Local instability and oscillations of trajectories in a diluted symmetric neural network

Using a generalized version of the signal-to-noise analysis, we study local instabilities of trajectories for the parallel dynamics of an extremely diluted, symmetric, Hopfield neural network. In order to reach a better understanding of the structure of the attractors of this model, a revision of the asymmetric version is performed in the case of zero and non-zero temperatures. New unexpected dynamical behaviours are found. Moreover, despite accepted beliefs, both analytical and numerical deviations between the dynamical properties of the two models (symmetric and asymmetric) can be exhibited. We show that, in some range of parameters, the diluted symmetric network exhibits strong dynamical oscillations of the neuronal activity, similar to those observed in synchronized networks. Furthermore, a deeper knowledge of the structure near attractors is achieved from this stability/instability analysis thanks to explicit analytical formulae for a two-step parallel dynamics for a symmetric network.