  One important question in neuroscience is how global behavior in a brain
network emerges from the interplay between network connectivity and the neural
dynamics of individual nodes. To better understand this theoretical
relationship, we have been exploring a simplified modeling approach in which we
equip each node with discrete quadratic dynamics in the complex plane, and we
study the emerging behavior of the resulting complex quadratic network (CQN).
The long-term behavior of CQNs can be represented by asymptotic fractal sets
with specific topological signatures going far beyond those described in
traditional single map iterations.
  In this study, we illustrate how topological measures of these asymptotic
sets can be used efficiently as comprehensive descriptors and classifiers of
dynamics in tractography-derived connectomes for human subjects. We investigate
to what extent the complex geometry of these sets is tied to network
architecture (on one hand) and to the network behavior (on the other). This
helps us understand the mechanics of the relationship between the subject's
brain function, physiology and behavior and their underlying connectivity
architecture.
