We study the initial-boundary value problem for McKean's caricature of the nerve equation$$\eqalign{u\sb{t}(x,t) &=u\sb{xx}-u+H(u -a)+v;\ x>0,t>0\cr v\sb{t}(x,t) &=bu-cv;\ x>0,t>0\cr u(x,0) &=v(x,0)=0;\ x\ge0\cr u\sb{x}(0,t) &=h(t)$ or $u(0,t)=h(t)\ t\ge0.\cr}$$ In Chapter 2, we show the existence and uniqueness of solutions of the Neumann problem. We prove that there is a sharp threshold such that a stimulus (h(t)) below the threshold decays exponentially in space (sub-threshold), but a stimulus above the threshold will rapidly approach a traveling wave (super-threshold) In Chapter 3, the existence and uniqueness of solutions of the Dirichlet problem for nondecreasing stimulus is proved. The existence of a sharp threshold is also proved Finally, computational results are given to illustrate the theory developed