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The neocortex of the brain plays a most important role in achieving functions of the brain via the electrical activities of neurons. Understanding the transition regularity of firing patterns and underlying dynamics of firing patterns of neurons can help to identify the brain functions and to treat some brain diseases. Different neocortical neurons exhibit regular spiking (RS), fast spiking (FS), intrinsic bursting (IB), and continuous bursting (CB), which play vital roles and wide range of functions. Fast-slow variable dissection method combined with bifurcation analysis has been an effective method to identify the underlying dynamical mechanism of spiking and bursting modulated by a single slow variable. The spiking is related to the stable limit cycle of the fast subsystem, and the bursting is associated with the transitions or bifurcations between the stable limit cycle and resting state of the fast subsystem. Such underlying dynamics of bursting has been widely used to distinguish different bursting patterns and identify complex dynamics of bursting modulated by various different factors such as synaptic current, autaptic current, and stimulations applied at a suitable phase related to the bifurcations, which play important roles in the real nervous system to regulate neural firing behaviors. Unfortunately, the bursting of neocortical neuronal model (wilson model) is modulated by two slow variables, i.e. the gating variable of calcium-activated potassium channel H and the gating variable of T-type calcium channel T, with H being slower than T. Then, the underlying dynamical mechanism of the IB and CB of the neocortical neurons cannot be acquired by the fast-slow variable dissection method when H is taken as the sole slow variable, due to the fact that the fast-subsystem contains the slow variable T. In the present paper, we use the fast-slow variable dissection method with two slow variables (H and T ) to analyze the bursting patterns. The bifurcations of the fast subsystem, and the intersections between the bifurcation curves and the phase trajectory of bursting in the parameter plane (H, T ) are acquired. Owing to the fact that neither of the two slow variables of the bursting is sufficiently slow, the bifurcations of only some intersections are related to the bursting behaviors, but others not. Then, the position relationship between the bifurcation curves and bursting trajectory in the three-dimensional space (H, T, V ) (V is membrane potential of bursting) is further acquired, from which the bifurcations related to bursting behaviors are acquired and bifurcations unrelated to bursting behaviors are excluded. The start phase and the termination phase of the burst of the IB are related to the saddle-node on invariant circle (SNIC) bifurcation, but not to the saddle-node (SN) bifurcation. The start phase and termination phase of the burst of the CB are related to the SNIC bifurcation and the supercritical Andronov-Hopf (SupHopf) bifurcation, respectively, but not to the SN bifurcation. The results present a comprehensive and in-depth understanding of the underlying dynamics of bursting patterns in the neocortical neurons, thereby laying the foundation for regulating the firing patterns of the neocortical neurons.
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Keywords:
- bifurcation /
- bursting /
- fast-slow variable dissection /
- neocortical
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图 1 Wilson模型在不同参数下的膜电位 (a) RS, 其中
$ I $ =10,$ {g_T} $ =0.1 mS/cm2,$ {g_H} $ = 5 mS/cm2,$ {\tau _R} $ = 4.2 ms; (b) FS, 其中$ I $ = 3.5,$ {g_T} $ = 0.25 mS/cm2,$ {g_H} $ = 0 mS/cm2,$ {\tau _R} $ = 1.5 ms; (c) CB, 其中$ I $ = 8.5,$ {g_T} $ = 2.25 mS/cm2,$ {g_H} $ = 9.5 mS/cm2,$ {\tau _R} $ = 4.2 ms; (d) IB, 其中$ I $ = 3.5,$ {g_T} $ = 1.2 mS/cm2,$ {g_H} $ = 3.4 mS/cm2,$ {\tau _R} $ = 4.2 ms; 图 (c), (d) 中, 点A和点B分别代表第一个簇(蓝色)的起始与结束相位Figure 1. Membrane potential of Wilson model at different parameter values: (a) RS,
$ I $ =10,$ {g_T} $ =0.1 mS/cm2,$ {g_H} $ = 5 mS/cm2,$ {\tau _R} $ = 4.2 ms; (b) FS,$ I $ = 3.5,$ {g_T} $ = 0.25 mS/cm2,$ {g_H} $ = 0 mS/cm2,$ {\tau _R} $ = 1.5 ms; (c) CB,$ I $ = 8.5,$ {g_T} $ = 2.25 mS/cm2,$ {g_H} $ = 9.5 mS/cm2,$ {\tau _R} $ = 4.2 ms; (d) IB,$ I $ = 3.5,$ {g_T} $ = 1.2 mS/cm2,$ {g_H} $ = 3.4 mS/cm2,$ {\tau _R} $ = 4.2 ms. In Fig. 1 (c) , (d), the points A and B represent the start and termination phases of first burst (blue), respectively.
