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超前同步(anticipated synchronization, AS)是一种普遍存在的违反直觉的非线性行为: 被驱动系统的响应会早于驱动出现, 在神经系统的实验中也被发现. 本研究揭示了抑制性自突触诱发Morris-Lecar神经元模型产生AS, 给出了产生AS的条件. 在单向兴奋性驱动的双神经元耦合系统, 无论神经元是I型兴奋性还是II型兴奋性, 都只会产生驱动行为在响应之前的滞后同步(delayed synchronization, DS). 在被驱动神经元引入抑制性自突触, II型兴奋性神经元构成的耦合系统会表现出驱动在响应之后的AS; 随着自突触电导的增大, DS会转迁到AS; 而I型兴奋性神经元构成的耦合系统则只会产生DS. 进一步, 提示了AS产生与不产生分别与II型和I型兴奋性神经元的放电响应特性有关: II型神经元在抑制性脉冲刺激下放电提前而I型兴奋性神经元不易产生放电提前. 研究结果给出了抑制性自反馈诱发AS的神经元的兴奋性类型, 有助于理解违反直觉的动力学行为—AS, 给出了调控AS的可能手段, 为进一步研究AS提供了方向.Anticipated synchronization, the response of the driven subsystem which appears earlier than the stimulation from the driving subsystem, is a universally counterintuitive nonlinear behavior. This behavior is also observed in the experiment on the nervous system in different brain regions. In the present paper, the anticipated synchronization phenomenon evoked by the inhibitory autapse is simulated in the coupled system composed of Morris-Lecar model, and the condition of excitability of single neurons and parameter ranges for the anticipated synchronization is presented. For a coupled system composed of two neurons, whether both neurons are either type-I excitability/Hopf bifurcation or type-II excitability/saddle-node bifurcation on an invariant cycle, in a driven neuron unidirectionally receiving the excitatory synaptic current from a driving neuron the delayed synchronization (the response of the driven neuron appears after the drive of the driving neuron) instead of the anticipated synchronization is simulated. After the inhibitory autapse is introduced into the driven neuron, the anticipated synchronization can be simulated in the coupled neurons when both neurons are of type-II excitability. With the increase of the conductance of the inhibitory autapse, the transition from delayed synchronization to anticipated synchronization is simulated. The time interval between spike of the driving and driven neuron is acquired, and the parameter ranges of anticipated synchronization in the plane of conductance of the inhibitory autapse and excitatory synapse are obtained. However, if both neurons are of type-I excitability, only delayed synchronization is simulated for the driven neuron with inhibitory autapse. Furthermore, the appearance of anticipated synchronization for type-II neurons and no anticipated synchronization for type-I neurons are suggested to have a relationship between the different responses of firing to external inhibitory stimulation for neurons with type-II excitability and type-I excitability. For spiking of type-II neuron, when an inhibitory pulse stimulation is applied, the spike following the pulse appears earlier than the one in the absence of stimulation in a wide range of the stimulation phase. However, for spiking behavior of type-I excitability, the spike following an inhibitory pulse stimulation appears later than the spike in the absence of stimulation. The results present the condition of single neurons for the appearance of anticipated synchronization induced by the inhibitory self-feedback mediated by autapse, which is helpful for understanding the dynamics of the counter-intuitive behavior, anticipated synchronization, presenting possible measures to modulate the anticipated synchronization, and proving directions for further study of anticipated synchronization.
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Keywords:
- anticipated synchronization /
- excitability /
- bifurcation /
- inhibitory autapse
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图 1 II型兴奋性ML模型在抑制性刺激下的放电 (a)与无刺激的放电(黑线)相比, 负向方波脉冲(虚线, 幅值A = –7 μA/cm2, 宽度d = 4 ms)诱发的放电(红线)提前; (b)不同的负向方波脉冲(宽度d = 4 ms)诱发的PRC; (c)图(b)的局部放大图; (d)黑线和红线分别对应图(a)的黑线和红线(V, dV/dt)的相轨迹, 从脉冲刺激结束到各自的动作电位峰值; (e)图(d)的局部放大图(从脉冲刺激开始到V = –20 mV, 箭头代表刺激结束); (f)图(d)的局部放大图( 动作电位峰值前)
Fig. 1. Firing of ML model with type II excitability under the action of inhibitory stimulation: (a) Compared with no stimulations (black solid line), firing (red line) induced by negative square pulse (dashed line, amplitude A = –7 μA/cm2, width d = 4 ms) is earlier; (b) PRC induced by negative square pulses with different strengths (width d = 4 ms); (c) locally enlargement of panel (b); (d) black and red curve correspond to trajectory in (V, dV/dt) plane of black and red curve of panel (a), respectively (from begging time of the pulse stimulation to peak of the action potential); (e) enlargement of panel (d) (from begging time of the pulse stimulation to –20 mV); (f) enlargement of panel (d) (phase before the peak of action potential).
