威尔逊考恩模型

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In computational neuroscience, the Wilson–Cowan model describes the dynamics of interactions between populations of very simple excitatory and inhibitory model neurons. It was developed by Hugh R. Wilson and Jack D. Cowan[1][2] and extensions of the model have been widely used in modeling neuronal populations.[3][4][5][6] The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal populations to stimuli. Because the model neurons are simple, only elementary limit cycle behavior, i.e. neural oscillations, and stimulus-dependent evoked responses are predicted. The key findings include the existence of multiple stable states, and hysteresis, in the population response.

在计算神经科学computational neuroscience中，威尔逊-考恩模型描述了非常简单的兴奋和抑制模型神经元neuron之间的交互动态。该模型由Hugh R. Wilson和Jack D. Cowan[1][2]开发，其扩展已被广泛用于神经元群体建模[3][4][5][6]。该模型具有重要的历史意义，因为它使用相平面方法和数值解来描述神经元群体对刺激的反应。由于模型神经元简单，只能预测基本的极限环行为，即神经振荡neural oscillations和刺激依赖的诱发反应。主要发现包括种群响应中存在多个稳定状态和迟滞。

数学描述Mathematical description

The Wilson–Cowan model considers a homogeneous population of interconnected neurons of excitatory and inhibitory subtypes. All cells receive the same number of excitatory and inhibitory afferents, that is, all cells receive the same average excitation, x(t). The target is to analyze the evolution in time of number of excitatory and inhibitory cells firing at time t, [math]\displaystyle{ E(t) }[/math] and [math]\displaystyle{ I(t) }[/math] respectively.

威尔逊-考恩模型考虑的是一个同质的、相互连接的兴奋和抑制亚型神经元群体。所有细胞接收相同数量的兴奋性和抑制性传入信号，即所有细胞接收相同的平均兴奋，x(t)。目的是分析兴奋性和抑制性细胞在时间t、[math]\displaystyle{ E(t) }[/math]和[math]\displaystyle{ I(t) }[/math]放电的数量在时间上的演化。

The equations that describes this evolution are the Wilson-Cowan model:

描述这种演变的方程组是威尔逊-考恩模型:

[math]\displaystyle{ E(t+\tau)=\left[1-\int_{t-r}^{t}E(t')dt'\right] \; S_e\left( \int_{-\infty}^{t}\alpha(t-t')[c_1E(t')-c_2I(t')+P(t')]dt'\right ) }[/math]

[math]\displaystyle{ I(t+\tau)=\left[ 1-\int_{t-r}^{t}I(t')dt'\right] \; S_i \left( \int_{-\infty}^{t}\alpha(t-t')[c_3E(t')-c_4I(t')+Q(t')]dt'\right) }[/math]

where:

其中：

  • [math]\displaystyle{ S_e\{\} }[/math] and [math]\displaystyle{ S_i\{\} }[/math] are functions of sigmoid form that depends on the distribution of the trigger thresholds (see below)
  • [math]\displaystyle{ \alpha(t) }[/math] is the stimulus decay function
  • [math]\displaystyle{ c_1 }[/math] and [math]\displaystyle{ c_2 }[/math] are respectively the connectivity coefficient giving the average number of excitatory and inhibitory synapses per excitatory cell; [math]\displaystyle{ c_3 }[/math] and [math]\displaystyle{ c_4 }[/math] its counterparts for inhibitory cells
  • P(t) and [math]\displaystyle{ Q(t) }[/math] are the external input to the excitatory/inhibitory populations.
  • [math]\displaystyle{ S_e\{\} }[/math]和 [math]\displaystyle{ S_i\{\} }[/math]是依赖于触发阈值分布的 sigmoid函数(见下文)
  • [math]\displaystyle{ \alpha(t) }[/math]是刺激衰减函数
  • [math]\displaystyle{ c_1 }[/math]和 [math]\displaystyle{ c_2 }[/math]分别为连接系数，表示每个兴奋性细胞的兴奋性突触和抑制性突触的平均数量，[math]\displaystyle{ c_3 }[/math]和 [math]\displaystyle{ c_4 }[/math]是抑制性细胞的对应物
  • P(t)和[math]\displaystyle{ Q(t) }[/math]是兴奋性/抑制性群体的外部输入。