图 5 峰放电的快子系统随H或T变化的分岔和峰放电的相轨迹 (蓝色实线) (a) RS, SNIC代表H ≈ 0. 568757处不变环上的鞍结分岔; (b) FS, SNIC和SupHopf分别代表H ≈ –0.365159处不变环上的鞍结分岔和H ≈ 7.012722处超临界霍普夫分岔. 红色实线、黑色虚线分别代表稳定和不稳定平衡点. 上 (
$ {V_{\max }} $ ) 和下 ($ {V_{\min }} $ ) 绿线代表稳定极限环的最大和最小幅值Figure 5. Bifurcation of the fast subsystem of the spiking with respect to H or T and the phase trajectory of spiking (solid blue curves): (a) RS, SNIC represents saddle-node on invariant circle bifurcation at H ≈ 0.568757; (b) FS, SNIC and SupHopf represent saddle-node on invariant circle bifurcation at H ≈ -0.365159 and the supercritical Andronov-Hopf bifurcation at H ≈ 7.012722, respectively. The red solid curves and the black dashed curves represent the stable and unstable equilibrium points, respectively. Upper (
$ {V_{\max }} $ ) and lower ($ {V_{\min }} $ ) green curves represent the maximum and minimum amplitudes of the stable limit cycle.
图 6 H为慢变量时, CB的快慢变量分离 (a) 快子系统的平衡点和极限环的分岔, 在H ≈ 0.287483和H ≈ 0.718103处的SN1和SN2代表鞍结分岔点, 红色实线和黑色虚线分别代表稳定和不稳定平衡点; (b) CB相轨迹与图 (a) 的叠加, 浅蓝色和灰色实线分别代表簇和休止期的相轨迹, 点A和点B分别代表簇的起始和结束
Figure 6. The fast-slow variable dissection to CB with H taken as the slow variable: (a) The bifurcations of the equilibrium point and the limit cycle of the fast subsystem. SN1 at H ≈ 0.287483 and SN2 at H ≈ 0.718103 represent saddle-node bifurcation points; respectively, the red solid curves and the black dashed curves represent stable and unstable equilibrium points, respectively; (b) phase trajectory of CB plotted with Fig. 6(a), the light blue and gray solid curves represent the phase trajectory of the burst and the quiescent state, respectively, and points A and B represent the start and termination phases of burst, respectively.
图 7 H为慢变量时, IB的快慢变量分离 (a) 快子系统的平衡点和极限环的分岔; (b) IB相轨迹与图 7 (a) 的叠加, 上、下支红色实线和中支黑色短划线、点线分别代表稳定和不稳定平衡点. 上、下绿色 (深蓝色) 实线分别代表稳定 (不稳定) 极限环, 浅蓝色和灰色实线代表簇和休止期的相轨迹, 点A和B分别代表簇的起始与结束. SN, SubHopf, SupHopf和FLC分别代表鞍结分岔、亚临界霍普夫分岔、超临界霍普夫分岔和极限环的鞍结分岔
Figure 7. The fast-slow variable dissection to IB with H taken as the slow variable: (a) Bifurcation of the equilibrium point and limit cycle of fast subsystem; (b) phase trajectory of IB plotted with Fig. 7(a). The upper and lower red solid curves represent stable equilibrium points, and the middle black dashed and dotted curves represent unstable equilibrium points, respectively. The upper and lower green (dark blue) solid curves represent the stable (unstable) limit cycles. The light blue and gray solid curves represent the phase trajectory of the burst and the quiescent state, and points A and B represent the start and termination phases of burst, respectively. SN, SubHopf, SupHopf, and FLC represent the saddle-node bifurcation, the subcritical Andronov-Hopf bifurcation, the supercritical Andronov-Hopf bifurcation, and the fold limit cycle bifurcation, respectively.
