

    Fig. 2.  . Phase response curve (PRC) of the ML neuron model (Equation (1) and (2)) to pulse current stimulations with different effects: (a) Excitatory current pulse; (b) inhibitory current pulse.

    图 3  ${I_{{\text{app}}}} = 42.{\text{6}}\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$时, 不同自突触诱导的频率降低的周期峰放电 (a) ${g_{{\text{exc}}}} = 2.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = 1.0\;{\text{m}}{{\text{s}}^{ - 1}}$的兴奋性自突触; (b) 图(a)的局部放大图; (c) ${g_{{\text{inh}}}} = 1.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = 0.3\;{\text{m}}{{\text{s}}^{ - 1}}$的抑制性自突触; (d) 图(c)的局部放大图. 黑和红色实线分别为ML神经元模型(方程(1)和方程(2))和自突触耦合ML神经元模型(方程(3)—(5))的膜电位, 红色虚线为自突触电流, 灰色虚线为黑色实线第一个峰峰间隙PRC的相位滞后和提前的临界点, 图(a)和图(b)中灰色虚线的左右两侧分为兴奋性刺激PRC的相位滞后和提前区域, 图(c)和图(d)中灰色虚线的左右两侧分为抑制性刺激PRC的相位提前和滞后区域

    Fig. 3.  Periodic spiking with reduced firing frequency induced by different autapses when ${I_{{\text{app}}}} = 42.6\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$: (a) Excitatory autapse with ${g_{{\text{exc}}}} = 2.0\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 1.0\;{\text{m}}{{\text{s}}^{ - 1}}$; (b) enlargement of Figure (a); (c) inhibitory autapse with ${g_{{\text{inh}}}} = 1.0\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 0.3\;{\text{m}}{{\text{s}}^{ - 1}}$; (d) enlargement of Figure (c). Black and red solid curves are the membrane potential for the ML neuron model (Equation (1) and (2)) and the ML neuron model with autapse (Equation (3)–(5)), respectively. Red dashed curve is the autaptic current, gray dashed vertical lines is critical border between phase delay and phase advance regions of the PRC for the first interspike interval of the black solid curve, the left and right sides of the gray dashed line in Figure (a) and Figure (b) respectively are the phase delay and advance regions of the PRC, the left and right sides of the gray dashed line in Figure (c) and Figure (d) respectively are phase advance and delay regions of the PRC.

    图 4  ${I_{{\text{app}}}} = 42.6\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$时, 不同自突触诱发共存峰放电(黑色实线)变为静息(红色实线) (a1) ${g_{{\text{exc}}}} = 2.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = $$ 0.4\;{\text{m}}{{\text{s}}^{ - 1}}$的兴奋性自突触; (a2) 图(a1)的局部放大; (a3) 图(a1)的相平面图; (a4) 图(a3)的局部放大; (b1) ${g_{{\text{inh}}}} = 1.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = 0.1\;{\text{m}}{{\text{s}}^{ - 1}}$的抑制性自突触; (b2) 图(b1)的局部放大; (b3) 对应于(b1)的相平面图; (b4) 图(b3)的局部放大. 在图(a1)和图(b1)中, 黑色实线为ML神经元模型(方程(1)和方程(2))膜电位, 蓝色和红色实线为自突触耦合ML神经元模型(方程(3)—(5))膜电位, 蓝色虚线为突触电流, 蓝色实线是对应于突触电流的膜电位. 在图(a3)和图(b3)中, 蓝色和红色实线分别对应于图(a1)和图(b1)中蓝色和红色实线的相轨迹. 灰色虚线和点线分别为方程(1)和方程(2)的$w$$V$零等倾斜线, 黑色实线和虚线分别为方程(1)和方程(2)的稳定和不稳定极限环, 红色实心点为方程(1)和方程(2)的稳定平衡点, 蓝色箭头表示相轨迹的方向, 青色实心点为兴奋性和抑制性突触电流改变相轨迹方向的关键相位

