  We establish center manifold theorems that allow one to study the bifurcation
of small solutions from a trivial state in systems of functional equations
posed on the real line. The class of equations includes most importantly
nonlinear equations with nonlocal coupling through convolution operators as
they arise in the description of spatially extended dynamics in neuroscience.
These systems possess a natural spatial translation symmetry but local
existence or uniqueness theorems for a spatial evolution associated with this
spatial shift or even a well motivated choice of phase space for the induced
dynamics do not seem to be available, due to the infinite range forward- and
backward-coupling through nonlocal convolution operators. We perform a
reduction relying entirely on functional analytic methods. Despite the nonlocal
nature of the problem, we do recover a local differential equation describing
the dynamics on the set of small bounded solutions, exploiting that the
translation invariance of the original problem induces a flow action on the
center manifold. We apply our reduction procedure to problems in mathematical
neuroscience, illustrating in particular the new type of algebra necessary for
the computation of Taylor jets of reduced vector fields.
