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 $\Gamma$-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.

[1]  C. M. Place,et al.  Ordinary Differential Equations , 1982 .

[2]  E. D. Giorgi,et al.  Existence theorem for a minimum problem with free discontinuity set , 1989 .

[3]  N. Fusco,et al.  Regularity of Minimizers for a Class of Anisotropic Free Discontinuity Problems , 2001 .

[4]  Cristina Trombetti.  Existence of minimizers for a class of anisotropic free discontinuity problems , 1999 .

[5]  A. Chambolle,et al.  Discrete approximation of the Mumford-Shah functional in dimension two , 1999, ESAIM: Mathematical Modelling and Numerical Analysis.

[6]  David J. Field,et al.  Contour integration by the human visual system: Evidence for a local “association field” , 1993, Vision Research.

[7]  G. Ermentrout,et al.  Symmetry and phaselocking in chains of weakly coupled oscillators , 1986 .

[8]  Irene Fonseca,et al.  Regularity results for anisotropic image segmentation models , 1997 .

[9]  L. Ambrosio.  Existence theory for a new class of variational problems , 1990 .

[10]  Massimo Gobbino.  Finite Difference Approximation of the Mumford-Shah Functional , 1998 .

[11]  G. Bellettini,et al.  Anisotropic motion by mean curvature in the context of Finsler geometry , 1996 .

[12]  Annalisa Baldi,et al.  WEIGHTED BV FUNCTIONS , 2001 .

[13]  G. Ermentrout,et al.  Phase transition and other phenomena in chains of coupled oscilators , 1990 .

[14]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[15]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[16]  H. Fédérer.  Geometric Measure Theory , 1969 .

[17]  Massimo Gobbino,et al.  Finite-difference approximation of free-discontinuity problems , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[18]  Luigi Ambrosio,et al.  ON THE APPROXIMATION OF FREE DISCONTINUITY PROBLEMS , 1992 .

[19]  Francoise Demengel.  Some Remarks on variational problems on bv(ω,S1) and W1,1(ω,S1) , 1993 .

[20]  Guy David,et al.  C1-Arcs for Minimizers of the Mumford-Shah Functional , 1996, SIAM J. Appl. Math..

[21]  J. Jost.  Riemannian geometry and geometric analysis , 1995 .

[22]  M. Gelli.  From Discrete to Continuum: a Variational Approach the One-dimensional Case , 2000 .

[23]  Adriana Garroni,et al.  Variational Formulation of Softening Phenomena in Fracture Mechanics: The One‐Dimensional Case , 1999 .

[24]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

[25]  A. Chambolle.  FINITE-DIFFERENCES DISCRETIZATIONS OF THE MUMFORD-SHAH FUNCTIONAL , 1999 .

[26]  A. Bonnet,et al.  On the regularity of edges in image segmentation , 1996 .

[27]  Andrea Braides,et al.  Continuum Limits of Discrete Systems without Convexity Hypotheses , 2002 .

[28]  H. Schuster,et al.  A model for neuronal oscillations in the visual cortex , 1990, Biological Cybernetics.

[29]  P. Hartman.  Ordinary Differential Equations , 1965 .

[30]  R Malladi,et al.  Subjective surfaces: a method for completing missing boundaries. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[31]  Luigi Ambrosio,et al.  Partial regularity of free discontinuity sets, II , 1997 .

[32]  Alexis Bonnet,et al.  Cracktip is a global Mumford-Shah minimizer , 2018, Astérisque.

[33]  Guido Cortesani,et al.  Sequences of Non‐Local Functionals Which Approximate Free‐Discontinuity Problems , 1998 .

[34]  Andrea Braides,et al.  Non-local approximation of the Mumford-Shah functional , 1997 .

[35]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[36]  H. Schuster,et al.  A model for neuronal oscillations in the visual cortex , 1990, Biological Cybernetics.

[37]  Massimo Gobbino,et al.  Gradient flow for the one-dimensional Mumford-Shah functional , 1998 .