# 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
• $\displaystyle{ J_m = C_m\frac{dV_m}{dt}+G_m(V_m-V_m^o) }$
• Cable Equation: $\displaystyle{ v_m+\tau_m\frac{\partial v_m}{\partial t}-\lambda_c^2\frac{\partial^2v_m}{\partial z^2}=r_o\lambda_c^2K_e }$
• $\displaystyle{ \tau_m=\frac{c_m}{g_m} }$ (independent of cell size)
• $\displaystyle{ \lambda_c = \frac{1}{\sqrt{g_m(r_o+r_i)}} }$
• $\displaystyle{ v_m = V_m - V_m^o }$
• 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
• $\displaystyle{ v_m-\lambda_c^2\frac{\partial^2v_m}{\partial z^2}=r_o\lambda_c^2K_e }$
• solution: $\displaystyle{ v_m(z) = \frac{r_o\lambda_c}{2}I_e e^{-|z|/\lambda_c} }$