Question

4. The cell membrane can be modeled like an electrical circuit. The phospholipid mem- brane can be represented by a capacitance CM. Each ion gradient can be represented as a current -GiVi where i represents a given ion, I is current, V is voltage, and G is conductance. (As a reminder, conductance is the inverse of resistance.) First, consider a model with one ion: (a) Find the transfer function for the figure shown above (b) In the 1950's Alan Hodgkin and Andrew Huxley developed the mathematical model of a nerve cell working on the squid giant axon. The following is a circuit model for a tvpical cell membrane with Nat, Kt, and Cl- channels. Find the transfer function ▽oth for figure shown below. V0m(s) membrane resting potential of the voltages of the three ions shown in the figure below (s) is the CI

Answer 1

R_k = 1/G_k v_m = Q/c using kirchoff voltage rate v_k - i R_k - v_m = 0 i = d phi/dt d phi/dt R_k + v_m - v_k = 0 d phi/dt R_k + Q/c - v_k = 0 dphi/dt + Q/R_k c = v_k/R_k Multiplying both sides by integrating factor e^t/R_k c e^t_R_k c dphi/dt + e^t_R_k c Q/R_k c = v_k/R_k e^t_R_k c \r\n d/dt [Q e^t_R_k c] = v_k/R_k e^t_R_k c Q e^t_R_k c = v_k/R_k integral e^t_R_k c dt Q e^t_R_k c = v_k/R_k [R_k c e^t_R_k c] + c_0 Q = c v_k + c_0 e^-t/R_k c at t = 0, Q = 0 0 = cv_k + c_0 c_0 = -cv_k Q = cv_k - cv_k e^-t_R_k c Q = c v_k [1- e^-t_R_k c] v = Q/c = c v_k/c [1- e^-t_R_k c] v_m/v_k = 1 - e^-t_R_k c