Question 2 We began our discussion with the 'hope' that a multicompartment model could indeed adequately capture the fiber's true potential and current profiles. In order to check this one should run fib1.m with increasing values of N until one can no longer detect changes in the computed potentials. (a) Please run fib1.m with N = 8, 16, 32, and 64. Plot all of the potentials on the same (use hold) graph, using different line types for each. (You may wish to alter fib1.m so that it accepts N as an argument). Let us now interpret this convergence. The main observation is that the difference equation, , approaches a differential equation. We can see this by noting that ⅆ z l N acts as a spatial 'step' size and that x k , the potential at k 1 ⅆ z , is approximately the value of the true potential at k 1 ⅆ z . In a slight abuse of notation, we denote the latter x k 1 ⅆ z Applying these conventions to and recalling the definitions of R i and R m we see become a 2 ρ i x 0 2 x ⅆ z x 2 ⅆ z ⅆ z 2 a ⅆ z ρ m x ⅆ z 0 or, after multiplying through by ρ m a ⅆ z , a ρ m ρ i x 0 2 x ⅆ z x 2 ⅆ z ⅆ z 2 2 x ⅆ z 0 . We note that a similar equation holds at each node (save the ends) and that as N and therefore ⅆ z 0 we arrive at z2 x z 2 ρ i a ρ m x z 0 (b) With μ 2 ρ i a ρ m show that x z α 2 μ z β 2 μ z satisfies regardless of α and β. We shall determine α and β by paying attention to the ends of the fiber. At the near end we find a 2 ρ i x 0 x ⅆ z ⅆ z i 0 which, as ⅆ z 0 becomes z x 0 ρ i i 0 a 2 At the far end, we interpret the condition that no axial current may leave the last node to mean z x l 0 (c) Substitute into and and solve for α and β and write out the final x z . (d) Substitute into x the l, a, ρ i , and ρ m values used in fib1.m, plot the resulting function (using, e.g., ezplot) and compare this to the plot achieved in part (a).