图 2 抑制性自突触诱发II型兴奋性ML神经元模型产生的3种动力学行为(
${g_{\text{I}}}$ = 0.3 μS/cm2). DS (${g_{\text{E}}}$ = 1.8 μS/cm2): (a1)驱动(黑)和被驱动(红)神经元的膜电位; (a2)两神经元放电时间间隔的变化. AS (${g_{\text{E}}}$ = 0.1 μS/cm2): (b1)驱动(黑)和被驱动(红)神经元的膜电位; (b2)两神经元放电时间间隔的变化. PD (${g_{\text{E}}}$ = 0.03 μS/cm2): (c1)驱动(黑)和被驱动(红)神经元的膜电位; (c2)两神经元放电时间间隔的变化Fig. 2. Three dynamic behaviors induced by inhibitory autapse (
${g_{\text{I}}}$ = 0.3 μS/cm2) in the ML neuron model with type II excitability. DS (${g_{\text{E}}}$ = 1.8 μS/cm2): (a1) Membrane potential of driving (black) and driven (red) neurons; (a2) change of time interval between spikes of two neurons. AS (${g_{\text{E}}}$ = 0.1 μS/cm2): (b1) Membrane potential of driving (black) and driven (red) neurons; (b2) change of time interval between spikes of two neurons. PD (${g_{\text{E}}}$ = 0.03 μS/cm2): (c1) Membrane potential of driving (black) and driven (red) neurons; (c2) change of time interval between spikes of two neurons.
图 3 单向耦合II型兴奋性ML模型在没有抑制性自突触(
${g_{\text{I}}}$ = 0)时产生DS (两神经元放电时差在不同兴奋性电导下大于0, 即$\tau $ > 0)Fig. 3. DS of type II ML model with unidirectional excitatory coupling and without inhibitory autapse (
${g_{\text{I}}}$ = 0) (time interval between spikes of the two neurons is larger than 0 at different values of conductance of excitatory synapse, i.e.$\tau $ > 0).
图 4 不同抑制性电导
${g_{\text{I}}}$ 下兴奋性耦合的II型兴奋性ML模型的AS (两神经元放电时间间隔$\tau $ 随${g_{\text{E}}}$ 的变化).$\tau $ > 0表示DS状态,$\tau $ < 0代表AS状态. 三角符号标记以左出现PDFig. 4. Anticipated synchronization of type II ML model with excitatory coupling at different values of the conductance of inhibitory autapse (changes of time interval
$\tau $ between spikes of two neurons with respect to${g_{\text{E}}}$ ).$\tau $ > 0 and$\tau $ < 0 represent DS and AS states, respectively. PD locates left to the triangle.
图 5 单向兴奋性耦合的II型兴奋性ML模型的3类行为在参数平面(
${g_{\text{I}}}$ ,${g_{\text{E}}}$ )的分布 (a)两神经元放电间隔$\tau $ 的分布, 红、绿和黑色分别表示DS ($\tau $ > 0), AS ($\tau $ < 0)和PD ($\tau $ 的值不稳定); (b)两神经元放电间隔$\tau $ 的量值Fig. 5. Distribution of three behavior in parameter plane (
${g_{\text{I}}}$ ,${g_{\text{E}}}$ ) of the type II ML model with unidirectional excitatory coupling: (a) Distribution of time interval$\tau $ between spikes of two neurons, red, green, and black indicate DS ($\tau $ > 0), AS ($\tau $ < 0), and PD (the value of$\tau $ is unstable), respectively; (b) values of time interval$\tau $ .
图 6 (a)不同抑制性方波脉冲(宽度d = 4 ms)刺激诱发I型兴奋性ML模型的PRC; (b)图(a)的局部放大图
Fig. 6. (a) PRC induced by inhibitory square pulses (width d = 4 ms) stimulation in ML model with type I excitability; (b) locally enlarged of panel (a).
图 7 单向耦合I型兴奋性ML模型在没有抑制性自突触电导(
${g_{\text{I}}}$ = 0 μS/cm2)时产生DS (两神经元放电时差在不同兴奋性电导下大于0, 即$\tau $ > 0)Fig. 7. DS of type I ML model with unidirectional excitatory coupling and without inhibitory autapse (
${g_{\text{I}}}$ = 0) (time interval between spikes of the two neurons is larger than 0 at different values of conductance of excitatory autapse, i.e.$\tau $ > 0).
图 8 抑制性自突触诱发I型兴奋性ML神经元模型产生的2种动力学行为(
${g_{\text{I}}}$ = 0.2 μS/cm2). DS (${g_{\text{E}}}$ = 0.02 μS/cm2): (a1)驱动(黑)和被驱动(红)神经元的膜电位; (a2)两神经元放电时间间隔的变化. PD (${g_{\text{E}}}$ = 0 μS/cm2): (b1)驱动(黑)和被驱动(红)神经元的膜电位; (b2)两神经元放电时间间隔的变化Fig. 8. Two dynamical behaviors induced by inhibitory autapse in the ML neuron model with type I excitability (
${g_{\text{I}}}$ = 0.2 μS/cm2). DS (${g_{\text{E}}}$ = 0.02 μS/cm2): (a1) Membrane potential of driving (black) and driven (red) neurons; (a2) change of time interval between spikes of two neurons. PD (${g_{\text{E}}}$ = 0.0 μS/cm2): (b1) Membrane potential of driving (black) and driven (red) neurons; (b2) change of time interval between spikes of two neurons.
图 9 不同抑制性电导
${g_{\text{I}}}$ 下兴奋性耦合I型兴奋性ML模型的DS (两神经元放电时间间隔$\tau $ 随${g_{\text{E}}}$ 的变化).$\tau $ > 0表示DS状态, 三角符号标记处出现PDFig. 9. DS of the type I ML model with excitatory coupling at different values of conductance of inhibitory autapse (changes of time interval
$\tau $ between spikes of the two neurons was with respect to${g_{\text{E}}}$ ).$\tau $ > 0 represents DS state, and the phase-drift locates to the triangle.
图 10 单向兴奋性耦合的I型兴奋性ML模型的DS在参数空间(
${g_{\text{I}}}$ ,${g_{\text{E}}}$ )的分布 (a)两神经元放电间隔$\tau $ 的分布; 红和黑分别表示DS ($\tau $ >0)和PD ($\tau $ 的值不稳定)区域; (b)两神经元放电间隔$\tau $ 的量值Fig. 10. Distribution of DS in parameter plane (
${g_{\text{I}}}$ ,${g_{\text{E}}}$ ) of type I ML model with unidirectional excitatory coupling: (a) Distribution of time interval$\tau $ between spikes of two neurons, red and black indicate DS ($\tau $ > 0) and PD (the value of$\tau $ is unstable), respectively; (b) values of time interval$\tau $ .表 1 ML模型的参数值
Table 1. Parameter values of ML model.
参数 第1组 第2组 $C$/(μF·cm–2) 5 20 ${g_{{\text{Ca}}}}$/(μS·cm–2) 4 4 ${V_{{\text{Ca}}}}$/mV 120 120 ${g_{\text{K}}}$/(μS·cm–2) 8 8 ${V_{\text{K}}}$/mV –80 –84 ${g_{\text{L}}}$/(μS·cm–2) 2 2 ${V_{\text{L}}}$/mV –60 –60 ${V_1}$/mV –1.2 –1.2 ${V_2}$/mV 18 18 ${V_3}$/mV 4 12 ${V_4}$/mV 17.4 17.4 $\phi $ 0.066667 0.066667 
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表 2 突触的参数值
Table 2. Parameter values of synapse.
