6.021/Notes/2006-10-20

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:
 * $$\frac{\partial I_i(z,x)}{\partial z}=-K_m(z,t)$$
 * $$\frac{\partial I_o(z,x)}{\partial z}=K_m(z,t)-K_e(z,t)$$
 * $$K_e$$ is externally applied current
 * $$\frac{\partial V_i(z,t)}{\partial z}=-r_iI_i(z,t)$$
 * $$\frac{\partial V_o(z,t)}{\partial z}=-r_oI_o(z,t)$$
 * 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 $$V_m$$ 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 $$\nu$$
 * $$V_m(z,t)=f(t-\frac{z}{\nu})$$
 * $$\frac{\partial^2 V_m(z,t)}{\partial z^2}=\frac{1}{\nu^2}\frac{\partial^2 V_m(z,t)}{\partial t^2}$$ (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