Human facial emotion recognition (FER) is a vibrant research field. This research proposes a novel, biologically inspired hybrid FER framework that uniquely connects event-driven Spiking Neural Networks (SNNs) with deep learning, specifically a Spike-based Support Vector Machine (S-SVM), which is designed for its event-driven processing and energy efficiency. The article proposes a novel SNN-based framework for FER that extracts robust image features using a Vision Transformer (ViT). Spikes are generated from the features using the rate-and-threshold encoding technique. The core algorithmic novelty lies in the integration of Leaky Integrate-and-Fire (LIF) and Quadratic Integrate-and-Fire (QIF) neurons, which are used in S-SVM, creating a highly optimized decision boundary in the spike domain. The goal is to determine which neuron performs best, as confirmed by the CK+ dataset. The same technique was validated on RAF-CE, a unprocessed dataset derived from real-world events and movie scenes. The approach was validated on the CK+ dataset, achieving 99.14% and 99.94% accuracy for the QIF and LIF neurons, respectively. For the compound emotions in the unprocessed dataset, which are hard to distinguish, the accuracy rates achieved with QIF and LIF neurons are 78.87% and 98.91%, respectively. In the FER system, the LIF neuron with an S-SVM classifier performs better than the LIF neuron. This hybrid design's capacity to provide cutting-edge FER accuracy while also enabling previously unheard-of energy efficiency gains of 99.52% for CK+ and 99.60% for RAF-CE when compared to traditional deep learning models is its primary significance.