# 6.021/Notes/2006-12-14

## Cable model

$\displaystyle{ J_m = C_m\frac{dV_m}{dt}+G_m(V_m-V_m^o) }$

Cable Equation: $\displaystyle{ v_m+\tau_m\frac{\partial v_m}{\partial t}-\lambda_c^2\frac{\partial^2v_m}{\partial z^2}=r_o\lambda_c^2K_e }$

$\displaystyle{ \tau_m=\frac{c_m}{g_m} }$

$\displaystyle{ \lambda_c = \frac{1}{\sqrt{g_m(r_o+r_i)}} }$

$\displaystyle{ v_m = V_m - V_m^o }$

Steady state solution of cable equation to impulse stimulus: $\displaystyle{ v_m(z) = \frac{r_o\lambda_c}{2}I_e e^{-|z|/\lambda_c} }$

Dynamics: $\displaystyle{ v_m(z,t)=w(z,t) e^{-t/\tau_m} }$ where $\displaystyle{ \frac{\partial w}{\partial t} = \frac{\lambda_c^2}{\tau_m} \frac{\partial^2 w}{\partial z^2} }$ (Diffusion equation with $\displaystyle{ D=\frac{\lambda_c^2}{\tau_m} }$)

## Ion channels

$\displaystyle{ I = \gamma (V_m-V_n) }$

$\displaystyle{ E[\tilde{s}(t)] = x }$, $\displaystyle{ E[\tilde{g}(t)]=\gamma x=g }$, $\displaystyle{ E[\tilde{i}(t)]= g(V_m - V_n) }$

$\displaystyle{ G = \frac{N}{A} g }$, $\displaystyle{ J = \frac{N}{A} g(V_m-V_n) }$

$\displaystyle{ x(t) = x_\infty+(x(0)-x_\infty)e^{-t/\tau_x}, \tau_x=\frac{1}{\alpha+\beta}, x_\infty=\frac{\alpha}{\alpha+\beta} }$

$\displaystyle{ \tilde{i}_g = \frac{d}{dt}\tilde{q}_g }$

$\displaystyle{ i_g = E[\tilde{i}_g] = Q\frac{dx}{dt} }$