Pre-Bötzinger复合体的从簇到峰放电的同步转迁及分岔机制

杨永霞; 李玉叶; 古华光

doi:10.7498/aps.69.20191509

摘要

Pre-Bötzinger复合体是兴奋性耦合的神经元网络, 通过产生复杂的放电节律和节律模式的同步转迁参与调控呼吸节律. 本文选用复杂簇和峰放电节律的单神经元数学模型构建复合体模型, 仿真了与生物学实验相关的多类同步节律模式及其复杂转迁历程, 并利用快慢变量分离揭示了相应的分岔机制. 当初值相同时, 随着兴奋性耦合强度的增加, 复合体模型依次表现出完全同步的“fold/homoclinic”, “subHopf/subHopf”簇放电和周期1峰放电. 当初值不同时, 随耦合强度增加, 表现为由“fold/homoclinic”, 到“fold/fold limit cycle”、到“subHopf/subHopf”与“fold/fold limit cycle”的混合簇放电、再到“subHopf/subHopf”簇放电的相位同步转迁, 最后到反相同步周期1峰放电. 完全(同相)同步和反相同步的周期1节律表现出了不同分岔机制. 反相峰同步行为给出了与强兴奋性耦合容易诱发同相同步这一传统观念不同的新示例. 研究结果给出了pre-Bötzinger复合体的从簇到峰放电节律的同步转迁规律及复杂分岔机制, 反常同步行为丰富了非线性动力学的内涵.

关键词:

Abstract

The pre-Bötzinger complex is a neuronal network with excitatory coupling, which participates in modulation of respiratory rhythms via the generation of complex firing rhythm patterns and synchronization transitions of rhythm patterns. In the present paper, a mathematical model of single neuron that exhibits complex transition processes from bursting to spiking is selected as a unit, the network model of the pre-Bötzinger complex composed of two neurons with excitatory coupling is constructed, multiple synchronous rhythm patterns and complex transition processes of the synchronous rhythm patterns related to the biological experimental observations are simulated, and the corresponding bifurcation mechanism is acquired with the fast-slow variable dissection method. When the initial values of two neurons of the pre-Bötzinger complex are the same, with increasing the excitatory coupling strength, the theoretical model of the pre-Bötzinger complex shows complete synchronization transition processes from "fold/homoclinic" bursting, to "subHopf/subHopf" bursting, and at last to period-1 spiking. When the initial values are different, with the increases of the excitatory coupling intensity, the rhythm transition processes begin from phase synchronization behaviors including "fold/homoclinic" bursting, "fold/fold limit cycle" bursting, mixed bursting composed of "subHopf/subHopf" bursting and "fold/fold limit cycle" bursting, and "subHopf/ subHopf" bursting in sequence, and to anti-phase synchronous behavior of the period-1 spiking. The complete (in-phase) synchronous period-1 spiking for the same initial values exhibits bifurcation mechanism different from the anti-phase synchronous period-1 spiking for different initial values. The anti-phase synchronous period-1 spiking presents a novel and abnormal example of the synchronization at large excitatory coupling strength, which is different from the traditional viewpoint that large excitatory coupling often induces in-phase synchronous behavior. The results present the synchronization transition process and complex bifurcation mechanism from bursting to period-1 spiking of the pre-Bötzinger complex, and the abnormal synchronization example enriches the contents of nonlinear dynamics.

Keywords:

作者及机构信息

1.
赤峰学院, 数学与计算机科学学院, 赤峰　024000

2.
赤峰学院, 应用数学研究所, 赤峰　024000

3.
同济大学, 航空航天与力学学院, 上海　200092

通信作者: 李玉叶, liyuye2000@163.com

基金项目: 国家级-国家自然科学基金面上项目(11762001)