    Fig. 4.  Resting state (solid red curve) evoked from coexisting spiking (solid black curve) by different autapses when ${I_{{\text{app}}}} = $$ 42.6\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$: (a1) Excitatory autapse with ${g_{{\text{exc}}}} = 2.0\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 0.4\;{\text{m}}{{\text{s}}^{ - 1}}$; (a2) enlargement of Figure (a1); (a3) phase portrait corresponds to Figure (a1); (a4) enlargement of Figure (a3); (b1) inhibitory autapse with ${g_{{\text{inh}}}} = 1.0\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = $$ 0.1\;{\text{m}}{{\text{s}}^{ - 1}}$; (b2) enlargement of Figure (b1); (b3) phase portrait corresponds to Figure (b1); (b4) enlargement of Figure (b3). In Figure (a1) and Figure (b1), black solid line is the membrane potential of the ML neuron (Equation (1) and (2)), blue and red solid lines are membrane potential of the ML neuron with autapse (Equation (3)-(5)), blue dashed curve is the autaptic current and the blue solid curve is membrane potential effected by the autaptic current. In Figure (a3) and Figure (b3), blue and red solid curves are the trajectories corresponding to the blue and red solid parts in Figure (a1) and Figure (b1), respectively, gray dashed (dotted) curve is the nullcline $\dot w = 0$($\dot V = 0$) of Equation (1) and (2), black solid (dashed) curve is the stable (unstable) limit cycle of Equation (1) and (2), red solid circle is the stable focus of Equation (1) and (2), blue arrow denotes the direction of the trajectory, cyan solid circle is a critical point that determines the direction of trajectory affected by the excitatory or inhibitory autaptic current.

    图 5  ${I_{{\text{app}}}} = 42.{\text{6}}\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$时, 不同自突触诱发共存峰放电变为混合振荡(蓝、绿实线) (a1) ${g_{{\text{exc}}}} = 2.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = 0.{\text{56}}\;{\text{m}}{{\text{s}}^{ - 1}}$的兴奋性自突触; (a2) 图(a1)的局部放大; (a3) 对应于图(a1)的相平面; (a4) 图(a3)的局部放大; (b1) ${g_{{\text{inh}}}} = {\text{1}}{\text{.0}}\;{\text{μS/c}}{{\text{m}}^2}$$\beta = $$ 0.{\text{263}}\;{\text{m}}{{\text{s}}^{ - 1}}$的抑制性自突触; (b2) 图(b1)的局部放大; (b3) 对应于(b1)的相平面; (b4) 图(b3)的局部放大. 在图(a1)和(b1)中, 黑色实线为ML神经元模型(方程(1)和方程(2))膜电位, 蓝色和绿色实线为自突触耦合ML神经元模型(方程(3)—(5))膜电位, 蓝色虚线为突触电流, 蓝色实线是对应于突触电流的部分. 在图(a3)和图(b3)中, 蓝色和绿色实线分别对应于图(a1)和图(b1)中蓝色和绿色实线的相轨迹. 灰色虚线和点线分别为方程(1)和方程(2)的$w$$V$零等倾斜线, 黑色实线和虚线分别为方程(1)和方程(2)的稳定和不稳定极限环, 红色实心点为方程(1)和方程(2)的稳定平衡点, 蓝色箭头表示相轨迹的方向

    Fig. 5.  Mixed-mode oscillations (blue and green solid curves) evoked from coexisting spiking by different autapses when ${I_{{\text{app}}}} = 42.6\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$: (a1) Excitatory autapse with ${g_{{\text{exc}}}} = 2.0\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 0.{\text{56}}\;{\text{m}}{{\text{s}}^{ - 1}}$; (a2) enlargement of Figure (a1); (a3) phase portrait corresponds to Figure (a1); (a4) enlargement of (a3); (b1) inhibitory autapse with ${g_{{\text{inh}}}} = {\text{1}}{\text{.0}}\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 0.{\text{263}}\;{\text{m}}{{\text{s}}^{ - 1}}$; (b2) enlargement of Figure (b1); (b3) phase portrait corresponds to Figure (b1); (b4) enlargement of Figure (b3). In Figure (a1) and Figure (b1), black solid line is the membrane potential of the ML neuron (Equation (1) and (2)), blue and green solid lines are membrane potential of the ML neuron with autapse (Equation (3)—(5)), blue dashed curve is the autaptic current and the blue solid curve is membrane potential effected by autaptic current. In Figure(a3) and Figure (b3), blue and green solid curves respectively are the trajectories corresponding to the blue and green solid parts in Figure (a1) and Figure (b1), gray dashed (dotted) curve is the nullcline $\dot w = 0$($\dot V = 0$) of Equation (1) and (2), black solid (dashed) curve is the stable (unstable) limit cycle of Equation (1) and (2), red solid circle is the stable focus of Equation (1) and (2), blue arrow denotes the direction of the trajectory.

