6.021/Notes/2006-12-14

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
$$I = \gamma (V_m-V_n)$$

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

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

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

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

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