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Edited by: Pedro Antonio Valdes-Sosa, Clinical Hospital of Chengdu Brain Science Institute, China

Reviewed by: Feng Liu, Tianjin Medical University General Hospital, China; Yifeng Wang, University of Electronic Science and Technology of China, China

*Correspondence: Lishan Qiao

Dinggang Shen

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) and the copyright owner 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.

Functional connectivity (FC) network has been becoming an increasingly useful tool for understanding the cerebral working mechanism and mining sensitive biomarkers for neural/mental disease diagnosis. Currently, Pearson's Correlation (PC) is the simplest and most commonly used scheme in FC estimation. Despite its empirical effectiveness, PC only encodes the low-order (i.e., second-order) statistics by calculating the pairwise correlations between network nodes (brain regions), which fails to capture the high-order information involved in FC (e.g., the correlations among different edges in a network). To address this issue, we propose a novel FC estimation method based on Matrix Variate Normal Distribution (MVND), which can capture both low- and high-order correlations simultaneously with a clear mathematical interpretability. Specifically, we first generate a set of BOLD subseries by the sliding window scheme, and for each subseries we construct a temporal FC network by PC. Then, we employ the constructed FC networks as samples to estimate the final low- and high-order FC networks by maximizing the likelihood of MVND. To illustrate the effectiveness of the proposed method, we conduct experiments to identify subjects with Mild Cognitive Impairment (MCI) from Normal Controls (NCs). Experimental results show that the fusion of low- and high-order FCs can generally help to improve the final classification performance, even though the high-order FC may contain less discriminative information than its low-order counterpart. Importantly, the proposed method for simultaneous estimation of low- and high-order FCs can achieve better classification performance than the two baseline methods, i.e., the original PC method and a recent high-order FC estimation method.

Functional connectivity (FC) network, calculated by resting-state functional magnetic resonance imaging (rs-fMRI) (Liu et al.,

In view of its great potential, how to construct high-quality FC networks comes to a key issue. Theoretically, we can treat the FC network as a graph, where the nodes correspond to different brain regions or, more generally, the regions-of-interest (ROIs), while the edges correspond to the pairwise FCs between these nodes. In other words, FC network can be seen as a combination of the node set and the edge set. Currently, researchers have proposed a series of FC network modeling methods (Smith et al.,

In practice, some high-order statistics (e.g., the correlations among different edges) may also offer additional and useful information for FC analysis (Plis et al.,

Recently, some high-order methods have been developed for estimating FCs (Plis et al.,

To address these problems, in this paper we put forward a novel high-order FC network estimation method based on Matrix Variate Normal Distribution (MVND) (Gupta and Nagar,

To our best knowledge, this is the first work that adopts MVND for estimating high-order FC networks. MVND

A new finding of this paper is that the low-order FC tends to contain more discriminative information than its high-order counterpart, which is exactly the opposite conclusion of Chen et al. (

The rest of this paper is organized as follows. In section Materials and Methods, we introduce the materials and propose our method. In section Results, we evaluate the proposed method in identifying MCI subjects from NCs. In section Discussion, we discuss our findings based on the experimental results. In section Conclusion, we conclude the whole paper.

Totally, 137 participants, including 68 MCI patients and 69 NCs from Alzheimer's Disease Neuroimaging Initiative (ADNI)^{1}

The acquired rs-fMRI data was processed by SPM8^{2}^{137 × 116}, which will be used for FC network estimation.

According to a recent review (Smith et al.,

where _{i}. Without loss of generality, in this paper we suppose that _{i} is centralized by

where

Given the fact that the BOLD time series signals commonly contain noises, the original PC-based FC network tends to be dense (Fornito et al.,

In this section, we introduce the new FC estimation scheme based on MVND that can encode both low- and the high-order correlations in a single framework. As a result, we can model FC from two different views.

In particular, we suppose the low-order FC between the _{ij} that follows the normal distribution, and thus the corresponding FC network is a random matrix _{wij)P×P} that has the multivariate normal distribution. That is,

where ^{P×P} is the population mean or mathematical expectation of ^{P2} × ^{2} is the population covariance matrix of

An intuitive explanation for the low- and high-order FC. Note that the low-order FC measures the traditional correlation between nodes, while the high-order FC measures the correlation between edges (i.e., the correlation's correlation).

