神经场

生理学动机

Beurle[1]和 Wilson & Cowan[5][4]是最早从数学上推导出神经场模型方程的人之一。最近 Faugeras 等人[11] 和 Bressloff [12]考虑到单个神经元的统计特性和随机动力学，给出了不同的推导。有趣的是，Faugeras等人[12][11]提出了一个扩展模型，同时考虑了平均活动及其方差的动态，而以前的模型只考虑了平均活动。以下段落给出了神经场模型更具启发性的推导，其中考虑了神经作用的三个主要要素：输入脉冲在化学突触处引起的树突电流、受到树突电流影响的神经元的放电以及沿轴突分支 的动作电位传输。诱发的树突电流 I(t) 服从

$\displaystyle{ I(t) = \int_{-\infty}^t {\rm d}\tau^\prime h(t-t^\prime) P(t^\prime)\label{eqn_1} }$ (1)

$\displaystyle{ \begin{array}{lcl} \frac{dI(t)}{dt}&=&-\frac{1}{\tau}\int_{-\infty}^t {\rm d}t^\prime \exp\left(-(t-t^\prime)/\tau\right)/\tau P(t^\prime) + \int_{-\infty}^t \left({\rm d}t^\prime/dt\right)|_{\infty}^t \exp\left(-(t-t^\prime)/\tau\right)/\tau P(t^\prime)\\ &=&-\frac{1}{\tau}I(t) + \frac{1}{\tau} P(t) \end{array} }$

$\displaystyle{ \hat{L} I(t)=P(t)\quad,\quad \hat{L}=\tau\frac{d}{dt}+1 . }$

$\displaystyle{ h(t)=\frac{1}{\tau_1-\tau_2}\left(e^{-t/\tau_1}-e^{t/\tau_2}\right) \quad , \quad \hat{L}=\tau_1\tau_2\frac{d^2}{dt^2}+\left(\tau_1+\tau_2\right)\frac{d}{dt}+1 }$

$\displaystyle{ h(t)=te^{-t/\tau}/\tau\quad , \quad \hat{L}=\tau^2\frac{d^2}{dt^2}+2\tau\frac{d}{dt}+1~. }$

数学框架

$\displaystyle{ \frac{1}{\alpha} \frac{\partial u(x,t)}{\partial t} = -u + \int_{-\infty}^\infty {\rm d} y w(y) f(u(x-y,t - |y|/v)) . }$

动力学行为

模式形成

$\displaystyle{ u(x) = \int_0^\Delta w(x-y) {\rm d y}, \qquad u(0)=h=u(\Delta) , }$

行波

$\displaystyle{ c = \frac{v(2h-1)}{2h-1 - 2 h v/\alpha} }$

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• Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
• John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
• S. Murray Sherman (2006) Thalamus. Scholarpedia, 1(9):1583.

推荐阅读

• Electric Fields of the Brain: The Neurophysics of EEG, by Paul L. Nunez and Ramesh Srinivasan, Oxford University Press, 2006.

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