Neuronal Oscillations in the Visual Cortex: Γ-Convergence to the Riemannian Mumford-Shah Functional

The aim of this paper is to provide a formal link between an oscillatory neural model, whose phase is represented by a difference equation, and the Mumford and Shah functional. A Riemannian metric is induced by the pattern of neural connections, and in this setting the difference equation is studied. Its Euler–Lagrange operator Γ-converges as the dimension of the grid tends to 0 to the Mumford and Shah functional in the same Riemannian space. Correspondingly, the solutions of the phase equation converge to a BV function, which is interpreted as the flow associated with the Mumford and Shah functional. In this way we provide a biological motivation to this celebrated functional.