The brain is an extraordinarily complex system that facilitates the optimal integration of information from different regions to execute its functions. With the recent advances in technology, researchers can now collect enormous amounts of data from the brain using neuroimaging at different scales and from numerous modalities. With that comes the need for sophisticated tools for analysis. The field of network neuroscience has been trying to tackle these challenges, and graph theory has been one of its essential branches through the investigation of brain networks. Recently, topological data analysis has gained more attention as an alternative framework by providing a set of metrics that go beyond pairwise connections and offer improved robustness against noise. In this hands-on tutorial, our goal is to provide the computational tools to explore neuroimaging data using these frameworks and to facilitate their accessibility, data visualisation, and comprehension for newcomers to the field. We will start by giving a concise (and by no means complete) overview of the field to introduce the two frameworks and then explain how to compute both well-established and newer metrics on resting-state functional magnetic resonance imaging. We use an open-source language (Python) and provide an accompanying publicly available Jupyter Notebook that uses the 1000 Functional Connectomes Project dataset. Moreover, we would like to highlight one part of our notebook dedicated to the realistic visualisation of high order interactions in brain networks. This pipeline provides three-dimensional (3-D) plots of pairwise and higher-order interactions projected in a brain atlas, a new feature tailor-made for network neuroscience.

Network theory is often based on pairwise relationships between nodes, which is not necessarily realistic for modeling complex systems. Importantly, it does not accurately capture non-pairwise interactions in the human brain, often considered one of the most complex systems. In this work, we develop a multivariate signal processing pipeline to build high-order networks from time series and apply it to resting-state functional magnetic resonance imaging (fMRI) signals to characterize high-order communication between brain regions. We also propose connectivity and signal processing rules for building uniform hypergraphs and argue that each multivariate interdependence metric could define weights in a hypergraph. As a proof of concept, we investigate the most relevant three-point interactions in the human brain by searching for high-order “hubs” in a cohort of 100 individuals from the Human Connectome Project. We find that, for each choice of multivariate interdependence, the high-order hubs are compatible with distinct systems in the brain. Additionally, the high-order functional brain networks exhibit simultaneous integration and segregation patterns qualitatively observable from their high-order hubs. Our work hereby introduces a promising heuristic route for hypergraph representation of brain activity and opens up exciting avenues for further research in high-order network neuroscience and complex systems.

In this work, we use methods and concepts of applied algebraic topology to comprehensively explore the recent idea of topological phase transitions (TPTs) in complex systems. TPTs are characterized by the emergence of nontrivial homology groups as a function of a threshold parameter. Under certain conditions, one can identify TPTs via the zeros of the Euler characteristic or by singularities of the Euler entropy. Recent works provide strong evidence that TPTs can be interpreted as the intrinsic fingerprint of a complex network. This work illustrates this possibility by investigating various networks from a topological perspective. We first review the concept of TPTs in brain networks and discuss it in the context of high-order interactions in complex systems. We then investigate TPTs in protein–protein interaction networks using methods of topological data analysis for two variants of the duplication–divergence model. We compare our theoretical and computational results to experimental data freely available for gene co-expression networks ofS. cerevisiae, also known as baker’s yeast, as well as of the nematodeC. elegans. Supporting our theoretical expectations, we can detect TPTs in both networks obtained according to different similarity measures. We then perform numerical simulations of TPTs in four classical network models: the Erdős–Rényi, the Watts–Strogatz, the random geometric, and the Barabasi–Albert models. Finally, we discuss the relevance of these insights for network science. Given the universality and wide use of those network models across disciplines, our work indicates that TPTs permeate a wide range of theoretical and empirical networks, offering promising avenues for further research.

Temporal lobe epilepsy (TLE) patients are at risk of memory deficits, which have been linked to functional network disturbances, particularly of integration of the default mode network (DMN). However, the cellular substrates of functional network integration are unknown. We leverage a unique cross-scale dataset of drug-resistant TLE patients (n = 31), who underwent pseudo resting-state functional magnetic resonance imaging (fMRI), resting-state magnetoencephalography (MEG) and/or neuropsychological testing before neurosurgery. fMRI and MEG underwent atlas-based connectivity analyses. Functional network centrality of the lateral middle temporal gyrus, part of the DMN, was used as a measure of local network integration. Subsequently, non-pathological cortical tissue from this region was used for single cell morphological and electrophysiological patch-clamp analysis, assessing integration in terms of total dendritic length and action potential rise speed. As could be hypothesized, greater network centrality related to better memory performance. Moreover, greater network centrality correlated with more integrative properties at the cellular level across patients. We conclude that individual differences in cognitively relevant functional network integration of a DMN region are mirrored by differences in cellular integrative properties of this region in TLE patients. These findings connect previously separate scales of investigation, increasing translational insight into focal pathology and large-scale network disturbances in TLE.

We apply the phenomenological renormalization group to resting-state fMRI time series of brain activity in a large population. By recursively coarse graining the data, we compute scaling exponents for the series variance, log probability of silence, and largest covariance eigenvalue. The scaling exponents clearly exhibit linear interdependencies in the form of scaling relations and inherent variability of values closely related to the structure of correlations of brain activity. The scaling relations between the exponents are derived analytically. We find a significant correlation of exponents with clinical (gray matter volume) and behavioral (cognitive performance) traits. Akin to scaling relations near critical points in thermodynamics, our results suggest that this interdependency is intrinsic to brain organization, and may also exist in other complex systems.

Signal processing is a very useful field of study in the interpretation of signals in many everyday applications. In the case of applications with time-varying signals, one possibility is to consider them as graphs, so graph theory arises, which extends classical methods to the non-Euclidean domain. In addition, machine learning techniques have been widely used in pattern recognition activities in a wide variety of tasks, including health sciences. The objective of this work is to identify and analyze the papers in the literature that address the use of machine learning applied to graph signal processing in health sciences. A search was performed in four databases (Science Direct, IEEE Xplore, ACM, and MDPI), using search strings to identify papers that are in the scope of this review. Finally, 45 papers were included in the analysis, the first being published in 2015, which indicates an emerging area. Among the gaps found, we can mention the need for better clinical interpretability of the results obtained in the papers, that is not to restrict the results or conclusions simply to performance metrics. In addition, a possible research direction is the use of new transforms. It is also important to make new public datasets available that can be used to train the models.