The Student Project tions S(z) which varied from 0 to 1 as z went from-co to +o. These functions, called compared several func- response functions, are used to model the growth of activity in a collection of nerve cells. One of the simplest functions of this type is the piecewise linear function ifz < -1/2, if-1/2 z 1/2, ifz > 1/2. A simple neural model using the function S is the equation - f + 1/2 S(z) = x' (t) = f(t, x) = -x(t) + S(20(x(t)-0.5 -0.4 cos (2t))). (2.32) To study the behavior of solutions of this equation in the square B = {(t, x) | 0I 1,0 x 1} we need to know if there exist unique solutions through any given initial point in B. 1. Show that the derivative is NOT a continuous function in B. 3x 2. Show that f does satisfy a Lipschitz condition in B and find a Lipschitz, constant M. 3. Plot five solutions of x' = f(t, x) in B. If you are using Maple, the function S(z) can be defined by the command: S:=z->piecewise (z