6.021/Notes/2006-10-20

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Core conductor model

  • Look at impact of topology on electrical properties
  • Vm(z,t): 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 Ri = ridz. Ri is in ohms and ri is in ohms/m.
  • Outer conductor: resistance Ro = rodz (similar to inner conductor)
  • Current through membrane: Im = kmdz Im is in amps and km is in A/m.
  • Assume topology, Ohm's law, but nothing about the membrane
  • Core conductor equations:
    1. \frac{\partial I_i(z,x)}{\partial z}=-K_m(z,t)
    2. \frac{\partial I_o(z,x)}{\partial z}=K_m(z,t)-K_e(z,t)
      • Ke is externally applied current
    3. \frac{\partial V_i(z,t)}{\partial z}=-r_iI_i(z,t)
    4. \frac{\partial V_o(z,t)}{\partial z}=-r_oI_o(z,t)
    • The first 2 equations are continuity of current, the second two are Ohm's law
  • Combining equations, we get THE core conductor equation:
    • \frac{\partial^2 V_m(z,t)}{\partial z^2}=(r_o+r_i)K_m(z,t)-r_oK_e(z,t)
    • We still have assumed nothing about the membrane
  • Suppose no external current. K_e = 0 \rightarrow I_i+I_o=0 (otherwise charge would build up)
  • If we know Vm for all space and time:
    • K_m = \frac{1}{r_o+r_i}\frac{\partial^2 V_m(z,t)}{\partial z^2}
    • \frac{\partial V_m(z,t)}{\partial z} = -r_iI_i + r_oI_o = -(r_o+r_i)I_i
  • For action potential traveling at constant speed ν
    • V_m(z,t)=f(t-\frac{z}{\nu})
    • \frac{\partial^2 V_m(z,t)}{\partial z^2}=\frac{1}{\nu^2}\frac{\partial^2 V_m(z,t)}{\partial t^2} (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
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