Spectral analysis of synaptic connection matrix of brain utilizing the Random matrix theory for large scale neural network.

In Random Matrix Theory, we study spectral properties like the eigenvalues and eigenvectors of large random matrices and their influence on overall system dynamics. Many neural networks, both artificial and biological, can model the synaptic weight matrix as a random matrix. Each entry in the matrix signifies the strength of the connection between two neurons, either randomly initialized in artificial neural networks or with the addition of randomness due to neural plasticity. In this study, we analyzed the framework of the neuronal network, considering the synaptic matrices as large eigen matrices with independent and identically distributed entries, which are typically from a Gaussian distribution. Here we observed that the eigenvalue distribution tends to follow a circular shape, with most of the eigenvalues concentrated near zero and a few found near larger magnitudes. In this article, we examine the eigenvalue spectrum of a random synaptic connection matrix, giving insight into the behavior of neural networks in various regimes: ordered, chaotic, and critical. We have studied parameters like memory efficiency, which is crucial for learning and generalizing based on data. The study establishes RMT as an essential tool for the study of large-scale neural networks, guiding their design and analyzing emergent properties.