Problem $1$: A cardiac fiber is being studies using two intracellular electrodes. The first, at $z=0cm,$ supplies a step$-$function subthreshold stimulus current $I_0=0\mathrm{.}5A.$ The second electrode records the intracellular potential at various positions along the fiber. The radius of the fiber is $a=0\mathrm{.}03cm.$

$($a$)$ Give expressions for the steady$-$state transmembrane voltage $V_m(z)$ and its spatial derivative $\frac{del{V}_m}{\text{d}\text{e}\text{l}\text{z}}.$

$($b$)$ Let us assume that the intracellular potential ${}_i$ is related to the transmembrane potential $V_m$ via the following expression ${}_i(z,t)=\frac{r_i}{r_i+r_e}{V}_m(z,t)($this is true only under certain conditions$).$ From the data below for the steady$-$state ${}_i(z),$ estimate the steady$-$state $\frac{\text{d}\text{e}\text{l}{}_i}{\text{d}\text{e}\text{l}\text{z}}$ at $z=0\mathrm{.}2cm$ and $z=0\mathrm{.}4cm.$ Using the corresponding expression from $($a$),$ calculate the length constant $.$

$\backslash$table$[[z(cm),{}_i(mV)$