Construction of embedded fMRI resting-state functional connectivity networks using manifold learning

Over the past years, functional magnetic resonance imaging (fMRI) has been widely used for the identification of brain regions that are related to both functional segregation and integration. Regarding functional segregation, the conventional analysis relies on the identification of the activated voxels based on functional response models and multivariate statistics between experimental conditions (e.g. resting-state vs. task-stimulated activity). A representative example is the General Linear Model (GLM) that is implemented in well established software packages such as SPM (Friston et al. 1994) and FSL (Smith et al. 2004). On the other hand, for the assessment of functional integration, there is a distinction between functional and effective connectivity (Friston 2011). Functional connectivity (FC) analysis looks for statistical dependencies (e.g. correlations, coherence) between brain regions. Effective connectivity (EC) analysis tries to reveal the influence that one neural system exerts on another. A detailed review on the differences between FC and EC approaches can be found in Friston (2011).

Here, we focus on the construction of functional connectivity networks (FCN) based on resting-state fMRI (rsfMRI) recordings. In rsfMRI, there is no stimuli and thus the assessment of functional integration is more complex and not so straightforward compared to task-related experiments (Khosla et al. 2019). Furthermore, spontaneous/ resting-state brain activity as measured with fMRI has been considered as a potential biomarker in psychiatric disorders (see e.g. the review of Zhou et al. 2010). In general, two basic frameworks are explored for the construction of FCN: (a) seed-based analysis (SBA) and (b) independent component-based analysis (ICA). In the SBA (Cole et al. 2010), the (averaged) fMRI signals of the regions of interest (ROIs) are correlated with each other; correlations above a threshold are considered functional connections between seeds/ROIs. Even though the SBA has been proved extremely useful in identifying functional networks of specific brain regions (Greicius et al. 2003; Fox et al. 2005; Margulies et al. 2007), its major disadvantage is the requirement of the a-priori knowledge of the functional organization of the brain, while possible correlations between seeds can be due to structured spatial confounds (e.g. scanner artifacts) (Cole et al. 2010). Furthermore, the definition of a seed is based on standard coordinates, while at the subject level, anatomical differences may lead to the consideration of functionally irrelevant voxels at the group level. Despite the use of normalization techniques, the accuracy of this approach is limited especially for brain regions, such as the hippocampus, where neurogenesis continues even in the adult life (Saxe et al. 2006). On the other hand, ICA (Hyvärinen and Oja 2000) has arisen as an alternative approach since the early 2000s (Beckmann et al. 2005; Beckmann and Smith 2005; Kim et al. 2010). ICA decomposes the 4D fMRI data to a set of spatial components with maximum statistical independence and their associated time series. Smith et al. (2009) in a meta-analytic study of 30,000 rsfMRI scans with the aid of ICA revealed a functional “partition” of the brain into resting-state networks (RSNs), such as the sensorimotor, default mode and auditory networks. Applications of ICA include also data pre-processing, where noise-related components are regressed out from the original fMRI signals (Pruim et al. 2015). However, while ICA produces spatial components that are statistically independent to each other, there is no clear link between the spatial components and specific brain functions and spatial components cannot in general be ordered by relative importance (Cole et al. 2010). Another issue is that most of the standard algorithms that compute independent components (ICs) utilize gradient based optimization algorithms that use an iterative scheme; the initial guesses in these algorithms are generated randomly making the whole process stochastic. As a consequence, the obtained spatial components may differ significantly for the same dataset over repeated runs (Himberg et al. 2004). Hence, the reproducibility of the ICA results over repeated runs may be questioned.

In order to tackle the above issues, several techniques have been proposed for the classification of ICs and the construction of subject-specific ROIs (Pamplona et al. 2020; Yang et al. 2008). Advances have also been made regarding the selection of the model order of the ICA decomposition, such as the Bayesian dimensionality estimation technique (Beckmann et al. 2005) and the use of theoretic information criteria for model order selection (Li et al. 2007). Finally, the so-called ranking and averaging ICA by reproducibility (RAICAR) (Yang et al. 2008; Himberg et al. 2004) (see also Cole et al. (2010) for a critical discussion) aims at resolving issues regarding stochasticity and robustness of the ICA decomposition. RAICAR utilizes a sufficient number of ICA realizations and based on the reproducibility of the ICs aims to rank them in terms of the most “reliable” components. Reliable ICs among realizations are assessed via correlations and the final estimate of each component is averaged.

Alternatively and/or complementary to the above analysis, linear manifold learning algorithms such as Principal Component Analysis (PCA) (Jollife 2002; Worsley et al. 2005; Baumgartner et al. 2000) and classical Multidimensional Scaling (MDS) (Kruskal 1964; Friston et al. 1996) have been also exploited. PCA has been succesfully applied in the pre-processing routine for dimensionality reduction (often prior to ICA) (Iraji et al. 2016). Applications of PCA include also the recovery of signals of interest (Viviani et al. 2005) and the construction of FCN from fMRI scans in task-related experiments (Worsley et al. 2005; Baumgartner et al. 2000). In these studies, the performance of PCA with respect to the detection of regions of correlated voxels has been shown to be satisfactory but not without problems. For example, a study by Baumgartner et al. (2000) highlighted the limits of PCA to correctly identify activation of brain regions in cases of low contrast-to-noise ratios (CNR) appearing when signal sources of e.g. physiological noise are present.

MDS has been also widely used in fMRI (mostly for task-based studies) mainly for the identification of similarities between brain regions in terms of voxel-wise connectivity (Shinkareva et al. 2012, 2013; Tzagarakis et al. 2009; O’Toole et al. 2007; Haxby et al. 2001; de Beeck et al. 2010). The implementation of MDS in neuroimaging dates back to the work of Friston et al. (1996), where embedded (voxel-wise) connectivity from PET data was investigated during word generation tasks between healthy and schizophrenia subjects. Salvador et al. (2005) used MDS to investigate the embedded connectivity of anatomical regions of the brain from rsfMRI data. Benjaminsson et al. (2010) used MDS to embed high-dimensional rsfMRI data from the mutual information space to a low dimensional Euclidean space for the identification of RSNs. Hervé et al. (2012) used MDS to acquire a low dimensional approximation of interregional correlations for the investigation of the affective speech comprehension. Finally, in a meta-analytic study by Etkin and Wager (2007), MDS was exploited to provide a low-dimensional visualization of co-activation interrelations of Regions of Interest (ROIs). MDS has been also used in studies investigating the functional (dys)connectivity associated with schizophrenia (Welchew et al. 2002) and Asperger’s Syndrome (Welchew et al. 2005).

However, thus far, only a few studies have exploited non-linear manifold learning algorithms such as Locally Linear Embedding (LLE) (Roweis and Saul 2000), Isometric Feature Mapping (ISOMAP) (Tenenbaum et al. 2000), diffusion maps (Coifman and Lafon 2006) and kernel PCA (kPCA) (Schölkopf et al. 1997) for the analysis of fMRI data and particularly for the construction of FCN. The LLE method has been applied in rsfMRI studies for the improvement of predictions in ageing (Qiu et al. 2015), for the classification of healthy subjects and patients with schizophrenia (Shen et al. 2010) and as an alternative method for dimensionality reduction before the application of ICA in task-related fMRI, where non-linear relationships in the BOLD signal are introduced (Mannfolk et al. 2010). The kPCA method has been recently applied to a fMRI study for non-linear feature extraction (Tsatsishvili et al. 2018). In this study, it was shown that certain important features could not be found by the standard PCA. kPCA has been also used for feature extraction towards the automated diagnosis of (Attention-Deficit Hyperactivity Disorder) ADHD (Sidhu et al. 2012). In Anderson and Cohen (2013), ISOMAP was employed to a benchmark rsfMRI dataset of 146 subjects for the construction of embedded low-dimensional FCN for the classification of controls and schizophrenic subjects. ROIs were selected using single-subject ICA and the similarities between the ICs were assessed using a pseudo-distance measure based on cross correlation. Graph-theoretic measures were then used for the discrimination between patients and healthy controls. Another study based on single-subject ICA exploited ISOMAP to classify spatially unaligned fMRI scans (Anderson et al. 2010). The study focused on comparisons between patients with schizophrenia versus healthy controls and different age groups of healthy controls versus patients with alzheimer’s disease. Despite the relatively low sample sizes, results were promising with good classification rates. Recently, Haak et al. (2018) utilized ISOMAP for the construction of individualised connectopies from rsfMRI recordings taken from the WU-Minn Human Connectome Project in a fully data-driven manner. Only a handful of studies have used diffusion maps for the analysis of fMRI data. These studies have been focused mainly on the clustering of spatial maps of task-related experiments (Shen and Meyer 2005; Sipola et al. 2013). Shen and Meyer (2005), and Sipola et al. (2013) used diffusion maps with a Gaussian kernel to cluster selected fMRI spatial maps that are derived by ICA. The approach was demonstrated using fMRI recordings acquired from healthy participants listening to a stimulus with a rich musical structure. Other applications of diffusion maps in neuroimaging include predicitions of epileptic seizures and the identification of the pre-seizure state in EEG timeseries (Lian et al. 2015; Duncan et al. 2013). A review on the intersection between manifold learning methods and the construction of FCN can be found in Siettos and Starke (2016), and Richiardi et al. (2013).

