神经系统疾病与认知动力学 (I) : 癫痫发作的动力学与控制

韩芳; 樊登贵; 张丽媛; 王青云

doi:10.6052/1000-0992-21-064

神经系统疾病与认知动力学 (I) : 癫痫发作的动力学与控制

doi: 10.6052/1000-0992-21-064

1.
东华大学信息科学与技术学院, 上海 201620
2.
北京科技大学数理学院, 北京 100083
3.
北京工业大学环境与生命学部, 北京 100124
4.
北京航空航天大学航空科学与工程学院, 北京 100191

基金项目: 国家自然科学基金资助 (11932003, 12072021, 12102014, 11972115).

详细信息

作者简介:
王青云, 北京航空航天大学动力学与控制学科责任教授, 博导. 国家杰出青年科学基金获得者, 教育部“长江学者”特聘教授, 2011年入选教育部新世纪人才支持计划. 2020被评为爱思唯尔中国高被引学者, 2019年获得中国力学学会科学技术奖自然科学二等奖, 2012年获教育部高等学校科学研究优秀成果自然科学二等奖, 2009年获全国百篇优秀博士论文提名奖, 2009年获内蒙古自治区第七届青年科技奖, 2008年获北京大学优秀博士后奖. 中国力学学会第十、十一届动力学与控制专业委员会副主任, 神经动力学专业组组长, 中国力学学会理性力学专业委员会副主任, 北京市力学学会理事, 中国振动工程学会非线性振动专业委员会副主任, 生物控制和生物医学工程专业委员会委员, 国际杂志《International Journal of Bifurcation and Chaos》的Associate Editor, 《Applied Mathematics and Mechanics》特约编委, 《Cognitive Neurodynamics》编委, 北京航空航天大学学报编委, 高等教育出版社出版的《网络科学与工程丛书》编委. 长期从事神经动力学和调控方面的研究工作. 研究成果主要结合生物神经系统电生理活动和脑疾病临床表征的非线性特征, 建立理论模型揭示神经系统呈现的非线性动力学行为的本质机制, 发展精准定位脑疾病病灶的理论方法, 发展控制神经系统疾病的调控方法和技术, 为脑神经疾病临床诊疗提供可靠的理论基础

通讯作者:
nmqingyun@163.com

中图分类号: R338.1,O322
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出版历程
- 收稿日期: 2021-12-13
- 录用日期: 2022-01-26
- 网络出版日期: 2022-01-27
- 刊出日期: 2022-06-25

Neurological disease and cognitive dynamics (I): Dynamics and control of epileptic seizures

1.
College of Information Science and Technology, Donghua University, Shanghai 201620, China
2.
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
3.
Faculty of Environment and Life, Beijing University of Technology, Beijing 100124, China
4.
School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

