  In dynamical systems theory, a fixed point of the dynamics is called
nonhyperbolic if the linearization of the system around the fixed point has at
least one eigenvalue with zero real part. The center manifold existence theorem
guarantees the local existence of an invariant subspace of the dynamics, known
as a center manifold, around such nonhyperbolic fixed points. A growing number
of theoretical and experimental studies suggest that some neural systems
utilize nonhyperbolic fixed points and corresponding center manifolds to
display complex, nonlinear dynamics and to flexibly adapt to wide-ranging
sensory input parameters. In this paper, we present a technique to study the
statistical properties of high-dimensional, nonhyperbolic dynamical systems
with random connectivity and examine which statistical properties determine
both the shape of the center manifold and the corresponding reduced dynamics on
it. This technique also gives us constraints on the family of center manifold
models that could arise from a large-scale random network. We demonstrate this
approach on an example network of randomly coupled damped oscillators.