Authors and contacts

1.
College of Mathematics and Computer Science, Chifeng University, Chifeng 024000, China

2.
Institute of Applied Mathematics, Chifeng University, Chifeng 024000, China

3.
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China

Corresponding author: Li Yu-Ye, liyuye2000@163.com

文章全文 : translate this paragraph

参考文献

[1]	Braun H A, Wissing H, Schäfer K, Hirsch M C 1994 Nature 367 270 Google Scholar
[2]	Gu H G, Pan B B 2015 Nonlinear Dyn. 81 2107 Google Scholar
[3]	谢勇, 程建慧 2017 物理学报 66 090501 Google Scholar Xie Y, Cheng J H 2017 Acta Phys. Sin. 66 090501 Google Scholar
[4]	Sun X J, Perc M, Kurths J, Lu Q S 2018 Chaos 28 106310 Google Scholar
[5]	徐莹, 王春妮, 靳伍银, 马军 2015 物理学报 64 198701 Xu Y, Wang C N, Jin W Y, Ma J 2015 Acta Phys. Sin. 64 198701
[6]	李国芳, 孙晓娟 2017 物理学报 66 240501 Google Scholar Li G F, Sun X J 2017 Acta Phys. Sin. 66 240501 Google Scholar
[7]	Bianchi A L, Denavit-Saubie M, Champagnat J 1995 Physiol. Rev. 75 1 Google Scholar
[8]	Cohen M I 1979 Physiol. Rev. 59 1105 Google Scholar
[9]	Funk G D, Smith J C, Feldman J L 1995 J. Neurosci. 15 4046 Google Scholar
[10]	Richter D W, Ballanyi K, Schwarzacher S 1992 Curr. Opin. Neurobiol. 2 788 Google Scholar
[11]	严亨秀, 张承武, 郑煜 2004 生理学报 56 665 Google Scholar Yan H X, Zhang C W, Zheng Y 2004 Acta Physiol. Sin. 56 665 Google Scholar
[12]	宋刚 1999 生理科学进展 3 237 Song G 1999 Prog. Physiol. Sci. 3 237
[13]	Smith J C, Ellenberger H H, Ballanyi K, Richter D W, Feldman J L 1991 Science 254 726 Google Scholar
[14]	Feldman J L, Negro C A D 2006 Nat. Rev. Neurosci. 7 232 Google Scholar
[15]	Smith J C 1997 Neurons, Networks, and Motor Behavior (Cambridge, MA: MIT Press) p97
[16]	Johnson S M, Smith J C, Funk G D, Feldman J L 1994 J. Neurophysiol. 72 2598 Google Scholar
[17]	Ramirez J M, Richter D W 1996 Curr. Opin. Neurobiol. 6 817 Google Scholar
[18]	Rekling J C, Feldman J L 1998 Annu. Rev. Physiol. 60 385 Google Scholar
[19]	Koshiya N, Smith J C 1998 28^th Annual Meeting of the Society for Neuroscience Los Angeles, California, USA, November 7-12, 1998 p531
[20]	Koshiya N, Smith J C 1999 Nature 400 360 Google Scholar
[21]	Negro C A D, Morgado V C, Hayes J A, Mackay D D, Pace R W, Crowder E A, Feldman J L 2005 J. Neurosci. 25 446 Google Scholar
[22]	Smith J C, Butera R J, Koshiya N, Del Negro C, Wilson C G, Johnson S M 2000 Resp. Physiol. 122 131 Google Scholar
[23]	Gray P A, Rekling J C, Bocchiaro C M, Feldman J L 1999 Science 286 1566 Google Scholar
[24]	Butera R J, Rinzel J, Smith J C 1999 J. Neurophysiol. 82 382 Google Scholar
[25]	Dunmyre J R, Negro C A D, Rubin J E 2011 J. Comput. Neurosci. 31 305 Google Scholar
[26]	Negro C A D, Johnson S M, Butera R J, Smith J C 2001 J. Neurophysiol. 86 59 Google Scholar
[27]	Butera R J, Rinzel J, Smith J C 1999 J. Neurophysiol. 82 398 Google Scholar
[28]	Purvis L K, Smith J C, Koizumi H, Butera R J 2007 J. Neurophysiol. 97 1515 Google Scholar
[29]	Best J, Borisyuk A, Rubin J E, Terman D, Wechselberger M 2005 SIAM J. Appl. Dyn. Syst. 4 1107 Google Scholar
[30]	Rubin J E 2006 Phys. Rev. E 74 021917 Google Scholar
[31]	Dunmyre J R, Rubin J E 2010 SIAM J Appl. Dyn. Syst. 9 154 Google Scholar
[32]	Guo D D, Lü Z S 2019 Chin. Phys. B 28 110501 Google Scholar
[33]	Rybak I A, Molkov Y I, Jasinski P E, Shevtsova N A, Smith J C 2014 Prog. Brain. Res. 209 1 Google Scholar
[34]	张应腾, 熊冬生, 刘深泉 2015 中国医学物理学杂志 32 115 Google Scholar Zhang Y T, Xiong D S, Liu S Q 2015 Chin. J. Med. Phys. 32 115 Google Scholar
[35]	刘义, 刘深泉 2011 动力学与控制学报 9 257 Google Scholar Liu Y, Liu S Q 2011 J. Dynam. Cont. 9 257 Google Scholar
[36]	Duan L X, Zhai D H, Tang X H 2012 Int. J. Bifurcation Chaos 22 1250114 Google Scholar
[37]	Lü Z S, Chen L N, Duan L X 2019 Appl. Math. Model. 67 234 Google Scholar
[38]	Lü Z S, Zhang B Z, Duan L X 2017 Cogn. Neurodynamics 11 443 Google Scholar
[39]	Wang Z J, Duan L X, Cao Q Y 2018 Chin. Phys. B 27 070502 Google Scholar
[40]	Duan L X, Liu J, Chen X, Xiao P C, Zhao Y 2017 Cogn. Neurodynamics 11 91 Google Scholar
[41]	Rubin J E, Shevtsova N A, Ermentrout G B, Smith J C, Rybak I A 2009 J. Neurophysiol. 101 2146 Google Scholar
[42]	Rubin J E, Bacak B J, Molkov Y I, Shevtsova N A, Smith J C, Rybak I A 2011 J. Comput. Neurosci. 30 607 Google Scholar
[43]	平小方, 刘深泉, 任会霞 2015 动力学与控制学报 13 215 Google Scholar Ping X F, Liu S Q, Ren H X 2015 J. Dynam. Cont. 13 215 Google Scholar
[44]	Belykh I, Shilnikov A 2008 Phys. Rev. Lett. 101 078102 Google Scholar
[45]	Wu F Q, Gu H G, Li Y Y 2019 Commun. Nonlinear Sci. Numer. Simul. 79 104924 Google Scholar
[46]	丁学利, 李玉叶 2016 物理学报 65 210502 Google Scholar Ding X L, Li Y Y 2016 Acta Phys. Sin. 65 210502 Google Scholar
[47]	Jia B, Wu Y C, He D, Guo B H, Xue L 2018 Nonlinear Dyn. 93 1599 Google Scholar
[48]	Li Y Y, Gu H G, Ding X L 2019 Nonlinear Dyn. 97 2091 Google Scholar
[49]	曹奔, 关利南, 古华光 2018 物理学报 67 240502 Google Scholar Cao B, Guan L N, Gu H G 2018 Acta Phys. Sin. 67 240502 Google Scholar
[50]	Uzuntarla M, Torres J J, Calim A, Barreto E 2019 Neural Networks 110 131 Google Scholar
[51]	埃门创特 B 著 (孝鹏程, 段丽霞, 苏建忠译) 2002 动力系统仿真, 分析与动画—XPPAUT使用指南 (北京: 科学出版社) 第155−167页 Ermentrout B (translated by Xiao P C, Duan L L, Su J Z) 2002 Simulating, Analyzing, and Animating Dynamical systems: A Guide to XPPAUT for Researchers and Students (Beijing: Science Press) p155−167 (in Chinese)
[52]	Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171 Google Scholar
[53]	Hashemi M, Valizadeh A, Azizi Y 2012 Phys. Rev. E 85 021917 Google Scholar
[54]	Jia B, Gu H G 2017 Int. J. Bifurcation Chaos 27 1750113 Google Scholar
[55]	Zhao Z G, Gu H G 2017 Sci. Rep. 7 6760 Google Scholar
[56]	Li J J, Du M M, Wang R, Lei J Z, Wu Y 2016 Int. J. Bifurcation Chaos 26 1650138 Google Scholar