More Information

Corresponding author: nmqingyun@163.com

摘要

摘要: 研究表明癫痫发作过程与神经系统本身的非线性动力学行为密切相关. 因此, 开展癫痫发作的非线性网络动力学建模与调控问题的研究, 有助于理解癫痫临床表征的动力学机理和定位致痫灶网络, 进而设计有效的网络调控策略. 本文回顾了癫痫脑神经疾病网络动力学与控制方面的研究进展, 系统总结了本文作者近年来在癫痫发作动力学建模分析及其调控等方面取得的研究成果. 首先, 基于海马齿状回CA3区环路神经元网络模型, 分析了影响颞叶癫痫发作的分子和网络结构因素, 阐释了癫痫发作转迁的动力学机制. 其次, 由于脑神经系统的集群编码特性, 基于神经场模型和平均场模型建模方法完善了皮质−基底节−丘脑环路网络动力学理论框架, 并基于此框架分析了失神癫痫发作转迁的动力学分岔机制, 探讨了不同类型癫痫发作的转迁路径, 发现了失神癫痫发作转迁的多稳态共存现象, 揭示了时滞对失神癫痫同步发作的控制效果, 设计了丰富有效的癫痫深脑刺激调控策略, 给出了电刺激调控失神癫痫发作的动力学解释. 最后, 通过数据驱动的统计建模和神经元群模型动力学建模分析, 提出了局灶癫痫致痫灶定位及寻找有效控制癫痫发作网络关键节点的理论新方法. 这些研究成果为理解难治性癫痫发作动力学本质及在临床诊疗的应用方面提供重要理论支撑. 最后对进一步研究给出若干建议.
- 癫痫发作动力学 /
- 平均场模型 /
- 时滞 /
- 同步 /
- 神经反馈与调控
Abstract: Studies have shown that the process of epileptic seizures is closely related to the nonlinear dynamics of the nervous system itself. Therefore, the study of nonlinear network dynamics modeling and regulation of epileptic seizures is helpful in understanding the dynamic mechanism of clinical manifestations of epilepsy, locating the epileptic foci network, and then designing effective network regulation strategies. This article reviews the research progress in network dynamics and control of epileptic neurological diseases and systematically summarizes our research results in recent years in the modeling and analysis of epileptic seizure dynamics and their regulation. Firstly, based on the neuron network model of the hippocampal dentate gyrus-CA3 loop, the molecular and network structural factors that affect temporal lobe seizures were analyzed, and the dynamic mechanism of seizure transition was explained. Secondly, due to the cluster coding characteristics of the brain nervous system, based on the methods of both the neural field model and mean field model, the network dynamics framework of the basal ganglia-thalamocortical (BGCT) circuit was improved. Based on this framework, the dynamic bifurcation mechanism of absence epileptic seizure transition was analyzed, the transition path of different types of seizures was explored, and the multi-stable coexistence phenomenon of absence seizure transition was discovered. The effect of time delay on the synchronization seizures was also revealed. We also designed rich and effective deep brain stimulation (DBS) control strategies for epilepsy and gave a dynamic explanation of electrical stimulation to control absence epileptic seizures. Finally, based on the data-driven statistical modeling and the dynamics analysis of the neuronal population model, new theoretical methods for the foci localization of focal epileptics and finding the key network nodes for effectively controlling seizures were proposed. These results provide important theoretical support for understanding the dynamic nature of refractory seizures and their application in clinical diagnosis and treatment. Lastly, some suggestions are given for further research.
- epileptic seizure dynamics /
- mean field model /
- delay /
- synchronization /
- neurofeedback and modulation

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图 1 DG-CA3网络模型连接示意图. (a)反向投射位置从CA3区锥体神经元到DG区苔藓纤维细胞的胞体, (b)反向投射位置从CA3区锥体神经元到DG区中间神经元 (Zhang et al. 2017)

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图 2 (a)锥体神经元的放电时序图, (b)随时间变化的能量谱图 (Zhang et al. 2017)

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图 3 DG-CA3区神经元放电主频随反向投射强度变化示意图. (a)反向投射位置从P到MS, (b)反向投射位置从P到DI (Zhang et al. 2017)

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图 4 改进型DG-CA3网络模型结构图 (Zhang et al. 2018a)

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图 5 网络平均膜电位在BDNF和神经噪声联合作用下的转迁情况. (a)噪声强度为2 μA/cm², (b)噪声强度为8 μA/cm² (Zhang et al. 2018a)

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图 6 在固定噪声强度下, 网络同步性指数随BDNF浓度变化的演化图 (Zhang et al. 2018a)

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图 7 节点随外部刺激变化的频带能量直方图 (Zhang et al. 2018b)

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图 8 (a)基于DG-CA3系统模体结构的详细原理图, (b)是基于(a)得出10种可能的连接结构示意图 (Zhang et al. 2018b)

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图 9 HFOs对于模体连接结构和外部刺激响应的箱状图. (a)为γ频带, (b)为连波频带, (c)为快连波频带 (Zhang et al. 2018b)