图 8 CB的快子系统在 (H, T ) 平面上的双参分岔和相轨迹 (a) SNIC (绿色实线)、SN (红色实线) 和SupHopf (黑色实线) 分别代表不变环上的鞍结分岔曲线、鞍结分岔曲线和超临界霍普夫分岔曲线, BT代表余维–2分岔点; 浅蓝色和灰色实线代表簇和休止期的相轨迹; (b) (a) 中BT分岔点附近的放大. P2是退化分岔点, SHC (蓝色实线) 代表鞍-同宿轨分岔
Figure 8. The bifurcations of fast subsystem of CB in two-parameter plane (H, T ) and the phase trajectory: (a) SNIC (green solid line), SN (red solid line) and SupHopf (black solid line) represent the saddle-node on invariant circle bifurcation curve, the saddle-node bifurcation curve and the supercritical Andronov-Hopf bifurcation curve respectively, BT represents the codimension-2 bifurcation point; the light blue and gray solid curves represent the phase trajectory of the burst and the quiescent state; (b) the enlargement near BT bifurcation point in Fig. (a). P2 is a degenerate bifurcation point, and SHC (blue solid line) represents the Saddle-Homoclinic orbit bifurcation.
图 9 在对应A, B和C三个T值处的CB的快慢变量分离 (a) 对应CB的起始点A (T ≈ 0.00522), A点对应SNIC分岔点; (b) 对应CB的结束点B (T ≈ 0.535852), B点对应SupHopf分岔点; (c) 对应点C (T ≈ 0.004636); 点C与SN分岔点距离较远. 浅蓝色和灰色实线代表簇和休止期的相轨迹. 红色实线、黑色虚线和绿色实线分别代表稳定平衡点、不稳定平衡点和稳定极限环. SN, SNIC和SupHopf分别代表鞍结分岔、不变环上的鞍结分岔和超临界霍普夫分岔
Figure 9. The fast-slow variable dissection to CB at three T values corresponding to points A, B and C: (a) Corresponding to initial point A of CB (T ≈ 0. 00522). Point A corresponds to the SNIC bifurcation point; (b) corresponding to termination point B of CB (T ≈ 0. 535852). Point B corresponds to the SupHopf bifurcation point; (c) corresponding to point C (T ≈ 0. 004636). Point C is far away from the SN bifurcation point. The light blue and gray solid curves represent the phase trajectory of burst and quiescent state, respectively. The red solid curves, black dashed curves, and green solid curves represent the stable equilibrium points, the unstable equilibrium points, and the stable limit cycle, respectively. SN, SNIC, and SupHopf represent the saddle-node bifurcation, the saddle-node on invariant circle bifurcation, and the supercritical Andronov-Hopf bifurcation, respectively.
图 10 (H, T, V ) 空间中CB的双参分岔和相轨迹. 蓝色、黄色和粉色面分别由不稳定焦点、鞍点和稳定结点构成, 红色、绿色、黑色粗线分别代表鞍结 (SN) 分岔、不变环上的鞍结 (SNIC) 分岔和超临界霍普夫 (SupHopf) 分岔, 上、下绿面代表稳定极限环的极大值 (
$ {V_{{\max}}} $ ) 和极小值 ($ {V_{{\min}}} $ ), 黑色细线代表簇的相轨迹 (a) 全图; (b) 图(a)中B点附近的放大图, B点位于SupHopf分岔曲线正下方, 对应簇的结束相位; (c) 图(a)中BT分岔点附近的放大; (d) 图(a)中A点和C点附近的放大, A点在SNIC分岔曲线正下方, 对应CB轨迹的起始相位, C点距离SN分岔曲线较远Figure 10. Two-parameter bifurcation of fast system and phase trajectory of CB in the (H, T, V ) space. The blue, yellow, and pink surfaces are composed of the unstable focus, the saddle and the stable node, respectively. The red, green, and black thick curves represent the saddle-node (SN) bifurcation, the saddle-node on invariant circle (SNIC) bifurcation, and the supercritical Andronov-Hopf (SupHopf) bifurcation. The upper and lower green surfaces represent the maximum value (
$ {V_{{\max}}} $ ) and the minimum value ($ {V_{{\min}}} $ ) of the stable limit cycle, respectively, and the black thin curve represents the phase trajectory of the bursting: (a) Global view; (b) the enlargement near point B in Fig. (a), point B is located just below the SupHopf bifurcation curve, corresponding to the termination phase of the burst; (c) the enlargement near BT bifurcation point in Fig. (a); (d) the enlargement near points A and C in Fig. (a). The point A is just below the SNIC bifurcation curve, corresponding to the start phase of burst, and point C is far from the SN bifurcation curve.
