Mental stress is found to be strongly connected with human cognition and wellbeing. As the complexities of human life increase, the effects of mental stress have impacted human health and cognitive performance across the globe. This highlights the need for effective non-invasive stress detection methods. In this work, we introduce a novel, artificial spiking neural network model called Online Neuroplasticity Spiking Neural Network (O-NSNN) that utilizes a repertoire of learning concepts inspired by the brain to classify mental stress using Electroencephalogram (EEG) data. These models are personalized and tested on EEG data recorded during sessions in which participants listen to different types of audio comments designed to induce acute stress. Our O-NSNN models learn on the fly producing an average accuracy of 90.76% (σ = 2.09) when classifying EEG signals of brain states associated with these audio comments. The brain-inspired nature of the individual models makes them robust and efficient and has the potential to be integrated into wearable technology. Furthermore, this article presents an exploratory analysis of trained O-NSNNs to discover links between perceived and acute mental stress. The O-NSNN algorithm proved to be better for personalized stress recognition in terms of accuracy, efficiency, and model interpretability.