On well-posedness of generalized neural field equations with impulsive control
We consider nonlinear nonlocal integral equation generalizing equations typically used in mathematical neuroscience. We investigate solutions tending to zero at any fixed moment with unbounded growth of the spatial variable (these solutions correspond to normal brain functioning). We consider an impulsive control problem, which models electrical stimulation used in the presence of various diseases of central nervous system. We define suitable complete metric space, where we obtain conditions for existence, uniqueness and extendability of solution to the problem as well as continuous dependence of this solution on the impulsive control.