# 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