Let the transfer function H(s) of an excitable neuron be given by: s+ B H(s) = output input Y (s + x) ³. a) Using partial fraction decomposition, write H(s) as the sum of two partial fractions, i.e., solve for A and B in this equation in terms of a and B: A B H(s) = Y(s + a)² + (s + a)³ b) Write the time-domain impulse response h(t) for the neuron. Please consult a table of Laplace transforms, or use this relationship: L{teat} = n! (s-a)n+1 c) Using the expression derived in part (b) and the code provided, plot in Matlab the response of the neuron to an impulse Vimpo(t), where Vimp = 150 mV. You will need to modify the code to include an anonymous function that defines your h(t). Use the constants provided in the code, gamma = y = 500 s¹, beta = ß = 0.1 s´¹, and alpha =α= 104 s¹. Assume the baseline membrane potential is Vbas = -70 mV, and h(t) represents a positive change above Vbas. What is the maximum value of the membrane potential (or change above Vbas) in the single impulse response? d) Now assume the input is an impulse train, or a sequence of impulses separated by a fixed delay T. The same code provided will also plot the response of the neuron to a sequence of Vimp [8(t) + 8(t − T) + 8(t − 2T) + ··· ], for up to 15 impulses, using four different values of T: 1.00 ms, 0.30 ms, 0.10 ms, and 0.05 ms. The neuron will activate when its membrane potential becomes more positive than its activation threshold Vthr = -50 mV. Which excitation signals (i.e., which values of T) lead to neuron activation? What is the time delay between the start of the impulse train (t = 0) and the activation of the neuron?

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