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图 10 同步性指数对不同模体结构、外部刺激和抑制性连接强度变化的响应曲线. FFI和FBI抑制性连接电导强度分别为 (a) 0.0001 mS/cm², (b) 0.01 mS/cm² (Zhang et al. 2018b)

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图 11 单个神经场振子模型, $Ex$表示兴奋性神经元集群, $In$表示抑制性神经元集群, ${C_1},{C_2},{C_3}$是两个神经元集群之间的相互作用强度, “+”和“−”分别表示兴奋性和抑制性突触作用

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图 12 (a)慢变量调控的双耦合振子结构示意图 (Zhang L Y et al. 2020b) , (b)由全局慢变量调控的无标度局灶性致痫网络结构示意图, (c)为网络连接矩阵, (d)为耦合强度矩阵图 (Zhang et al. 2020a)

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图 13 具有相同时间尺度的耦合振子模型模拟癫痫发作状态 (红色局部极大值, 蓝色局部极小值) 关于耦合参数演化的转迁动力学 (Zhang et al. 2020b)

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图 14 网络振子 (上) 和慢变量 (下) 的时间序列图. (a1)和(b1)为间歇性癫痫的慢速高幅值发作状态, (a2)和(b2)为间歇性癫痫的快速低幅值发作状态, (a3)和(b3)为持续性癫痫发作状态 (Zhang et al. 2020a)

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图 15 局灶性癫痫致痫网络的状态转迁空间. 横坐标为平均耦合强度, 纵坐标为网络结构参数的比例, 不同的颜色代表不同的状态. 从(a1)到(a5), 初始节点的个数分别为3,4,5,9和10. 横坐标为耦合强度C, 纵坐标为网络结构系数的比例m₀/m (Zhang et al. 2020a)

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图 16 平均PLV值的矩阵热图(a), 耦合强度分别为0.1, 0.1444和0.1888, 与对应的平均网络度分布直方图(b), CC为节点PLV与度的相关系数 (Zhang et al. 2020a)

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图 17 牵制控制策略的示例. (a)为时间序列, (b)为刺激信号. (a1)(b1)为无刺激节点, (a2)(b2)为刺激PLV值排名前六的节点, (a3)(b3)为刺激PLV值排名前十的节点 (Zhang et al. 2020a)

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图 18 基于多样网络结构的牵制刺激策略效果图. (a) ~ (d)分别为刺激20%, 16%, 12%和8%的节点 (Zhang et al. 2020a)

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图 19 (a)来自三名不同癫痫患者发作的脑电图记录 (Marten et al. 2009) , 上图和中图为儿童失神癫痫患者 (CAE) 脑电图, 下图为青少年失神癫痫患者 (JAE) 脑电图. (b)皮质−丘脑网络框架: 皮质子系统由兴奋性 (PY) 锥体神经元集群和抑制性 (IN) 中间神经元集群组成; 丘脑子系统由丘脑网状核 (RE) 和中继核 (TC) 组成 (Fan & Duan 2017) . 皮层-丘脑环路 (PY-TC) 形成的递归兴奋性微回路和 (TC-IN-PY) 形成的前馈抑制性微回路 (Paz et al. 2015). (c)皮质−丘脑环路中皮质子网络神经元集群PY (P) 和IN (I) 平均动力学 ( (P+I) /2) 时间序列图: 单脉冲刺激 (绿竖直线) 诱导的SWD振荡和单脉冲刺激 (红竖直线) 诱导的SWD振荡终止 (Fan & Liu 2016)

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图 20 丘脑中继核与皮质组成的递归兴奋性回路作用下皮质网络中集群平均动力学关于参数状态变化分岔分析. (a) 状态分岔图, 包括失神癫痫SWD振荡 (III,IV) , 强直 (tonic) (I) 和阵挛性 (clonic) 振荡 (V) , 以及低/高 (II/VI) 饱和放电 (saturated firing) 等; (b) 对应状态的主频演化, A/C对应高/低频振荡, B和D对应背景正常状态 (Fan & Liu 2016)

