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
• [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] (independent of cell size)
  • [math]\displaystyle{ \lambda_c = \frac{1}{\sqrt{g_m(r_o+r_i)}} }[/math]
  • [math]\displaystyle{ v_m = V_m - V_m^o }[/math]
• 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
  • [math]\displaystyle{ v_m-\lambda_c^2\frac{\partial^2v_m}{\partial z^2}=r_o\lambda_c^2K_e }[/math]
  • solution: [math]\displaystyle{ v_m(z) = \frac{r_o\lambda_c}{2}I_e e^{-|z|/\lambda_c} }[/math]