In this study, we investigate if phase-locking of fast oscillatory activity relies on the anatomical skeleton and if simple computational models informed by structural connectivity can help further to explain missing links in the structure-function relationship. We use diffusion tensor imaging data and alpha band-limited EEG signal recorded in a group of healthy individuals. Our results show that about 23. 4% of the variance in empirical networks of resting-state functional connectivity is explained by the underlying white matter architecture. Simulating functional connectivity using a simple computational model based on the structural connectivity can increase the match to 45. 4%. In a second step, we use our modeling framework to explore several technical alternatives along the modeling path. First, we find that an augmentation of homotopic connections in the structural connectivity matrix improves the link to functional connectivity while a correction for fiber distance slightly decreases the performance of the model. Second, a more complex computational model based on Kuramoto oscillators leads to a slight improvement of the model fit. Third, we show that the comparison of modeled and empirical functional connectivity at source level is much more specific for the underlying structural connectivity. However, different source reconstruction algorithms gave comparable results. Of note, as the fourth finding, the model fit was much better if zero-phase lag components were preserved in the empirical functional connectome, indicating a considerable amount of functionally relevant synchrony taking place with near zero or zero-phase lag. The combination of the best performing alternatives at each stage in the pipeline results in a model that explains 54. 4% of the variance in the empirical EEG functional connectivity. Our study shows that large-scale brain circuits of fast neural network synchrony strongly rely upon the structural connectome and simple computational models of neural activity can explain missing links in the structure-function relationship.
In this study we probed this assumption of a strong structure-function relationship by simulating local node dynamics based on SC and comparing the phase relationships emerging from the simulated neural activity with empirically measured phase relationships. To this end, we combined SC from DTI data using probabilistic fiber tracking and FC from EEG data recorded during wakeful rest in 17 healthy individuals. We then used computational modeling approaches to link SC and empirical FC at the alpha frequency range. We demonstrate that empirical networks of resting-state fast oscillations are strongly determined by the underlying SC and that additional variance between structure and function can be explained by modeling dynamic activity based on white matter architecture. Specifically, the simulated FC explained 28. 5% of the variance in the empirical FC that was left unexplained by SC alone. To further understand the explanatory power of our model we investigated its performance at the local level by assessing specific properties of ROIs (nodes) or connections (edges). We found that the model error was highest for large highly interacting ROIs. However, modeling large-scale brain dynamics based on structural priors brings up several methodological alternatives, not only regarding the modeling itself, but also regarding the comparison of simulated and empirical data. Especially with resting-state MEG/EEG activity, the specificity of analytic routines requires methodological decisions which potentially lead to tremendous differences in modeling outcomes. We systematically assessed the effect of technical variations on results and their influence on the interpretation of structure-function relations. Specifically, we used our modeling framework to explore several technical alternatives along the modeling path and evaluate the alternative processing steps based on their effect on the performance of the model in simulating empirical FC. Specifically, we addressed the effects of five critical aspects in the modeling pipeline: We used DTI and probabilistic tracking algorithms to compile a whole-brain structural connectome [37]. However, several studies suggested that current fiber tracking algorithms fail at capturing particularly transcallosal motor connections that are observed in non-human primate tracer studies [38,39]. In addition, structural connection strength modeled by probabilistic tractography algorithms is influenced by fiber length due to the progressive dispersion of uncertainty along the fiber tract [15,40]. Therefore, we evaluateed the effect of normalizations for fiber length of the SC and examined the effect of weighting homotopic connections in our model. Our results show that the correction for fiber distance leads to a small decrease in the performance of our model. The additional weighting of homotopic transcallosal connections, however, increased the model fit [24,25]. Several alternative computational models of neural dynamics are available. In the choice of a more abstract version to a more realistic description of cortical interactions, these models vary in the complexity of their formulation and therefore might explain more or less variance in the observed FC. The downside of complex models, however, is the increased number of free parameters. These have to be approximated, need to be known a priori, or explored systematically. All these aproaches are problematic. For an assessment of the factor of model complexity, we compared a simple spatial autoregressive (SAR) model to the Kuramoto model of coupled oscillators. We