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A system of partial differential equations representing stochastic neuralfields was recently proposed with the aim of modelling the activity of noisygrid cells when a mammal travels through physical space. The system wasrigorously derived from a stochastic particle system and its noise-drivenpattern-forming bifurcations have been characterised. However, due to itsnonlinear and non-local nature, standard well-posedness theory for smoothtime-dependent solutions of parabolic equations does not apply. In thisarticle, we transform the problem through a suitable change of variables into anonlinear Stefan-like free boundary problem and use its representation formulaeto construct local-in-time smooth solutions under mild hypotheses. Then, weprove that fast-decaying initial conditions and globally Lipschitz modulationfunctions imply that solutions are global-in-time. Last, we improve previouslinear stability results by showing nonlinear asymptotic stability ofstationary solutions under suitable assumptions.

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation