MRI protocol

Five volunteers all right-handed, 4 males, 1 female, mean age 38, were recruited at Hospital Los Madroños in Madrid, Spain. A previous questionnaire, a general medical and neurological examination were performed for volunteer selection. Those volunteers were taken from a previous unpublished study (Tractografía Funcional de Imagen de Resonancia Magnética) whose protocol was previously approved from the Local Ethics Committee. As the current individuals have been selected from the former study, Los Madroños Board of Director waived any new requirement for the current study. The study was performed using a Siemens-Avanto 1.5 Tesla imaging system with a 12-element head matrix coil. The finger-tapping test (FTT) is a neuropsychological test that examines motor functioning, specifically, motor speed and lateralized coordination. Subjects, while they were comfortably lying down on the scanner stretcher, were asked to perform FTT with their index finger by flexing-extending the metacarpo-phalangeal joint. During the procedure, the subject’s palm was immobile and flat on the board, with fingers extended, and the index finder placed on the counting device. One hand at a time, subjects tap their index finger on the lever as quickly as possible within a 10 s time interval, in order to increase the number on the counting device with each tap.

Onset and end of a task were indicated by a brief light signal (≈ 1 s) generated by light-emitting diodes. For head fixation, a vacuum head holder was used. Subjects had ear plugs and were advised to keep their eyes closed during the whole examination. Task execution was controlled by on-line video monitoring.

fMRI Data Processing Gradient-echo EPI acquired during fMRI examinations were exported to a workstation with Brain Voyager software [10] for fMRI data processing. Prior to statistical analyses, the time series of the functional images were aligned for each slice to minimize the effects of head motion and linear trends. Data analyses for all studies included spatial and temporal Gaussian smoothing, removal of linear trend and nearest-neighbour cluster analysis. Correlation maps were computed with the stimulation protocol as reference function reflecting the temporal sequence of stimulation and control condition. On a pixel by pixel basis, the signal time course was cross-correlated with the reference function. Significant activation was reported at P < 0.001. We registered the mean increase of the perfusion in each region of interest (ROI). We used a minimal cluster size of six pixels considering that the subcortical ROI were smaller than the cortical motor areas.

Cubic spline interpolation was used for slice scan time correction. Images from the first dynamics were used as a reference, and translation and rotation of images in subsequent dynamics were plotted in all directions to illustrate motion [11]. Trilinear estimation and interpolation were used for 3D-motion correction. An 8-mm full width at half maximum Gaussian filter was used for spatial smoothing. Linear trend removal and a high-pass filter with 3 cycles/points were used for temporal filtering. The T1-weighted 3D images were also exported to BrainVoyager to create an anatomic image series, and the processed fMRI was coregistered to the anatomic images automatically. The 3D dataset with anatomic images and fMRI information was then transformed to the Talairach atlas to create a 3D-aligned time course dataset. A stimulation protocol was then created in BrainVoyager to represent the block design (with hemodynamic response function -HRF- refinement) used in the fMRI scans. General linear model analysis was performed to calculate activation maps for the 3D-aligned time course dataset for each subject [12]. Finger tapping paradigm was design as a sequence of activation-deactivation cycles starting with an activation state [13]. Blood oxygenation level-dependent single-shot T2* with TR 3000ms, TE 150ms, Slice thickness 8, and 64x64, 20 planes were acquired. Sagittal 3d T1 MPRAGE isotropic 256x256x1mm FOV 25.6 sequence was acquired as an anatomical basis for subsequent fusion with BOLD results. Each paradigm-cycle had a duration of 15 seconds and the finger tapping task lasted a total of 195 seconds. Thus the temporal signal for each voxel V(x_i, t) consisted of 60 points with a sampling rate for the BOLD signal of 0.3077 Hz.

Graph implementation

Graph theory is the mathematical study of networks. A graph is a mathematical representation of a network of elements connected to each other. A simple graph is formed by a set of nodes, or vertices, and a set of edges, or links, connecting pairs of nodes. The total number of nodes is called the order of the graph, the total number of edges is called the size of the graph. The nodes represent the elements of the system. In our case they are the voxels of a functional magnetic resonance imaging (fMRI) [2]. The links represent the connections between each pair of voxels. In our case, they mark the existence of a correlation between the time-signals of the two voxels with value above a given threshold.

This correlation has been measured by the Pearson coefficient according to the expression: [5] (1) where r(x_i, x_j) is the correlation coefficient between voxels i and j, V(x_i, t) is the BOLD signal of voxel i as a function of time t and 〈⋅〉 represents the average over the time values in the series, defined as (2) N being the total number of values in the temporal series. The standard deviation, σ, is given by: (3) The short time span of the signal implies that no subsampling (such as windowing or analogue techniques) was relevant to the analysis. Longer signals such as those obtained in Resting State studies may demand considering the time evolution of the signal [14, 15]. Our work focus on the influence of the different threshold values on the network not the temporal dynamics as shown in other approaches [16–19]. Since the coefficient r(x_i, x_j) is symmetric, we consider undirected links: in a pair of correlated voxels there is not a source and a destination.

The links between voxels i and j are represented in a matrix with elements r(x_i, x_j). This is the adjacency matrix. In our case, when the correlation is above the given threshold, r_c, such that r(x_i, x_j) > r_c, we will assign the value 1, and 0 otherwise. In this way, our effective matrices will always be considered as binary ones with all the accepted links having the same weight [20].

Once the network is, thus, created, its main topological graph measures are computed, namely the degree distribution and its Shannon Entropy S, the clustering coefficient C, the characteristic path L and the network efficiency E [21, 22].

The characteristic path of a network, L, is the average of the minimun paths lenghts between all pairs of nodes in the graph. These are the number of links to be traveled on the shortest route from one node to the other. Although relevant, the value of L can be biased by a small number of nodes very remote or even disconnected, since the logical topological distance between disconnected nodes is infinite. A more reliable measure is the network efficiency, E, computed from the inverses of the minimum path lengths between pairs of nodes. This avoids the excessive influence of disconnected nodes, as their efficiency has null value [2]. For node k, we define its characteristic length as: (4) where n is the number of nodes in the graph and d(k, i) is the length of the shortest path connecting nodes k and i. Using L_k we can define the (global) characteristic path of the graph as: (5) Equivalently we can define the Efficiency of node k as (6) Using E_k we define the (global) efficiency E of the graph as: (7)

The topological proximity between the nodes has a positive effect on their interaction in many networks. There are several graph measures that evaluate the organization of the network in logical topological neighbourhoods, known as clusters. One of these parameters is the clustering coefficient, C, that measures the density of connections between the neighbours of a node. The graph clustering coefficient is computed as the average of the clustering coefficients for each node in the graph. We define the clustering coefficient for node k as: (8) where Γ(k) is the number of edges between the neighbours of node k (thus, the number of triangles that have node k at one of its vertices) and n_k is the number of nodes that are neighbours of node k. The clustering coefficient of a graph with n nodes can be defined now as: (9)

A significative description of a graph can be obtained considering the degree distribution. As mentioned above, in the case of scale-free networks the degree distribution follows a power law P(k) ∝ k^−γ, where k represents here the degree. A simple way to illustrate this is to represent P(k) as a function of k in a double-logarithmic scale since it corresponds, then, to a linear dependence with (−γ) as linear coefficient: (10)

A measure associated to the uncertainty of a probability distribution is the Shannon entropy, S [23]. It measures the uncertainty in the degree of a node picked randomly. A distribution with maximun Shannon entropy is a uniform distribution, in which all events occur with the same probability. A distribution with minimal Shannon’s entropy is a Dirac delta, in which the uncertainty is zero because the result of the event is always the same. We compute the Shannon’s entropy of the degree distribution with the following expression: (11) where P(k) is the number of occurrences of the degree k. We may view P(k) as a probability distribution given by the number of times that a given degree appears in the graph if we divide by the total number of nodes.

Due to the huge size of the order of the resulting graphs (≈ 2 ⋅ 10⁵) and of their size (≈ 4 ⋅ 10⁹) a new C library for graph creation and computation has been built. This library supports different types of data structures (adjacency matrix, adjacency list, hash table and double linked list) for graph manipulation, selecting the optimal type of data for the measure to be computed. The library (C source code) has been designed having as main objective to obtain a fast computing time with a low memory consumption [24]. The more demanding computations were carried out on a workstation with 3.00GHz Intel Xeon E5-2687W v4 processor and 512-GB RAM.

Voxel-paradigm filtering

To validate the voxel-paradigm filtering as a tool with a possible clinical character, our procedure was compared to standard software, namely Brain Voyager and FSL [25]. Parametric maps were generated with selected voxels with a correlation with HRF compensated paradigm p < 0.012 for all the subjects with Brain Voyager software. Activation regions were located on sagittal, coronal and axial planes for healthy volunteers under a right-handed finger tapping fMRI study. Pearson correlation values between the temporal BOLD signal from the 81920 voxels obtained from the scanner divided in 20 planes of 64 × 64 cells and the HRF compensated paradigm function were performed with threshold r_c = 0.7 for all the subjects. Similar results were obtained when correlating BOLD-voxel signal with paradigm and from Brain Voyager postprocess.

Voxel correlation in frequency domain

To avoid artifacts and any influence due to time shifting when correlating temporal voxel-voxel signals, the study was repeated in the frequency domain. Using the Fourier transform (FT) with real and imaginary components, as well as the power spectrum, all relevant measures were computed for all the subjects. Namely, the degree distribution fitting parameter γ, the clustering coefficient C, the Shannon entropy S, the characteristic path L and the network efficiency E. The results obtained were similar to those in the time domain.

Graph parameters comparison

In a similar manner than Eguíluz et al., our present work include voxel-voxel correlation matrices of BOLD signals which were constructed from fMRI studies under right-handed finger tapping tasks for all subjects. As shown is Fig 2, a free-scale network was identified, where we considered a range of threshold values up to r_c = 0.90. However, the exponent of the degree distribution shows a linear dependence with threshold ranging from γ = −0.74 to γ = −1.79 as shown in averaged γ values in Fig 3(a) and 3(b) shows the dependence of the clustering coefficient with the threshold for subject 1. Similar results are obtained for the rest of the subjects. The average Efficiency E shows a minimum for an intermediate value of r_c. As to the average Shanon entropy S, it shows a linear decrease with r_c in accordance to the degree distribution showing less dispersion as the threshold increases.

When the graphs are constructed with a prior filtering by correlation with the task paradigm the linear dependence of the distribution is lost for lower values of r_c. As the value of the threshold increases the linear dependence seems to be recovered as shown in Fig 4 for the same subject 1 shown in Fig 2. For lower values of the threshold the distribution is similar to that of a random network with the average of the distribution located at the mean degree value. This results is consistent for all the subjects. The change of form in the distributions seems to appears close to the value of the threshold where E has a minimum.

All the network characterization parameters namely C, E and S behave in a similar manner as in the non-filtered case as can be oberved in Fig 5(a), 5(b) and 5(c). The log-log linear fitting of the degree distributions could only be achieve for those obtained with the highest values of the threshold with an average value of γ = −0.83 ± 0.13 for r_c = 0.85.