施引文献

图 1 不同${g_{\rm{K}}}$下单神经元放电在(h, V)相平面的轨迹　(a) $ {g_{\rm{K}}} = {\rm{7}}{\rm{.1}}\;{\rm{nS}} $; (b) $ {g_{\rm{K}}} = {\rm{7}}{\rm{.8}}\;{\rm{nS}} $; (c) $ {g_{\rm{K}}} = {\rm{10}}{\rm{.0}}\;{\rm{nS}} $; (d) $ {g_{\rm{K}}} = {\rm{25}}{\rm{.0}}\;{\rm{nS}} $

Fig. 1. The (h, V) trajectory of the single neuron at different ${g_{\rm{K}}}$ values: (a) $ {g_{\rm{K}}} = {\rm{7}}{\rm{.1}}\;{\rm{nS}} $; (b) $ {g_{\rm{K}}} = {\rm{7}}{\rm{.8}}\;{\rm{nS}} $; (c) $ {g_{\rm{K}}} = {\rm{10}}{\rm{.0}}\;{\rm{nS}} $; (d) $ {g_{\rm{K}}} = {\rm{25}}{\rm{.0}}\;{\rm{nS}} $.

下载: 全尺寸图片幻灯片

图 2 单神经元模型的随${g_{\rm{K}}}$的分岔　(a) ISIs分岔序列; (b)图(a)左下角方框的局部放大

Fig. 2. Bifurcation of the single neuron model with increasing ${g_{\rm{K}}}$: (a) Bifurcations of ISIs; (b) the enlargement of ISIs within the square at the down-left corner of fig (a).

下载: 全尺寸图片幻灯片

图 3 当$ {g_{\rm{K}}} = 7.1\;{\rm{nS}} $时, 单神经元模型的快子系统随着慢变量h变化的分岔

Fig. 3. Bifurcations of the fast-subsystem of the single neuron with respect to h when $ {g_{\rm{K}}} = 7.1\;{\rm{nS}} $.

