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Edited by: Maria V. Sanchez-Vives, Consorci Institut D'Investigacions Biomediques August Pi i Sunyer, Spain

Reviewed by: Preston E. Garraghty, Indiana University Bloomington, USA; Adenauer Girardi Casali, Federal University of São Paulo, Brazil

*Correspondence: Morten L. Kringelbach

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In recent years, the application of network analysis to neuroimaging data has provided useful insights about the brain's functional and structural organization in both health and disease. This has proven a significant paradigm shift from the study of individual brain regions in isolation. Graph-based models of the brain consist of vertices, which represent distinct brain areas, and edges which encode the presence (or absence) of a structural or functional relationship between each pair of vertices. By definition, any graph metric will be defined upon this dyadic representation of the brain activity. It is however unclear to what extent these dyadic relationships can capture the brain's complex functional architecture and the encoding of information in distributed networks. Moreover, because network representations of global brain activity are derived from measures that have a continuous response (i.e., interregional BOLD signals), it is methodologically complex to characterize the architecture of functional networks using traditional graph-based approaches. In the present study, we investigate the relationship between standard network metrics computed from dyadic interactions in a functional network, and a metric defined on the

The application of graph theoretical analysis to neuroimaging data has provided important new insights about the functional organization of the human brain in health and disease. Graph measures considering the global properties of brain networks have notably helped shape our understanding of the system-wide functional architectures which enable the brain to balance the segregation and integration of information in macro-scale networks (Bullmore and Sporns,

Whilst standard graph metrics are powerful descriptive means to characterize functional neuroimaging data at the whole-brain scale, they also involve significant conceptual and methodological limitations. First, these measures are exclusively based on

Secondly, the adjacency matrices which form the basis for constructing network representations are derived from measures that have a continuous response and are therefore typically weighted, fully connected, and signed. That is, the value of the pair-wise measure of association (i.e., bivariate/partial correlation, phase synchrony, transfer entropy, mutual information) between the activity signals across brain areas is non-zero, varies considerably across region pairs, and may include both positive and negative values. Therefore,

An alternative to traditional network analysis methods is the use of the

The network organization of the human brain is characterized by a large number of distributed network modules which perform segregated local computations (Power et al.,

The present study investigates the relationship between standard network metrics computed from dyadic interactions in a functional brain network, and a novel metric computed on the

Neuroimaging data were collected at CFIN, Aarhus University Hospital, Denmark, from 16 healthy right-handed participants (11 men and 5 women, mean age: 24.7 ± 2.5). Participants with a history of psychiatric or neurological disorders were excluded from participation in the study. The study was previously approved by the Center of Functionally Integrative Neuroscience internal research board. The study was performed in accordance with the Declaration of Helsinki ethical principles for medical research and ethics approval was granted by the Research Ethics Committee of the Central Denmark Region (De Videnskabsetiske Komiter for Region Midtjylland). Informed consent was obtained from all participants.

MRI data were collected in one session on a 3T Siemens Skyra scanner. The parameters for the structural MRI T1 scan were as follows: voxel size of 1 mm^{3}; reconstructed matrix size 256 × 256; echo time (TE) of 3.8 ms and repetition time (TR) of 2300 ms. The resting-state fMRI data were collected using whole-brain echo planar images (EPI) with TR = 3030 ms, TE = 27 ms, flip angle = 90^{o}, reconstructed matrix size = 96 × 96, voxel size 2 × 2 mm with slice thickness of 2.6 mm and a bandwidth of 1795 Hz/Px. Seven minutes of resting state fMRI data were acquired for each subject.

We used the automated anatomical labeling (AAL) template (Tzourio-Mazoyer et al.,

We used FSL to extract and average the time courses from all voxels within each AAL cluster. We then used Matlab (The MathWorks Inc.) to compute the pairwise Pearson correlation between all 90 regions.

