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

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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}} $.

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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).

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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}} $.

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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}} $.

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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.

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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}} $.

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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.

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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.

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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.

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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.

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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.

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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).

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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

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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]

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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]

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[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
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