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