Topology-preservation emergence by the Hebb rule with infinitesimal short-range signals

Topology preservation is a ubiquitous phenomenon in the mammalian nervous system. What are the necessary and sufficient conditions for the self-organized formation of topology preservation due to a Hebb mechanism? A relatively realistic Hebb rule and neurons with stochastic fluctuations are modeled. Moreover, the reasonable growth law is used for coupling growth that the biomass increase is proportional to the present biomass under the constraint that the biomass is limited at a neuron. It is proven for such general Hebb-type networks that infinitesimal lateral signal transfer to neighboring neurons is necessary and sufficient for the emergence of topology preservation. As a consequence, observed topology preservation in nervous systems may emerge with or without purpose as a byproduct of infinitesimal lateral signal transfer to neighboring neurons due to ubiquitous chemical and electrical leakage.