Location and stability of the equilibria of nonlinear neural networks

The number, location and stability behavior of the equilibria of arbitrary nonlinear neural networks are analyzed without resorting to energy arguments based on assumptions of symmetric interactions or no self-interactions. The following results are proved. Let H=(0,1)ⁿ denote the open n-dimensional hypercube on which the neural network evolves, and let I denote the vector of external inputs. As the neural sigmoid characteristics become steeper, the following statements are true for all I except for those belonging to a set of measure zero. (i) There are only finitely many equilibria in any compact subset of H. (ii) There are only finitely many equilibria in any face of H. (iii) A systematic procedure is given for determining which corners of H contain equilibria, and it is shown that all equilibria in the corners of H are asymptotically stable. These results continue to hold even if the network dynamics are slightly perturbed