下载: 全尺寸图片幻灯片

图 4 单神经元在不同的${g_{\rm{K}}}$下簇放电模式的快慢变量分离　(a) $ {g_{\rm{K}}} = {\rm{7}}.1\;{\rm{nS}} $; (b) $ {g_{\rm{K}}} = {\rm{7}}.8\;{\rm{nS}} $; (c) $ {g_{\rm{K}}} = {\rm{10}}.0\;{\rm{nS}} $; (d) ${g_{\rm{K}}} =$ 25.0 nS

Fig. 4. The fast-slow variable dissection of bursting of single neuron at different ${g_{\rm{K}}}$ values: (a) $ {g_{\rm{K}}} = {\rm{7}}.1\;{\rm{nS}} $; (b) $ {g_{\rm{K}}} = {\rm{7}}.8\;{\rm{nS}} $; (c) $ {g_{\rm{K}}} = {\rm{10}}.0\;{\rm{nS}} $; (d) $ {g_{\rm{K}}} = {\rm{25}}.0\;{\rm{nS}} $.

下载: 全尺寸图片幻灯片

图 5 随着耦合强度${g_{{\text{syn-e}}}}$增大, 耦合神经元模型的同步转迁过程. 相同初值　(a1)耦合电流平均值$\bar I$; (a2)峰相位差的最大值$\max (\Delta (\phi (t)))$; (a3)簇相位差的最大值$ \max (\Delta (\varPhi (t))) $; (a4)相关系数ρ; (a5)神经元1的ISIs序列. 不同初值: (b1)耦合电流平均值$\bar I$; (b2)峰相位差的最大值$\max (\Delta (\phi (t)))$; (b3)簇相位差的最大值$ \max (\Delta (\varPhi (t))) $; (b4)相关系数ρ; (b5)神经元1的ISIs序列

Fig. 5. Transitions with respect to ${g_{{\text{syn-e}}}}$ of coupled neurons model. The same initial values: (a1) The mean values of coupling current $\bar I$; (a2) maximum spike phase difference $\max (\Delta (\phi (t)))$; (a3) maximum burst phase difference $ \max (\Delta (\varPhi (t))) $; (a4) coefficient ρ; (a5) ISIs of neuron 1. Different initial values: (b1) The mean values of coupling current $\bar I$; (b2) maximum spike phase difference $\max (\Delta (\phi (t)))$; (b3) maximum burst phase difference $ \max (\Delta (\varPhi (t))) $; (b4) coefficient ρ; (b5) ISIs of neuron 1.

下载: 全尺寸图片幻灯片

图 6 初值相同时, 不同耦合强度下神经元1(红)和2(蓝)的膜电位$V$(上)及耦合电流${I^{{\text{syn-e}}}}$ (下), 插图是局部放大　(a) $ {g_{{\text{syn-e}}}} = {\rm{0}}{\rm{.35}}\;{\rm{nS}} $; (b) $ {g_{{\text{syn-e}}}} = {\rm{2}}{\rm{.5}}\;{\rm{nS}} $; (c) $ {g_{{\text{syn-e}}}} = {\rm{5}}{\rm{.0}}\;{\rm{nS}} $; (d) $ {g_{{\text{syn-e}}}} = {\rm{18}}{\rm{.0}}\;{\rm{nS}} $

Fig. 6. Membrane potential $V$ (top) and coupling current ${I^{{\text{syn-e}}}}$ (low) of neurons 1 (red) and 2 (blue) with the same initial values at different ${g_{{\text{syn-e}}}}$ values (Insert figure: the enlargement of bursting): (a) $ {g_{{\text{syn-e}}}} = {\rm{0}}{\rm{.35}}\;{\rm{nS}} $; (b) $ {g_{{\text{syn-e}}}} = {\rm{2}}{\rm{.5}}\;{\rm{nS}} $; (c) $ {g_{{\text{syn-e}}}} = {\rm{5}}{\rm{.0}}\;{\rm{nS}} $; (d) $ {g_{{\text{syn-e}}}} = {\rm{18}}{\rm{.0}}\;{\rm{nS}} $.

下载: 全尺寸图片幻灯片

图 7 初值不同时, 不同耦合强度下神经元1(红)和2(蓝)的膜电位V(上)及耦合电流${I^{{\text{syn-e}}}}$ (下), 插图是局部放大　(a) $g_\text{syn-e}$ = 0.35 nS; (b) $g_\text{syn-e}$ = 1.5 nS (c) $g_\text{syn-e}$ = 2.5 nS; (d) $g_\text{syn-e}$ = 5.0 nS; (e) $g_\text{syn-e}$ = 18.0 nS

Fig. 7. Membrane potential V (top) and coupling current ${I^{{\text{syn-e}}}}$ (low) of neurons 1 (red) and 2 (blue) with different initial values at different $g_\text{syn-e}$ (Insert figure: the enlargement of bursting): (a) $g_\text{syn-e}$ = 0.35 nS; (b) $g_\text{syn-e}$ = 1.5 nS; (c) $g_\text{syn-e}$ = 2.5 nS; (d) $g_\text{syn-e}$ = 5.0 nS; (e) $g_\text{syn-e}$ = 18.0 nS.

