Solutions of the Haeussler-von der Malsburg Equations in Manifolds with Constant Curvatures

Abstract

We apply generic order parameter equations for the emergence of retinotopy between manifolds of different geometry to one- and two-dimensional Euclidean and spherical manifolds. To this end we elaborate both a linear and a nonlinear synergetic analysis which results in order parameter equations for the dynamics of connection weights between two cell sheets. Our results for strings are analogous to those for discrete linear chains obtained previously by Haeussler and von der Malsburg. The case of planes turns out to be more involved as the two dimensions do not decouple in a trivial way. However, superimposing two modes under suitable conditions provides a state with a pronounced retinotopic character. In the case of spherical manifolds we show that the order parameter equations provide stable stationary solutions which correspond to retinotopic modes. A further analysis of higher modes furnishes proof that our model describes the emergence of a perfect one-to-one retinotopy between two spheres.