A self-organizing network in the weak-coupling limit
We prove the existence of spatially localized ground states of the diffusive Haken model.
This model describes a self-organizing network whose elements are arranged on a $d$-dimensional lattice with short-range
diffusive coupling. The network evolves according to a competitive gradient dynamics in which the effects of diffusion are
counteracted by a localizing potential that incorporates an additional global coupling term. In the absence of diffusive
coupling, the ground states of the system are strictly localized, that is, only one lattice site is excited. For
sufficiently small non-zero diffusive coupling $\alpha$, it is shown analytically that localized ground states persist in the
network with the excitations exponentially decaying in space. Numerical results establish that localization occurs
for arbitrary values of $\alpha$ in one dimension but vanishes beyond a critical coupling $\alpha_c(d)$ when $d > 1$. The
one-dimensional localized states are interpreted in terms of instanton solutions of a continuum version of the model.
University of Utah
| Department of Mathematics
|
bressloff@math.utah.edu
Aug 2001.