If [math]\displaystyle{ \theta }[/math] denotes a cell's threshold potential and [math]\displaystyle{ D(\theta) }[/math] is the distribution of thresholds in all cells, then the expected proportion of neurons receiving an excitation at or above threshold level per unit time is:

如果 [math]\displaystyle{ \theta }[/math]表示一个细胞的阈值电位threshold potential，[math]\displaystyle{ D(\theta) }[/math]表示所有细胞中阈值的分布，那么每单位时间接受阈值水平或以上兴奋的神经元的预期比例是:

[math]\displaystyle{ S(x)=\int_{0}^{x}D(\theta)d\theta }[/math],

that is a function of sigmoid form if [math]\displaystyle{ D() }[/math] is unimodal.

如果 [math]\displaystyle{ D() }[/math]是单峰形式，则为sigmoid函数。

If, instead of all cells receiving same excitatory inputs and different threshold, we consider that all cells have same threshold but different number of afferent synapses per cell, being [math]\displaystyle{ C(w) }[/math] the distribution of the number of afferent synapses, a variant of function [math]\displaystyle{ S() }[/math] must be used:

如果不考虑所有细胞接受相同的兴奋性输入和不同的阈值，我们认为所有细胞的阈值相同，但每个细胞的传入突触数目不同，作为传入突触数目分布的[math]\displaystyle{ C(w) }[/math]，必须使用函数[math]\displaystyle{ S() }[/math]的变体:

[math]\displaystyle{ S(x)=\int_{\frac{\theta}{x}}^{\infty}C(w)dw }[/math]

模型推导 Derivation of the model

If we denote by [math]\displaystyle{ \tau }[/math] the refractory period after a trigger, the proportion of cells in refractory period is[math]\displaystyle{ \int_{t-r}^{t}E(t')dt' }[/math] and the proportion of sensitive (able to trigger) cells is [math]\displaystyle{ 1-\int_{t-r}^{t}E(t')dt' }[/math].

如果我们用[math]\displaystyle{ \tau }[/math]表示触发后的不应期(性) ，不应期(性)中细胞的比例是[math]\displaystyle{ \int_{t-r}^{t}E(t')dt' }[/math]，而敏感(能够触发)细胞的比例是[math]\displaystyle{ 1-\int_{t-r}^{t}E(t')dt' }[/math]。

The average excitation level of an excitatory cell at time [math]\displaystyle{ t }[/math] is:[math]\displaystyle{ \int_{t-r}^{t}E(t')dt' }[/math]

时间[math]\displaystyle{ t }[/math]时兴奋细胞的平均兴奋水平为:

[math]\displaystyle{ x(t) = \int_{-\infty}^{t}\alpha(t-t')[c_1 E(t')-c_2 I(t')+P(t')]dt' }[/math]

Thus, the number of cells that triggers at some time [math]\displaystyle{ E(t+\tau) }[/math] is the number of cells not in refractory interval, [math]\displaystyle{ 1-\int_{t-r}^{t}E(t')dt' }[/math] AND that have reached the excitatory level, [math]\displaystyle{ S_e(x(t)) }[/math], obtaining in this way the product at right side of the first equation of the model (with the assumption of uncorrelated terms). Same rationale can be done for inhibitory cells, obtaining second equation.

因此，在某一时刻触发[math]\displaystyle{ E(t+\tau) }[/math]的细胞数是不处于不可逆区间的细胞数，[math]\displaystyle{ 1-\int_{t-r}^{t}E(t')dt' }[/math]AND 达到兴奋水平，即[math]\displaystyle{ S_e(x(t)) }[/math] ，由此得到模型第一方程右边的乘积(假定不相关项)。同样的原理也适用于抑制细胞，得到第二个方程。

时间粗粒化条件下模型的简化Simplification of the model assuming time coarse graining

When time coarse-grained modeling is assumed the model simplifies, being the new equations of the model::

当采用时间粗粒度coarse-grained建模时，模型就简化了，即模型的新方程组为: :

[math]\displaystyle{ \tau\frac{d\bar{E}}{dt}=-\bar{E}+(1-r\bar{E})S_e[kc_1\bar{E}(t)-kc_2\bar{I}(t)+kP(t)] }[/math]

[math]\displaystyle{ \tau'\frac{d\bar{I}}{dt}=-\bar{I}+(1-r'\bar{I})S_i[k'c_3\bar{E}(t)-k'c_4\bar{I}(t)+k'Q(t)] }[/math]

where bar terms are the time coarse-grained versions of original ones.

