# 6.021/Notes/2006-12-14

(diff) ←Older revision | Current revision (diff) | Newer revision→ (diff)

## Cable model

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

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

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

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

$v_m = V_m - V_m^o$

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

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

## Ion channels

I = γ(VmVn)

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

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

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

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

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