Nonlocal Ginzburg-Landau equation for cortical pattern formation

Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Oct;78(4 Pt 1):041916. doi: 10.1103/PhysRevE.78.041916. Epub 2008 Oct 24.

Abstract

We show how a nonlocal version of the real Ginzburg-Landau (GL) equation arises in a large-scale recurrent network model of primary visual cortex. We treat cortex as a continuous two-dimensional sheet of cells that signal both the position and orientation of a local visual stimulus. The recurrent circuitry is decomposed into a local part, which contributes primarily to the orientation tuning properties of the cells, and a long-range part that introduces spatial correlations. We assume that (a) the local network exists in a balanced state such that it operates close to a point of instability and (b) the long-range connections are weak and scale with the bifurcation parameter of the dynamical instability generated by the local circuitry. Carrying out a perturbation expansion with respect to the long-range coupling strength then generates a nonlocal coupling term in the GL amplitude equation. We use the nonlocal GL equation to analyze how axonal propagation delays arising from the slow conduction velocities of the long-range connections affect spontaneous pattern formation.

MeSH terms

  • Animals
  • Humans
  • Models, Neurological*
  • Nerve Net / physiology*
  • Visual Cortex / physiology*
  • Visual Perception / physiology