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Conceived and designed the experiments: DSM AH BP JL HS MS AV RT. Performed the experiments: DSM AH BP JL HS MS AV RT. Analyzed the data: GL DG. Contributed reagents/materials/analysis tools: GL DG. Wrote the paper: GL.

The authors have declared that no competing interests exist.

Functional magnetic resonance data acquired in a task-absent condition (“resting state”) require new data analysis techniques that do not depend on an activation model. In this work, we introduce an alternative assumption- and parameter-free method based on a particular form of node centrality called eigenvector centrality. Eigenvector centrality attributes a value to each voxel in the brain such that a voxel receives a large value if it is strongly correlated with many other nodes that are themselves central within the network. Google's PageRank algorithm is a variant of eigenvector centrality. Thus far, other centrality measures - in particular “betweenness centrality” - have been applied to fMRI data using a pre-selected set of nodes consisting of several hundred elements. Eigenvector centrality is computationally much more efficient than betweenness centrality and does not require thresholding of similarity values so that it can be applied to thousands of voxels in a region of interest covering the entire cerebrum which would have been infeasible using betweenness centrality. Eigenvector centrality can be used on a variety of different similarity metrics. Here, we present applications based on linear correlations and on spectral coherences between fMRI times series. This latter approach allows us to draw conclusions of connectivity patterns in different spectral bands. We apply this method to fMRI data in task-absent conditions where subjects were in states of hunger or satiety. We show that eigenvector centrality is modulated by the state that the subjects were in. Our analyses demonstrate that eigenvector centrality is a computationally efficient tool for capturing intrinsic neural architecture on a voxel-wise level.

Functional magnetic resonance data (fMRI) of the human brain acquired in a task-absent (“resting state”) condition has attracted increasing interest in recent years. Due to the absence of an experimental paradigm, analysis procedures based on an activation model are not applicable. New types of techniques have been developed focusing on functional connectivity rather than task activation. For instance, correlation of time series between a pre-specified seed region and all other voxels of the brain is robust and conceptually clear. However, it can only be successfully applied if some prior knowledge exists for identifying seed regions. Another widely used technique is based on independent component analysis (ICA) whose primary advantage is its freedom from hypotheses preceding the analysis and the need for selecting seed regions

More recently however, graph-based methods have been proposed for the analysis of functional and structural magnetic resonance data of the human brain. Their main feature is that they take brain regions as nodes in a graph. Some of these methods have also been applied to the analysis of resting state fMRI data

In the present study, we focus on a particular type of graph-based method that identifies nodes which play central roles within the network structure. Such nodes are characterized by a measure called “node centrality”. Node centrality is a key concept in social network analysis of which several competing definitions exist and some of which have been applied to fMRI data analysis in the past

Thus far, centrality measures have been applied to a pre-selected set of nodes consisting of at most several hundred elements (e.g.

However, due to computational complexity, closeness and betweenness centrality measures are not suited for compiling brain maps with thousands of voxels. Therefore in this study, we will focus primarily on ‘eigenvector centrality’

Eigenvector centrality can be used on a variety of different similarity metrics. Here, we present applications based on linear correlations and on spectral coherences between times series. This latter approach allows us to draw conclusions about connectivity patterns in different spectral bands. The motivation for choosing spectral measures came from Salvador et al.

We propose to use eigenvector centrality as a mapping tool for the entire brain or parts of it. Such maps can be subjected to statistical tests to detect groupwise differences in centrality between experimental states. For abbreviation, we will call this method ECM (Eigenvector Centrality Mapping).

Several definitions of node centrality exist - each having a slightly different interpretation. Common to all of these definitions is that they are based on a symmetric matrix containing pairwise similarity measures. Let

The matrix

The simplest centrality measure is called “degree centrality”. The degree

Eigenvector centrality was first introduced by Bonacich

As before let

Uniqueness of this definition is ensured by the Perron-Frobenius theorem which states that any square matrix with strictly positive entries has a unique largest real eigenvalue with strictly positive components. This is also true for irreducible square matrices with non-negative entries. An irreducible matrix has at least one non-zero off-diagonal element in each row and column.

Since we assume that

Eigenvector centrality is related to principal components analysis (PCA) in that both methods are based on eigenvector decompositions of similarity matrices. However, PCA differs from eigenvector centrality in that it only allows linear correlations as a similarity metric. But linear correlations may be negative so that the first principal component is not uniquely defined because of possible multiplicities of eigenvalues.

In our experiments we used linear correlations which were re-scaled to be non-negative and also a spectral coherence metric which is non-negative by definition (see below). Other similarity metrics such as mutual information or wavelet transform coherence (WTC)

Many algorithms for computing eigenvectors of symmetric matrices are known. In the present context, it suffices to find the eigenvector belonging to the largest eigenvalue. For this special case, the power iteration method

The betweenness centrality

Betweennnesss centrality is computationally expensive. For weighted graphs, its complexity is

Linear correlation has been proposed as a metric for analysing functional connectivity

Let

Because the similarity matrix

Salvador et al.

