Self-organizing neural networks near the anti-integrable limit
Paul C. Bressloff
Department of Mathematical Sciences
Loughborough University
Leics.
LE11 3TU, U.K.
Exploiting certain similarities between pattern formation in synergetic systems and pattern recognition, Haken [1] has constructed a neural network model that implements a form of competitive gradient dynamics. The ground states of the system consist of strictly localised states in which only one neuron is excited and the remainder quiescent. In other words, the network dynamically realises a winner-take-all strategy. An interesting extension of this type of network is the diffusive Haken model in which the introduction of a diffusive interaction between the neurons leads to a delocalization of the original model's ground states [2]. When there exists a balance between the effects of diffusion and localization it is possible to obtain new ground states that are localized excitations (or bubbles) distributed over many neurons. Such ground states are more robust than those of the simple Haken model and also better reflect the kind of coherent structures found in neurobiological systems. The underlying dynamical equation of the diffusive Hakens model is
where 
is the state of the neuron at lattice site n and the
diffusive coupling term is over all nearest
neighbours on the lattice.
One of the major claims of [2] is that for a network arranged on a
d-dimensional square lattice with standard nearest
neighbour diffusive coupling, localized ground states only exist when d =
1; such states cannot be sustained for d ;SPMgt;
1 even for arbitrarily small diffusive coupling 
. This claim is
based on numerical simulations of the
lattice model together with some variational calculations of a continuum
model obtained in the limit of large diffusive
coupling, which possesses instanton-like solutions.
We prove analytically that the above claim is false. In other words, we
show that the (standard) diffusive Haken model
supports localised ground states (lattice instantons) in any finite
dimension provided that the coupling is
sufficiently small. Our approach is based on perturbation theory about the
limiting case of zero diffusive
coupling (anti-integrable limit) [3], and is an extension of the work of
Aubry and collaborators on the
Frenkel-Kontorova model [4].
We also discuss some other applications of the anti-integrable limit to studying the behaviour of nonlinear discrete systems in physics and biology. Examples include multistability and wave propagation failure in discrete coupled cells [5], [6] and localized breathers in nonlinear oscillator networks [7].