Reciprocal connections in brain networks

A strong hypothesis in neuroscience is that many aspects of brain function are determined by the “map of the brain" and that its computational power relies on its connectivity architecture. Impressive scientific and engineering advances in recent years generated a plethora of large brain networks of incredibly complex architectures. A crucial aspect of the architecture is its inherent directionality reflecting the direction of information flow. One of the stark differences between directed and undirected networks is the presence of reciprocal connections. It has been shown that reciprocal connections are an overrepresented motif in neural networks and that these are formed selectively rather than randomly.

This brings forward the mathematical question: how to build appropriate null-models for directed brain networks and how the amount and location of reciprocal connections affect these? We take this question to its core and ask: how does the presence of reciprocal connections in a graph change the number of directed simplex counts (a relevant motif in neuroscience)? This inquiry is linked to deep mathematical questions in the fields of combinatorics and computational complexity as it can be phrased as counting the number of possible linear extensions of certain posets, or the number of inversions of certain permutations.

This is joint work with Jason Smith and Matteo Santoro