  A system of partial differential equations representing stochastic neural
fields was recently proposed with the aim of modelling the activity of noisy
grid cells when a mammal travels through physical space. The system was
rigorously derived from a stochastic particle system and its noise-driven
pattern-forming bifurcations have been characterised. However, due to its
nonlinear and non-local nature, standard well-posedness theory for smooth
time-dependent solutions of parabolic equations does not apply. In this
article, we transform the problem through a suitable change of variables into a
nonlinear Stefan-like free boundary problem and use its representation formulae
to construct local-in-time smooth solutions under mild hypotheses. Then, we
prove that fast-decaying initial conditions and globally Lipschitz modulation
functions imply that solutions are global-in-time. Last, we improve previous
linear stability results by showing nonlinear asymptotic stability of
stationary solutions under suitable assumptions.