Here, we employed MDS, ISOMAP, diffusion maps, kPCA and LLE to construct embedded FCN from rsfMRI data taken from healthy controls and schizophrenia patients. For our demonstrations, we used the Center for Biomedical Research Excellence (COBRE) rsfMRI dataset that is publicly available and has been used recently in many studies (Calhoun et al. 2012; Mayer et al. 2013; Anderson and Cohen 2013; Qureshi et al. 2017). Based on key global graph-theoretic measures of the embedded graphs, we assessed their classification efficiency using several machine learning algorithms, namely linear standard Support vector machines (LSVM), radial (radial basis function kernel) support vector machines (RSVM), k-nearest neighbours (k-NN) classifier, and artificial neural networks (ANN). We also investigated their performance considering two commonly used distance metrics, namely the cross correlation and the Euclidean distance. Our analysis showed that diffusion maps with the cross correlation outperformed all other combinations.

At this point, we should note, that our study does not aim at extracting the best classification performance by trying to find the best possible pre-processing pipe-line of the raw fMRI data and/or the selection of “best” subjects and/or the selection of the best set of graph-theoretic measures that provide the maximum classification. Yet, we aim at using state-of-the-art manifold learning methods for the construction of embedded FCN and compare their classification efficiency using only the three fundamental global graph measures, i.e. the average path length, the global clustering coefficient and the degree. Furthermore, our results can be compared to those obtained by similar studies (see e.g. Anderson and Cohen 2013) using the same pipe-line for data pre-processing and single-subject ICA. To the best of our knowledge, this paper is the first to perform such a thorough comparative analysis of both linear and nonlinear manifold learning on rsfMRI data. It is also the first study to show how diffusion maps can be used for the construction of FCN from rsfMRI, assessing also the efficiency of two basic distance metrics, the cross correlation and the Euclidean distance.

In this study, we constructed embedded FCN from rsfMRI data using linear and non-linear manifold learning techniques. Based on fundamental graph theoretic measures of the constructed FCN, we then used machine learning algorithms for classification purposes. We also compared the performance of two widely used metrics in neuroimaging, namely the cross correlation and the Euclidean distance. For our demonstrations, we used a publicly available dataset of resting-state fMRI recordings taken from healthy and patients with schizophrenia. This is the first study that performs such a systematic comparative analysis between various manifold learning algorithms, machine learning algorithms and metrics. To the best of our knowledge, it is also the first study that shows how the algorithm of diffusion maps can be exploited to construct FCN from rsfMRI data.

At this point we should note that our intention was not to try to obtain the best possible classification performance by “optimising” the pre-processing of the raw fMRI data and/or by trying to find the best set of graph-theoretic measures [other studies have already shed light towards this direction (Čukić et al. 2020; Xiang et al. 2020; Vergara et al. 2017)]. For example Čukić et al. (2020) showed that successful discrimination of depression from EEG could be attributed to proper feature extraction and not to a particular classification method. Thus, here we aimed at comparing mainly the manifold learning methods by factoring out the influence of a specific feature selection method; classification was based only on three key global graph-theoretic measures that are widely used in neuroscience (Stam and Reijneveld 2007; Bullmore and Sporns 2009), namely, the average path length, the global clustering coefficient and the median degree of the embedded binary networks. Indeed as also discussed in Bullmore and Sporns (2009) the general brain network can be described at a global scale by the shortest path length which is associated with the transfer of information, the clustering coefficient associated with robustness to random error, and the degree associated with the existence of hubs, and a modular community structure. Even though, we could consider a few more global graph measures, we decided not to do so, as other measures like the global efficiency (in essence, the inverse of average path length) or the diameter of a graph (the largest path length of a graph from one vertex to another) are highly correlated with one (or more) of the three basic measures stated above (e.g. the global efficiency is the inverse of characteristic path length and the diameter of a graph is likely to be higher as the average path length gets larger). Based on the above fundamental graph measures, our best reported accuracy obtained with diffusion maps and cross correlation was 79.3 % (evaluated with a 10 fold cross validation scheme, repeated 100 times). The conventional methodology was outperformed in both overall performance and with respect to the maximum classification accuracy as there was an 8.3% difference in classification accuracy in favor of the diffusion maps.

