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

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.

arXiv comment25 pages + 5 pages of refs, 9 figures

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Cited by in corpus (10)

• Edge of chaos and avalanches in neural networks with heavy-tailed synaptic weight distribution
• Network structural origin of instabilities in large complex systems
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Universal transient behavior in large dynamical systems on networks
• The Efficiency of Synchronization Dynamics and the Role of Network Syncreactivity
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Real spectra of large real asymmetric random matrices
• Different eigenvalue distributions encode the same temporal tasks in recurrent neural networks
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge