On well-posedness of a mathematical model of evoked activity in the primary visual cortex

We propose a mathematical model that formalizes the macro- and meso-level dynamics of electrical potentials in the primary visual cortex of subjects, which corresponds to the presentation of visual stimuli to them. The mathematical framework is based on a two-layer neural field model, represented by a system of integro-differential equations, where the deep layer of the neural field models electrical activity that does not depend directly on the spatial orientation of the visual stimuli, whereas the activity of the superficial layer is sensitive to spatially oriented stimuli. The experimental design of presenting a series of visual stimuli is formalised in the present study in terms of an impulse control problem for the aforementioned two-layer neural field model. We propose a special metric space for construction of a unique solution to the control problem under standard assumptions for mathematical neurobiology regarding the functions involved in the modeling equations. We formulate sufficient conditions for continuous dependence of the solutions on the impulse control.