其中条形条款是原始条款的时间粗粒度版本。

应用于癫痫Application to epilepsy

The determination of three concepts is fundamental to an understanding of hypersynchronization of neurophysiological activity at the global (system) level:[7]

以下三个概念的确定是理解神经生理活动在整体(系统)水平的超同步的基础::

  1. The mechanism by which normal (baseline) neurophysiological activity evolves into hypersynchronization of large regions of the brain during epileptic seizures
  2. The key factors that govern the rate of expansion of hypersynchronized regions
  3. The electrophysiological activity pattern dynamics on a large-scale
  1. 癫痫发作期间，正常(基线)神经生理活动演变为大脑大区域超同步化的机制
  2. 控制超同步区域扩展速度的关键因素
  3. 大尺度电生理活动模式动力学研究

A canonical analysis of these issues, developed in 2008 by Shusterman and Troy using the Wilson–Cowan model,[7] predicts qualitative and quantitative features of epileptiform activity. In particular, it accurately predicts the propagation speed of epileptic seizures (which is approximately 4–7 times slower than normal brain wave activity) in a human subject with chronically implanted electroencephalographic electrodes.[8][9]

2008年，舒斯特曼和特洛伊利用威尔逊-考恩模型对这些问题进行了典型的分析，[7] 预测了癫痫样活动的定性和定量特征。特别是，它精确地预测了在长期植入脑电图的人体中癫痫发作的传播速度(大约比正常脑电波活动慢4-7倍)。[8][9]

转换到超同步 Transition into hypersynchronization

The transition from normal state of brain activity to epileptic seizures was not formulated theoretically until 2008, when a theoretical path from a baseline state to large-scale self-sustained oscillations, which spread out uniformly from the point of stimulus, has been mapped for the first time.[7]

直到2008年，从正常的大脑活动状态到癫痫病发作的转变才在理论上得到阐述。2008年，从基线状态到从刺激点均匀扩散的大规模自我持续振荡的理论路径首次被绘制出来。[7]

A realistic state of baseline physiological activity has been defined, using the following two-component definition:[7]

基线生理活动的现实状态已被定义，使用以下双分量two-component definition定义:[7]

(1) A time-independent component represented by subthreshold excitatory activity E and superthreshold inhibitory activity I.

(1)以阈下兴奋性活性E和超阈抑制活性I为代表的时间无关分量。

(2) A time-varying component which may include singlepulse waves, multipulse waves, or periodic waves caused by spontaneous neuronal activity.

(2)一种时变分量，可包括单脉冲波、多脉冲波或由神经元自发活动引起的周期性波。

This baseline state represents activity of the brain in the state of relaxation, in which neurons receive some level of spontaneous, weak stimulation by small, naturally present concentrations of neurohormonal substances. In waking adults this state is commonly associated with alpha rhythm, whereas slower (theta and delta) rhythms are usually observed during deeper relaxation and sleep. To describe this general setting, a 3-variable [math]\displaystyle{ (u,I,v) }[/math] spatially dependent extension of the classical Wilson–Cowan model can be utilized.[10] Under appropriate initial conditions,[7] the excitatory component, u, dominates over the inhibitory component, I, and the three-variable system reduces to the two-variable Pinto-Ermentrout type model[11]

这种基线状态代表大脑处于放松状态时的活动，在这种状态下，神经元会受到一定程度的自发的、微弱的刺激，这种刺激来自于小量的、自然存在的神经激素物质。在清醒的成年人中，这种状态通常与阿尔法节律alpha rhythm有关，而较慢的节律(θ 和 δ)通常在深度放松和睡眠中观察到。为了描述这个一般的设置，一个3分量变量[math]\displaystyle{ (u,I,v) }[/math]的空间相关的扩展经典的 Wilson-Cowan 模型可以被利用。[10] 在适当的初始条件下[7]，兴奋成分 u 主导抑制成分 i，三变量系统降低为二变量 Pinto-Ermentrout 型模型[11]

