# 6.021/Notes/2006-10-23

• The core-conductor equation: $\displaystyle{ \frac{\partial^2 V_m(z,t)}{\partial z^2}=(r_o+r_i)K_m(z,t)-r_oK_e(z,t) }$
• action potential in neurons
• spatial extent
• positive membrane potential for about 1ms
• speed of propagation about 30 m/s
• over a space of about 30 mm (large)
• transmembrane current is inward at action potential peak
• transmembrane current is outward ahead of action potential peak
• outward current hels depolarize membrane and can help AP to propagate
• but same logic would have AP propagating in other direction also (if it weren't refractory)
• dependence of speed on geometry
• AP, $\displaystyle{ k_e=0 }$
• $\displaystyle{ \frac{\partial^2 V_m(z,t)}{\partial z^2}=(r_o+r_i)K_m(z,t) }$
• $\displaystyle{ \frac{1}{\nu^2}\frac{\partial^2 V_m(z,t)}{\partial t^2}=(r_o+r_i)2\pi a J_m }$
• we converted $\displaystyle{ K_m }$ (per length) to $\displaystyle{ J_m }$ (per area)
• $\displaystyle{ \frac{\frac{\partial^2 V_m(z,t)}{\partial t^2}}{J_m}=\nu^2(r_o+r_i)2\pi a = C }$
• this is a constitutive relationship
• right hand side is constant and independent of the network topology
• $\displaystyle{ \nu = \sqrt{\frac{C}{(r_o+r_i)2\pi a}} }$
• we can determine how the speed of an action potential depends on $\displaystyle{ r_i, r_o, a }$, e.g. increasing external resistance slows AP
• a space clamp shorts the internal resistance with a wire so that $\displaystyle{ r_i=0 }$. As the external resistance is usually very small, the speed of the action potential becomes very large (thus changing the cell to be 1D)
• Assume external resistance is small, $\displaystyle{ r_i = \frac{\rho}{A} = \frac{\rho}{\pi a^2} }$, so $\displaystyle{ \nu = \sqrt{\frac{Ca}{2\rho}} \propto \sqrt{a} }$
• this only holds true for unmyelinated neurons
• can also infer transmembrane potential using the outside potential (which is easier to measure)
• $\displaystyle{ v_o(z)=-\frac{r_o}{r_o+r_i}(v_m(z) - V_m^o) }$