下载: 全尺寸图片幻灯片

图 8 当$g_\text{syn-e}$ = 1.5 nS时, 两耦合神经元的快子系统的分岔, 插图是局部放大　(a)平衡点分岔; (b)平衡点分岔和极限环的分岔

Fig. 8. Bifurcations of the fast-subsystem of the two coupled neurons with respect to h when $g_\text{syn-e}$ = 1.5 nS (Insert figure: the enlargement): (a) Equilibrium points; (b) equilibrium points and limit cycle.

下载: 全尺寸图片幻灯片

图 9 初值相同时, 神经元1在不同耦合强度下簇放电模式的快慢变量分离, 插图是局部放大　(a) $g_\text{syn-e}$ = 0.35 nS; (b) $g_\text{syn-e}$ = 2.5 nS; (c) $g_\text{syn-e}$ = 5.0 nS; (d) $g_\text{syn-e}$ = 18.0 nS

Fig. 9. The fast-slow variable dissection of neuron 1 for different initial values at different $g_\text{syn-e}$ values (Insert figure: the enlargement): (a) $g_\text{syn-e}$ = 0.35 nS; (b) $g_\text{syn-e}$ = 2.5 nS; (c) $g_\text{syn-e}$ = 5.0 nS; (d) $g_\text{syn-e}$ = 18.0 nS.

下载: 全尺寸图片幻灯片

图 10 初值不同时, 神经元1在不同耦合强度下簇放电模式的快慢变量分离, 插图是局部放大　(a) ${g_{{\rm{syn - e}}}}$ = 0.35 nS; (b) $g_\text{syn-e}$ = 1.5 nS; (c)和(d) $g_\text{syn-e}$ = 2.5 nS; (e) $g_\text{syn-e}$ = 5.0 nS; (f) $g_\text{syn-e}$ = 18.0 nS

Fig. 10. The fast-slow variable dissection of neuron 1 for different initial values at different $g_\text{syn-e}$ values (Insert figure: the enlargement): (a)$g_\text{syn-e}$ = 0.35 nS; (b) $g_\text{syn-e}$ = 1.5 nS; (c) and (d) $g_\text{syn-e}$ = 2.5 nS; (e) $g_\text{syn-e}$ = 5.0 nS; (f) $g_\text{syn-e}$ = 18.0 nS.

下载: 全尺寸图片幻灯片

图 11 反相同步(紫色)和同相同步(绿色)周期1峰放电节律　(a) (h, V₁)相平面上的相轨迹图; (b)耦合电流随时间t的变化

Fig. 11. The anti-phase (purple) and in-phase (green) period-1 spiking: (a) The V-h trajectory; (b) coupling current.

下载: 全尺寸图片幻灯片

图 12 (a)快子系统的平衡点和极限环的分岔; (b)图(a)中极限环分岔处的放大; (c)反相同步(紫色)和同相同步(绿色)周期1峰放电的快慢变量分离; (d)图(c)中反向同步(紫色)和同向(绿色)同步周期1峰放电的放大

Fig. 12. (a) Bifurcations of equilibrium points and limit cycle of the fast-subsystem; (b) enlargement of (a); (c) fast-slow variable dissection of anti-phase (purple) and in-phase (green) period-1 spiking; (d) enlargement of anti-phase (purple) and in-phase (green) period-1 spiking in Fig. (c).

下载: 全尺寸图片幻灯片

表 1 理论模型中的参数值

Table 1. Parameter values used in the theoretical model.

参数	参数值	参数	参数值	参数	参数值	参数	参数值
C	21 pF	$ {\sigma _{ {\rm{m_p} }} } $	–6 mV	$ {g_{ {\rm{Nap} }} } $	2.8 nS	${E_{{\rm{Na}}}}$	50 mV
$ {\theta _{ {\rm{m_p} }} } $	–40 mV	${\sigma _{\rm{m}}}$	–5 mV	${g_{{\rm{Na}}}}$	28 nS	${E_{\rm{K}}}$	–85 mV
${\theta _{\rm{m}}}$	–34 mV	$\sigma {}_{\rm{h}}$	6 mV	${g_{\rm{L}}}$	2.8 nS	${E_{\rm{L}}}$	–65 mV
${\theta _{\rm{h}}}$	–48 mV	${\sigma _{\rm{n}}}$	–4 mV	${g_{ {\text{tonic-e} } } }$	0.4 nS	${\bar \tau _{\rm{h}}}$	10000 ms
${\theta _{\rm{n}}}$	–29 mV	${\sigma _{\rm{s}}}$	–5 mV	${\varepsilon _{}}$	6	${\bar \tau _{\rm{n}}}$	5 ms
$\theta {}_{\rm{s}}$	–10 mV	${\alpha _{\rm{s}}}$	–5 mV