The next two sections will introduce fundamental notions needed to understand persistent homology, which is presented in the third section. Homological scaffolds are then defined and a toy example is presented in the penultimate section. The last section exposes the open problem and implications of the choice of a cycle's representative in the filtration. The workflow is illustrated in Figure

A

There are many types of simplicial complexes. In this study, we focus on _{k} ⊂ _{k}. Each _{k} that identify and can be used to construct the hole.

One of the most studied problems in mathematics is that of defining a notion of similarity between spaces. Intuitively, two spaces can be thought to be similar if we can transform one into the other via a well-behaved transformation. In particular, if there exists a continuous bijective map, a homeomorphism, that transforms one space into the other, then the two spaces are said to be homeomorphic. Such spaces are, informally, topologically the same, and any of their properties that are conserved by homeomorphism are are thus called

The homology group, or simply

The process of adding simplices to form a simplicial complex is called a filtration, and the filtration we use in this paper is the _{i} and when it disappears, it is tagged with a “death time,” δ_{i}. The difference between the two time points defines its persistence π_{i}. It is important to note that when the starting network is fully connected, all the cycles eventually die along the filtration. While it is true that the order in which edges are introduced can depend on very small differences in the weights, the same small differences would alter the persistence or appearance of generators by a similarly small value hence ultimately producing small variations in the scaffold. This is a consequence of the robustness theorems for persistent homology, where one substitutes the usual metric with an extended semi-metric (Cohen-Steiner et al.,

The homological scaffolds are secondary networks and were introduced in Petri et al. (

Two scaffolds are introduced to highlight different aspects of the importance of an edge in the network: the number of cycles an edge belongs to and the total persistence of the cycles it belongs to. The weights of the edges are defined as:

for the frequency scaffold

for the persistence scaffold

The information given by the scaffolds has to be interpreted with care, see Section 2.2.6 below for a full description of the limitations. The python library we developed for persistent homology analysis, that includes the weight rank clique filtration and the scaffolds generation is available at:

Persistent homology and the computation of the scaffolds can be illustrated by a simple toy example, which is described in the following lines and shown graphically in Figure

As illustrated by the present paper and Petri et al. (

In practice, however, this will have an impact. We used the software package javaplex (Tausz et al.,

A cycle shrinks by triadic closure,

a cycle is split into 2 smaller cycles.

These two possibilities are illustrated in Figure

Practically, this means exploring the statistics of the holes and verify how they close. It is also important to note that the aforementioned phenomena are more likely to occur in cycles that are long lived.

By construction, the graphs that we have considered for the standard graph analysis are unweighted, undirected, and do not contain self-loops. Their adjacency matrix

We now briefly introduce the standard local centrality measures that were applied to the networks: degree centrality (

The degree centrality is a measure of the total number of connections that a node has. It therefore depends on the direct neighborhood of the node. For a node

The betweenness-centrality of a node measures how many of the shortest paths between all other node pairs pass through it and is a measure of its importance when routing information in the network. By contrast to the degree,

where

The local efficiency of a node

where

In addition, a community detection algorithm based on modularity (

where _{i} is the degree of node _{Ci}

As a follow-up analysis, we explored the relationship between the

For the weighted versions of betweenness centrality and efficiency, the difference resides in the definition of the shortest path. In the BCT implementation, the shortest path is computed via a breadth-first search algorithm that follows the links with the smallest weight (Brandes,

Lastly, we define a new centrality measure for the homological scaffolds, the nodal

The

Node-level values were calculated for the

The main objective of this analysis was to examine the relationship between standard topological centrality measures described above;

