Spiking neural networks (SNN) represent the third generation of the neural networks. They are inspired by the information processing principles in the human brain. The theory and applications of SNN can further benefit from: a) The use of new brain-inspired learning principles for more efficient learning in SNNs. b) The use of quantum computation principles for novel learning mechanisms, improvement in the SNNs’ performance, and parameter optimisation. c) The use of the integration of the above. To address point (a) outlined above, novel mathematical and computational models of spiking neural networks (SNN) are introduced in this thesis as generic SNN models, to perform both off-line and on-line prediction and classification tasks. These generic models are unified into one single framework called Evolving Predictive Unsupervised-Supervised deep learning algorithms for Spike Streams (EPUSSS) to perform both prediction and classification tasks in a hierarchical fashion. To address point (b) outlined above, the Quantum-Inspired Evolutionary Algorithm (QIEA) is improved using Chirikov chaotic map and used as the learning rule for SNNs. A search mechanism is proposed to recall, associatively, a pattern stored in the memory. A new parameter optimisation method to improve models’ performance is proposed. A novel Quantum Inspired Spiking Neural Network (QISNN) framework is introduced that combines a neuron’s macro level structural functionality with its micro level physical and structural functionality to demonstrate a biological behaviour and to reinforce the computational power of SNNs. To address point (c) outlined above, a novel Quantum Inspired Associative Memory for Spiking Neural Network (QIASM-SNN) is introduced to preserve the spiking activities produced by the proposed models in SNNs and to recall the stored memories in the presence of partial noisy data. Several methods for pre-processing and feature extraction of Spatio-temporal EEG data are proposed and illustrated on a real-world problem related to brain neurodegenerative disease. Also, a novel feature extraction method is introduced based on a combination of the proposed Chaotic Quantum-Inspired Evolutionary Algorithm and fuzzy rough set theory for clinical static data related to the EEG recordings. The purpose of this approach is to provide a potential for weighted learning of Spatio-temporal data based on the clinical and demographical observations of the subjects in the proposed methods. The thesis has mainly a theoretical contribution to the area of SNN and future investigations and applications will follow. To produce precise spiking activities, the idea of a novel neuron model inspired by the nuclear physic concept of Microtron Accelerator is presented as a future study to add a self-adaptable delay mechanism to the SNNs. As a future study, several pre-processing techniques are also suggested for real-world datasets. Four papers are under preparation to submit in order to publish the main contributions elaborated in this thesis.