Synchronization transition from bursting to spiking and bifurcation mechanism of the pre-Bötzinger complex

Yang Yong-Xia; Li Yu-Ye; Gu Hua-Guang

doi:10.7498/aps.69.20191509

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:

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

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References

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Cited By

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}} $

Figure 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

Figure 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

Figure 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

Figure 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

Figure 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}} $

Figure 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

Figure 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

Figure 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

Figure 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

Figure 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

Figure 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

Figure 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
[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
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