图 11 (H, T ) 平面上的IB的相轨迹与快子系统的分岔, 浅蓝色和灰色实线代表簇和休止期的相轨迹 (a) SNIC (绿色实线)、SN (红色实线) 和SupHopf (黑色实线) 分别代表不变环上的鞍结分岔曲线、鞍结分岔曲线和超临界霍普夫分岔曲线, BT代表余维–2分岔点 (Bogdanov-Takens); (b)图11(a) 中IB相轨迹附近的放大图, 蓝色区域对应放电区域, 空白区域对应静息态
Figure 11. The phase trajectory of IB bursting and bifurcations of the fast subsystem in phase plane (H, T ). The light blue and gray solid curves represent the phase trajectory of burst and quiescent state, respectively: (a) SNIC (green solid curves), SN (red solid curves), and SupHopf (black solid curves) represent the saddle-node on invariant circle bifurcation curve, the saddle-node bifurcation curve, and the supercritical Andronov-Hopf bifurcation curve, respectively, and BT represents the codimension-2 bifurcation point (Bogdanov-Takens); (b) the enlargement around the phase trajectory of IB in Fig. 11(a), the blue area corresponds to the firing behavior, and the blank area corresponds to the resting state.
图 12 在对应A, B, C和D四个T值处IB的快慢变量分离, 红色实线、黑色虚线和绿色实线分别代表稳定平衡点、不稳定平衡点和稳定极限环, SN, SNIC和SupHopf分别代表鞍结分岔、不变环上的鞍结分岔和超临界霍普夫分岔. 浅蓝色和灰色实线代表IB的相轨迹 (a) 对应点A (T ≈ 0.003489, 簇的起始相位); (b) 对应点B (T ≈ 0.178687, 簇的结束相位); (c) 对应点C (T ≈ 0.138212, 距离SN分岔曲线较远); (d) 对应点D (T ≈ 0.025683, 距离SN分岔曲线较远)
Figure 12. The fast-slow variable dissection to IB bursting at four T values corresponding to points A, B, C and D. The red solid curve, the black dashed curves, and the green solid curves represent the stable equilibrium point, the unstable equilibrium point, and the stable limit cycle, respectively. SN, SNIC, and SupHopf represent the saddle-node bifurcation, the saddle-node on invariant circle bifurcation, and the supercritical Andronov-Hopf bifurcation, respectively. The light blue, and gray solid curves represent the phase trajectory of IB: (a) Corresponding to point A (T ≈ 0.003489, the start phase of burst); (b) corresponding to point B (T ≈ 0.178687, the termination phase of burst); (c) corresponding to point C (T ≈ 0.138212, far from the SN bifurcation curve); (d) corresponding to point D (T ≈ 0.025683, far from the SN bifurcation curve).
图 13 (H, T, V ) 空间中快子系统的双参分岔和IB的相轨迹, H和T为慢变量, 蓝色、黄色和粉色面分别由不稳定焦点、鞍点和稳定结点构成, 红色和绿色粗线分别代表鞍结 (SN) 分岔和不变环上的鞍结分岔 (SNIC), 上、下绿面代表稳定极限环的极大值和极小值, 黑色细线代表IB的相轨迹 (a)全图; (b) IB相轨迹附近的放大, 点A和点B对应簇的起始相位和结束相位, 点C和点D距离SN分岔曲线较远
Figure 13. Two-parameter bifurcations of fast subsystem in the (H, T, V ) space and phase trajectory of IB with H and T regarded as slow variables. The blue, yellow, and pink surfaces are composed of unstable focus, saddle, and stable node, respectively. The red and green thick curves represent the saddle-node (SN) bifurcation and the saddle-node on invariant circle (SNIC) bifurcation, respectively. The upper and lower green surfaces represent the maximum and minimum values of the stable limit cycle, and the thin black line represents the phase trajectory of IB: (a) Global view; (b) the enlargement around the phase trajectory of IB in Fig. 13(a). Points A and B correspond to initial and termination phases of the burst, and points C and D are far from the SN bifurcation curve.
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Google Scholar
[2] Keener J, Sneyd J 2009 Mathematical Physiology: II: Systems Physiology (New York: Springer) pp523–626
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Google Scholar
[4] Cowansage K K, Shuman T, Dillingham B, Chang A, Golshani P, Mayford M 2014 Neuron 84 432
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
[9] Izhikevich E M 2007 Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Cambridge: The MIT Press) pp159–378
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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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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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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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