    图 6  ${I_{{\text{app}}}} = 42.9\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$时, 不同自突触引起的单稳峰放电到混合振荡(蓝和绿实线)的转变 (a1) ${g_{{\text{exc}}}} = 2.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = 0.35\;{\text{m}}{{\text{s}}^{ - 1}}$的兴奋性自突触; (a2) 图(a1)的局部放大; (a3) 对应于(a1)的相平面图; (a4) 图(a3)的局部放大; (b1) ${g_{{\text{inh}}}} = $$ 1.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = 0.1\;{\text{m}}{{\text{s}}^{ - 1}}$的抑制性自突触; (b2) 图(b1)的局部放大; (b3) 对应于(b1)的相平面图; (b4) 图(b3)的局部放大. 在图(a1)和图(b1)中, 蓝色和绿色实线为自突触耦合ML神经元模型(方程(3)—(5))膜电位, 蓝色虚线为突触电流, 蓝色实线是对应于突触电流的部分. 在图(a3)和图(b3)中, 蓝色和绿色实线分别对应于图(a1)和图(b1)中蓝色和绿色实线的相轨迹. 灰色虚线和点线分别为方程(1)和方程(2)的$w$$V$零等倾斜线, 黑色实线和虚线分别为方程(1)和方程(2)的稳定和不稳定极限环, 红色空心点为方程(1)和方程(2)的稳定平衡点, 蓝色箭头表示相轨迹的方向

    Fig. 6.  The mixed-mode oscillations (blue and red curves) evoked from the monostable spiking by different autapses when ${I_{{\text{app}}}} = 42.9\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$: (a1) Excitatory autapse with ${g_{{\text{exc}}}} = 2.0\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 0.35\;{\text{m}}{{\text{s}}^{ - 1}}$; (a2) enlargement of Figure (a1); (a3) phase portrait corresponding to Figure (a1); (a4) enlargement of Figure (a3); (b1) inhibitory autapse with ${g_{{\text{inh}}}} = {\text{1}}{\text{.0}}\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 0.1\;{\text{m}}{{\text{s}}^{ - 1}}$; (b2) enlargement of Figure (b1); (b3) phase portrait corresponds to Figure (b1); (b4) enlargement of Figure (b3). In Figure (a1) and Figure (b1), blue and green solid lines are membrane potential of the ML neuron with autapse (Equation (3)—(5)), blue dashed curve is the autaptic current and the blue solid curve is membrane potential effected by the autaptic current. In Figure (a3) and Figure (b3), blue and green solid curves respectively are the trajectories corresponding to the blue and green solid parts in Figure (a1) and Figure (b1), gray dashed (dotted) curve is the nullcline $\dot w = 0$($\dot V = 0$) of Equation (1) and (2), black solid curve is the stable limit cycle of Equation (1) and (2), red hollow circle is the stable focus of Equation (1) and (2), blue arrow denotes the direction of the trajectory.

