ON THE OVERLAP DYNAMICS OF MULTISTATE NEURAL NETWORKS WITH A FINITE NUMBER OF PATTERNS

Neural networks with multi-state neurons are studied in the case of low loading. For symmetric couplings satisfying a certain positivity condition, a Lyapunov function is shown to exist in the space of overlaps between the instantaneous microscopic state of the system and the learned patterns. Furthermore, an algorithm is derived for zero temperature to determine all the fixed points. As an illustration, the three-state model is worked out explicitly for Hebbian couplings. For finite temperature the time evolution of the overlap is studied for couplings which need not be symmetric. The stability properties are discussed in detail for the three-state model. For asymmetric couplings limit-cycle behaviour is shown to be possible.