Neural networks are applied to a theoretical subject in developmental psychology: modeling developmental transitions. Two issues that are involved will be discussed: discontinuities and acquiring qualitatively new knowledge. We will argue that by the appearance of a bifurcation, a neural network can show discontinuities and may acquire qualitatively new knowledge. First, it is shown that biological principles of neurite outgrowth result in self-organization in a neural network, which is strongly dependent on a bifurcation in the activity dynamics. Second, the effect of a bifurcation due to morphological change is investigated in an Adaptive Resonance Theory (ART) network. Exact ART networks with quantitative differences in network structure at the category level show qualitatively different dynamical regimes, which are separated by bifurcations. These qualitative differences in dynamics affect the cognitive function of Exact ART: Representations of learned categories are local or distributed.