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Homework 4.2 1. (dV 4.5.5) Consider the following model for signal tranduction in the axon: ut = uxx + u(1 − u)(u − 1/2), where u represents the membrane potential. We study this model on 0 ≤ x ≤ 1 with homogeneous Neumann boundary conditions: ux (0, t) = 0, ux (1, t) = 0. a) Determine the system of two ODEs which describe the steady-state solution of the PDE b) Find equilibrim points of the system of (a) and study their stability c) Show that 1 1 1 1 H(u, ux ) = (ux )2 − u4 + u3 − u2 2 4 2 4 is a Hamiltonian function for the system you found in (a) d) Sketch a phase portrait in the (u, ux ) plane e) Find the steady state solution that satisfies the Neumann boundary conditions and sketch them as a function of x f) Give a biological interpretation of these steady-state solutions 1