6.021/Notes/2006-11-08

Cable model

 * The cable model is linear approximation of Hodgkin-Huxley model for small signals
 * If the change in membrane voltage is small, than the change in m, n, and h are all small so we can ignore
 * Thus all conductances are constant
 * $$J_m = C_m\frac{dV_m}{dt}+G_m(V_m-V_m^o)$$
 * Cable Equation: $$v_m+\tau_m\frac{\partial v_m}{\partial t}-\lambda_c^2\frac{\partial^2v_m}{\partial z^2}=r_o\lambda_c^2K_e$$
 * $$\tau_m=\frac{c_m}{g_m}$$ (independent of cell size)
 * $$\lambda_c = \frac{1}{\sqrt{g_m(r_o+r_i)}}$$
 * $$v_m = V_m - V_m^o$$
 * Lord Kelvin (1855)
 * problem of putting cable under Atlantic for telegraphy
 * made with copper and tar surrounded by sea water
 * Same problem as axon + membrane
 * Assume infintesmal electrode to a remote electrode
 * Given a pulse of current stimulus, look at steady state response
 * $$v_m-\lambda_c^2\frac{\partial^2v_m}{\partial z^2}=r_o\lambda_c^2K_e$$
 * solution: $$v_m(z) = \frac{r_o\lambda_c}{2}I_e e^{-|z|/\lambda_c}$$