| layout | title | date | author | summary | weight |
|---|---|---|---|---|---|
notes |
06.Cable Equation and Its Solutions |
2016-04-02 |
OctoMiao |
Cable Equation and Its Solutions |
6 |
Checkout the attachment: 06.1.cable_equation_green_function.pdf
The stationary equation is \begin{equation} \left(\frac{d^2}{dx^2} -1\right) u(x) = - i_e(x). \end{equation}
The form solution using Green's function method is \begin{equation} u(x) = \int G(x,x') (-i_e(t,x') ) dx', \end{equation} where \begin{equation*} G(x,x') = \begin{cases} \frac{1}{2}e^{x-x'}, & \qquad x<x';\ \frac{1}{2}e^{x' - x}, & \qquad x>x'. \end{cases} \end{equation*}
So far we have been dealing with math. What is the actual meaning of GF? To dive into this question we need to review the equation for GF, in this case, \begin{equation*} \left(\frac{d^2}{dx^2} -1\right) u(x) = \delta(x'-x). \end{equation*}
In a stimulation-response system, one of the most important properties is the resonance width, or reaction width, which means the deviation required for the amplitude to drop to
Just to build a picture, this length is around\footnote{Since opening of ion channesl can significantly change the transverse conductivity, this estimation can change significantly in different situations.}
\begin{equation*} \lambda = \sqrt{ \frac{r_T}{r_L}} = \sqrt{ \frac{30\mathrm{k\Omega\cdot cm^2}/(2\pi \rho)}{ 100 \mathrm{k\Omega\cdot cm}/(2\pi \rho) } } = \sqrt{ \frac{5\times 10^{11} \mathrm{\Omega \cdot \mu m} }{ 3\times 10^{5} \mathrm{\Omega \cdot \mu m^{-1}} } } = 1.2\mathrm{mm} \end{equation*}
To include the time-dependent source we need a two dimensional Dirac delta distribution, \begin{equation} \frac{\partial}{\partial t} G(t,t';x,x') - \frac{\partial^2}{\partial x^2} G(t,t';x,x') + G(t,t';x,x') = \delta(t'-t)\delta(x'-x). \end{equation}
The Green's function is solved out \begin{equation} G(t,0;x,0) = \frac{\Theta(t)}{\sqrt{4\pi t}} \exp\left( -t - \frac{x^2}{4t} \right). \end{equation}
The boundary condition is given by \begin{equation} i(t,x=0) = i(t,x=L) = 0. \end{equation}
The current for the ion channels is not simply proportional to the membrane potential which we used for the previous cable equations. Introducing such a time dependent conductivity for the transverse current will significantly increase the complexity of the equations.