Solving degenerate optimization problems using networks of neural oscillators

Abstract

Optimization problems containing multiple, identical solutions, in particular the travelling salesman problem (TSP), are solved using networks of neural oscillators. Processing units are described by two state variables representing fast and slow membrane events, much like the Fitzhugh-Nagumo model neurons, and are passive oscillators in that at least two processing units with mutually inhibitory couplings are required to generate sustained oscillations. The emergent behavior of the network is to produce stable travelling waves (limit cycles) of firing rate impulses in the position coordinate of the network with the phase relationship between active (suprathreshold) processing units determining the TSP solution. This approach has the advantage of avoiding most local energy minima with the best or near-best solution being found every time for small networks (<10 2 processing units). As network size is enlargened, the time required to reach a stable travelling wave configuration increases. To improve the convergence rate, the strength of the network couplings encoding the cost function and the strength of the network couplings encoding the syntactic constraints are alternately reduced such that network evolution is dictated by only one set of couplings at any given time. Good solutions are found for networks containing up to 10 4 processing units without a large expenditure of computational effort.