6.021/Notes/20061020
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Core conductor model
 Look at impact of topology on electrical properties
 V_{m}(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 R_{i} = r_{i}dz. R_{i} is in ohms and r_{i} is in ohms/m.
 Outer conductor: resistance R_{o} = r_{o}dz (similar to inner conductor)
 Current through membrane: I_{m} = k_{m}dz I_{m} is in amps and k_{m} is in A/m.
 Assume topology, Ohm's law, but nothing about the membrane
 Core conductor equations:


 K_{e} is externally applied current


 The first 2 equations are continuity of current, the second two are Ohm's law
 Combining equations, we get THE core conductor equation:

 We still have assumed nothing about the membrane
 Suppose no external current. (otherwise charge would build up)
 If we know V_{m} for all space and time:
 For action potential traveling at constant speed ν

 (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