[math]\displaystyle{ {\partial u \over \partial t}=u-v+ \int_{R^2}\omega(x-x',y-y')f(u-\theta)\,dxdy + \zeta(x,y,t), }[/math]
[math]\displaystyle{ {\partial v \over \partial t}=\epsilon (\beta u-v). }[/math]

The variable v governs the recovery of excitation u; [math]\displaystyle{ \epsilon\gt 0 }[/math] and [math]\displaystyle{ \beta\gt 0 }[/math] determine the rate of change of recovery. The connection function [math]\displaystyle{ \omega(x,y) }[/math] is positive, continuous, symmetric, and has the typical form [math]\displaystyle{ \omega=Ae^{-\lambda\sqrt {-(x^2+y^2)}} }[/math].[11] In Ref.[7] [math]\displaystyle{ (A,\lambda)=(2.1,1). }[/math] The firing rate function, which is generally accepted to have a sharply increasing sigmoidal shape, is approximated by [math]\displaystyle{ f(u-\theta)=H(u-\theta) }[/math], where H denotes the Heaviside function; [math]\displaystyle{ \zeta(x,y,t) }[/math] is a short-time stimulus. This [math]\displaystyle{ (u,v) }[/math] system has been successfully used in a wide variety of neuroscience research studies.[11][12][13][14][15] In particular, it predicted the existence of spiral waves, which can occur during seizures; this theoretical prediction was subsequently confirmed experimentally using optical imaging of slices from the rat cortex.[16]

变量v控制激励u的恢复[math]\displaystyle{ \epsilon\gt 0 }[/math]和[math]\displaystyle{ \beta\gt 0 }[/math]决定恢复的变化率。连接函数[math]\displaystyle{ \omega(x,y) }[/math]是正的，连续的，对称的，并且具有典型的形式[math]\displaystyle{ \omega=Ae^{-\lambda\sqrt {-(x^2+y^2)}} }[/math]。[11]在参考文献7中，[7]射速函数，一般认为是一个急剧增加的s形，近似为[math]\displaystyle{ f(u-\theta)=H(u-\theta) }[/math]，其中H表示Heaviside函数;[math]\displaystyle{ \zeta(x,y,t) }[/math]是一个短时刺激。这个[math]\displaystyle{ (u,v) }[/math]系统已经成功地应用于各种各样的神经科学研究。[11][12][13][14][15]特别是，它预测了癫痫发作时可能出现的螺旋波的存在;这一理论预测随后通过大鼠皮质切片的光学成像实验得到了证实。[16]

Rate of expansion

The expansion of hypersynchronized regions exhibiting large-amplitude stable bulk oscillations occurs when the oscillations coexist with the stable rest state [math]\displaystyle{ (u,v)=(0,0) }[/math]. To understand the mechanism responsible for the expansion, it is necessary to linearize the [math]\displaystyle{ (u, v) }[/math] system around [math]\displaystyle{ (0,0) }[/math] when [math]\displaystyle{ \epsilon\gt 0 }[/math] is held fixed. The linearized system exhibits subthreshold decaying oscillations whose frequency increases as [math]\displaystyle{ \beta }[/math] increases. At a critical value [math]\displaystyle{ \beta^{*} }[/math] where the oscillation frequency is high enough, bistability occurs in the [math]\displaystyle{ (u,v) }[/math] system: a stable, spatially independent, periodic solution (bulk oscillation) and a stable rest state coexist over a continuous range of parameters. When [math]\displaystyle{ \beta\ge\beta^{*} }[/math] where bulk oscillations occur,[7] "The rate of expansion of the hypersynchronization region is determined by an interplay between two key features: (i) the speed c of waves that form and propagate outward from the edge of the region, and (ii) the concave shape of the graph of the activation variable u as it rises, during each bulk oscillation cycle, from the rest state u=0 to the activation threshold. Numerical experiments show that during the rise of u towards threshold, as the rate of vertical increase slows down, over time interval [math]\displaystyle{ \Delta t, }[/math] due to the concave component, the stable solitary wave emanating from the region causes the region to expand spatially at a Rate proportional to the wave speed. From this initial observation it is natural to expect that the proportionality constant should be the fraction of the time that the solution is concave during one cycle." Therefore, when [math]\displaystyle{ \beta\ge\beta^{*} }[/math], the rate of expansion of the region is estimated by