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图 21 (a)皮质平均动力学分岔分析 (对应图20竖直线虚线箭头) , 其中粉色/白线代表稳定/不稳定的平衡点, 红/蓝线代表稳定/不稳定的极限环. 系统依次经历超临界和亚临界霍普夫分岔 (HB) 以及极限环分岔 (LPC) . 双稳定吸引子及其吸引域示意图: (b) SWD诱导刺激扰动, (c) SWD终止刺激扰动. 扰动使得系统 (小球) 从一个稳定状态越过临界状态 (三角形) 转迁到另外一个稳定状态 (Fan & Liu 2016)

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图 22 由前馈抑制和前馈兴奋组成的2D参数平面上可以确定癫痫失神发作与强直阵挛性发作之间的相互转迁路径. (a)状态演化, 包括高饱和 (HS) 和低饱和 (LS) 放电, (多) 棘慢波SWD, 2-SWD和3-SWD振荡, 反向棘慢波 (r-SWD) , 强直 (TO) 和低/高频阵挛性 (l/h-CO) 以及反向阵挛性 (r-CO) 振荡等; (b)对应的主频演化 (Fan & Duan 2017)

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图 23 (a)实验观察到的睡眠纺锤波(spindle)向癫痫棘慢波 (SWD) 转迁 (Kostopoulos 1981, Kostopoulos et al. 2000), (b)癫痫患者睡眠期间颞叶电极的脑电图记录显示10 s的睡眠纺锤波和棘慢波振荡 (Fan et al. 2017c)

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图 24 (a) (b) (c)环式连接的3室耦合模型网络, 周期单脉冲刺激 (红色竖直线) 从$t = 20\;{\text{s}}$开始间隔20 s依次施加刺激在第一室的RE, 三室的不同核团都呈现SWD与纺锤波振荡的周期性转迁(c)右侧为局部放大图. (d) (e)纺锤波吸引子 (spindle, 黑色) , 极限环吸引子 (低幅度强直振荡, 背景正常状态, 蓝色) 和SWD吸引子 (红色) 共存状态. 粉色箭头代表对RE施加的刺激. (e)右侧的局部放大图 (Fan et al. 2017b)

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图 25 皮质−丘脑环路时滞网络动力学模型示意图, $ {D_{\text{1}}} $为丘脑中继核到皮质时滞, ${D_2}$为皮质到丘脑中继核时滞, ${D_{\text{3}}}$为丘脑网状核到中继核时滞, ${D_4}$是耦合时滞 (Fan et al. 2018)

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图 26 (a)皮质平均动力学关于D₃的分岔图, 系统状态连续地从纺锤波振荡 (spindle) 转迁到SWD, 2-SWD和3-SWD; (b) D₁诱导的SWD精确发作, D₁与其诱导SWD的发作时刻呈线性增长关系 (白色虚线); (c)随着D₂增加 (向下箭头) 和D₃增加 (向右箭头) , 在三维空间 (D₁^X, D₁^Y, R)及其投影平面(D₁^X,D₁^Y) (等高线图) 上D₃诱导的SWD (左列) , 2SWD (中列) 和3SWD (右列) 的同步 (R₁, R₂, R₃) 演变, 其中发现D₂能有效规范同步发作模式. (d)驱动系统驱动响应系统的癫痫多棘慢波放电终止, 绿线表示控制输入 (Fan et al. 2018)

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图 27 皮质−基底节−丘脑环路网络框架图: 皮质由兴奋性椎体神经元(PY)和抑制性中间神经元 (IN) 组成; 丘脑由中继核 (TC) 和网状核 (RE) 组成; 基底节由纹状体 (D1/D2) , 丘脑底核 (STN) , 内苍白球 (GPi) 和外苍白球 (GPe) 组成. 箭头代表神经元集群之间的兴奋性投射作用, 带有实心圆端点的射线代表由GABA_A (实线) 和GABA_B (虚线) 调节的抑制性投射作用. ${\phi _n}$代表到TC的非特定外部输入. 红色箭头代表新引入的STN自突触作用, ${V_{stim}}$为注入的外部刺激干扰. 阴影线分别代表从STN到皮质的前驱 (L0) 和后驱 (L1) 投射路径 (Fan & Wang 2018)