    图 7  不同${I_{{\text{app}}}}$下, 不同自突触诱发的不同动力学行为及其频率在参数平面$\beta $${g_{{\text{exc}}}}$(${g_{{\text{inh}}}}$)上的分布. ${I_{{\text{app}}}} = 42.6\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$(共存峰放电) (a) 兴奋性自突触; (b) 抑制性自突触. ${I_{{\text{app}}}} = 43\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$(单稳峰放电) (c) 兴奋性自突触; (d) 抑制性自突触. 黑线为不同电活动的分界线, 区域①, ②和③分别为频率降低的周期峰放电、混合振荡和静息, 区域④为频率增加的周期峰放电. 色标代表放电频率

    Fig. 7.  Distribution of different dynamical behaviors and firing frequency induced by different autapses on the parameter plane $({g_{{\text{exc}}}}, \beta )$ or $({g_{{\text{inh}}}}, \beta )$ when ${I_{{\text{app}}}}$ is fixed at different values, ${I_{{\text{app}}}} = 42.6\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$ (coexisting spiking): (a) Excitatory autapse; (b) inhibitory autapse. ${I_{{\text{app}}}} = 43\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$ (monostable spiking): (c) Excitatory autapse; (d) inhibitory autapse. Black curve is the border of different firing activities. Regions ①, ② and ③ are periodic spiking with reduced firing frequency, mixed-mode oscillations, and resting state, respectively, region ④ is periodic spiking with enhanced firing frequency. The color scale represents firing frequency.

    图 8  ${I_{{\text{app}}}} = 42.{\text{6}}\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$时, 不同自突触诱导出的频率增加的周期峰放电 (a) ${g_{{\text{exc}}}} = 1.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = 0.26\;{\text{m}}{{\text{s}}^{ - 1}}$的兴奋性自突触; (b) 图(a)的局部放大; (c) ${g_{{\text{inh}}}} = 1.0\;{\text{μS/c}}{{\text{m}}^2}$$\beta = 1.0\;{\text{m}}{{\text{s}}^{ - 1}}$的抑制性自突触; (d) 图(c)的局部放大. 黑色实线为无自突触耦合神经元模型(方程(1)和方程(2))的膜电位, 红色实线为自突触耦合神经元模型(方程(3)—(5))的膜电位, 红色虚线为相对应的突触电流, 灰色虚线为黑色实线第一个峰峰间隙PRC的相位滞后和提前的临界点, 图(a)和图(b)中灰色虚线的左右两侧分为兴奋性刺激PRC的相位滞后和提前区域, 图(c)和图(d)中灰色虚线的左右两侧分为抑制性刺激PRC的相位提前和滞后区域

    Fig. 8.  Periodic spiking with increased firing frequency induced by different autapses when ${I_{{\text{app}}}} = 42.6\;{\text{μA/c}}{{\text{m}}^{\text{2}}}$: (a) Excitatory autapse with ${g_{{\text{exc}}}} = 1.0\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 0.26\;{\text{m}}{{\text{s}}^{ - 1}}$; (b) enlargement of (a); (c) inhibitory autapse with ${g_{{\text{inh}}}} = 1.0\;{\text{μS/c}}{{\text{m}}^2}$ and $\beta = 1.0\;{\text{m}}{{\text{s}}^{ - 1}}$; (d) enlargement of Figure (c). Black and red solid curves are the membrane potential for the ML neuron (Equation (1) and (2)) and the ML neuron with autapse (Equation (3)-(5)), respectively, red dashed curve is the autaptic current, gray dashed vertical lines is critical border between phase delay and phase advance regions of the PRC for the first interspike interval (the black solid line), the left and right sides of the gray dashed line in Figure (a) and Figure (b) respectively are phase delay and advance regions of the PRC, the left and right sides of the gray dashed line in Figure (c) and Figure (d) respectively are phase advance and delay regions of the PRC.

  • [1]

    Izhikevich E M 2000 Int. J. Bifurcat. Chaos 10 1171Google Scholar

    [2]

    Izhikevich E M 2004 IEEE Trans. Neural Networks 15 1063Google Scholar

    [3]

    Bean B P 2007 Nat. Rev. Neurosci. 8 451Google Scholar

    [4]

    Bartos M, Vida I, Jonas P 2007 Nat. Rev. Neurosci. 8 45Google Scholar

    [5]

    Tateno T, Pakdaman K 2004 Chaos 14 511Google Scholar

    [6]

    Lee S G, Neiman A, Kim S 1998 Phys. Rev. E 57 3292Google Scholar

    [7]

    Prescott S A, De Koninck Y, Sejnowski T J 2008 PloS Comput. Biol. 4 e1000198Google Scholar

    [8]