= = = 膨胀速率 = = =

当振荡与稳定的静止态[math]\displaystyle{ (u,v)=(0,0) }[/math]共存时，表现出大振幅稳定体振荡的超同步区域会膨胀。要理解负责展开的机制，有必要在[math]\displaystyle{ \epsilon\gt 0 }[/math]固定时将[math]\displaystyle{ (u, v) }[/math]系统在[math]\displaystyle{ (0,0) }[/math]附近线性化。线性化系统表现出亚阈值衰减振荡，其频率随着[math]\displaystyle{ \beta }[/math]的增加而增加。在振荡频率足够高的临界值[math]\displaystyle{ \beta^{*} }[/math]，双稳态发生在[math]\displaystyle{ (u,v) }[/math]系统中：在连续的参数范围内，一个稳定的、空间独立的周期解(体振荡)和一个稳定的静止态共存。当[math]\displaystyle{ \beta\ge\beta^{*} }[/math]时体积振荡发生，[7]“超同步区域的膨胀率是由两个关键特征之间的相互作用决定的：(i)从区域边缘形成并向外传播的波的速度c，和(ii)激活变量u在每个体积振荡周期中从静止状态u=0到激活阈值上升时的凹形曲线。”数值实验表明，在u向阈值上升的过程中，随着垂直上升速率的减慢，由于凹分量时间间隔[math]\displaystyle{ \Delta t, }[/math]，从该区域发出的稳定孤立波导致该区域以与波速成正比的速率在空间上扩展。从这个初始观察可以很自然地预期，比例常数应该是溶液在一个周期内凹下去的时间的比例。”因此，当[math]\displaystyle{ \beta\ge\beta^{*} }[/math]时，该区域的扩张速率估计为[7]

[math]\displaystyle{ Rate =(\Delta t/T)*c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1) }[/math]

where [math]\displaystyle{ \Delta t }[/math] is the length of subthreshold time interval, T is period of the periodic solution; c is the speed of waves emanating from the hypersynchronization region. A realistic value of c, derived by Wilson et al.,[17] is c=22.4 mm/s.

其中[math]\displaystyle{ \Delta t }[/math] 是亚阈值时间间隔的长度，T 是周期解的周期，c 是从超同步区域发出的波的速度。Wilson 等人[17]推导出的 c 的实际值是 c = 22.4 mm/s。

How to evaluate the ratio

[math]\displaystyle{ \Delta t/T? }[/math] To determine values for [math]\displaystyle{ \Delta t/T }[/math] it is necessary to analyze the underlying bulk oscillation which satisfies the spatially independent system

如何计算比值？

[math]\displaystyle{ \Delta t/T? }[/math]为了确定[math]\displaystyle{ \Delta t/T }[/math]的值，有必要对满足空间独立系统的潜在体振荡进行分析

[math]\displaystyle{ {{du} \over {dt}}=u-v+H(u-\theta), }[/math]
[math]\displaystyle{ {{dv} \over {dt}}=\epsilon (\beta u-v). }[/math]

This system is derived using standard functions and parameter values [math]\displaystyle{ \omega=2.1e^{-\lambda\sqrt {-(x^2+y^2)}} }[/math], [math]\displaystyle{ \epsilon=0.1 }[/math] and [math]\displaystyle{ \theta=0.1 }[/math][7][11][12][13] Bulk oscillations occur when [math]\displaystyle{ \beta \ge \beta^{*}=12.61 }[/math]. When [math]\displaystyle{ 12.61 \le \beta \le 17 }[/math], Shusterman and Troy analyzed the bulk oscillations and found [math]\displaystyle{ 0.136 \le \Delta t/T \le 0.238 }[/math]. This gives the range

这个系统是用标准函数和参数值从[math]\displaystyle{ \omega=2.1e^{-\lambda\sqrt {-(x^2+y^2)}} }[/math]，[math]\displaystyle{ \epsilon=0.1 }[/math]和[math]\displaystyle{ \theta=0.1 }[/math]导出的。[7][11][12][13][math]\displaystyle{ \beta \ge \beta^{*}=12.61 }[/math]时发生体振荡。当[math]\displaystyle{ 12.61 \le \beta \le 17 }[/math]时，Shusterman和Troy分析了体积振荡，发现[math]\displaystyle{ 0.136 \le \Delta t/T \le 0.238 }[/math]。这就给出了范围
[math]\displaystyle{ 3.046 mm/s \le Rate \le 5.331 mm/s~~~~~~~~~~~~(2) }[/math]
Since [math]\displaystyle{ 0.136 \le \Delta t/T \le 0.238 }[/math], Eq. (1) shows that the migration Rate is a fraction of the traveling wave speed, which is consistent with experimental and clinical observations regarding the slow spread of epileptic activity.[18] This migration mechanism also provides a plausible explanation for spread and sustenance of epileptiform activity without a driving source that, despite a number of experimental studies, has never been observed.[18]

