Ring networks, a particular form of Hopfield neural networks, can be used in computational neurosciences in order to model the activity of place cells or head-direction cells. The behaviour of these models is highly dependent on their recurrent synaptic connectivity matrix and on individual neurons' activation function, which must be chosen appropriately to obtain physiologically meaningful conclusions. In this article, we propose some simpler ways to tune this synaptic connectivity matrix compared to existing literature so as to achieve stability in a ring attractor network with a piece-wise affine activation functions, and we link these results to the possible stable states the network can converge to.