For the same benchmark fMRI dataset, Anderson and Cohen (2013) used ISOMAP for the construction of embedded FCN for the classification between heatlhy controls and schizophrenia patients. ROIs were acquired as here using single subject ICA and functional connectivity was accessed using the cross correlation distance. The analysis revealed differences in small-world properties among groups and 13 graph theoretic features led to a reported 65% accuracy rate. Xiang et al. (2020) reported a 93.1% accuracy (with sparse group Lasso and 78.6% with a Welch’s t test) testing more than 1000 graph-based features. An anatomical atlas was used for signal extraction and ROI selection, while an SVM-based classifier was used along with a “leave one out” scheme to evaluate its performance. In the same study, the authors compare their method with other novel approaches that have been previously proposed (Cheng et al. 2015; Huang et al. 2018) by applying them on the COBRE dataset. Cheng et al. (2015) calculated the betweenness centrality of nodes and used their ranks to classify patients with schizophrenia and healthy controls. For the COBRE dataset, this approach yielded a 74.4% classification accuracy. Finally, Huang et al. (2018) used a tree-guided group sparse learning method to extract key information in four frequency bands. Applied on the COBRE dataset, the classification performance peaked at 77.3%.

Our best reported accuracy (79.3% evaluated with tenfold cross validation repeated 100 times) is still higher than some of the previously proposed methods applied to the same dataset.

Some recent studies have suggested that the correlation matrices lie on a non-linear manifold (Venkatesh et al. 2020). Regarding our results, the diffusion maps algorithm (based on the diffusion distance) and the ISOMAP (based on the geodesic distance) managed to outperform the correlation matrix (that is most frequently used in constructing FCN from fMRI data) in terms of classification accuracy. However, other techniques such as the LLE (locally preserving the distance among neighbours), the gaussian kernel PCA (the non linear extension of PCA) and the MDS (which preserves the Euclidean distances on the embedded manifold) performed similarly or poorer compared to the cross correlation matrix. On the other hand, when using the Euclidean distance as a metric of functional connectivity none of the methods used in this study exhibited much higher results than the Euclidean matrix (which was slightly outperformed by ISOMAP and MDS). However, LLE provided a more robust classification pattern with most of the classifiers reaching a 70% accuracy. Though in general, there is no single best manifold learning method outperforming all the others for both metrics, our study showed that the diffusion maps result in higher classification accuracy when using the cross correlation distance (10.5% difference with respect to the Euclidean distance) outperforming the “conventional” method for constructing FCN from fMRI data.

However, this study does not come without limitations. For example, the method chosen for signal extraction is the single-subject ICA (Anderson and Cohen 2013). While this methodology holds the advantage of yielding subject-specific ICs (taking into account the within-subject variation), yet, due to this fact, we could not utilize local features in our graph theoretic analysis as the ICs (the nodes in the constructed graphs) were not the same across participants (even the number of ICs were different just like in Anderson and Cohen 2013). Thus, in order to factor out the influence of specific features, as discussed above, we used the three fundamental global theoretical measures to quantify differences among groups in the global topology of the network (Stam and Reijneveld 2007; Bullmore and Sporns 2009). An alternative method for the signal extraction could be the use of group-ICA. For example, a group-ICA analysis has been recently applied in a large fMRI dataset (151 healthy controls and 163 schizophrenia patients) for classification purposes (Salman et al. 2019); under this methodology the authors reached a maximum of 76.4% classification accuracy. While studies have shown that group-ICA can capture inter-subject spatial variability, it is not without limitations (Allen et al. 2012). For example group-ICA makes the assumption that each subject makes the same contribution to the observed “group” ICs, discarding random subject to subject variations. Thus, one cannot generalize the conclusions to the population (Friston et al. 1999). Hence, different approaches (such as the Independent Vector Analysis) have been suggested that seek for an optimal trade-off between group and individual representation, trying to preserve subject’s variability within a group (Michael et al. 2014).