Nonlinear mechanism of excitatory autapse-induced reduction or enhancement of firing frequency of neuronal bursting

Hua Hong-Tao; Lu Bo; Gu Hua-Guang

doi:10.7498/aps.69.20191709

Abstract

Excitatory and inhibitory effect always induces the enhancement and inhibitory effect of neural electronic activities, which is the common viewpoint of the modulations to the neural firing and plays important roles in the information processing of the nervous system. In the present paper, the Homoclinic/Homoclinic bursting pattern with alternation behavior between burst containing multiple spikes and subthreshold oscillations and the tough value of the burst lower than that of the subthreshold oscillations is chosen as representative, and the excitatory effect on the complex nonlinear dynamics of the representative bursting pattern is studied. For the excitatory autapse with suitable autaptic time delay and strength, the autaptic current pulse applied to the trough of the burst can induce the number of spikes within a burst to decrease and then the average firing frequency to decline, which presents a novel example different from the common viewpoint of the excitatory effect. The excitatory autapse induces the average firing frequency to increase in the remained parameter region of two-parameter plane of the autaptic time delay and strength. With bifurcations acquired by the fast/slow variable dissection method and phase trajectory, the subthreshold oscillations of the bursting correspond to a subthreshold limit cycle of the fast subsystem and the spike within burst corresponds to a suprathreshold limit cycle, and excitatory autaptic current can induce the transition from suprathreshold limit cycle to subthreshold limit cycle, which leads the spike to terminate in advance and is the cause for reducing the average firing frequency. The results is the present paper are compared with the phenomenon and bifurcation mechanism that the excitatory autapse can induce the spike number to decrease within a burst but the average firing frequency to increase as indicated in a recent study on the Fold/Homoclinic bursting. These results enrich the uncommon phenomenon of the neuronal electrical activities, reveal the underlying nonlinear mechanism, provide a new way to regulate the bursting pattern, and disclose the potential functions of the excitatory autapse.

Keywords:

Authors and contacts

1.
School of Mathematical Science, Henan Institute of Science and Technology, Xinxiang 453003, China
2.
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China

Corresponding author: Lu Bo, cheersnow@163.com

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Cited By

图 1 改进FHN模型簇放电模式及其快慢变量分离　(a) 周期8簇(圆圈对应V = –0.3396, t = 141.15); (b) 快子系统的分岔, 其中黑色实线、黑色虚线、蓝色点线和蓝色虚线分别表示稳定结点、稳定焦点、鞍点和不稳定焦点; L₁和L₂为两个鞍结分岔点、H₁和H₂为两个Hopf分岔点; 从H₁开始的上下两个洋红色实线分别表示阈下稳定极限环的最大值与最小值, SH₁为鞍同宿轨分岔点, 对应u ₁ = –0.760022; 从H₂开始的上、下两个红色实线分别表示阈上稳定极限环的最大值与最小值, SH₂为鞍同宿轨分岔点, 对应u ₂ = –0.959267; (c) 图(b)与周期8簇放电轨线(蓝色实线)的叠加; (d) 图(c)的局部放大

Figure 1. Bursting pattern and fast/slow variable dissection of the modified FHN model. (a) Period-8 bursting pattern (the position of the hollow circle corresponds to V = –0.3396 and t = 141.15). (b) The bifurcations of the fast subsystem. The black solid line, black dashed line, blue dotted line, and blue dashed line represent the stable node, the stable focus, the saddle, and the unstable focus. There are two fold bifurcation points of equilibrium point L₁ and L₂ and two Hopf bifurcation points H₁ and H₂. The maximum (minimum) value of the subthreshold stable limit cycle is represented by the upper (lower) magenta solid line, and the homoclinic bifurcation point SH₁ corresponds to u ₁ = –0.760022. The maximum (minimum) value of the suprathreshold stable limit cycle corresponds to upper (lower) solid red line, and the homoclinic bifurcation point SH₂ corresponds to u ₂ = –0.959267. (c) The trajectory of period-8 bursting (solid blue line) and panel (b) plotted in one figure. (d) The partial enlargement of panel (c).

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图 2 改进FHN模型的快子系统和全系统在相平面(w, V)的动力学行为　(a) u = –0.8513时快子系统的阈上极限环(红色虚线)和阈下极限环(洋红色虚线), 箭头表示轨线运行方向, $\dot V$的零值线(蓝色点线)和$\dot w$的零值线(蓝色虚线), “□”表示不稳定焦点, “○”表示鞍点; (b)全系统周期8簇放电的轨线在相平面(w, V)的投影(蓝色实线, 箭头为运行方向)与图(a)中阈上(红色虚线)和阈下极限环(洋红色虚线)的叠加

Figure 2. Dynamical behavior of fast subsystem and whole system in plane (w, V) of the modified FHN model: (a) The suprathreshold stable limit cycle (red dashed line), the subthreshold stable limit cycle (magenta dashed line), the direction of the trajectory (arrow), and the nullcline of $\dot V$ (blue dotted line) and $\dot w$ (blue dashed line) of the fast subsystem corresponds to u = –0.8513; “□” represents the unstable focus, and “○” represents the saddle; (b) the projection (solid blue line) of period-8 bursting of the whole system onto the phase plane (w, V), and suprathreshold stable limit cycle (red dashed line) and the subthreshold stable limit cycle (magenta dashed line) in panel (a) plotted in one figure.

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图 3 兴奋性脉冲(绿色实线)作用在周期8簇放电(蓝色点线)的簇内第1—7个谷值附近, 使系统跃迁到阈下极限环(洋红色虚线)提前结束放电(黑色实线), 然后分别经过2, 3, 4, 4, 5, 6和7个阈下振荡后先是经过9个放电峰, 然后恢复到周期8簇放电 (a1), (a2) Δt = 69.7, 作用在第1谷值附近; (b1), (b2) Δt = 78.35, 作用在第2谷值附近; (c1), (c2) Δt = 87.15, 作用在第3谷值附近; (d1), (d2) Δt = 96.2, 作用在第4谷值附近; (e1), (e2) Δt = 105.55, 作用在第5谷值附近; (f1), (f2) Δt = 115.25, 作用在第6谷值附近; (g1), (g2) Δt = 125.5, 作用在第7谷值附近; 左列, 放电; 右列, 对应左列图中“►”到“■”之间部分在相平面(w, V)的轨迹(黑实线)和快子系统的阈上(红虚线)和阈下(洋红虚线)极限环的轨迹图(u = –0.8513); “►”表示脉冲作用前放电峰的峰值的位置和轨线的顺时针运行方向, “●”表示脉冲作用相位, “■”表示恢复簇放电后的第1个峰值

Figure 3. Transition from suprathreshold stable limit cycle (dashed red line) to subthreshold stable limit cycle (dashed magenta line) to terminate the firing (solid black line) in advance induced by excitatory impulse (solid green line) applied at suitable phase near the 1^st to 8^th trough within burst of the period-8 bursting (dotted blue line), and recover to period-8 bursting after 2, 3, 4, 4, 5, 6 and 7 subthreshold oscillations and one period-9 bursting. (a1), (a2) Δt = 69.7, application phase near the 1^st trough; (b1), (b2) Δt = 78.35, application phase near the 2^nd trough; (c1), (c2) Δt = 87.15, application phase near the 3^rd trough; (d1), (d2) Δt = 96.2, application phase near the 4^th trough; (e1), (e2) Δt = 105.55, application phase near the 5^th trough; (f1), (f2) Δt = 115.25, application phase near the 6^th trough; (g1), (g2) Δt = 125.5, application phase near the 7^th trough. Left column: bursting; Right column: The projections (black solid line) in the phase plane (w, V) corresponding to the part between “►” and “■” of the corresponding left figure, and the suprathreshold stable limit cycle (red dashed line) and the subthreshold stable limit cycle (magenta dashed line) corresponding to u = –0.8513 plotted in one figure; “►” represents the peak of the spike within burst before the pulse stimulation and clockwise direction of the trajectory, “●” represents the application phase of the pulse, and “■” represents the first peak after the recovery of bursting.

