How well-connected is the cerebral cortex? How random is its wiring design? (Existing) data alone do not allow a comprehensive answer to these questions but they do suggest simple models that, in turn, may suggest new experiments. As a first attempt at a theoretical approach, I shall introduce a class of networks in which connections exist at all scales but are more probable at short scales than at long scales, in analogy with data on neural connections in the cortex. I shall then discuss the connectedness of such networks; specifically, I shall determine how many connections (synapses) are present in a typical (neural) path that spans a large distance in the network (cortex). This question allows one to delineate sub-classes of well-connected, marginally connected, and locally connected networks, depending on the probability distribution of connection lengths. In this talk, I shall emphasize mathematical aspects of this question and variants, rather than biological aspects. If time allows, I shall illustrate some implications for the dynamics of neural activity in the cortex as well as some experiments that may complement our theoretical approach.

LPS ENS