D = 0.10 | 0.0001 | 0.0405 | 0.0001 | 0.0003 | 0.0001 | 0.8792 |

D = 0.15 | 0.0016 | 0.0306 | 0.0016 | 0.7669 | 0.0001 | 0.0170 |

D = 0.20 | 0.0001 | 0.0812 | 0.0001 | 0.4625 | 0.0057 | 0.0004 |

D = 0.25 | 0.0031 | 0.0001 | 0.0031 | 0.0725 | 0.6113 | 0.0001 |

D = 0.30 | 0.0001 | 0.0001 | 0.0001 | 0.1602 | 0.0165 | 0.0003 |

D = 0.35 | 0.0003 | 0.0003 | 0.0003 | 0.0113 | 0.0786 | 0.0041 |

D = 0.40 | 0.0001 | 0.0001 | 0.0001 | 0.0104 | 0.3999 | 0.0001 |

D = 0.45 | 0.0001 | 0.0001 | 0.0001 | 0.0009 | 0.7328 | 0.0001 |

D = 0.50 | 0.0001 | 0.0001 | 0.0001 | 0.0024 | 0.4340 | 0.0001 |

D = 0.55 | 0.0009 | 0.0001 | 0.0009 | 0.0051 | 0.6192 | 0.0001 |

D = 0.60 | 0.0023 | 0.0001 | 0.0023 | 0.3332 | 0.5165 | 0.0001 |

As a follow-up analysis, the relationships between the

For the network with an intermediate density of

We now explain the results shown in Figures

Persistent homology provides a window into the global organization of the edges' weights fabric of a graph. The present results indicate that persistence homological scaffolds may be useful objects to consider in functional neuroimaging research. The persistence scaffold notably circumvents the need for

In order to study the relationship between standard network metrics and on the persistence homological scaffold, we calculated the strength of each node in the persistence scaffold and termed this novel measure the persistence scaffold strength (

Of the binary graph metrics under study,

Taken together, these observations lead to an understanding of the meaning of this new centrality measure and on the interpretation of persistent homological scaffold. The tendency of high

When bypassing the thresholding step and instead comparing the

Finally, we note that the nodal

The highest-ranking regions on the

We also paid attention to the special case of high-ranking

It has now become well recognized that the brain performs local computations in segregated modules that become seamlessly integrated over space and time to support high-level functions necessary for survival. Some brain regions are likely to play a more critical role than others toward enabling the global integration of information. The exact identities of these regions and the optimal experimental approaches for identifying them remain unclear. However, recent evidence would suggest that integrative nodes, such as those potentially identified via the persistence homological scaffold, require metastability for maximal exploration of the full dynamic repertoire of the brain (Kringelbach et al.,

The application of computational topology analysis to functional neuroimaging data is a novel avenue of research, and the physiological significance of homological scaffolds and related measures remains unclear. Given that high

Whilst the present results suggest that high-ranking

Another limitation of this study, as mentioned in Section 2.2.6, is the choice of the representative cycles for homology classes, which could result in selecting edges that do not belong to the shortest cycle around a certain hole. A possible way around this limitation would be to perform an

In summary, the present study has explored the relationship between standard network metrics in functional brain network and the persistence homological scaffold derived from the same fMRI dataset. The computation of a local graph measure on the

LDL, PE, MK, FT designed the study. TV, HF, MK collected and processed the fMRI data. PE, GP, FV developed and implemented the persistence homological scaffolds methodology essential to this study. LDL, HF performed the graph theoretical analysis of the data. PE, LDL, TV made the figures. PE, GP, FV, and LDL wrote the methods section. LDL wrote the results section. LDL and PE wrote the introduction and discussion sections, with editorial guidance from MK, FT, and GD.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

LDL is supported by the Canadian Institutes of Health Research (LDL), the Canadian Centennial Scholarship Fund, and a scholarship award from Hertford College (University of Oxford). MK is supported by European Research Council (ERC) Consolidator Grant: CAREGIVING (615539). FT and PE are supported by a PET Methodology Programme grant from the Medical Research Council UK (ref no. G1100809/1). GD is supported by ERC Advanced Grant: DYSTRUCTURE (295129) and by the Spanish Research Project PSI2013-42091-P. GP and FV are supported by the TOPDRIM project supported by the Future and Emerging Technologies programme of the European Commission under Contract IST-318121.

The Supplementary Material for this article can be found online at:

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