Applications of Random Matrix Theory in Neuroscience and Neural Network Analysis: Unraveling High-Dimensional Connectivity

Random Matrix Theory (RMT) has emerged as a powerful mathematical framework to study large-scale, high-dimensional data across various scientific fields. In neuroscience, it provides unparalleled tools to elucidate the functional connectivity and dynamic properties underlying neural activity. By modeling connectivity matrices, synaptic weight distributions, and neural correlation structures through random matrices, researchers can dissect large-scale neural systems’ stability, detect phase transitions in neural firing patterns, and explain emergent phenomena such as criticality in brain networks. Similarly, in the realm of artificial neural networks, RMT guides the understanding of weight initialization, spectral properties of the network’s Jacobian matrices, generalization, and optimization landscapes. This article provides a comprehensive introduction to Random Matrix Theory’s fundamental principles, their application in contemporary neuroscience, and their intersection with modern neural network paradigms. We introduce the theoretical underpinnings and methodological steps for applying RMT to neuroscience data, including sequentially numbered didactic equations. We further prese