Universality and synchronization in complex quadratic networks*

Abstract: Dynamic rhythms in networks of neural oscillators is an important point in neuroscience. In a brain circuit, different asymptotic patterns (e.g., steady state versus oscillations) may reflect into different functional and cognitive outcomes and could mark differences between health and pathology. The preference for one behavior versus others depends not only on the dynamic behavior of each networked unit, but also on the arrangement and strengths of synaptic pathways between them. Patterns like synchronization and clustering -- which reflect at a functional level the type and efficiency of network function -- are also highly dependent on the network interconnectivity and architecture. Establishing mathematical ties between connectivity and dynamics of a network is notoriously difficult. We start by investigating this relationship in a canonical framework (using complex quadratic node dynamics). We build upon historic research in complex iterations, to define network extensions for traditional objects (like the Mandelbrot set). We use these to quantify the effects of network architecture on dynamic patterns (e.g., node "synchronization"), and we search for principles which may universally apply to larger classes of networks and dynamics. We harness the idea of encoding network dynamics using geometry, so that asymptotic patterns can be visualized as topological properties of fractals. We use this novel approach to analyze and classify fractals for tractography-derived brain networks.

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