The frustrated and compositional nature of chaos in small Hopfield networks

Frustration in a network described by a set of ordinary differential equations induces chaos when the global structure is such that local connectivity patterns responsible for stable oscillatory behaviours are intertwined, leading to mutually competing attractors and unpredictable itinerancy among brief appearance of these attractors. Frustration destabilizes the network and provokes an erratic 'wavering' among the periodic saddle orbits which characterize the same network when it is connected in a non-frustrated way. The characterization of chaos as some form of unpredictable 'wavering' among repelling oscillators is rather classical but the originality here lies in the identification of these oscillators as the stable regimes of the 'non-frustrated' network. In this paper, a simple and small 6-neuron Hopfield network is treated, observed and analyzed in its chaotic regime. Given a certain choice of the network parameters, chaos occurs when connecting the network in a specific way (said to be 'frustrated') and gives place to oscillatory regimes by suppressing whatever connection between two neurons. The compositional nature of the chaotic attractor as a succession of brief appearances of orbits (or parts of orbits) associated with the non-frustrated networks is evidenced by relying on symbolic dynamics, through the computation of Lyapunov exponents, and by computing the autocorrelation coefficients and the spectrum.