6.021/Notes/2006-10-20

Core conductor model

• Look at impact of topology on electrical properties
• [math]\displaystyle{ V_m(z,t) }[/math]: different potentials along the cell
• Break into lumps/nodes
• Treat as internal resistors, outer resistors, and unknown boxes connecting inside/outside (membrane potential)
• Inner conductor: resistance [math]\displaystyle{ R_i = r_i dz }[/math]. [math]\displaystyle{ R_i }[/math] is in ohms and [math]\displaystyle{ r_i }[/math] is in ohms/m.
• Outer conductor: resistance [math]\displaystyle{ R_o = r_o dz }[/math] (similar to inner conductor)
• Current through membrane: [math]\displaystyle{ I_m = k_m dz }[/math] [math]\displaystyle{ I_m }[/math] is in amps and [math]\displaystyle{ k_m }[/math] is in A/m.
• Assume topology, Ohm's law, but nothing about the membrane
• Core conductor equations:
  1. [math]\displaystyle{ \frac{\partial I_i(z,x)}{\partial z}=-K_m(z,t) }[/math]
  2. [math]\displaystyle{ \frac{\partial I_o(z,x)}{\partial z}=K_m(z,t)-K_e(z,t) }[/math]
     • [math]\displaystyle{ K_e }[/math] is externally applied current
  3. [math]\displaystyle{ \frac{\partial V_i(z,t)}{\partial z}=-r_iI_i(z,t) }[/math]
  4. [math]\displaystyle{ \frac{\partial V_o(z,t)}{\partial z}=-r_oI_o(z,t) }[/math]
  • The first 2 equations are continuity of current, the second two are Ohm's law
• Combining equations, we get THE core conductor equation:
  • [math]\displaystyle{ \frac{\partial^2 V_m(z,t)}{\partial z^2}=(r_o+r_i)K_m(z,t)-r_oK_e(z,t) }[/math]
  • We still have assumed nothing about the membrane
• Suppose no external current. [math]\displaystyle{ K_e = 0 \rightarrow I_i+I_o=0 }[/math] (otherwise charge would build up)
• If we know [math]\displaystyle{ V_m }[/math] for all space and time:
  • [math]\displaystyle{ K_m = \frac{1}{r_o+r_i}\frac{\partial^2 V_m(z,t)}{\partial z^2} }[/math]
  • [math]\displaystyle{ \frac{\partial V_m(z,t)}{\partial z} = -r_iI_i + r_oI_o = -(r_o+r_i)I_i }[/math]
• For action potential traveling at constant speed [math]\displaystyle{ \nu }[/math]
  • [math]\displaystyle{ V_m(z,t)=f(t-\frac{z}{\nu}) }[/math]
  • [math]\displaystyle{ \frac{\partial^2 V_m(z,t)}{\partial z^2}=\frac{1}{\nu^2}\frac{\partial^2 V_m(z,t)}{\partial t^2} }[/math] (wave equation)
• From this model alone, we find that the current at the peak of the action potential is predicted to be inwards!
  • For all standard electrical elements (resistor, capacitor, inductor), we would predict outward current
  • This model makes no assumption about the membrane, only that Ohm's law holds