    Ermentrout B 1996 Neural Comput. 8 979Google Scholar

    [9]

    Smeal R M, Ermentrout G B, White J A 2010 Philos. Trans. R. Soc. London, Ser. B 365 2407Google Scholar

    [10]

    Goldwyn J H, Slabe B R, Travers J B, Terman D 2018 PloS Comput. Biol. 14 e1006292Google Scholar

    [11]

    Guo D Q, Wang Q Y, Perc M 2012 Phys. Rev. E 85 061905Google Scholar

    [12]

    Silver R A 2010 Nat. Rev. Neurosci. 11 474Google Scholar

    [13]

    Ferrante M, Shay C F, Tsuno Y, Chapman G W, Hasselmo M E 2017 Cereb. Cortex 27 2111

    [14]

    Zhao Z G, Li L, Gu H G, Gao Y 2020 Nonlinear Dyn. 99 1129Google Scholar

    [15]

    Dodla R, Rinzel J 2006 Phys. Rev. E 73 010903Google Scholar

    [16]

    Dodla R, Svirskis G, Rinzel J 2006 J. Neurophysiol. 95 2664Google Scholar

    [17]

    Van Vreeswijk C, Abbott L F, Bard Ermentrout G 1994 J. Comput. Neurosci. 1 313Google Scholar

    [18]

    Wang X J, Buzsaki G 1996 J. Neurosci. 16 6402Google Scholar

    [19]

    Vida I, Bartos M, Jonas P 2006 Neuron 49 107Google Scholar

    [20]

    Li L, Zhao Z G, Gu H G 2019 Int. J. Bifurcat. Chaos 29 1950147Google Scholar

    [21]

    Uzuntarla M, Torres J J, Calim A, Barreto E 2019 Neural Networks 110 131Google Scholar

    [22]

    Van Der Loos H, Glaser E M 1972 Brain Res. 48 355Google Scholar

    [23]

    Bacci A, Huguenard J R 2006 Neuron 49 119Google Scholar

    [24]

    Cobb S R, Halasy K, Vida I, Nyiri G, Tamas G, Buhl E H, Somogyi P 1997 Neuroscience 79 629Google Scholar

    [25]

    Yin L P, Zheng R, Ke W, He Q S, Zhang Y, Li J L, Wang B, Mi Z, Long Y S, Rasch M J, Li T F, Luan G M, Shu Y S 2018 Nat. Commun. 9 4890Google Scholar

    [26]

    Saada R, Miller N, Hurwitz I, Susswein A J 2009 Curr. Biol. 19 479Google Scholar

    [27]

    Bacci A, Huguenard J R, Prince D A 2003 J. Neurosci. 23 859Google Scholar

    [28]

    Wang H T, Chen Y 2015 Chin. Phys. B 24 128709Google Scholar

    [29]

    Yilmaz E, Ozer M, Baysal V, Perc M 2016 Sci. Rep. 6 30914Google Scholar

    [30]

    Yang X L, Yu Y H, Sun Z K 2017 Chaos 27 083117Google Scholar

    [31]

    Qin H X, Wu Y, Wang C N, Ma J 2015 Commun. Nonlinear Sci. Numer. Simul. 23 164Google Scholar

    [32]

    Yilmaz E, Baysal V, Perc M, Ozer M 2016 Sci. China Technol. Sci. 59 364Google Scholar

    [33]

    Qin H X, Ma J, Wang C N, Chu R T 2014 Sci. China Phys. Mech. Astron. 57 1918Google Scholar

    [34]

    Ma J, Song X L, Tang J, Wang C N 2015 Neurocomputing 167 378Google Scholar

    [35]

    Yilmaz E, Baysal V, Ozer M, Perc M 2016 Physica A 444 538Google Scholar

    [36]

    Wang H T, Wang L F, Chen Y L, Chen Y 2014 Chaos 24 043113Google Scholar

    [37]

    Li Y Y, Schmid G, Hanggi P, Schimansky-Geier L 2010 Phys. Rev. E 82 061907Google Scholar

    [38]

    Wang H T, Ma J, Chen Y L, Chen Y 2014 Commun. Nonlinear Sci. Numer. Simul. 19 3242Google Scholar

