6.021/Notes/2006-10-23


 * The core-conductor equation: $$\frac{\partial^2 V_m(z,t)}{\partial z^2}=(r_o+r_i)K_m(z,t)-r_oK_e(z,t)$$
 * 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, $$k_e=0$$
 * $$\frac{\partial^2 V_m(z,t)}{\partial z^2}=(r_o+r_i)K_m(z,t)$$
 * $$\frac{1}{\nu^2}\frac{\partial^2 V_m(z,t)}{\partial t^2}=(r_o+r_i)2\pi a J_m$$
 * we converted $$K_m$$ (per length) to $$J_m$$ (per area)
 * $$\frac{\frac{\partial^2 V_m(z,t)}{\partial t^2}}{J_m}=\nu^2(r_o+r_i)2\pi a = C$$
 * this is a constitutive relationship
 * right hand side is constant and independent of the network topology
 * $$\nu = \sqrt{\frac{C}{(r_o+r_i)2\pi a}}$$
 * we can determine how the speed of an action potential depends on $$r_i, r_o, a$$, e.g. increasing external resistance slows AP
 * a space clamp shorts the internal resistance with a wire so that $$r_i=0$$. 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, $$r_i = \frac{\rho}{A} = \frac{\rho}{\pi a^2}$$, so $$\nu = \sqrt{\frac{Ca}{2\rho}} \propto \sqrt{a}$$
 * this only holds true for unmyelinated neurons
 * can also infer transmembrane potential using the outside potential (which is easier to measure)
 * $$v_o(z)=-\frac{r_o}{r_o+r_i}(v_m(z) - V_m^o)$$