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 di luted, symmetric, Hopfield neural network. In order to reach a better under standing of the structure of the attractors of this model, a revision of th e asymmetric version is performed in:the case of zero and non-zero temperat ures. New unexpected dynamical behaviours are found. Moreover, despite acce pted beliefs, both analytical and numerical deviations between the dynamica l properties of the two models (symmetric and asymmetric) can be exhibited. We show that, in some range of parameters, the diluted symmetric network e xhibits 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 dyn amics for a symmetric network.