This paper investigates the event-triggered control problem for perturbed systems under neural network controllers. We propose a novel event-triggering mechanism, based on local sector conditions related to the activation functions, to reduce the computational cost associated with the neural network evaluation. It avoids redundant computations by updating only a portion of the layers instead of evaluating periodically the whole neural network. Sufficient conditions in terms of matrix inequalities are established to design the parameters of the event-triggering mechanism and compute an inner-approximation of the region of attraction for the perturbed feedback system. The theoretical conditions are obtained by using a quadratic Lyapunov function and an abstraction of the activation functions via quadratic constraints to decide whether the outputs of the layers should be transmitted through the network or not. Such conditions allow us to reduce the computational activity on the neural network while preserving the stability and performance level of the perturbed feedback system. To illustrate the efficacy of our approach, we consider the nonlinear inverted pendulum system stabilized by a trained neural network.