Dynamical behaviors of Hopfield neural network with multilevel activation functions

When the activation function possesses multilevel property, the Hopfield neural network has some novel dynamical behaviors, and it is worthwhile to study. First, some properties about the activation function are obtained, on this foundation, some theoretical analysis about the quasi-equilibrium points has been made. From local and global view, some theorems about the boundedness are presented. Finally, two theorems about the first derivative of trajectory with respect to time are found, the first theorem indicates that the trajectory cannot keep increasing or decreasing for time t > t₀, the second theorem is about the complete stability of the trajectory.