Equivalence between local exponential stability of the unique equilibrium point and global stability for Hopfield-type neural networks with two neurons

Fang and Kincaid (1996) proposed an open problem about the relationship between the local stability of the unique equilibrium point and the global stability for a Hopfield-type neural network with continuously differentiable and monotonically increasing activation functions. As a partial answer to the problem, in the two-neuron case it is proved that for each given specific interconnection weight matrix, a Hopfield-type neural network has a unique equilibrium point which is also locally exponentially stable for any activation functions and for any other network parameters if and only if the network is globally asymptotically stable for any activation functions and for any other network parameters. If the derivatives of the activation functions of the network are bounded, then the network is globally exponentially stable for any activation functions and for any other network parameters