6.021/Notes/2006-10-23

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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, $k e = 0$
- $\frac{\partial^2 V_m(z,t)}{\partial z^2}=(r_o+r_i)K_m(z,t)$
- - we converted $K m$ (per length) to $J m$ (per area)
- - 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, , so
  - this only holds true for unmyelinated neurons
can also infer transmembrane potential using the outside potential (which is easier to measure)
- $v_o(z)=-\frac{r_o}{r_o+r_i}(v_m(z) - V_m^o)$