    [39]

    Guo D Q, Wu S D, Chen M M, Perc M, Zhang Y S, Ma J L, Cui Y, Xu P, Xia Y, Yao D Z 2016 Sci. Rep. 6 26096Google Scholar

    [40]

    Guo D Q, Chen M M, Perc M, Wu S D, Xia C, Zhang Y S, Xu P, Xia Y, Yao D Z 2016 Europhys. Lett. 114 30001Google Scholar

    [41]

    Zhao Z G, Gu H G 2017 Sci. Rep. 7 6760Google Scholar

    [42]

    Song X L, Wang H T, Chen Y 2019 Nonlinear Dyn. 96 2341Google Scholar

    [43]

    Zhao Z G, Li L, Gu H G 2020 Sci. Rep. 10 3646Google Scholar

    [44]

    曹奔, 关利南, 古华光 2018 物理学报 67 240502Google Scholar

    Cao B, Guan L N, Gu H G 2018 Acta Phys. Sin. 67 240502Google Scholar

    [45]

    丁学利, 李玉叶 2016 物理学报 65 210502Google Scholar

    Ding X L, Li Y Y 2016 Acta Phys. Sin. 65 210502Google Scholar

    [46]

    Li Y Y, Gu H G, Jia Bing, Ding X L 2021 Sci. China Technol. Sci. 64 1459Google Scholar

    [47]

    Tikidji-Hamburyan R A, Martinez J J, White J A, Canavier C C 2015 J. Neurosci. 35 15682Google Scholar

    [48]

    谢勇, 程建慧 2017 物理学报 66 090501Google Scholar

    Xie Y, Cheng J H 2017 Acta Phys. Sin. 66 090501Google Scholar

    [49]

    Wilson C J 2017 J. Neurophysiol. 118 855Google Scholar

    [50]

    Canavier C C, Wang S G, Chandrasekaran L 2013 Front. Neural Circuits 7 194

    [51]

    丁学利, 贾冰, 李玉叶 2019 物理学报 68 180502Google Scholar

    Ding X L, Jia B, Li Y Y 2019 Acta Phys. Sin. 68 180502Google Scholar

    [52]

    Zhao Z, Li L, Gu H 2020 Commun. Nonlinear Sci. Numer. Simul. 85 105250Google Scholar

    [53]

    Moss S J, Smart T G 2001 Nat. Rev. Neurosci. 2 240

    [54]

    Prescott S A, Ratte S, De Koninck Y, Sejnowski T J 2008 J. Neurophysiol. 100 3030Google Scholar

    [55]

    Ermentrout B 2002 Simulating, Analyzing, and Animating Dynamical Systems: A guide to XPPAUT for Researchers and Students (Philadelphia: SIAM) pp77–173

    [56]

    Liu C M, Liu X L, Liu S Q 2014 Biol. Cybern. 108 75Google Scholar

    [57]

    Drion G, O'Leary T, Marder E 2015 Proc. Natl. Acad. Sci. U. S. A. 112 E5361Google Scholar

    [58]

    Coggan J S, Ocker G K, Sejnowski T J, Prescott S A 2011 J. Neural Eng. 8 065002Google Scholar

    [59]

    Rho Y A, Prescott S A 2012 PloS Comput. Biol. 8 e1002524Google Scholar

    [60]

    Dipoppa M, Gutkin B S 2013 Proc. Natl. Acad. Sci. U. S. A. 110 12828Google Scholar

