  We extend the theory of neural fields which has been developed in a
deterministic framework by considering the influence spatio-temporal noise. The
outstanding problem that we here address is the development of a theory that
gives rigorous meaning to stochastic neural field equations, and conditions
ensuring that they are well-posed. Previous investigations in the field of
computational and mathematical neuroscience have been numerical for the most
part. Such questions have been considered for a long time in the theory of
stochastic partial differential equations, where at least two different
approaches have been developed, each having its advantages and disadvantages.
It turns out that both approaches have also been used in computational and
mathematical neuroscience, but with much less emphasis on the underlying
theory. We present a review of two existing theories and show how they can be
used to put the theory of stochastic neural fields on a rigorous footing. We
also provide general conditions on the parameters of the stochastic neural
field equations under which we guarantee that these equations are well-posed.
In so doing we relate each approach to previous work in computational and
mathematical neuroscience. We hope this will provide a reference that will pave
the way for future studies (both theoretical and applied) of these equations,
where basic questions of existence and uniqueness will no longer be a cause for
concern.