find that the SAR model explains already a large portion of the variance and that the Kuramoto model only gives a slight improvement. The comparatively few existing studies on large-scale modeling of MEG/EEG data differ systematically with respect to the comparison with empirical data. Some approaches project the observed time series onto the cortex using an inverse solution, whereas others project the simulated cortical signals into sensor space using the forward model [21,41,42]. We used our analytic framework to compare empirical and simulated FC at different spatial levels. We found that the importance of structural information is dramatically reduced if the higher spatial resolution obtained by source reconstruction is bypassed. Estimating the spatiotemporal dynamics of neuronal currents in source space generating the EEG and MEG signals is an ill-posed problem, due to the vastly larger number of active sources compared to the number of sensors. Therefore, we assess the impact of specific source reconstruction algorithms on the match of simulated and empirical FC. We compared three routinely used algorithms that differ regarding the assumptions made about the source signal, such as smoothness, sparsity, norms, correlation between source signals. However, we found no compelling superiority of one algorithm over another. Functional connectivity describes statistical dependencies between two signals often based on undirected temporal averages such as correlation. In the last decades, various additional FC metrics have been introduced. These differ with regard to the relative weighting of phase and amplitude or concerning the removal of zero-phase lag components prior to correlation. The theoretical superiority of one approach over another is debated [43]. However, no consensus appears achieved and currently no single metric is dominantly used over the others. Therefore, we compared several widely used metrics to compare empirical and simulated FC. We found that the model fit was much better if zero-phase lag components were preserved in the empirical functional connectome. In the following sections, we first present a reference procedure for modeling FC based on DTI and the comparison with empirical FC as measured by EEG. After an initial short overview of the modeling approach (see the Workflow section), we guide the reader step by step through the model details with the resulting outputs of each processing stage (see the Reference Procedure section). From there, the impact of technical alternatives on the performance of the model is presented (see the Alternative Modeling Approaches section).
We compared the simulated FC based on SC with the empirical FC derived from EEG data (Fig 1). Our model includes the processing steps as shown in Fig 1 with the DTI measurements on the left and the EEG measurements on the right. We address preprocessing of DTI data in the form of homotopic reweighting. Then, the 66 ROIs of the cerebral cortex according to the ‘Desikan-Killiany’ cortical atlas made available in the Freesurfer toolbox, were individually registered for 17 healthy subjects using Freesurfer (surfer. nmr. mgh. harvard. edu) [44]. The SAR model used in the reference procedure was selected based on simplicity (low number of parameters) and performance (computationally very efficient). Furthermore, the SAR model allows to systematically evaluate the full parameter space with a high resolution grid-search, which is necessary for an unbiased comparison of all alternatives along the modeling path. We reconstructed source activity at the geometric center of each ROI based on the EEG time series by a linear constraint minimum variance spatial beam former (LCMV). Then we assessed FC between source time series band pass filtered at 8 Hz where the averaged coherence showed a peak (see supporting material S1 Fig). Finally, we evaluated the match of simulated and empirical FC based on the correlation between all pairs of ROIs [17]. Following this modeling approach, several alternative ways at each processing stage arise. Choices exist, for example, for the level of abstraction of the model type [45], metrics to compare functional connectivity and the approach to the inverse problem in interpreting EEG data. The modeling of large-scale brain dynamics based on structural priors brings up several methodological alternatives. As a principal choice, the model may be evaluated either in source or in sensor space. In the baseline model that was presented above, we made specific choices at each processing stage based on simplicity and good explanatory performance. Especially with resting-state EEG activity, a lack of analytic routines requires methodological decisions to be made heuristically, which could potentially lead to substantial differences in the conclusions drawn. In the following section we systematically compare different alternatives of the procedural stages delineated above and compare the outcome regarding global correlation between simulated and empirical FC. First, we assessed the influence of distance normalization and weighting of homotopic connections in the structural connectome on simulated FC. Second, we tested if a more complex simulation model of coupled oscillators is able to capture a larger part of the variance of the empirical data that is not explained by the simple SAR model. Third, we evaluated an alternative comparison in the sensor space using a forward projection of the source time series in contrast to source reconstruction. Then, we compared different source reconstruction methods. Finally, we tested the impact of removing zero-phase lags in functional interactions.