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图 4 具有兴奋性自突触的改进FHN模型在g = 0.02和τ = 3.75时的新放电模式及其快慢变量分离　(a) 周期8簇(蓝色点线)在兴奋性自突触电流(上黑色实线)作用下诱导出新的放电模式(下黑色实线); (b)自突触的作用下新模式(►到■时段)在相平面(w, V )的投影, 轨线在相位●从阈上极限环(红色虚线)跃迁到阈下极限环(洋红色虚线); (c) 新放电模式(►到■时段) 在相平面(u, V )的投影; (d) 图(c)与快子系统分岔图1(b)的叠加

Figure 4. A novel bursting pattern and the corresponding fast/slow variable dissection of the modified-FHN model with excitatory autapse when g = 0.02 and τ = 3.75: (a) The new bursting pattern (lower solid black line) induced by excitatory autaptic current (upper solid black line) acted on the period-8 bursting (dotted blue line); (b) projection of the new pattern (from ► to ■ in panel (a)) on phase plane (w, V ); at the phase ●, the trajectory of the new pattern (from ► to ■ in panel (a)) jumps into the subthreshold stable limit cycle (dashed magenta line) from suprathreshold stable limit cycle (dashed red line); (c) the projection of the trajectory of the novel bursting pattern (from ► to in panel (a)) on the plane (u, V ); (d) the panel (c) and the bifurcation of the fast subsystem Fig. 1(b) plotted in one figure.

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图 5 具有兴奋性自突触的改进FHN模型在g = 0.02和τ = 12.6时的新放电模式及其快慢变量分离　(a) 周期8簇(蓝色点线)在兴奋性自突触(上黑色实线)作用下诱导出新的放电模式(下黑色实线); (b) 自突触的作用下新模式(►到■时段)在相平面(w, V )的投影, 轨线在相位●从阈上极限环(红色虚线)跃迁到阈下极限环(洋红色虚线); (c) 新放电模式(►到■时段) 在相平面(u, V )的投影; (d) 图(c)与快子系统分岔图1(b)的叠加

Figure 5. A new bursting pattern and the corresponding fast/slow variable dissection of the modified-FHN model with excitatory autapse when g = 0.02 and τ = 12.6: (a) The new bursting pattern (lower solid black line) induced by excitatory autaptic current (upper solid black line) acted on the period-8 bursting (dotted blue line); (b) projection of the trajectory of the new pattern (from ► to ■ in panel (a)) on phase plane (w, V ); at the phase ●, the trajectory of the new pattern (from ► to ■ panel (a)) jumps into the subthreshold stable limit cycle (dashed magenta line) from suprathreshold stable limit cycle (dashed red line); (c) the projection of the trajectory of the novel bursting pattern (from ► to ■ in panel (a)) is plotted on the plane (u, V ); (d) the panel (c) and the bifurcation of the fast subsystem Fig. 1(b) plotted in one figure.

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图 6 具有兴奋性自突触的改进FHN模型在g = 0.02和τ = 20.65时的新放电模式及其快慢变量分离　(a) 周期8簇(蓝色点线)在兴奋性自突触(上黑色实线)作用下诱导出新的放电模式(下黑色实线); (b) 自突触的作用下新模式(►到■时段)在相平面(w, V )的投影, 轨线在相位●从阈上极限环(红色虚线)跃迁到阈下极限环(洋红色虚线); (c) 新放电模式(►到■时段) 在相平面(u, V )的投影; (d) 图(c)与快子系统分岔图1(b)的叠加

Figure 6. A new bursting pattern and the corresponding fast/slow variable dissection of the modified-FHN model with excitatory autapse when g = 0.02 and τ = 20.65: (a) The new bursting pattern (lower solid black line) is induced by excitatory autaptic current (upper solid black line) acted on the period-8 bursting (dotted blue line); (b) projection of the trajectory of the new pattern (from ► to ■ in panel (a)) on phase plane (w, V ); at the phase ●, the trajectory of the new pattern (from ► to ■ in panel (a)) jumps into the subthreshold stable limit cycle (dashed magenta line) from suprathreshold stable limit cycle (dashed red line); (c) the projection of the trajectory of the novel bursting pattern (from ► to ■ in panel (a)) is plotted on the plane (u, V ); (d) the panel (c) and the bifurcation of the fast subsystem Fig. 1(b) and plotted in one figure.

