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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