On the low dimensional dynamics of structured random networks

Abstract

Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach [J. Aljadeff, M. Stern, T. Sharpee, Phys. Rev. Lett. 114, 088101 (2015)], we study the relationship between the network connectivity structure and its low dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and post-synaptic neurons through a sufficiently smooth function $g$ of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of $g$. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.

16 pages, 4 figures

References in corpus (4)

• Accurate path integration in continuous attractor network models of grid cells
• Transition to chaos in random neuronal networks
• Transition to chaos in random networks with cell-type-specific connectivity
• Cavity approach to the spectral density of non-Hermitian sparse matrices

Cited by in corpus (12)

• Intrinsically-generated fluctuating activity in excitatory-inhibitory networks
• Linking structure and activity in nonlinear spiking networks
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Interpreting recurrent neural networks behaviour via excitable network attractors
• Does the brain behave like a (complex) network? I. Dynamics
• Linear stability analysis for large dynamical systems on directed random graphs
• Eigenvalue spectra of large correlated random matrices
• Eigenvalue Repulsion and Eigenfunction Localization in Sparse Non-Hermitian Random Matrices
• Instabilities of complex fluids with partially structured and partially random interactions
• Input correlations impede suppression of chaos and learning in balanced rate networks
• Learning unidirectional coupling using echo-state network
• Effect of Synaptic Heterogeneity on Neuronal Coordination