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图 7 具有兴奋性自突触的改进FHN模型在g = 0.02和τ = 70.6时的新放电模式及其快慢变量分离　(a) 周期8簇(蓝色点线)在兴奋性自突触(上黑色实线)作用下诱导出新的放电模式(下黑色实线); (b) 自突触的作用下新模式(图(a)中►到■时段)在相平面(w, V)的投影; 第1个兴奋脉冲作用在相位●1, 在放电峰上没有引起放电峰大的变化; 第2个兴奋脉冲作用在相位●2, 使本来应该产生的阈下振荡变成阈上放电; (c) 新放电模式在相平面(u, V)的投影; (d) 图(c)和原放电轨线(蓝虚线)及快子系统分岔图1(b)的叠加

Figure 7. A new bursting pattern and the corresponding fast/slow variable dissection of the modified-FHN model with excitatory autapse when g = 0.02 and τ = 70.6: (a) The new bursting pattern (lower solid black line) induced by excitatory autaptic current (upper solid black line) acted on the period-8 bursting (dotted blue line); (b) the projection of the trajectory of the new pattern (from ► to ■ in panel (a)) on phase plane (w, V) under the action of autapse; The nearly unchanged spike induced by the 1^st excitatory autaptic current pulse acting on phase ●1 within the spike; The expected subthreshold oscillation changes to suprathreshold firing induced by the 2^nd excitatory autaptic current pulse acting on phase ●2; (c) the projection of the trajectory of the novel bursting pattern is plotted on the plane (u, V); (d) the panel (c), original period-8 bursting (blue dotted line), and bifurcations of the fast subsystem Fig. 1(b) plotted in one figure.

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图 8 平均放电频率随不同参数增长的变化　(a) 固定g时随τ增长, 其中g = 0 (黑色实线), g = 0.02 (蓝色点线)和g = 0.07 (绿色星线); (b) 固定τ时随g的增长, 其中τ = 3 (蓝色实线), τ = 4 (绿色点线)和τ = 5 (红色点虚线); 黑色实线对应平均放电频率f₀ = 0.0567

Figure 8. Changes of the average firing frequency with increasing different parameter values: (a) With increasing τ values when g is fixed at g = 0 (solid black line), g = 0.02 (dotted blue line), g = 0.07 (asterisk green line); (b) with increasing g when τ is fixed at τ = 3 (solid blue line), τ = 4 (dotted green line), and τ = 5 (dash-dot red line); the solid black line represents f₀ = 0.0567.

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图 9 兴奋性自突触作用下的改进FHN模型的放电频率在平面(τ, g)上的分布　(a) 平均放电频率分布, 彩色代表频率高低; (b) 平均放电频率与f₀ = 0.0567的差的分布, 其中黑色区域, 频率低于内在频率f₀ = 0.0567, 白色区域, 频率高于内在频率f₀ = 0.0567; 绿色、洋红色和蓝色实心圆点对应图4 (τ = 3.75, g = 0.02)、图5 (τ = 12.6, g = 0.02)和图6 (τ = 20.65, g = 0.02), 红色实心圆点对应图7 (τ = 70.6, g = 0.02)

Figure 9. Distribution of the average firing frequency on the (τ, g)-plane of the modified-FHN model with excitatory autapse. (a) The average firing frequency. Color scale represents the value of firing frequency. (b) The difference between the average firing frequency and f₀ = 0.0567. Black area: average frequency is lower than f₀ = 0.0567; white area: average frequency is higher than f₀ = 0.0567. The green, magenta, and blue solid cycles correspond to Fig. 4 (τ = 3.75, g = 0.02), Fig. 5 (τ = 12.6, g = 0.02), and Fig. 6 (τ = 20.65, g = 0.02), and the red solid cycle corresponds to Fig. 7 (τ = 70.6, g = 0.02).

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表 1 本文与文献[21]的主要的差别(兴奋性自突触)

Table 1. Main differences between this paper and Ref. [21] (excitatory autapse).

	结果相同部分	不同部分
	结果相同部分	条件	结果	机制
文献[21]	放电个数降低	Fold/Homoclinic簇	无结论	休止期跃迁到放电
本文	放电个数降低	Homoclinic/Homoclinic簇	频率降低	放电跃迁到阈下振荡
今后	待研究	其他典型类型的簇	待研究

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[1]	Braun H A, Wissing H, Schäfer K, Hirsch M C 1994 Nature 367 270 Google Scholar
[2]	Glass L 2001 Nature 410 277 Google Scholar
[3]	Izhikevich E M, Desai N S, Walcott E C, Hoppensteadt F C 2003 Trends Neurosci. 26 161 Google Scholar
[4]	Jia B, Gu H G 2017 Int. J. Bifurcation Chaos 27 1750113 Google Scholar
[5]	Rinzel J, Lee Y S 1987 J. Math. Biol. 25 653 Google Scholar
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