The cortical V1 transform as a heterogeneous Poisson problem

TL;DR

This paper models the V1 cortical cell responses as a heterogeneous Poisson problem, linking neural connectivity to solving inverse image reconstruction problems with heterogeneity and demonstrating convergence to homogeneous solutions.

Contribution

It introduces a novel mathematical framework connecting cortical heterogeneity to Poisson inverse problems and applies homogenisation techniques to analyze convergence.

Findings

Neural connectivity weights form the fundamental solution of the Poisson problem.

Receptive field heterogeneity influences image reconstruction.

Homogenisation leads to convergence towards homogeneous solutions.

Abstract

Receptive profiles of V1 cortical cells are very heterogeneous and act by differentiating the stimulus image as operators changing from point to point. A lightness and color constancy image can be reconstructed as the solution of the associated inverse problem, that is a Poisson equation with heterogeneous differential operators. At the neural level the weights of short range connectivity constitute the fundamental solution of the Poisson problem adapted point by point. A first demonstration of convergence of the result towards homogeneous reconstructions is proposed by means of homogenisation techniques.