Solving variational inequality problems by using neural networks are of great interest in recent years. To date, most work in this direction focus on solving monotone variational inequalities. In this paper, we show that an existing recurrent neural network proposed originally for solving monotone variational inequalities can be used to solve pseudomonotone variational inequalities with proper choice of a system parameter. The global convergence, global asymptotic stability and global exponential stability of the neural network are discussed under various conditions. The existing stability results are thus extended in view of the fact that pseudomonotonicity is a weaker condition than monotonicity.