Bifurcations of neural networks with almost symmetric interconnection matrices

The symmetry of the neuron interconnection matrix ensures that additive neural networks are completely stable, i.e., each trajectory converges towards some equilibrium point. However, the crucial point is that in any practical realization it is not possible to implement perfectly symmetric interconnections, and therefore robustness of complete stability with respect to small perturbations of the nominal symmetric neuron interconnections is an extremely important issue. This paper addresses such a basic problem showing that symmetry of the interconnection matrix is not sufficient to ensure robustness of complete stability. More precisely, some third-order neural network configurations are discussed through which the existence of non-convergent dynamics as close to the symmetry condition as one pleases is proved. These include stable limit cycles generated via local Hopf bifurcations or global heteroclinic bifurcations. Moreover, numerical simulations show the presence of period-doubling bifurcations which originate more complicated attractors.