THE FRUSTRATED AND COMPOSITIONAL NATURE OF CHAOS IN SMALL HOPFIELD NETWORKS

Frustration in a network described by a set of ordinary differential e quations induces chaos when the global structure is such that local co nnectivity patterns responsible for stable oscillatory behaviours are intertwined, leading to mutually competing attractors and unpredictabl e itinerancy among brief appearance of these attractors. Frustration d estabilizes the network and provokes an erratic 'wavering' among the p eriodic saddle orbits which characterize the same network when it is c onnected in a non-frustrated way. The characterization of chaos as som e form of unpredictable 'wavering' among repelling oscillators is rath er classical but the originality here lies in the identification of th ese oscillators as the stable regimes of the 'non-frustrated' network. In this paper, a simple and small 6-neuron Hopfield network is treate d, observed and analyzed in its chaotic regime. Given a certain choice of the network parameters, chaos occurs when connecting the network i n a specific way (said to be 'frustrated') and gives place to oscillat ory regimes by suppressing whatever connection between two neurons. Th e compositional nature of the chaotic attractor as a succession of bri ef appearances of orbits (or parts of orbits) associated with the non- frustrated networks is evidenced by relying on symbolic dynamics, thro ugh the computation of Lyapunov exponents, and by computing the autoco rrelation coefficients and the spectrum. (C) 1998 Elsevier Science Ltd . All rights reserved.