下载: 导出CSV

表 2 不同${g_{\rm{K}}}$下快子系统中关键点的慢变量h的值

Table 2. The values of slow variable h of the bifurcation or key points at different ${g_{\rm{K}}}$ values.

关键点	h的值
关键点	F1	F2	subh	HC	LPC	共存区域
$ {g_{\rm{K} }} = 7.1\;{\rm{nS}} $	0.4928	–1.6780	0.2128	0.3265	0.4308	[0.3265, 0.4308]
$ {g_{\rm{K} }} = 7.8\;{\rm{nS}} $	0.4928	–1.6680	0.2858	0.3476	0.4973	[0.3476, 0.4928]
$ {g_{\rm{K} }} = 10.0 \;{\rm{nS}} $	0.4928	–1.6390	0.5072	0.3941	0.7025	[0.3941, 0.4928]
$ {g_{\rm{K} }} = 25.0 \;{\rm{nS}} $	0.4928	–1.4800	1.7880	0.4849	1.9240	[0.4849, 0.4928]

下载: 导出CSV

表 3 不同${g_{{\rm{syn\text-e}}}}$下快子系统中关键点的慢变量h的值

Table 3. The slow variable h values of the bifurcation or key points at different ${g_{{\rm{syn\text-e}}}}$ values.

关键点	h的值
关键点	$g_\text{syn-e}$ = 0.35 nS	$g_\text{syn-e}$ = 2.5 nS	$g_\text{syn-e}$ = 5.0 nS	$g_\text{syn-e}$ = 18.0 nS
F1	0.4874	0.4918	0.4908	0.4856
F2	–1.6695	–1.6759	–1.6685	–1.7212
subh1	0.2817	0.2565	0.2259	0.0746
subh2	0.2858	0.2852	0.2274	0.0794
LPC1	0.4927	0.4273	0.3598	0.0960
LPC2	\	0.3103	0.2406	–0.2504
LPC3	\	\	\	0.0890
LPC4	\	\	\	–0.099
HC	0.3398	\	\	\
共存区域	[0.3398, 0.4927]	[0.3103, 0.4273]	[0.2406, 0.3598]	[0.0960, 0.250]和[0.0890, 0.099]

