Differential Operators Heterogenous in Orientation and Scale in the $$V_1$$ Cortex

In the primary visual cortex $$V_1$$ of the human brain, cortical receptive profiles (RPs) changing from point to point act as a differential operator on the visual stimulus, while cortical connectivity inverts the problem and recovers an image called perceived image. We analyze the transform of the visual stimulus performed by the RPs on $$V_1$$ in order to reconstruct the image perceived by the subject. We consider a differential operator determined by a heterogenous distribution of orientation and scale, whose local representation can be a regular Laplacian, a degenerate Laplacian or a degenerate Laplacian with different scale at different points. We model the associated cortical dynamic via discrete differential operators whose coefficients have a stochastic distribution. We prove a homogenization result, showing that a large class of heterogenous operators H-converges to the classic Laplacian.