Attention-deficit/hyperactivity disorder (ADHD) is a developmental disorder characterized by difficulty to control the own behavior. Neuroimaging studies have related ADHD with the interplay of fronto-parietal attention systems with the default mode network (DMN; Castellanos and Aoki, 2016). However, some results have been inconsistent, potentially due to methodological differences in the analytical strategies when defining the brain functional network, i.e., the functional connectivity threshold and/or the brain parcellation scheme. Here, we make use of topological data analysis (TDA) to explore the brain connectome as a function of the filtration value (i.e., the connectivity threshold), instead of using a static connectivity threshold. Specifically, we characterized the transition from all nodes being isolated to being connected into a single component as a function of the filtration value. We explored the utility of such a method to identify differences between 81 children with ADHD (45 male, age: 7.26–17.61 years old) and 96 typically developing children (TDC; 59 male, age: 7.17–17.96 years old), using a public dataset of resting state (rs)fMRI in human subjects. Results were highly congruent when using four different brain segmentations (atlases), and exhibited significant differences for the brain topology of children with ADHD, both at the whole-brain network and the functional subnetwork levels, particularly involving the frontal lobe and the DMN. Therefore, this is a solid approach that complements connectomics-related methods and may contribute to identify the neurophysio-pathology of ADHD.

AbstractThe functional organization of the brain network (connectome) has been widely studied as a graph; however, methodological issues may affect the results, such as the brain parcellation scheme or the selection of a proper threshold value. Instead of exploring the brain in terms of a static connectivity threshold, this work explores its algebraic topology as a function of the filtration value (i.e., the connectivity threshold), a process termed the Rips filtration in Topological Data Analysis. Specifically, we characterized the transition from all nodes being isolated to being connected into a single component as a function of the filtration value, in a public dataset of children with attention-deficit/hyperactivity disorder (ADHD) and typically developing children. Results were highly congruent when using four different brain segmentations (atlases), and exhibited significant differences for the brain topology of children with ADHD, both at the whole brain network and at the functional sub-network levels, particularly involving the frontal lobe and the default mode network. Therefore, this approach may contribute to identify the neurophysio-pathology of ADHD, reducing the bias of connectomics-related methods.HighlightsTopological Data Analysis was implemented in functional connectomes.Betti curves were assessed based on the area under the curve, slope and kurtosis.The explored variables were robust along four different brain atlases.ADHD showed lower areas, suggesting decreased functional segregation.Frontal and default mode networks showed the greatest differences between groups.Graphical Abstract

Adolescence is a crucial developmental period in terms of behavior and mental health. Therefore, understanding how the brain develops during this stage is a fundamental challenge for neuroscience. Recent studies have modelled the brain as a network or connectome, mainly applying measures from graph theory, showing a change in its functional organization such as an increase in its segregation and integration. Topological Data Analysis (TDA) complements such modelling by extracting high-dimensional features across the whole range of connectivity values, instead of exploring a fixed set of connections. This study enquiries into the developmental trajectories of such properties using a longitudinal sample of typically developing participants (N = 98; 53/45 F/M; 6.7-18.1 years), applying TDA into their functional connectomes. In addition, we explore the effect of puberty on the individual developmental trajectories. Results showed that compared to random networks, the adolescent brain is more segregated at the global level, but more densely connected at the local level. Furthermore, developmental effects showed nonlinear trajectories for the integration of the whole brain and fronto-parietal networks, with an inflection point and increasing trajectories after puberty onset. These results add to the insights in the development of the functional organization of the adolescent.

a b s t r a c tThe problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The (p, q)-extremal problem is to maximize the number of edges in a k-uniform hypergraph H with given order subject to the constraint that any p edges contain q edges that have a vertex in common. In this paper we found a link between the fractional chromatic number of Kneser hypergraphs and the (p, q)-extremal problem. We solve the (p, q)-extremal problem for graphs and with the aid of this result we calculate the fractional chromatic number of Kneser hypergraphs when they are composed with sets of cardinality 2.

Adolescence is a crucial developmental period in terms of behavior and mental health. Therefore, understanding how the brain develops during this stage is a fundamental challenge for neuroscience. Recent studies have modeled the brain as a network or connectome, mainly applying measures from graph theory, showing a change in its functional organization, such as an increase in its segregation and integration. Topological Data Analysis (TDA) complements such modeling by extracting high-dimensional features across the whole range of connectivity values instead of exploring a fixed set of connections. This study inquires into the developmental trajectories of such properties using a longitudinal sample of typically developing human participants (N = 98; 53/45 female/male; 6.7–18.1 years), applying TDA to their functional connectomes. In addition, we explore the effect of puberty on individual developmental trajectories. Results showed that the adolescent brain has a more distributed topology structure compared with random networks but is more densely connected at the local level. Furthermore, developmental effects showed nonlinear trajectories for the topology of the whole brain and fronto-parietal networks, with an inflection point and increasing trajectories after puberty onset. These results add to the insights into the development of the functional organization of the adolescent brain.Significance StatementTopological Data Analysis (TDA) may be used to explore the topology of the brain along the whole range of connectivity values instead of selecting only a fixed set of connectivity thresholds. Here, we explored some properties of the topology of the brain’s functional connectome and how they develop in adolescence. We show that developmental trajectories are nonlinear and better adjusted by puberty status than chronological age, with an inflection point around the onset of puberty. In particular, the results show that the topology of the fronto-parietal network is the one that drives the functional connectome changes in the adolescent period.