On the Relation between Topology and Geometry in Biological Neural Networks

Coupling of topology and geometry, i.e. of connectivity between structures of the nervous tissue and their anatomy, is investigated in biological neural networks. It is shown that a function called the synaptic density-connectivity function plays a central role in the description of the dynamical processes. The topology of the neural network, i.e. the functional organization, is described in terms of non-symmetric functional interactions. The continuous geometry imposes non-local dynamics described in terms of fields at each level of the functional organization. Learning rules are discussed in the framework of this theory.