Nonlocal Ginzburg-Landau equation for cortical pattern formation.
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.
Bressloff
Paul C
PC
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA.
Kilpatrick
Zachary P
ZP
D000818:Animals; D006801:Humans; D008959:Models, Neurological; D009415:Nerve Net / Q000502:physiology*; D014793:Visual Cortex / Q000502:physiology*; D014796:Visual Perception / Q000502:physiology