Let

Since

Note that information about phase lags is not included in the above measure. Frequency-dependent phase coherence can be computed using the above definitions as follows

Functional MRI/EPI data were acquired of 35 normal volunteers on a 3T MRI scanner (Siemens Tim Trio) using TR = 2.3 sec, TE = 30ms, 3×3

All data sets were initially fieldmap corrected using the software system Lipsia

The mask covering the entire brain including the cerebellum containing about

Functional MRI/EPI data were acquired of 22 normal volunteers on a 3T MRI scanner (Siemens Tim Trio) using TR = 2.3 sec, TE = 30ms, 3×3

Data processing was done using the software system Lipsia

The mask used in experiment 2 containing about

For both experiments, we computed pairwise similarity matrices between time series of any two voxels inside the mask using scaled linear correlation and for experiment 2 also spectral coherence, and applied the ECM algorithm to these matrices. The resulting centrality maps were then transformed as described by van Albada et al.

The group average of the first scan is shown in the top row. The bottom row shows results of the second scan. The similarity metric was scaled linear correlation. MNI coordinates of slice positions are (−4,−71,−20).

Results are thresholded at

For comparison, we additionally computed another centrality map - this time using degree centrality instead of eigenvector centrality (

The similarity metric was scaled linear correlation. Note that degree centrality is larger almost everywhere in the brain during the second scan. MNI coordinates of slice positions are (0,−17,18).

The top row shows group averages of eigenvector centrality maps of subjects in the sated state. Below, the group average across the hungry state is shown. The similarity metric used here was scaled linear correlation. Talairach coordinates of slice positions are (4,−49,58).

Results are thresholded at

We next employed spectral coherence to investigate frequency based similarity metrics because of their known advantages in the observation of interregional dependencies

The maps show a t-test contrasting spectral ECMs of 1/10, 1/15, 1/20 Hz versus 1/25, 1/20, 1/35 Hz. Blue colors indicate regions were higher frequencies showed stronger centrality. Red colors indicate regions where very low frequencies dominate. The maps are thresholded at

The results are shown for three frequencies (0.1 Hz, 0.05 Hz, 0.033 Hz) thresholded at p

frequency | coordinates | ||

1/10 Hz | sup. front. sulc. | 3618 | (26, −7, 48) |

intrapariet. sulc. | 3186 | (35, −61, 18) | |

thalamus | 3429 | (8, −7, 21) | |

1/20 Hz | precuneus | 7857 | (8, −46, 48) |

sup. temp. sulc. | 324 | (56, −19, −12) | |

1/30 Hz | ventral striatum | 4725 | (−10, 26, 3) |

precuneus | 5589 | (−4, −40, 51) |

We propose eigenvector centrality as a new method for analyzing fMRI data. It is parameter-free, computationally fast and does not depend on prior assumptions. In contrast to previous studies using centrality measures

In the first experiment, we found significant differences between ECMs of two resting state scans following each other within the same session. In particular, left and right thalamus had higher eigenvector centrality scores during the first scan. Thalamus has been implicated in mediating attention and arousal in humans

On the other hand, posterior cingulate and anterior medial frontal cortex appeared stronger in the second scan - regions that are associated with the “default mode network”

For comparison, we also computed degree centrality and found that during a second resting state scan, degree centrality increased almost everywhere indicating a general increase in correlations across the brain. This may be due to a global physiological influence such as respiration or heart rate. Eigenvector centrality on the other hand did not show such a global effect. Rather it highlighted specific regions that were differentially affected by the prolonged duration of the experiment.

For the second experiment, we additionally used frequency instead of time based similarity metrics with the known advantages in the detection of interregional dependencies

The spectral coherence measure assumes that the coupling between fMRI time series is stationary over time. This assumption may sometimes be unrealistic. In such cases, the wavelet transform coherence (WTC)

For the present work, we have only used spectral coherence but not phase coherence. However, it might be advantageous to include phase coherence and use it in conjunction with spectral coherence. We plan to explore that possibility in future work.

In both experiments, we found high centrality values in cortical and subcortical areas, but also in white matter regions. This agrees with results found by Mezer et al.

It should be noted that low frequency fluctuations may also be caused by aliasing effects (undersampling) so that the actual sources of these signals need not be in that same low frequency range. Nonetheless, recent studies have confirmed that oscillations - even at very low frequencies - appear robust and reliable

The initial analyses presented in this study demonstrate that eigenvector centrality is a computationally efficient tool for capturing intrinsic neural architecture on a voxel-wise level. The independence of centrality approaches from a priori hypotheses, makes it a valuable methodological addition to the “model-free” analytic toolbox.

A symmetric matrix with non-unique eigenvalues.

(0.02 MB PDF)

Axial slices of ECM group averages in experiment 2.

(0.34 MB PDF)

We thank Karsten Müller, Franziska Busse and Stefan Kabisch for their support in acquiring the data for this work.