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In dynamical systems theory, a fixed point of the dynamics is callednonhyperbolic if the linearization of the system around the fixed point has atleast one eigenvalue with zero real part. The center manifold existence theoremguarantees the local existence of an invariant subspace of the dynamics, knownas a center manifold, around such nonhyperbolic fixed points. A growing numberof theoretical and experimental studies suggest that some neural systemsutilize nonhyperbolic fixed points and corresponding center manifolds todisplay complex, nonlinear dynamics and to flexibly adapt to wide-rangingsensory input parameters. In this paper, we present a technique to study thestatistical properties of high-dimensional, nonhyperbolic dynamical systemswith random connectivity and examine which statistical properties determineboth the shape of the center manifold and the corresponding reduced dynamics onit. This technique also gives us constraints on the family of center manifoldmodels that could arise from a large-scale random network. We demonstrate thisapproach on an example network of randomly coupled damped oscillators.

A Center Manifold Reduction Technique for a System of Randomly Coupled Oscillators