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Abstract:
Introduction:
Eigenvector centrality mapping (ECM) is a popular technique for analyzing fMRI data of the human brain (Lohmann et al , 2010). It is used to obtain maps of functional hubs in networks of the brain in a manner similar to Google's PageRank algorithm (Langville et al, 2006). Currently, there exist two different implementations ECM, one of which is very fast but limited to one particular type of correlation metric whose interpretation can be problematic. The second implementation supports many different metrics, but it is computationally costly and requires a very large main memory. Here we propose two new implementations of the ECM approach that resolve these issues. The first technique is based on a new correlation metric that we call 'ReLU correlation (RLC)'. The second technique is based on matrix projections. Below, we describe both methods. A more detailed description can be found in (Lohmann et al, 2018).
Methods:
ECM attributes a centrality score to each voxel based on its correlations with other voxels. In order to ensure uniqueness of the results, only nonnegative correlation metrics are allowed. The ECM score for the ith voxel is defined as the ith entry of the principal eigenvector of the correlation matrix f(X X^T) where X is the data matrix and f is a function that maps correlation values to the nonnegative range. For f(x) = x+1, a very efficient implementation of ECM is possible (Wink et al, 2012). However, this particular function f maps zero correlations to some intermediate value which makes interpretation of the resulting maps problematic.
Here we introduce two new implementations of ECM. The first is called 'ECMRLC'. It is based on a new correlation metric that we call 'ReLU correlation coefficient (RLC)'. We define the RLC between two time courses x, y of length m as RLC(x,y) = 1/(2m) Σ i xi yi +  xi yi . ECM can be efficiently implemented with this new metric by expanding the data matrix X so that Y = (X,X) where we add m columns to X containing componentwise absolute values of X. ECM is then defined as the principal eigenvector of Y Y^T. This data representation allows a very fast and memoryefficient implementation of ECM. Furthermore, it provides more informative ECM scores than the previous implementation that was based on f(x) = x+1, see (Fig. 1).
The second method  called 'ECMproject'  approximates the principal eigenvector using matrix projections. Matrix projections were introduced in (Halko et al, 2011) as a technique for computing singular value decompositions. Here we use it for estimating the principal eigenvector in a memoryefficient way. ECMproject requires more computation time than ECMRLC, but unlike ECMRLC is applicable for arbitrary correlation metrics. For more details regarding both methods, see (Lohmann et al, 2018).
Results:
Resting state fMRI data were acquired at a 9.4T Siemens Magnetom scanner of a single subject (female, 29 yrs). Acquision parameters: spatial resolution: (1.2mm)3, matrix size: 160x160, 55 slices, 330 volumes, TR=2.03 sec, TE=19ms, flipangle=70. The data were corrected for motion and the baseline drift was removed using a highpass filter with a cutoff frequency of 1/100 Hz. A ROI containing 466462 voxels was manually constructed. The corresponding correlation matrix would require about 405 GByte of main memory making it impracticably large for standard PCs. We applied ECMRLC and obtained the result shown in Fig 1.
Conclusions:
We have introduced two new implementations of ECM called ECMRLC and ECMproject. Both methods are memory efficient so that they are applicable to highresolution fMRI data acquired at field strength ≥7 Tesla. Even very large data sets containing more than 400,000 voxels can be handled easily. We also introduced a new correlation metric called 'RLC'. It applies a filter for each time point separately so that it may be better suited for capturing dynamically varying connectivity.