Complex network analysis has become a gold standard to investigate functional connectivity in the human brain. Popular approaches for quantifying functional coupling between fMRI time series are linear zerolag correlation methods; however, they might reveal only partial aspects of the functional links between brain areas. In this work, we propose a novel approach for assessing functional coupling between fMRI time series and constructing functional brain networks. A phase space framework is used to map couples of signals exploiting their cross recurrence plots (CRPs) to compare the trajectories of the interacting systems. A synchronization metric is extracted from the CRP to assess the coupling behavior of the time series. Since the functional communities of a healthy population are expected to be highly consistent for the same task, we defined functional networks of taskrelated fMRI data of a cohort of healthy subjects and applied a modularity algorithm in order to determine the community structures of the networks. The withingroup similarity of communities is evaluated to verify whether such new metric is robust enough against noise. The synchronization metric is also compared with Pearson’s correlation coefficient and the detected communities seem to better reflect the functional brain organization during the specific task.
The human brain, as many biological systems, can be seen as a complex network of interacting components whose integration leads to a hierarchical architecture of highly specialized modules [
In particular, over the past few years, there has been an increasing interest in inferring connectivity properties from fMRI data. Functional connectivity analysis aims at assessing the strength of functional coupling between the signal responses in distinct brain areas [
A number of important questions regarding the identification of networks have to be addressed before considering any analysis technique. Recent studies have demonstrated that different edge definitions could affect the topological properties of brain networks obtaining variable findings [
In this work, we propose a novel approach for quantifying functional coupling between fMRI time series and constructing functional brain networks. We use a phase space framework to map pairs of signals in their reconstructed phase space, that is, a topological representation of their behavior under all possible initial conditions [
The proposed metric and Pearson’s correlation coefficient are applied to the fMRI data of a cohort of healthy subjects acquired during performing a working memory task to construct weighted networks.
At macroscopic level, functional related brain regions exhibit similar BOLD responses. These groups of regions form dense communities that reflect the functional organization of the brain and whose properties can be linked to the topological features of the taskevoked network configuration [
We studied
Demographic data of the healthy cohort (mean ± standard deviation).
Demographic Data  

Age (years) 

Gender (M/F) 

Handedness 

Socioeconomic status 

IQ 

Participants performed the NBack working memory task, in which a sequence of stimuli is presented and the subject has to remember the stimulus from “N” steps earlier. The stimuli consisted of numbers (1–4) presented in random sequence and displayed at the points of a diamondshaped box. The control condition (0back) simply required the subjects to identify the current stimulus. In the working memory condition, the task required the collection of a stimulus seen two stimuli earlier (2back). The task was organized in a block design, consisting of eight alternating 0back and 2back conditions, each lasting 30 seconds. Each 30 sec. block includes 14
Echo planar imaging blood oxygenation level dependent fMRI data were acquired on a GE Signa 3T scanner (GE Healthcare) equipped with a standard quadrature head coil. A gradientecho planar imaging sequence (repetition time,
Images were preprocessed using Statistical Parametric Mapping 8 software (SPM8;
The brain volume of each subject was divided into 246 nonoverlapping anatomical regions of interest (ROIs) according to the Brainnetome Atlas [
by calculating their Pearson’s correlation coefficient;
by computing their CRP and then by calculating their synchronization (SYNC) index as described in the following subsection.
Finally, for each subject, we identified two undirected weighted networks, whose edges resulted from
the signed pairwise Pearson’s correlation coefficients;
the SYNC indexes.
A state of a system is defined by the values of the variables that describe it at a given time. When such system evolves in time, the sequence of all its states forms a trajectory in the phase space, that is, a multidimensional space whose dimension depends on the number of the variables of the system. Starting from different initial conditions, a real physical dissipative system tends to evolve in similar ways, such that its trajectories converge in a region of the phase space called attractor which represents the steadystate behavior of the system [
In experimental contexts, where the time series
Both parameters have to be properly selected to avoid redundancy in the phase space. The dimension
The trajectories of two distinct systems with the same embedded parameters can be compared in a CRP [
The value of the parameter
A CRP exhibits characteristic patterns that show local time relationships of the segments of the trajectories of the two interacting systems. Typical structures include single dots, diagonal lines, and vertical and horizontal lines. Diagonal lines occur when the evolution of the states is similar at different times and their lengths are related to the periods during which the two systems move in similar ways remaining close to each other [
For a visual reference, see Figure
Pairs of fMRI time series and their CRPs for (a) occipital inferior L and frontal medial orbital L (SYNC = 0.05); (b) occipital superior L and occipital superior R (SYNC = 1).
Several community detection methods have been proposed to find an optimum partition of the nodes into nonoverlapped communities, that is, clusters of nodes that are more densely connected to each other than to other nodes in the network [
A statistical framework was adopted in order to compare the partitions of all the subjects for each functional network [
The normalized mutual information (NMI) [
The statistical relevance of the withingroup community structure similarity was evaluated through a permutation test. First, a randomly rewired version of each functional network was generated preserving weights, density, and degree sequence, resulting in two groups of networks: the actual and its randomized matching network. Then, the NMI was calculated between all the possible pairs of network partitions within each group. A null distribution was generated by randomizing group labels
In order to inspect the consistency of node assignments to specific functional communities, we carried out further analyses on the networks. Since the labels of modules are arbitrarily assigned by the community detection algorithm at each iteration, it is necessary to match the partition values across the subjects for visualizing the group level community structure. This problem can be overcome by finding a template partition as a reference and by reassigning the labels of communities to match the template, while preserving the distinctions between different modules in each partition [
We randomly selected a subset of
Permutation tests reveal significant differences of modularity structures between all the functional networks and their randomly rewired versions (
Mean and median (interquartile range) quantities of NMI and
Distributions  NMI 


