This paper focuses on multistable analysis of one dimensional Hopfield neural networks, whose sigmoid activation function may not be bounded. Firstly, the condition for the existence of equilibria is established. Moreover, the conditions for exactly 1, 2, or 3 equilibria and their stability respectively are proposed with some constraints for network parameters. Then, the corresponding results about the equilibria features are supplemented in the remaining cases for the parameters. Thus, we obtain the relationships between all the different values of the parameters and the number of equilibria as well as their stability. Finally, by employing bounded and unbounded activation functions, two examples and numerical simulations are used to illustrate the theory developed in this paper.
