6.021/Notes/2006-12-14

Cable model

[math]\displaystyle{ J_m = C_m\frac{dV_m}{dt}+G_m(V_m-V_m^o) }[/math]

Cable Equation: [math]\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 }[/math]

[math]\displaystyle{ \tau_m=\frac{c_m}{g_m} }[/math]

[math]\displaystyle{ \lambda_c = \frac{1}{\sqrt{g_m(r_o+r_i)}} }[/math]

[math]\displaystyle{ v_m = V_m - V_m^o }[/math]

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

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

Ion channels

[math]\displaystyle{ I = \gamma (V_m-V_n) }[/math]

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

[math]\displaystyle{ G = \frac{N}{A} g }[/math], [math]\displaystyle{ J = \frac{N}{A} g(V_m-V_n) }[/math]

[math]\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} }[/math]

[math]\displaystyle{ \tilde{i}_g = \frac{d}{dt}\tilde{q}_g }[/math]

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