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图 28 具有 (b) (d) 和不具有 (a) (b) 脉冲间期 (IPG) 的电荷平衡的双相脉冲刺激模式, 其中 (a) (b) 为非对称形式, (c) (d) 为对称形式 (Fan & Wang 2018)

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图 29 具有不同相位间期的双相脉冲刺激以及丘脑底核自突触v_ss对SWD的控制效果. 为了定量观察SWD的控制效果, 将2D参数区域进行均匀网格划分, 得到n × n个网格点, 统计系统显示SWD的参数网格点数 (Fan & Wang 2018)

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图 30 (a) BGCT环路示意图, 由皮层兴奋性锥体神经元 (PY, p) 和抑制性中间神经元 (IN, i) ; 丘脑中继核 (TC, t) 和网状核 (RE, r) ; 基底节中纹状体D₁和D₂型神经元群、苍白球内侧部 (GPi, g₁) 和外侧部 (GPe, g₂) 、丘脑底核 (STN, s) 组成. 箭头代表由谷氨酸介导的兴奋性突触, 圆圈代表GABA_A (实线) / GABA_B (虚线) 介导的抑制性投射. 红色和蓝色线条分别表示BG接收来自皮质−丘脑环路的兴奋性输入和反馈回皮质−丘脑环路的抑制性输出. (b)简化的BGCT模型(SBGCT), 其中BG被看作一个2I:3O调制器, 即接收来自皮质−丘脑环路2个核团的兴奋性输入和反馈回皮质−丘脑环路3个核团的抑制性输出. (c)简化的CT模型 (RCT) , 这里将来自BG的3个反馈抑制信号作为皮质−丘脑环路的特定输入常数, $ {P}_{\text{p}}, {P}_{\text{t}}, {P}_{\text{r}} $. $ {S}_{\text{p}}, {S}_{\text{t}}, {S}_{\text{r}} $是用于神经调节的外部刺激输入 (Fan & Wang 2020)

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图 31 (a)在SBGCT模型上对PY(p),TC(t)和RE(r)施加的协调重置刺激 (CRS) , 其中PY和TC接收负相脉冲 (Cathodic, C) 刺激, RE接收正相脉冲 (Anodic, A) 刺激; (b) (1:0开-关) CCA-CRS刺激模式, T₀是在一个CRS刺激周期Tcrs内施加单个核团上的刺激脉冲的周期; (c)m:n = 3:2开−关CCA-CRS: 每进行3个周期的CRS刺激 (3开) 之后附带2个周期的刺激间歇期 (2关) , 其中每个周期的CRS (T_crs= T₂= 3T₁) 刺激过程中每个核团平均接收T₁= kT₀时长的脉冲刺激. (d) CCA规则并行刺激 (regular parallel stimulation-RPS) . (e)CCA随机重置刺激 (random resetting stimulation-RRS) (Fan et al. 2020)

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图 32 皮质−丘脑方向可控的刺激原理 (a) (b) 及其对SWD的最优控制效果(c). (a)由(RE, TC, PY)生成的三维空间中, 给出了SWD振荡的吸引子, 并建立了一个原点(0, 0, 0)的笛卡尔坐标系${O_{XYZ}}$, ${{ {\boldsymbol{V}}}}$表示有向刺激 (directional stimulation-DS) 的方向, ${\theta _p}$,$ {\theta }_{t} $,$ {\theta }_{r} $分别是有向刺激与三个坐标轴正向的夹角. (b) 笛卡尔3D空间直角坐标系中的单位球面, 球面上的散点是随机选取的并满足${\theta _p} \in \left( {{{90}^\circ },{{180}^\circ }} \right)$, ${\theta }_{t}\in \left({90}^\circ,{180}^\circ\right), {\theta }_{r}\in \left({0}^\circ,{90}^\circ\right)$, 代表对TC和PY的负相 (C,$\cos {\theta _{\text{p}}} \lt 0,\cos {\theta _{\text{t}}} \lt 0$) 脉冲刺激及对RE的正相 (A,$\cos {\theta _{\text{r}}} \gt 0$) 脉冲刺激. (c) 3:2开−关CCA有向刺激 (CCA-DS) 对SWD的控制效果, 其中标记①②③表示取得最好控制效果的刺激方向角、SWD参数格点数和相应的电流消耗 (Fan & Zheng 2020)