下载: 导出CSV

[1]	Braun H A, Wissing H, Schäfer K, Hirsch M C 1994 Nature 367 270 Google Scholar
[2]	Gu H G, Pan B B 2015 Nonlinear Dyn. 81 2107 Google Scholar
[3]	谢勇, 程建慧 2017 物理学报 66 090501 Google Scholar Xie Y, Cheng J H 2017 Acta Phys. Sin. 66 090501 Google Scholar
[4]	Sun X J, Perc M, Kurths J, Lu Q S 2018 Chaos 28 106310 Google Scholar
[5]	徐莹, 王春妮, 靳伍银, 马军 2015 物理学报 64 198701 Xu Y, Wang C N, Jin W Y, Ma J 2015 Acta Phys. Sin. 64 198701
[6]	李国芳, 孙晓娟 2017 物理学报 66 240501 Google Scholar Li G F, Sun X J 2017 Acta Phys. Sin. 66 240501 Google Scholar
[7]	Bianchi A L, Denavit-Saubie M, Champagnat J 1995 Physiol. Rev. 75 1 Google Scholar
[8]	Cohen M I 1979 Physiol. Rev. 59 1105 Google Scholar
[9]	Funk G D, Smith J C, Feldman J L 1995 J. Neurosci. 15 4046 Google Scholar
[10]	Richter D W, Ballanyi K, Schwarzacher S 1992 Curr. Opin. Neurobiol. 2 788 Google Scholar
[11]	严亨秀, 张承武, 郑煜 2004 生理学报 56 665 Google Scholar Yan H X, Zhang C W, Zheng Y 2004 Acta Physiol. Sin. 56 665 Google Scholar
[12]	宋刚 1999 生理科学进展 3 237 Song G 1999 Prog. Physiol. Sci. 3 237
[13]	Smith J C, Ellenberger H H, Ballanyi K, Richter D W, Feldman J L 1991 Science 254 726 Google Scholar
[14]	Feldman J L, Negro C A D 2006 Nat. Rev. Neurosci. 7 232 Google Scholar
[15]	Smith J C 1997 Neurons, Networks, and Motor Behavior (Cambridge, MA: MIT Press) p97
[16]	Johnson S M, Smith J C, Funk G D, Feldman J L 1994 J. Neurophysiol. 72 2598 Google Scholar
[17]	Ramirez J M, Richter D W 1996 Curr. Opin. Neurobiol. 6 817 Google Scholar
[18]	Rekling J C, Feldman J L 1998 Annu. Rev. Physiol. 60 385 Google Scholar
[19]	Koshiya N, Smith J C 1998 28^th Annual Meeting of the Society for Neuroscience Los Angeles, California, USA, November 7-12, 1998 p531
[20]	Koshiya N, Smith J C 1999 Nature 400 360 Google Scholar
[21]	Negro C A D, Morgado V C, Hayes J A, Mackay D D, Pace R W, Crowder E A, Feldman J L 2005 J. Neurosci. 25 446 Google Scholar
[22]	Smith J C, Butera R J, Koshiya N, Del Negro C, Wilson C G, Johnson S M 2000 Resp. Physiol. 122 131 Google Scholar
[23]	Gray P A, Rekling J C, Bocchiaro C M, Feldman J L 1999 Science 286 1566 Google Scholar
[24]	Butera R J, Rinzel J, Smith J C 1999 J. Neurophysiol. 82 382 Google Scholar
[25]	Dunmyre J R, Negro C A D, Rubin J E 2011 J. Comput. Neurosci. 31 305 Google Scholar
[26]	Negro C A D, Johnson S M, Butera R J, Smith J C 2001 J. Neurophysiol. 86 59 Google Scholar
[27]	Butera R J, Rinzel J, Smith J C 1999 J. Neurophysiol. 82 398 Google Scholar
[28]	Purvis L K, Smith J C, Koizumi H, Butera R J 2007 J. Neurophysiol. 97 1515 Google Scholar
[29]	Best J, Borisyuk A, Rubin J E, Terman D, Wechselberger M 2005 SIAM J. Appl. Dyn. Syst. 4 1107 Google Scholar
[30]	Rubin J E 2006 Phys. Rev. E 74 021917 Google Scholar
[31]	Dunmyre J R, Rubin J E 2010 SIAM J Appl. Dyn. Syst. 9 154 Google Scholar
[32]	Guo D D, Lü Z S 2019 Chin. Phys. B 28 110501 Google Scholar
[33]	Rybak I A, Molkov Y I, Jasinski P E, Shevtsova N A, Smith J C 2014 Prog. Brain. Res. 209 1 Google Scholar
[34]	张应腾, 熊冬生, 刘深泉 2015 中国医学物理学杂志 32 115 Google Scholar Zhang Y T, Xiong D S, Liu S Q 2015 Chin. J. Med. Phys. 32 115 Google Scholar
[35]	刘义, 刘深泉 2011 动力学与控制学报 9 257 Google Scholar Liu Y, Liu S Q 2011 J. Dynam. Cont. 9 257 Google Scholar
[36]	Duan L X, Zhai D H, Tang X H 2012 Int. J. Bifurcation Chaos 22 1250114 Google Scholar
[37]	Lü Z S, Chen L N, Duan L X 2019 Appl. Math. Model. 67 234 Google Scholar
[38]	Lü Z S, Zhang B Z, Duan L X 2017 Cogn. Neurodynamics 11 443 Google Scholar
[39]	Wang Z J, Duan L X, Cao Q Y 2018 Chin. Phys. B 27 070502 Google Scholar
[40]	Duan L X, Liu J, Chen X, Xiao P C, Zhao Y 2017 Cogn. Neurodynamics 11 91 Google Scholar
[41]	Rubin J E, Shevtsova N A, Ermentrout G B, Smith J C, Rybak I A 2009 J. Neurophysiol. 101 2146 Google Scholar
[42]	Rubin J E, Bacak B J, Molkov Y I, Shevtsova N A, Smith J C, Rybak I A 2011 J. Comput. Neurosci. 30 607 Google Scholar
[43]	平小方, 刘深泉, 任会霞 2015 动力学与控制学报 13 215 Google Scholar Ping X F, Liu S Q, Ren H X 2015 J. Dynam. Cont. 13 215 Google Scholar
[44]	Belykh I, Shilnikov A 2008 Phys. Rev. Lett. 101 078102 Google Scholar
[45]	Wu F Q, Gu H G, Li Y Y 2019 Commun. Nonlinear Sci. Numer. Simul. 79 104924 Google Scholar
[46]	丁学利, 李玉叶 2016 物理学报 65 210502 Google Scholar Ding X L, Li Y Y 2016 Acta Phys. Sin. 65 210502 Google Scholar
[47]	Jia B, Wu Y C, He D, Guo B H, Xue L 2018 Nonlinear Dyn. 93 1599 Google Scholar
[48]	Li Y Y, Gu H G, Ding X L 2019 Nonlinear Dyn. 97 2091 Google Scholar
[49]	曹奔, 关利南, 古华光 2018 物理学报 67 240502 Google Scholar Cao B, Guan L N, Gu H G 2018 Acta Phys. Sin. 67 240502 Google Scholar
[50]	Uzuntarla M, Torres J J, Calim A, Barreto E 2019 Neural Networks 110 131 Google Scholar
[51]	埃门创特 B 著 (孝鹏程, 段丽霞, 苏建忠译) 2002 动力系统仿真, 分析与动画—XPPAUT使用指南 (北京: 科学出版社) 第155−167页 Ermentrout B (translated by Xiao P C, Duan L L, Su J Z) 2002 Simulating, Analyzing, and Animating Dynamical systems: A Guide to XPPAUT for Researchers and Students (Beijing: Science Press) p155−167 (in Chinese)
[52]	Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171 Google Scholar
[53]	Hashemi M, Valizadeh A, Azizi Y 2012 Phys. Rev. E 85 021917 Google Scholar
[54]	Jia B, Gu H G 2017 Int. J. Bifurcation Chaos 27 1750113 Google Scholar
[55]	Zhao Z G, Gu H G 2017 Sci. Rep. 7 6760 Google Scholar
[56]	Li J J, Du M M, Wang R, Lei J Z, Wu Y 2016 Int. J. Bifurcation Chaos 26 1650138 Google Scholar