参数 第1组 第2组 ${T_{\max }}$/mM–1 1 1 ${V_{\rm{p}}}$/mV 30 20 ${K_{\rm{p}}}$/mV 5 0.8 ${E_{\rm{E}}}$/mV 45 35 ${E_{\rm{I}}}$/mV –60 –50 ${\alpha _{\rm{E}}}$/(mM–1·ms–1) 0.1 0.8 ${\beta _{\rm{E}}}$/ms–1 0.5 1 ${\alpha _{\rm{I}}}$/(mM–1·ms–1) 0.1 0.05 ${\beta _{\rm{I}}}$/ms–1 0.18 1
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[1] Boccaletti S, Kurths J, Osipov G, Valladares D L, Zhou C S 2002 Phys. Rep. 366 1
Google Scholar
[2] Izhikevich E M 2007 Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Cambridge: The MIT Press) pp301−348
[3] Voss H U 2000 Phys. Rev. E 61 5115
Google Scholar
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Google Scholar
[5] 丁学利, 贾冰, 李玉叶 2019 物理学报 68 180502
Google Scholar
Ding X L, Jia B, Li Y Y 2019 Acta Phys. Sin. 68 180502
Google Scholar
[6] He Z W, Yao C G, Shuai J W, Nakano T 2020 Chin. Phys. B 29 128702
Google Scholar
[7] Wu F, Gu H 2020 Int. J. Bifurcat. Chaos 30 2030009
Google Scholar
[8] Zhao Z G, Li L, Gu H G 2020 Commun. Nonlinear Sci. Numer. Simulat. 85 105250
Google Scholar
[9] Voss H U 2001 Phys. Rev. Lett. 87 014102
Google Scholar
[10] Voss H U 2001 Phys. Rev. E 64 039904
Google Scholar
[11] Voss H U 2016 Phys. Rev. E 93 030201
Google Scholar
[12] Voss H U 2018 Chaos 28 113113
Google Scholar
[13] Tang S, Liu J M 2003 Phys. Rev. Lett. 90 194101
Google Scholar
[14] Ciszak M, Calvo O, Masoller C, Mirasso C R, Toral R 2003 Phys. Rev. Lett. 90 204102
Google Scholar
[15] Simonov A Y, Gordleeva S Y, Pisarchik A, Kazantsev V 2014 JETP Lett. 98 632
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[19] Sausedo-Solorio J M, Pisarchik A N 2014 Phys. Lett. A 378 2108
Google Scholar
[20] Pyragienè T, Pyragas K 2013 Nonlinear Dyn. 74 297
Google Scholar
[21] Pinto M A, Rosso O A, Matias F S 2019 Phys. Rev. E 99 062411
Google Scholar
[22] Matias F S, Gollo L L, Carelli P V, Bressler S L, Copelli M, Mirasso C R 2014 NeuroImage 99 411
Google Scholar
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Google Scholar
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Google Scholar
[25] Brovelli A, Ding M, Ledberg A, Chen Y, Nakamura R, Bressler S L 2004 Proc. Natl. Acad. Sci. USA 101 9849
Google Scholar
[26] Matias F S, Carelli P V, Mirasso C R, Mirasso C R, Copelli M 2017 Phys. Rev. E 95 052410
Google Scholar
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Google Scholar
[28] Dima G C, Copelli M, Mindlin G B 2018 Int. J. Bifurcat. Chaos 28 1830025
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[29] Ciszak M, Marino F, Toral R, Balle S 2004 Phys. Rev. Lett. 93 114102
Google Scholar
[30] Montani F, Rosso O A, Matias F S, Bressler S L, Mirasso C R 2015 Philos. Trans. A Math. Phys. Eng. Sci. 373 20150110
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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
[37] Sivaprakasam S, Shahverdiev E M, Spencer P S, Shore K A 2001 Phys. Rev. Lett. 87 154101
Google Scholar
[38] Ciszak M, Mirasso C R, Toral R, Calvo O 2009 Phys. Rev. E 79 046203
Google Scholar
[39] Stepp N, Turvey M T 2017 J. Exp. Psychol. Hum. Percept. Perform. 43 914
Google Scholar
[40] Morris C, Lecar H 1981 Biophys. J. 35 193
Google Scholar
[41] 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
[42] Ermentrout B 1996 Neural. Comput. 8 979
Google Scholar
[43] Smeal R M, Ermentrout G B, White J A 2010 Philos. Trans. R. Soc. Lond. B:Biol. Sci. 365 2407
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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