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图 33 电刺激调控SWD的动力学解释. (a) PY(“■”), TC(“►”), RE(“★”)的AMFR随着-v_tr的增大而降低. 两条虚线之间的区域表示的是典型的2 ~ 4 Hz的SWD振荡区域, 两条虚线分别对应着-v_tr高和低2-4Hz的SWD触发平均放电率(TAMFR); (b)(c)(d)在AAC-CRS(黑色★), CCA-CRS(红色★),CCA-RPS(紫色★), 3:2开−关CCA-CRS(◄), 3:2开−关CCA-DS-CRS(►)以及控制组(○)下, PY, TC, RE的AMFR与对应-v_tr 的高 (红色■) 和低 (蓝色■) TAMFR随着v_pp的增大的演化图. 图(b)到图(d)中绿色区域表示由-v_tr 确定的典型2 ~ 4 Hz的SWD振荡区域 (Fan et al. 2020)

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图 34 (a) 基于STN的TAMFR设计的闭环刺激策略, 其中包括了调节器 (基底节) , 控制器和刺激器; (b)不同刺激模式对SWD的控制效果和电流消耗柱状图. (c) 开环和闭环3:2开−关CRS刺激脉冲序列图; (d)常规深脑刺激 (DBS) , 闭环DBS (DBS-CL) , 协调重置刺激 (CRS), 闭环CRS (CRS-CL) , 3:2开−关CRS和闭环3:2开−关CRS (3:2CRS-CL) 等六种不同刺激模式以及无刺激时 (控制组) STN的AMFR随着${v_{sp}}$的变化趋势, ${\varGamma _1}$和${\varGamma _2}$为GPi作用下确定的低触发TAMFR随着${v_{sp}}$增大的演化图 (Fan & Wang 2020)

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图 35 时滞耦合神经元群模型结构. 虚线矩形框展示了单个神经元群模型X的内在动力学机理, Y和Z与之类似. $ {h}_{e},{h}_{i},{h}_{d} $分别是兴奋性突触后电位EPSP和抑制性突触后电位IPSP的线性转迁函数, $S\left( v \right)$是非线性转迁函数, ${{{C}}_{\text{1}}}{\text{,}}{{{C}}_{\text{2}}}{\text{,}}{{{C}}_{\text{3}}}{\text{,}}{{{C}}_{\text{4}}}$表示关联突触的平均数量. $p\left( t \right)$是高斯白噪声, 是对环境影响的建模. ${x_0}$表示中间神经元EPSP转迁函数${h_e}$的输出, ${x_1}$表示锥体神经元EPSP转迁函数${h_e}$的输出, ${x_2}$表示锥体神经元IPSP转迁函数${h_i}$的输出, ${x_{\text{3}}}$表示EPSP转迁函数${h_d}$的输出.

