Dynamical Systems and Control
Vol.07 No.03(2018), Article ID:24880,7 pages
10.12677/DSC.2018.73016

Discharge Pattern of Neuron in the Pre-Bötzinger Complex under Magnetic Field

Qinyu Cao*, Fen Ma, Wangjuan Liang

School of Science, North China University of Technology, Beijing

Received: Apr. 15th, 2018; accepted: May 9th, 2018; published: May 16th, 2018

ABSTRACT

Magnetic field has important effects on the firing activities of neuron. The function of memristor is similar as that of plasticity of synapse. Based on the Butera dynamic model added with memristor, the discharge pattern of pre-Bӧtzinger complex under the magnetic flux is studied. By fast-slow analysis and two-parameter bifurcation analysis, the dynamic mechanisms of the discharge pattern of neurons in the pre-Bӧtzinger complex are studied with the potassium conductance varying.

Keywords:Memristor, Magnetic Flux, Bursting, Fast-Slow Bifurcation Analysis

1. 引言

2. 模型

$\left\{\begin{array}{l}\frac{\text{d}v}{\text{d}t}=\left(-{I}_{Nap}-{I}_{Na}-{I}_{K}-{I}_{L}-{I}_{tonic}-{I}_{app}\right)/C\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(1\right)\\ \frac{\text{d}h}{\text{d}t}=\epsilon \left({h}_{\infty }\left(v\right)-h\right)/{\tau }_{h}\left(v\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ 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}_{n}\left(v\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(3\right)\\ \frac{\text{d}\phi }{\text{d}t}=f\left(\phi \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(4\right)\\ \frac{\text{d}q\left(\phi \right)}{\text{d}\phi }=\rho \left(\phi \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(5\right)\end{array}$

${I}_{Nap}={g}_{Nap}{m}_{p,\infty }\left(v\right)h\left(v-{E}_{Na}\right)$${I}_{Na}={g}_{Na}{m}_{\infty }^{3}\left(v\right)\left(1-n\right)\left(v-{E}_{Na}\right)$

${I}_{K}={g}_{K}{n}^{4}\left(v\right)\left(v-{E}_{k}\right)$${I}_{L}={g}_{L}\left(v-{E}_{L}\right)$${I}_{tonic}={g}_{tonic\text{-}e}\left(v-{E}_{syn\text{-}e}\right)$

${I}_{app}={k}_{1}\ast v\ast \rho \left(\phi \right)$$f\left(\phi \right)=v-{k}_{2}\ast \phi$

$\rho \left(\phi \right)$ 是指磁通控制忆阻器，用来描述磁通量φ与v之间的关系，其表达式为： $\rho \left(\phi \right)=\alpha +3\beta {\phi }^{2}$ ，其中 $\alpha ,\beta$ 是给定的参数值。参数 ${k}_{1}$ 对膜电位上的电流起到调节作用， ${k}_{2}$ 用来抑制磁通量的增加。其它函数的具体表达式以及参数的取值详见附录。

3. 神经元放电行为转迁的分析

${g}_{K}=5$ 时，在不考虑和考虑磁通量并添加磁流忆阻器的条件下，神经元的放电模式分别如图1(a)，

(a) (b)(c) (d)

Figure 1. Neuronal discharge patterns and fast/slow bifurcation analysis with ${g}_{K}=5$ . (a) Membrane potential under conditions no magnetic flow with ${I}_{app}=0,\text{\hspace{0.17em}}f\left(\phi \right)=0,\text{\hspace{0.17em}}\rho \left(\phi \right)=0$ ; (b) The time series of membrane potential under magnetic flow with ${k}_{1}=0.05,\text{\hspace{0.17em}}{k}_{2}=2,\text{\hspace{0.17em}}\alpha =2,\text{\hspace{0.17em}}\beta =0.0005$ ; (c) Fast/slow bifurcation analysis of bursting under the same parameter set as that of (a); (d) fast/slow bifurcation analysis of bursting under the same parameter set as that of (b)

(a) (b)(c) (d)

Figure 2. Neuronal discharge patterns and fast/slow bifurcation analysis with ${g}_{K}=8.5$ . (a) Membrane potentialunder conditions no magnetic flow with ${I}_{app}=0,\text{\hspace{0.17em}}f\left(\phi \right)=0,\text{\hspace{0.17em}}\rho \left(\phi \right)=0$ ; (b) The time series of membrane potential under magnetic flow with ${k}_{1}=0.05,\text{\hspace{0.17em}}{k}_{2}=2,\text{\hspace{0.17em}}\alpha =2,\text{\hspace{0.17em}}\beta =0.0005$ ; (c) Fast/slow bifurcation analysis of bursting under the same parameter set as that of (a); (d) fast/slow bifurcation analysis of bursting under the same parameter set as that of (b)

(a) (b)(c) (d)

Figure 3. Neuronal discharge patterns and fast/slow bifurcation analysis with ${g}_{K}=20$ . (a) Membrane potentialunder conditions no magnetic flow with ${I}_{app}=0,\text{\hspace{0.17em}}f\left(\phi \right)=0,\text{\hspace{0.17em}}\rho \left(\phi \right)=0$ ; (b) The time series of membrane potential under magnetic flow with ${k}_{1}=0.05,\text{\hspace{0.17em}}{k}_{2}=2,\text{\hspace{0.17em}}\alpha =2,\text{\hspace{0.17em}}\beta =0.0005$ ; (c) Fast/slow bifurcation analysis of bursting under the same parameter set as that of (a); (d) fast/slow bifurcation analysis of spiking under the same parameter set as that of (b)

“fold/homoclinic”型簇放电，如图2(c)和图2(d)所示。

${g}_{K}=20$ 时，磁流作用使得神经元的放电模式由周期为2的簇放电转变为周期为1的峰放电，如图3(a)和图3(b)所示，对应的快慢分析分别如图3(c)和图3(d)所示。磁流作用使得神经元的放电周期变小，频率增大。同时，磁流作用使得双稳区域消失，此时只存在稳定的极限环吸引子(图3(d))，即神经元的放电模式为峰放电(图3(b))。也即，磁流作用使得神经元的放电模式由簇放电转变为峰放电。

4. 双参数分岔分析

5. 结论

(a) (b)

Figure 4. Two parameter bifurcation analysis of the fast subsystem in $\left(h,{g}_{K}\right)$ plane. (a) ${I}_{app}=0,\text{\hspace{0.17em}}f\left(\phi \right)=0,\text{\hspace{0.17em}}\rho \left(\phi \right)=0$ , discharge patterns of neuron are not affected by the magnetic flow; (b) ${k}_{1}=0.05,\text{\hspace{0.17em}}{k}_{2}=2,\text{\hspace{0.17em}}\alpha =2,\text{\hspace{0.17em}}\beta =0.0005$ , under the conditions of magnetic flow

Discharge Pattern of Neuron in the Pre-BO¨ tzinger Complex under Magnetic Field[J]. 动力系统与控制, 2018, 07(03): 147-153. https://doi.org/10.12677/DSC.2018.73016

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Table 1. Parameter Values in Articles