Consider the following passive membrane equation dV dt = (V EL) + R Iapp(t) (1) with EL = 60 mV, C = 1 F/cm2 , GL = 0.1 mS/cm2 , Write a Matlab code (or use the template code) to solve eq. (1). Use V (0) = 60 mV and the following units for V , t and Iapp respectively: [V ] = mV,[t] = msec, [Iapp] = A/cm2 . (a) Calculate the time constant . 1 (b) Consider the following two time-independent applied currents Iapp = 0.5 and Iapp = 0.5. For each value of Iapp (Iapp = 0.5 and Iapp = 0.5) compare the numerical and analytical solutions to the passive membrane equation (1) by (i) plotting superimposed graphs of these solutions, and (ii) plotting the error (absolute value of the difference between these two solutions). (c) Consider a square pulse of current Iapp(t) = I0 Heav(t ti) Heav(tf t) with I0 = 0.5, ti = 100 msec and tf = 200 msec. Plot both the numerical and analytical solutions to the passive membrane equation (1), and compare them as in (a). Run your simulation for 400 msec. (d) Consider the following current Iapp(t) = I0 sin(2 t/1000). with I0 = 0.5 nA. (i) Plot the numerical solution to the passive membrane equation (1) for = 1, = 5, = 10, = 20, and = 100. (ii) Plot a graph relating the output frequency (y-axis) vs. the input frequency (x-axis). (iii) Plot a graph relating the amplitude of the output oscillations (y-axis) vs. the input frequency (x-axis).

2) Build an integrate-and-fire model using Vth = 50 mV , Vrst = 65 mV and the parameters values for the passive membrane equation above. (a) Calculate (analytically) the interspike-interval firing rate (risi) for gsra = 0 (no spike-rate adaptation) and i. Iapp = 0.5 ii. Iapp = 1 iii. Iapp = 1.01 iv. Iapp = 2 Plot the corresponding numerical solutions and compare your analytical and numerical results. (b) Compute the numerical solutions and plot the corresponding graphs for for Iapp = 2, Ek = 85, gsra = 0.1 and i. sra = 10 msec. ii. sra = 100 msec.Type or paste question here