  Delayed neural field models can be viewed as a dynamical system in an
appropriate functional analytic setting. On two dimensional rectangular space
domains, and for a special class of connectivity and delay functions, we
describe the spectral properties of the linearized equation. We transform the
characteristic integral equation for the delay differential equation (DDE) into
a linear partial differential equation (PDE) with boundary conditions. We
demonstrate that finding eigenvalues and eigenvectors of the DDE is equivalent
with obtaining nontrivial solutions of this boundary value problem (BVP). When
the connectivity kernel consists of a single exponential, we construct a basis
of the solutions of this BVP that forms a complete set in $L^2$. This gives a
complete characterization of the spectrum and is used to construct a solution
to the resolvent problem. As an application we give an example of a Hopf
bifurcation and compute the first Lyapunov coefficient.

Dynamical Systems