Nonlocal Ginsburg-Landau equation for cortical dynamics
We show how a nonlocal version of the real Ginzburg-Landau equation arises in a large--scale recurrent network model of primary visual cortex (V1). 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.
University of Utah
| Department of Mathematics
|
bressloff@math.utah.edu
Jan 2004.