由于[math]\displaystyle{ 0.136 \le \Delta t/T \le 0.238 }[/math]，公式(1)表明迁移率migration Rate是行波速度的一个分数，这与癫痫活动缓慢传播的实验和临床观察一致。[18]这种迁移机制也为没有驱动源的癫痫样活动的传播和维持提供了一个合理的解释，尽管有许多实验研究，但从未观察到驱动源。[18]

比较理论迁移率和实验迁移率 Comparing theoretical and experimental migration rates

The rate of migration of hypersynchronous activity that was experimentally recorded during seizures in a human subject, using chronically implanted subdural electrodes on the surface of the left temporal lobe,[8] has been estimated as[7]

在人类癫痫发作过程中，利用在左侧颞叶表面使用慢性植入的硬膜下电极，实验记录了超同步活动的迁移率，[8] 估计为[7]
[math]\displaystyle{ Rate \approx 4 mm/s }[/math],

Rate \approx 4 mm/s,

速度大约每秒4毫米,

which is consistent with the theoretically predicted range given above in (2). The ratio [math]\displaystyle{ Rate/c }[/math] in formula (1) shows that the leading edge of the region of synchronous seizure activity migrates approximately 4–7 times more slowly than normal brain wave activity, which is in agreement with the experimental data described above.[8]

这与上文(2)中给出的理论预测范围是一致的。公式(1)中[math]\displaystyle{ Rate/c }[/math]的比率表明，同步癫痫活动区域前缘的移动速度比正常脑电波活动慢4ー7倍，这与上述实验数据一致。[8]

To summarize, mathematical modeling and theoretical analysis of large-scale electrophysiological activity provide tools for predicting the spread and migration of hypersynchronous brain activity, which can be useful for diagnostic evaluation and management of patients with epilepsy. It might be also useful for predicting migration and spread of electrical activity over large regions of the brain that occur during deep sleep (Delta wave), cognitive activity and in other functional settings.

综上所述，大规模电生理活动的数学建模和理论分析为预测超同步脑活动的传播和迁移提供了工具，可用于癫痫患者的诊断评估和治疗。这也可能有助于预测在深度睡眠deep sleep(Delta 波)、认知活动和其他功能环境中大脑大区域电活动的迁移和传播。

