The chapter deals with the design of event-triggering mechanisms (ETM) for discrete-time linear systems stabilized by neural network controllers. The proposed event-triggering mechanism is based on the use of local sector conditions related to the activation functions, to reduce the computational cost associated with the neural network evaluation. Such a mechanism avoids redundant computations by updating only a portion of the layers instead of evaluating periodically the complete neural network. Sufficient matrix inequality conditions are provided to design the parameters of the event-triggering mechanism and compute an inner-approximation of the region of attraction for the feedback system. The theoretical conditions are obtained by using a quadratic Lyapunov function and an adequate abstraction of the activation functions via generalised sector condition to decide whether the outputs of the layers should be transmitted through the network or not. Convex optimisation procedures can be associated to the theoretical conditions in order to maximise the approximation of the region of attraction or to minimise the number of updates. The advantages and the drawbacks of our approach are illustrated in an example borrowed from the literature, namely the nonlinear inverted pendulum system stabilized by a trained neural network.