Homological scaffolds of brain functional networks

Abstract

Networks, as efficient representations of complex systems, have appealed to scientists for a long time and now permeate many areas of science, including neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci. 10, 186–198. (doi:10.1038/nrn261810.1038/nrn2618)). Traditionally, the structure of complex networks has been studied through their statistical properties and metrics concerned with node and link properties, e.g. degree-distribution, node centrality and modularity. Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. This can be done by focusing on the features of a set of topological objects—homological cycles—associated with the weighted functional network. We leverage the detected topological information to define the homological scaffolds, a new set of objects designed to represent compactly the homological features of the correlation network and simultaneously make their homological properties amenable to networks theoretical methods. As a proof of principle, we apply these tools to compare resting-state functional brain activity in 15 healthy volunteers after intravenous infusion of placebo and psilocybin—the main psychoactive component of magic mushrooms. The results show that the homological structure of the brain's functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo.

What can topology tell us about the neural code?

“…Here I will only mention that topological network analyses have already been used in a variety of neuroscience applications, many of them medically motivated: fMRI networks in patients with ADHD [16]; FDG-PET based networks in children with autism and ADHD [25]; morphological networks in deaf adults [24]; metabolic connectivity in epileptic rats [7]; and functional EEG connections in depressed mice [23]. Other applications to fMRI data include human brain networks during learning [35] and drug-induced states [30]. At a finer scale, recordings of neural activity can also give rise to functional connectivity networks among neurons (which are not the same as the neural networks defined by synaptic connections).…”

Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets

“…This would allow us to study the evolution of a network "under its own pressure" and to detect and examine such catastrophic events as virus attacks and denial of service attempts. Given the basic numerical simplicity of our method, this approach might prove to be an effective alternative to the Persistent Homology method (see, e.g., [6]) for the 1-dimensional case of networks. Moreover, the Ricci flow does not need to make appeal to higher dimensional structures (namely simplicial complexes) that are necessary for the Persistent Homology based applications, with clear computational advantages (see, e.g., the code described in [45]), but also theoretically rigor.…”

“…Complex networks are by now ubiquitous, both in every day life and as mathematical models for a wide range of phenomena [1][2][3][4] with applications in such diverse fields as Biology [5,6], transportation and urban planning [4,7,8], social networks like Facebook and Twitter [9], and-in its relevance dating back to early work on networks-in communication and computer systems [2]. The latter belong to the class of peer-to-peer networks whose structure is characterized by information transfer between "peers".…”

Schützenberger Symmetries in Network Structures

“…Useful applications of these symmetries and invariants include understanding the consequences of link reconfiguration in communication and sensor networks, e.g., [13], identifying important relationships in social networks, e.g., [14], and providing classification and analysis of biological network data, e.g., [15]. In conclusion, it is noted that research associated with understanding the "internal dynamics" of L and R class evolutions and quantifying their complexities is reported in [5].…”