Although the ^{2} × ^{2} matrix is challenging, since it contains a consistent amount of parameters. For example, ^{2} × (^{2} − 1)/2 ≈ 9 × 10^{7} free parameters. In Chen et al. (

Therefore, in this paper we propose a new strategy to eliminate the difficulty of estimating _{1} ⊗ Ω_{2}

More specifically, the probability density function of MVND is defined (Gupta and Nagar,

where

In this study, we mainly focus on the _{1} = _{2}. Without loss of generality, we let _{1} = _{2} for simplifying the mathematical expression. Since the ^{2} × (^{2} − 1)/2 to

In order to estimate ^{(k)} is constructed as follows:

The two-step framework for estimating low- and high-order FC networks.

As a result, we can get

Based on the

and the MLE of

Note that, however, the estimation of

Algorithm of MVND-based low- and high-order FC estimation.

Input: |

Output: M and Ω //low- and high-order FC |

Apply sliding windows to obtain more samples ^{(k)} and PC to construct temporal |

low-order FC W^{(k)} = (X^{(k)})^{(k)}; |

Initialize Ω = |

while not converge |

end |

In order to verify the performance of the estimated FCs, we utilize the low-order FC network

Due to the limited subjects, in this paper, we use the nested leave-one-out cross validation (LOOCV) to estimate the classification performance, in which only one participant is left out for testing while the others are adopted for training a classifier and obtaining the optimal parameters. In terms of the thresholding parameter of FC networks, we empirically employ 11 sparsity levels ranging in [1%, 10%, ⋯, 90%, 100%] for all the methods. For instance, 10% means that 90% of the weak edges are filtered out from the FC networks, while 100% means all the edges are reserved. We determine the optimal thresholding parametric value using an inner LOOCV procedure on the training dataset.

In our experiments, we adopt accuracy, sensitivity and specificity (Sokolova et al.,

where

In Table

Comparison on MCI classification performance with different methods.

PC | 0.7956 | 0.7647 | 0.8261 |

HON (Chen et al., |
0.8207 | 0.8194 | 0.8377 |

LoM | 0.9051 | 0.9118 | 0.8986 |

HiO | 0.8394 | 0.8235 | 0.8551 |

FuMO | 0.8905 | 0.8676 | 0.9130 |

In general, the free parameters involved in the FC network estimation methods have a big influence on the ultimate classification performance. In the proposed framework, there are two free parameters, including the width of sliding windows (

The

In Figure

In this paper, we propose a new FC network estimation framework based on MVND that can simultaneously capture low- and high-order correlation information in data. The proposed method is validated on ADNI dataset by an MCI identification task. According to the experimental results, we have the following discussions:

In general, the performance of the proposed method outperforms the baseline methods, including the original PC method and the recently proposed high-order method (HON) in Chen et al. (

Interestingly, we find that the proposed low-order FC is generally more discriminative than its high-order counterpart, which is contrary to the conclusion in Chen et al. (

From the experimental results, we also find that the estimated FC networks (including the low-order, high-order, and their combination) consistently outperform HON in Chen et al. (

In this paper, we develop a novel PC-based FC estimation framework with the assumption that the network edge weights follow the MVND. The proposed method is simple, has a rigorous mathematical model, and can capture both low- and high-order FCs of the brain network simultaneously. The experiments on MCI identification show that our proposed method outperforms the original PC method and a recent high-order FC estimation method. On the other hand, although we design our method based on PC, the idea can be used in any correlation-based FC estimation methods. In the future, we plan to generalize the MVND-based scheme to the partial correlation-based FC estimation problem.

DS: Proposed the idea of high-order FC and provided the preprocessed rs-fMRI data; LQ: Proposed the mathematical models for simultaneously estimating low- and high-order FC networks; YZ, LZ, and WL: Designed the procedures of MCI evaluation experiments; All authors developed the estimation algorithm and contributed to the preparation of the article, figures, and charts.

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.

This work is partly supported by National Natural Science Foundation of China (61300154, 61402215), Chongqing Graduate Student Research Innovation Project (CYS16183), and NIH grants (EB022880, AG041721, AG049371, and AG042599).

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