From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior (1805.03592v2)

Abstract: The study of neuronal interactions is currently at the center of several big collaborative neuroscience projects (including the Human Connectome Project, the Blue Brain Project, the Brainome, etc.) which attempt to obtain a detailed map of the entire brain. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of the synaptic interaction matrix. This work explores the application of free random variables to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra in types of models of neural networks proposed by Rajan and Abbott, we extend them to heavy-tailed distributions of interactions. More importantly, we derive analytically the behavior of eigenvector overlaps, which determine the stability of the spectra. We observe that upon imposing the neuronal excitation/inhibition balance, despite the eigenvalues remaining unchanged, their stability dramatically decreases due to the strong non-orthogonality of associated eigenvectors. It leads us to the conclusion that the understanding of the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of free random variables theory in a wide variety of disciplines, we hope that the results presented here foster the additional application of these ideas in the area of brain sciences.