Linear systems in a saturated mode and convergence as gain becomes large of asymptotically stable equilibrium points of neural nets

We study solutions of the "linear system in a saturated mode" (M) ' epsilon Tx + c - partial derivative I(Dn)x. We show that a trajectory is in a constant face of the cube D-n on some interval (0, d]. We answer a question about comparing the two systems: (M) and (H) Cu' = Tv + c - R(-1)u, v = G(lambda u). As lambda --> infinity, limits of v corresponding to asymptotically stable equilibrium points of (H) are asymptotically stable equilibrium points of(M), and the converse is also true. We study the assumptions to see which are required and which may be weakened.