LOCAL CONDITIONS FOR PHASE-TRANSITIONS IN NEURAL NETWORKS WITH VARIABLE CONNECTION STRENGTHS

Qualitative changes in the behavior of a neural network may occur when parameters cross critical boundaries. These phase transitions may be triggered either by learning or by the intrinsic dynamics of the network, and must be understood in order to guarantee that the behavior of the model will be meaningful. This is particularly important for models with connection strengths that are permitted to vary rapidly. Such models, which have been applied to a variety of problems in computer science, neuroscience, and cognitive science, may display transitions from phases characterized by bounded total activation to phases during which total activation may grow explosively. It has been observed, however, that even when total network activation remains bounded, individual node activations can grow without limit. In this paper, local conditions for boundedness and divergence are derived, in the form of a balance between each node's decay vs the gain it imparts to the rest of the system. These conditions do not require that the connection matrix be symmetric. These results extend the range of models whose phase transitions are understood and, therefore, expand the choices of well-behaved models that may be selected for applications.