The connectivity of the brain: multi-level quantitative analysis.

We develop a mathematical formalism or calculating connectivity volumes generated by specific topologies with various physical packing strategies. We consider four topologies (full, random, nearest-neighbor, and modular connectivity) and three physical models: (i) interior packing, where neurons and connection fibers are intermixed, (ii) sheeted packing where neurons are located on a sheet with fibers running underneath, and (iii) exterior packing where the neurons are located at the surfaces of a cube or sphere with fibers taking up the internal volume. By extensive cross-referencing of available human neuroanatomical data we produce a consistent set of parameters for the whole brain, the cerebral cortex, and the cerebellar cortex. By comparing these inferred values with those predicted by the expressions, we draw the following general conclusions for the human brain, cortex, and cerebellum: (i) Interior packing is less efficient than exterior packing (in a sphere). (ii) Fully and randomly connected topologies are extremely inefficient. More specifically we find evidence that different topologies and physical packing strategies might be used at different scales. (iii) For the human brain at a macro-structural level, modular topologies on an exterior sphere approach the data most closely. (iv) On a mesostructural level, laminarization and columnarization are evidence of the superior efficiency of organizing the wiring as sheets. (v) Within sheets, microstructures emerge in which interior models are shown to be the most efficient. With regard to interspecies similarities and differences we conjecture (vi) that the remarkable constancy of number of neurons per underlying square millimeter of cortex may be the result of evolution minimizing interneuron distance in grey matter, and (vii) that the topologies that best fit the human brain data should not be assumed to apply to other mammals, such as the mouse for which we show that a random topology may be feasible for the cortex.