This paper is in the scope of emotion recognition by employing a brain-inspired recurrent spiking neural network (BI-SNN) architecture for modelling, mapping, learning, classifying, visualising, and understanding of spatio-temporal Electroencephalogram (EEG) data related to different emotional states. It further explores, develops, and applies a methodology based on the NeuCube BI-SNN, that includes methods for EEG data encoding, data mapping into a 3-dimensional BI-SNN model, unsupervised learning using spike-timing dependent plasticity (STDP) rule, spike-driven supervised learning, output classification, network analysis, and model visualisation and interpretation. The research conducted to model different emotional subtypes through mapping both space (brain regions) and time (brain dynamics) components of EEG brain data into brain-inspired SNN architecture. Here, a benchmark EEG dataset was used to design an empirical study that consisted of different experiments for classification of positive emotions (calmness and happiness) and negative emotions (fear and anger). The obtained accuracy of 94.83% for EEG classification of four types of emotions was superior when compared with traditional machine learning techniques, such as Multiple Linear Regression (MLR), Multi-Layer Perceptron (MLP), and Radial Basis Function (RBF). The BI-SNN models not only detected the brain activity patterns related to positive and negative emotions with a high accuracy, but also revealed new knowledge about the brain areas activated in relation to different emotions. The research confirmed that neural activation increased in the frontal sites of brain (F7, F3, AF4) associated with positive emotions, while in the case of the negative emotions, connectivity strength was concentrated in the frontal (F4, AF3, F7, F8) and parietal sites of the brain (P7, P8). The experiments also confirm the suitability of the BI-SNN for emotion recognition using EEG data.