For many years, topological data analysis (TDA) and deep learning (DL) have been considered separate data analysis and representation learning approaches, which have nothing in common. The root cause of this challenge comes from the difficulties in building, extracting, and integrating TDA constructs, such as barcodes or persistent diagrams, within deep neural network architectures. Therefore, the powers of these two approaches are still on their islands and have not yet combined to form more powerful tools for dealing with multiple complex data analysis tasks. Fortunately, we have witnessed several remarkable attempts to integrate DL-based architectures with topological learning paradigms in recent years. These topology-driven DL techniques have notably improved data-driven analysis and mining problems, especially within graph datasets. Recently, graph neural networks (GNNs) have emerged as a popular deep neural architecture, demonstrating significant performance in various graph-based analysis and learning problems. Explicitly, within the manifold paradigm, the graph is naturally considered as a topological object (e.g., the topological properties of the given graph can be represented by the edge weights). Therefore, integrating TDA and GNN is considered an excellent combination. Many well-known studies have recently presented the effectiveness of TDA-assisted GNN-based architectures in dealing with complex graph-based data representation analysis and learning problems. Motivated by the successes of recent research, we present systematic literature about this nascent and promising research direction in this article, which includes general taxonomy, preliminaries, and recently proposed state-of-the-art topology-driven GNN models and perspectives.

Electrocorticographic (ECoG) signals from the brain surface typically exhibit high synchrony across large cortical areas, interrupted by brief periods of desynchronization exhibiting propagating phase discontinuities, across which spatial patterns of phase emerge in selected frequency bands. Experiments with rabbits trained using classical conditioning paradigms indicated that such desynchronization periods demarcate cognitive processing in the subjects; the ECoG in the frames between such periods revealed spatial patterns of amplitude modulation that were classified with respect to sensory stimuli that the rabbits had been trained to recognize. The present work describes intermittent synchrony and desynchronization of ECoG signals measured over the visual cortex. We analyze the analytic amplitude (AA) and analytic phase (AP) of the signals bandpassed over the beta band (12.5–25Hz) and theta band (3–7Hz) using the Hilbert transform. The AP of analytic signals evaluated using a Shannon-based synchronization index in theta band exhibits phase synchronization for varying time periods averaging about 1s, interrupted by desynchronization periods of duration about 0.1s. Synchronization periods in the beta-band last <100ms, with interruptions by desynchronization lasting one-tenth that, in which the analytic amplitude drops drastically. During these “null spikes,” the analytic phase is undefined, and the spatial and temporal phase differences show high dispersion. Detailed examination of the bandpass filtered ECoG confirms the presence of a shared mean frequency in a frame of synchronized oscillation, at which frequency the spatial pattern of the AP has the form of a cone. Between frames the AA approaches zero. The form of the null spike resembles a tornado (a vortex), as shown in sequential frames by a rotating spatial pattern of amplitude in the filtered ECoG.