Synchronization 


Pearson 


Violin plots of (a) withingroup NMI distributions and (b)
In addition, we evaluated the modularity index
Since we computed connectivity measures on a time series derived from a working memory task, we expected to find modules related to working memory performance involving the frontoparietal network [
The five group level functional communities detected in SYNC networks. In each row, a single community is shown in four brain views (left side, right side, top side, and bottom side).
The two group level functional communities detected in Pearson’s networks. In each row, a single community is shown in four brain views (left side, right side, top side, and bottom side).
The consistency of the assignment of brain regions to functional modules for the SYNC networks is shown in Figure
Consistency of the assignment of brain regions to modules measured as the frequency of occurrence of the node with a specific label (in percent) for SYNC networks.
Consistency of the assignment of brain regions to modules measured as the frequency of occurrence of the node with a specific label (in percent) for Pearson’s networks.
In the current study, a modularity analysis is applied to networks defined with both the proposed SYNC index and Pearson’s correlation coefficient in order to investigate the taskrelated functional organization of the brain. Modularity is implicitly related to significant selfregulating mechanisms of the human brain: efficient dense withinmodule processing and sparse fast integration among subsystems reduce noise propagation and latency [
In our framework, the same community detection algorithm was applied to both kinds of networks. Since the algorithm generates a node partition of a connectivity matrix, some properties of the index used to identify the network such as sensitivity to noise and to complex interaction mechanisms occurring among the brain regions could affect the degree of partition of the network into communities. Several brain connectivity metrics have been proposed as alternatives to Pearson’s correlation coefficient. Coherence and partial coherence analysis were applied to fMRI data to extend linear metrics of zerolag correlation. These spectral measures estimate the linear timeinvariant relationship between time series by using phase and magnitude information for all the time lags [
These results are promising with respect to the value of the novel technique we are proposing, even though they are not free of limitations. For example, since we used taskdependent time series, we do not know yet whether these results extend to resting state data, and this will be the object of future studies. We chose to examine taskdriven functional connectivity as done in several other studies [
Another relevant issue concerns the modularity properties used to perform the comparison between the SYNC metric and Pearson’s correlation index. Indeed, in our analysis we found both higher modularity and higher consistency of taskrelated communities in the SYNC matrices. These features are related to a greater homogeneity of the functional organization across the subjects in response to the same task and although they are compatible with behaviors expected in a healthy cohort, a more rigorous assessment of the sensitivity of the proposed synchronization metric should require further analysis. Future studies could employ alternative topological properties of SYNC networks and their correlation with task performance or behavioral data to uncover additional insights into the suitability of the SYNC index as a functional connectivity metric for fMRI time series.
Finally, our study has focused on an alternative method to define functional connectivity between pairs of BOLD time series. Generally, functional connectivity refers to a larger spectrum of neuroimaging techniques including EEG, MEG, and NIRS. As discussed above, recurrence plots have been used to explore dynamical properties of EEG and MEG, providing interesting features on complex phenomena in human brain. Although the SYNC metric is extracted from cross recurrence plots, a separate and accurate analysis may be needed to assert the validity of the index in a broader context and extend its use to more functional imaging techniques.
In this work, a new synchronizationbased metric is proposed to assess functional connectivity in human brain. The metric is a generalized synchronization measure that takes into account both the amplitude and phase coupling between pairs of fMRI series. This method differs from the correlation measures used in the literature, as it is more sensitive to nonlinear coupling phenomena between time series and it is more robust against the physiological noise. In order to probe these latter two aspects, we performed a modularity analysis of taskrelated fMRI networks of a cohort of healthy subjects built with the new proposed metric. The aim was to verify whether the new metric was able to return networks whose functional modules were coherent with the actual organization of the brain regions during the taskbased activity. We considered unthresholded complete connectivity matrices to test the effectiveness of the synchronization against noise and spurious correlations. Indeed unthresholded networks have lower signaltonoise ratio as the most important links do not stand out among all the weights. By comparing the networks constructed by means of the proposed metric with those obtained through Pearson’s coefficient, it seems that the synchronization metric better reflects the taskrelated network structure for number of detected communities, for the functional organization of the ROIs, and for greater consistency of communities across the subjects.
The authors declare that there are no conflicts of interest regarding the publication of this paper.