6.021/Notes/2006-12-14

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Cable model

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

Cable Equation: [math]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]\tau_m=\frac{c_m}{g_m}[/math]

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

[math]v_m = V_m - V_m^o[/math]

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

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

Ion channels

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

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

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

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

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