参考文献References

  1. ↑ 1.0 1.1 Wilson, H.R.; Cowan, J.D. (1972). "Excitatory and inhibitory interactions in localized populations of model neurons". Biophys. J. 12 (1): 1–24. Bibcode:1972BpJ....12....1W. doi:10.1016/s0006-3495(72)86068-5. PMC 1484078. PMID 4332108.
  2. ↑ 2.0 2.1 Wilson, H. R.; Cowan, J. D. (1 September 1973). "A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue". Kybernetik. 13 (2): 55–80. doi:10.1007/BF00288786. PMID 4767470. S2CID 292546.
  3. ↑ 3.0 3.1 Jirsa, V. K.; Haken, H. (29 July 1996). "Field Theory of Electromagnetic Brain Activity". Physical Review Letters. 77 (5): 960–963. Bibcode:1996PhRvL..77..960J. doi:10.1103/PhysRevLett.77.960. PMID 10062950.
  4. ↑ 4.0 4.1 Robinson, P. A.; Rennie, C. J.; Wright, J. J. (1 July 1997). "Propagation and stability of waves of electrical activity in the cerebral cortex". Physical Review E. 56 (1): 826–840. Bibcode:1997PhRvE..56..826R. doi:10.1103/PhysRevE.56.826.
  5. ↑ 5.0 5.1 "D.T.J. Liley, P.J. Cadusch and J.J. Wright. A continuum theory of electrocortical activity. Neurocomputing. 26–27:795–800 (1999)" (PDF).
  6. ↑ 6.0 6.1 Wright, J. J. (1 August 1999). "Simulation of EEG: dynamic changes in synaptic efficacy, cerebral rhythms, and dissipative and generative activity in cortex". Biological Cybernetics. 81 (2): 131–147. doi:10.1007/s004220050550. PMID 10481241. S2CID 6558413.
  7. ↑ 7.00 7.01 7.02 7.03 7.04 7.05 7.06 7.07 7.08 7.09 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 Shusterman, V; Troy, WC (2008). "From baseline to epileptiform activity: a path to synchronized rhythmicity in large-scale neural networks". Physical Review E. 77 (6): 061911. Bibcode:2008PhRvE..77f1911S. doi:10.1103/PhysRevE.77.061911. PMID 18643304.
  8. ↑ 8.0 8.1 8.2 8.3 8.4 8.5 V. L. Towle, F. Ahmad, M. Kohrman, K. Hecox, and S. Chkhenkeli, in Epilepsy as a Dynamic Disease, pp. 69–81
  9. ↑ 9.0 9.1 Milton, J. G. (2010). "Mathematical Review: From baseline to epileptiform activity: A path to synchronized rhythmicity in large-scale neural networks by V. Shusterman and W. C. Troy (Phys. Rev. E 77: 061911)". Math. Rev. 2010: 92025.
  10. ↑ 10.0 10.1 Wilson, H. R.; Cowan, J. D. (1973). "A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue". Kybernetik. 13 (2): 55–80. doi:10.1007/BF00288786. PMID 4767470. S2CID 292546.
  11. ↑ 11.0 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Pinto, D.; Ermentrout, G. B. (2001). "Spatially Structured Activity in Synaptically Coupled Neuronal Networks: I. Traveling Fronts and Pulses". SIAM J. Appl. Math. 62: 206. doi:10.1137/s0036139900346453.
  12. ↑ 12.0 12.1 12.2 12.3 Folias, S. E.; Bressloff, P. C. (2004). "Breathing pulses in an excitatory neural network". SIAM J. Appl. Dyn. Syst. 3 (3): 378–407. Bibcode:2004SJADS...3..378F. doi:10.1137/030602629.
  13. ↑ 13.0 13.1 13.2 13.3 Folias, S. E.; Bressloff, P. C. (Nov 2005). "Breathers in two-dimensional neural media". Phys. Rev. Lett. 95 (20): 208107. Bibcode:2005PhRvL..95t8107F. doi:10.1103/PhysRevLett.95.208107. PMID 16384107.
  14. ↑ 14.0 14.1 Kilpatrick, Z. P.; Bressloff, P. C. (2010). "Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network". Physica D. 239 (9): 547–560. Bibcode:2010PhyD..239..547K. doi:10.1016/j.physd.2009.06.003.
  15. ↑ 15.0 15.1 Laing, C. R.; Troy, W. C. (2003). "PDE methods for non-local models". SIAM J. Appl. Dyn. Syst. 2 (3): 487–516. Bibcode:2003SJADS...2..487L. doi:10.1137/030600040.
  16. ↑ 16.0 16.1 Huang, X.; Troy, W. C.; Yang, Q.; Ma, H.; Laing, C. R.; Schiff, S. J.; Wu, J. Y. (Nov 2004). "Spiral waves in disinhibited mammalian neocortex". J. Neurosci. 24 (44): 9897–902. doi:10.1523/jneurosci.2705-04.2004. PMC 4413915. PMID 15525774.
  17. ↑ 17.0 17.1 Wilson, H. R.; Blake, R.; Lee, S. H. (2001). "Dynamics of travelling waves in visual perception". Nature. 412 (6850): 907–910. Bibcode:2001Natur.412..907W. doi:10.1038/35091066. PMID 11528478. S2CID 4431136.
  18. ↑ 18.0 18.1 18.2 18.3 模板:Page neededEpilepsy as a Dynamic Disease, edited by J. Milton and P. Jung, Biological and Medical Physics series Springer, Berlin, 2003.


Category:Computational neuroscience Category:Epilepsy Category:Neural circuits Category:Neurophysiology Category:Electrophysiology

类别: 计算神经科学类别: 癫痫类别: 神经回路类别: 神经生理学类别: 电生理学


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