[1]	梁艳美, 陆博, 古华光. 利用双慢变量的快慢变量分离分析新脑皮层神经元Wilson模型的复杂电活动. 物理学报, 2022, 71(23): 230502. doi: 10.7498/aps.71.20221416
[2]	黎丽, 赵志国, 古华光. 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. 物理学报, 2022, 71(5): 050504. doi: 10.7498/aps.71.20211829
[3]	谢盈, 朱志刚, 张晓锋, 任国栋. 光电流驱动下非线性神经元电路的放电模式控制. 物理学报, 2021, 70(21): 210502. doi: 10.7498/aps.70.20210676
[4]	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
[5]	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
[6]	赵雅琪, 刘谋天, 赵勇, 段利霞. 耦合前包钦格复合体神经元中复杂混合簇放电的动力学. 物理学报, 2021, 70(12): 120501. doi: 10.7498/aps.70.20210093
[7]	郑志刚, 翟云, 王学彬, 陈宏斌, 徐灿. 耦合相振子系统同步的序参量理论. 物理学报, 2020, 69(8): 080502. doi: 10.7498/aps.69.20191968
[8]	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
[9]	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
[10]	郑殿春, 丁宁, 沈湘东, 赵大伟, 郑秋平, 魏红庆. 基于分形理论的尖-板电极短空气隙放电现象研究. 物理学报, 2016, 65(2): 024703. doi: 10.7498/aps.65.024703
[11]	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
[12]	向俊杰, 毕闯, 向勇, 张千, 王京梅. 峰值电流模式控制同步开关Z源变换器的动力学研究. 物理学报, 2014, 63(12): 120507. doi: 10.7498/aps.63.120507
[13]	黄晨, 陈龙, 毕勤胜, 江浩斌. 机动车协商模型与分岔特性研究. 物理学报, 2013, 62(21): 210507. doi: 10.7498/aps.62.210507
[14]	王付霞, 谢勇. "Hopf/homoclinic"簇放电和"SubHopf/homoclinic"簇放电之间的同步. 物理学报, 2013, 62(2): 020509. doi: 10.7498/aps.62.020509
[15]	李群宏, 闫玉龙, 杨丹. 耦合电路系统的分岔研究. 物理学报, 2012, 61(20): 200505. doi: 10.7498/aps.61.200505
[16]	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
[17]	古华光, 朱洲, 贾冰. 一类新的混沌神经放电的动力学特征的实验和数学模型研究. 物理学报, 2011, 60(10): 100505. doi: 10.7498/aps.60.100505
[18]	陈章耀, 毕勤胜. Jerk系统耦合的分岔和混沌行为. 物理学报, 2010, 59(11): 7669-7678. doi: 10.7498/aps.59.7669
[19]	包伯成, 康祝圣, 许建平, 胡文. 含指数项广义平方映射的分岔和吸引子. 物理学报, 2009, 58(3): 1420-1431. doi: 10.7498/aps.58.1420
[20]	张维, 周淑华, 任勇, 山秀明. Turbo译码算法的分岔与控制. 物理学报, 2006, 55(2): 622-627. doi: 10.7498/aps.55.622

计量

文章访问数: 7786
PDF下载量: 113
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板