  • [1] 李瑞, 徐邦林, 周建芳, 姜恩华, 汪秉宏, 袁五届. 一种突触可塑性导致的觉醒-睡眠周期中突触强度变化和神经动力学转变. 物理学报, 2023, 72(24): 248706. doi: 10.7498/aps.72.20231037
    [2] 安新磊, 乔帅, 张莉. 基于麦克斯韦电磁场理论的神经元动力学响应与隐藏放电控制. 物理学报, 2021, 70(5): 050501. doi: 10.7498/aps.70.20201347
    [3] 丁学利, 古华光, 贾冰, 李玉叶. 抑制性自突触诱发耦合Morris-Lecar神经元电活动的超前同步. 物理学报, 2021, 70(21): 218701. doi: 10.7498/aps.70.20210912
    [4] 姜伊澜, 陆博, 张万芹, 古华光. 快自突触反馈诱发混合簇放电的反常变化及分岔机制. 物理学报, 2021, 70(17): 170501. doi: 10.7498/aps.70.20210208
    [5] 曲良辉, 都琳, 曹子露, 胡海威, 邓子辰. 化学自突触的电导扰动诱导相干或随机双共振现象. 物理学报, 2020, 69(23): 230501. doi: 10.7498/aps.69.20200856
    [6] 华洪涛, 陆博, 古华光. 兴奋性自突触引起神经簇放电频率降低或增加的非线性机制. 物理学报, 2020, 69(9): 090502. doi: 10.7498/aps.69.20191709
    [7] 杨永霞, 李玉叶, 古华光. Pre-Bötzinger复合体的从簇到峰放电的同步转迁及分岔机制. 物理学报, 2020, 69(4): 040501. doi: 10.7498/aps.69.20191509
    [8] 刘益春, 林亚, 王中强, 徐海阳. 氧化物基忆阻型神经突触器件. 物理学报, 2019, 68(16): 168504. doi: 10.7498/aps.68.20191262
    [9] 薛晓丹, 王美丽, 邵雨竹, 王俊松. 基于抑制性突触可塑性的神经元放电率自稳态机制. 物理学报, 2019, 68(7): 078701. doi: 10.7498/aps.68.20182234
    [10] 丁学利, 贾冰, 李玉叶. 利用相位响应曲线解释抑制性反馈增强神经电活动. 物理学报, 2019, 68(18): 180502. doi: 10.7498/aps.68.20190197
    [11] 曹奔, 关利南, 古华光. 兴奋性作用诱发神经簇放电个数不增反降的分岔机制. 物理学报, 2018, 67(24): 240502. doi: 10.7498/aps.67.20181675
    [12] 于文婷, 张娟, 唐军. 动态突触、神经耦合与时间延迟对神经元发放的影响. 物理学报, 2017, 66(20): 200201. doi: 10.7498/aps.66.200201
    [13] 谢勇, 程建慧. 计算相位响应曲线的方波扰动直接算法. 物理学报, 2017, 66(9): 090501. doi: 10.7498/aps.66.090501
    [14] 丁学利, 李玉叶. 具有时滞的抑制性自突触诱发的神经放电的加周期分岔. 物理学报, 2016, 65(21): 210502. doi: 10.7498/aps.65.210502
    [15] 任国栋, 武刚, 马军, 陈旸. 一类自突触作用下神经元电路的设计和模拟. 物理学报, 2015, 64(5): 058702. doi: 10.7498/aps.64.058702
    [16] 丁学利, 李玉叶. 相位噪声诱发神经放电的单次或两次相干共振. 物理学报, 2014, 63(24): 248701. doi: 10.7498/aps.63.248701
    [17] 夏小飞, 王俊松. 基于分岔理论的突触可塑性对神经群动力学特性调控规律研究. 物理学报, 2014, 63(14): 140503. doi: 10.7498/aps.63.140503
    [18] 李晓静, 陈绚青, 严静. 一类具时滞的厄尔尼诺-南方涛动充电-放电振子模型的Hopf分岔与周期解问题. 物理学报, 2013, 62(16): 160202. doi: 10.7498/aps.62.160202
    [19] 古华光, 惠磊, 贾冰. 一类位于加周期分岔中的貌似混沌的随机神经放电节律的识别. 物理学报, 2012, 61(8): 080504. doi: 10.7498/aps.61.080504
    [20] 陈军, 李春光. 具有自适应反馈突触的神经元模型中的混沌:电路设计. 物理学报, 2011, 60(5): 050503. doi: 10.7498/aps.60.050503
计量
  • 文章访问数:  3074
  • PDF下载量:  68
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-09-30
  • 修回日期:  2021-11-23
  • 上网日期:  2022-01-26
  • 刊出日期:  2022-03-05

/

返回文章
返回