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图 36 选取的癫痫患者病灶定位的计算结果. (a)上半部分是部分通道的SEEG数据, 下半部分是对应时刻每个通道的出度变化 (为了突出出度较大的通道, 仅将出度大于30的用红色点显示); (b)发作间期、发作前期和发作期的出度分布 (仅列举出度较大的通道) , 不同的片段计算的结果类似, 表明结果的可重复性; (c)电极位置的示意图; (d)手术切除区域的示意图 (Yang et al. 2018)

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图 37 神经元集群时空扩展网络模型的仿真结果. (a)选定患者某个通道的真实脑电数据, 发作间期和发作期各30 s; (b)切除前的仿真结果; (c)随机切除理论病灶之外的一个通道后的仿真结果; (d)切除一个理论致痫通道后的仿真结果; (e)统计三种条件下各重复30次仿真后发作间期和发作期的棘波数量; (f)另一个病人的真实数据 (上) 、切除前的仿真 (中) 和准确切除病灶后的仿真 (下) (Yang et al. 2018)

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图 38 选取的癫痫患者 (见表1中编号为1的病人) 大脑通道G05对其他9个通道信号 (F08, M09, C09, D03, D04, H10, J07, K01, K02) 的因果性随时间的演化. 垂直虚线标出了癫痫发作位置, 水平虚线分别标出了因果阈值${q^\tau }\left( n \right) = $0和${q^\tau }\left( n \right) = $ 10的位置. 选取${q^\tau }\left( n \right) = $10作为阈值来表示显著的因果关系. 其他每个通道与其余通道的信息流因果关系分析类似 (Fan et al. 2021)

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图 39 基于多通道信息流因果关系分析 (见图38) 进行癫痫发作的效应网络构建: 基于表1中编号为1的患者数据的因果关系与同步度分析构建的有向加权网络, 这里将每个通道视作一个网络节点 (10个通道对应10个节点, 1-F08, 2-M09, 3-C09, 4-D03, 5-D04, 6-G05, 7-H10, 8-J07, 9-K01, 10-K02) . (a)10个通道EEG信号假设为癫痫脑动力系统中的10个变量${x_1},{x_2}, \ldots ,{x_{10}}$, 从初始状态出发, 网络在癫痫发作间期的状态(b)和癫痫发作期的状态(c)可以通过控制到达任意预期的最终状态 ((a)中向右的黄色箭头) , 这个控制为对无匹配节点 (如: 节点1, 4, 6和8, 是针对(c)癫痫发作期计算所得) 的输入刺激扰动$u\left( t \right) = \left( {{u_1}\left( t \right),{u_2}\left( t \right),{u_3}\left( t \right),{u_4}\left( t \right)} \right)$. (b)和(c)左侧的网络呈现了多种形态的度分布, 节点大小表示网络的出度, 黄色的节点只有入度, 箭头的粗细表示因果性的大小 (只有$q\left( n \right) \gt 10$才会被绘制出来) . (c)右侧图展示了所有匹配路径 (由绿色箭头标出) , 通过粉色的箭头被依序连接, 其他连接用浅灰色表示 (Fan et al. 2021)

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图 40 数值模拟效果. (a)表1中编号为1的癫痫患者原始EEG时间序列; (b)对计算所得的关键控制节点F08, D03, G05, J07进行刺激扰动后模拟的时间序列; (c)对任意两条非病灶通道 (C09, K01为例) 进行刺激扰动后模拟的时间序列; (d)移除时间序列中的病灶通道F08, D03, G05, J07信号后模拟的时间序列 (Fan et al. 2021)

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表 1 基于10位局灶性癫痫患者计算的网络关键控制节点与临床定位的病灶节点对比


患者编号	年龄	选择通道数	临床定位的病灶节点	计算的网络关键控制节点
1	≤6	10	F08,D03	F08,D03,G05,J07
2	15 ~ 20	12	D11,D12	D12,M09,W03,C03,E02
3	5 ~ 10	10	L11,H13,L09	L11,L09
4	5 ~ 10	9	G10,H10	G10,H10,G06
5	≤6	9	P08,E07,G06	F11,E07,E09
6	25 ~ 30	9	A01,B01	A01
7	15 ~ 20	10	L04,H04	L04,H04,J02
8	10 ~ 15	10	K09,G11	G10,K09,D06,H01
9	30 ~ 35	12	C02,D01,E10	D01,E03
10	5 ~ 10	11	E04,J14,M05	M05,M04,J14,C08

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