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Voxel hierarchy on dynamic brain graphs is produced by k core percolation on functional dynamic amplitude correlation of resting-state fMRI. Directed graphs and their afferent/efferent capacities are produced by Markov modeling of the universal cover of undirected graphs simultaneously with the calculation of volume entropy. Positive and unsigned negative brain graphs were analyzed separately on sliding-window representation to underpin the visualization and quantitation of mental dynamic states with their transitions. Voxel hierarchy animation maps of positive graphs revealed abrupt changes in coreness k and kmaxcore, which we called mental state transitions. Afferent voxel capacities of the positive graphs also revealed transient modules composed of dominating voxels/independent components and their exchanges representing mental state transitions. Animation and quantification plots of voxel hierarchy and afferent capacity corroborated each other in underpinning mental state transitions and afferent module exchange on the positive directed functional connectivity graphs. We propose the use of spatiotemporal trajectories of voxels on positive dynamic graphs to construct hierarchical structures by k core percolation and quantified in- and out-flows of information of voxels by volume entropy/directed graphs to subserve diverse resting mental state transitions on resting-state fMRI graphs in normal human individuals.

Linear matrix factorizations (LMFs) such as independent component analysis (ICA), principal component analysis (PCA), and their extensions, have been widely used for finding relevant spatial maps in brain imaging data. The last step of an LMF before interpretation is usually to extract the activated brain regions from the map by thresholding. However, it is difficult to determine an appropriate threshold level. Thresholding can remove the underlying properties of spatial maps and their features imposed by the model. In this study, we propose a threshold-free activated region extraction method which involves simplifying a brain spatial map to a dendrogram through Morse filtration. Since a dendrogram is related to the change of clustering structure in Rips filtration, we first show the relationship between the Rips filtration of a graph and the Morse filtration of a function. Then, we dendrogramize a spatial map in order to visualize the activated brain regions and the range of their importance in a spatial map. The proposed method can be applied to any spatial maps that a user wants to threshold and interpret. In experiments, we applied the proposed method to independent component maps (ICMs) obtained from resting-state fMRI data, and the dominant subnetworks obtained by the PCA of a correlation-based functional connectivity of FDG PET Alzheimer's disease neuroimaging initiative (ADNI) data. We found that dendrogramization can help to understand a brain spatial map without thresholding.

Persistent homology has been applied to brain network analysis for finding the shape of brain networks across multiple thresholds. In the persistent homology, the shape of networks is often quantified by the sequence of $k$-dimensional holes and Betti numbers.The Betti numbers are more widely used than holes themselves in topological brain network analysis. However, the holes show the local connectivity of networks, and they can be very informative features in analysis. In this study, we propose a new method of measuring network differences based on the dissimilarity measure of harmonic holes (HHs). The HHs, which represent the substructure of brain networks, are extracted by the Hodge Laplacian of brain networks. We also find the most contributed HHs to the network difference based on the HH dissimilarity. We applied our proposed method to clustering the networks of 4 groups, normal control (NC), stable and progressive mild cognitive impairment (sMCI and pMCI), and Alzheimer's disease (AD). The results showed that the clustering performance of the proposed method was better than that of network distances based on only the global change of topology., Comment: The paper is accepted for publication at the 7th Workshop on Computational Topology in Image Context (CTIC) (http://www.ctic2019.uma.es)

Entropy is a classical measure to quantify the amount of information or complexity of a system. Various entropy-based measures such as functional and spectral entropies have been proposed in brain network analysis. However, they are less widely used than traditional graph theoretic measures such as global and local efficiencies because either they are not well-defined on a graph or difficult to interpret its biological meaning. In this paper, we propose a new entropy-based graph invariant, called volume entropy. It measures the exponential growth rate of the number of paths in a graph, which is a relevant measure if information flows through the graph forever. We model the information propagation on a graph by the generalized Markov system associated to the weighted edge-transition matrix. We estimate the volume entropy using the stationary equation of the generalized Markov system. A prominent advantage of using the stationary equation is that it assigns certain distribution of weights on the edges of the brain graph, which we call the stationary distribution. The stationary distribution shows the information capacity of edges and the direction of information flow on a brain graph. The simulation results show that the volume entropy distinguishes the underlying graph topology and geometry better than the existing graph measures. In brain imaging data application, the volume entropy of brain graphs was significantly related to healthy normal aging from 20s to 60s. In addition, the stationary distribution of information propagation gives a new insight into the information flow of functional brain graph.

Finding the underlying relationships among multiple imaging modalities in a coherent fashion is one of challenging problems in the multimodal analysis. In this study, we propose a novel multimodal network approach based on multidi- mensional persistent homology. In this extension of the previous threshold-free method of persistent homology, we visualize and discriminate the topological change of integrated brain networks by varying not only threshold but also mixing ratios between two different imaging modalities. Moreover, we also pro- pose an integration method for multimodal networks, called one-dimensional projection, with a specific mixing ratio between modalities. We applied the proposed methods to PET and MRI data from 21 autism spectrum disorder (ASD) children and 10 pediatric control subjects. From the results, we found that the brain networks of ASD children and controls differ significantly, with ASD showing asymmetrical changes of connected structures between PET and MRI. The integrated MRI and PET networks showed that ASD children had weaker connections than controls within the visual cortex, between dorsal and ventral parts of the temporal pole, between frontal and parietal regions, and between the left perisylvian and other brain regions. These results provide a multidimensional homological understanding of disease-related PET and MRI networks that discloses the network association with ASD.