  One of the central challenges facing modern neuroscience is to explain the
ability of the nervous system to coherently integrate information across
distinct functional modules in the absence of a central executive. To this end
Tononi et al. [Proc. Nat. Acad. Sci. USA 91, 5033 (1994)] proposed a measure of
neural complexity that purports to capture this property based on mutual
information between complementary subsets of a system. Neural complexity, so
defined, is one of a family of information theoretic metrics developed to
measure the balance between the segregation and integration of a system's
dynamics. One key question arising for such measures involves understanding how
they are influenced by network topology. Sporns et al. [Cereb. Cortex 10, 127
(2000)] employed numerical models in order to determine the dependence of
neural complexity on the topological features of a network. However, a complete
picture has yet to be established. While De Lucia et al. [Phys. Rev. E 71,
016114 (2005)] made the first attempts at an analytical account of this
relationship, their work utilized a formulation of neural complexity that, we
argue, did not reflect the intuitions of the original work. In this paper we
start by describing weighted connection matrices formed by applying a random
continuous weight distribution to binary adjacency matrices. This allows us to
derive an approximation for neural complexity in terms of the moments of the
weight distribution and elementary graph motifs. In particular we explicitly
establish a dependency of neural complexity on cyclic graph motifs.
