Multitasking attractor networks with neuronal threshold noise

We consider the multitasking associative network in the low-storage limit and we study its phase diagram with respect to the noise level T and the degree d of dilution in pattern entries. We find that the system is characterized by a rich variety of stable states, including pure states, parallel retrieval states, hierarchically organized states and symmetric mixtures (remarkably, both even and odd), whose complexity increases as the number of patterns P grows. The analysis is performed both analytically and numerically: Exploiting techniques based on partial differential equations, we are able to get the self-consistencies for the order parameters. Such self-consistency equations are then solved and the solutions are further checked through stability theory to catalog their organizations into the phase diagram, which is outlined at the end. This is a further step towards the understanding of spontaneous parallel processing in associative networks.