3.d Rint Now, we will re-derive an expression for the longitudinal resistance Rint for our hollow axon. Remember here that the thickness of the ring is much smaller than the diameter (t < d). To find the cross-sectional area for Rint using the form:
Rint = (px) / (cross-sectional area)
Your final expression can be in terms of resistivity and x.
3.e Cable equation
Finally, re-derive the Cable equation for the hollow axon. Your derivation will start with the ODEs below:
dV(x) / dx = (Rmt(x) / x) * dx
Your final result should be a single second-order equation for the membrane potential with respect to x. Solve here for 3, the characteristic distance over which voltage perturbation diminishes along the length of the axon. Please write all equations using variables (do not plug in numbers for this problem).
If thickness t is constant, how does 3 depend on changes in the diameter d of the hollow axon?
3 Cable equation with a hollow axon
In class and in your textbook, we derived the Cable Equation to incorporate the effect of spatial variation of membrane potential. This derivation was for a solid cylindrical axon. Now in this problem, we will re-derive the Cable Equation for a hollow axon (see Figure 1), which you can think of as a very thin shell, in order to determine the effect of the axon cross-sectional area on membrane potential. The hollow axon has a cross-sectional diameter d, and the shell has a small thickness t << d. The length of a patch of membrane is Ax.
Ion channels
Figure 1: A hollow axon
3.a Nernst potential
To begin, compute the Nernst potential for the two ions below. Use T = 25 mV
Table 1. Intracellular concentration (mM) Extracellular concentration (mM) V (mV) Na+ 33 44 K+ 8 29
3.b gpatch Calculate the conductance of a patch of membrane gpatch using the values below. The conductance per unit area is g = 4 S/m^2, the axon diameter is d = 0.6 mm, and the length of a patch of membrane is r = 0.1 mm. How does gpatch change with changes in the diameter d of the axon?
3.c Current Now using your results from parts 3.a and 3.b, calculate the radial current i(x) that flows perpendicular to the cell membrane. First, state the general expression for current. Then, solve for the currents iNa+ (x) and iCl- (x) at x = 0, given V(0) = -90 mV. Compare the current of Na+ to that of Cl-. Discuss if either ion is at equilibrium and how you know.