It has been assumed that for the resting-state networks based on fast dynamics the underlying anatomical skeleton is less important compared to the slow resting-state networks, but this issue has not yet been systematically investigated [10]. We calculated the performance of the reference model as the correlation between all modeled pairwise interactions and all empirical pairwise interactions in an empirical functional phase relation connectome of the alpha rhythm and found a good match of 45. 4% (Fig 2C). This finding is in contrast to the prior assumptions and shows that the anatomical skeleton is equally crucial for fast timescale functional interactions [29,36]. To better understand the reference model performance we investigated the model error in relation to node and edge characteristics (Fig 3). In general, the model error decreased with longer fiber distance and Euclidean distance. Specifically, for short fiber distances, the model overestimated FC (negative residuals blue in Fig 3). Why are short connections in general more difficult to model based on white matter tracts? The empirical connectome was extracted from resting-state alpha topographies in which propagating waves play an important role for adjacent and remote brain areas to communicate with each other. Cortico-cortical axons in the white matter tracts are considered as the major route for traveling waves. However, a recent study presented compelling evidence for intracortical axons accounting for spatial propagation of alpha oscillations [79]. Such a mechanism would enable high local synchrony in the relative absence of structural connectivity measured by DTI. We used a stepwise linear model to extract node characteristics explaining most of the model error and found that ROI size and betweenness centrality play an important role. Regarding ROI size, the smaller model error for larger ROIs could be attributed to the measurements of structural as well as functional connectivity being more reliable for larger ROIs: In the case of the SC measurements using DTI, a larger parcellated cortical region allows to track more streamlines with different initial conditions (i. e. for more voxels) and thereby allows a more reliable estimation of the connection probabilities between regions. In EEG as well as DTI, the localization and inter-subject registration of large ROIs can be assumed to be less effected by small deviations because a small spatial shift of a large ROI still allows a large overlap with the correct ROI volume whereas a small spatial shift of a small ROI could displace it completely outside of the original volume. For betweenness centrality, the opposite scenario was the case: the smaller the betweenness centrality the smaller was the model error. Central hubs in a structural network offer anatomical bridges which enable functional links between regions that are structurally not directly related [63]. Hard-wired connections do not necessarily contribute at all times to FC in the network and, vice-versa, functionally relevant connections do not necessarily have to be strongly hard-wired [13]. Possibly, the simple SAR model, which captures only stationary dynamics, has weaknesses at these central hub nodes. In order to capture the empirical FC at these nodes, a more complex dynamical model able to capture non-stationary dynamics with context switches at slower time scales is needed. Nodes with a high betweenness centrality can be expected to communicate with certain cortical modules only at certain times in specific dynamical regimes. We hypothesize that a more complex dynamical model of neural activity could capture this behavior more accurately. Therefore we suggest that further research could especially improve the model in these cases of dynamical context switches in central hub nodes, which cannot be captured by simple models such as the SAR model. Using our modeling framework to compare different alternatives of reconstructing the structural connectome, we found that the best match between simulated and empirical FC was obtained when an additional weighting of connections between homotopic transcallosal regions was applied. Additional weighting for fiber distances did not improve the simulation performance significantly. Overall, the differences were very small proving the modeling approach to be rather robust regarding the evaluated choices of reconstruction as long as the total input strength per region is normalization prior to the simulation. Currently, there is no common approach to correct for the influence of fiber distance on the probabilistic tracking algorithm [16,40,80]. Although we found that the model error was largest for small fiber distances (modeled FC higher than empirical FC), a correction for fiber lengths did not improve the result of the simulation. This suggests that the high local connection strength of SC obtained by DTI reflects actual structural connectivity. Methodically, this finding is supported by the fact that accuracy of probabilistic fiber reconstrunction decreases with distance, whereas short-distance connections are reconstructed with high reliability [38]. However, it remains a challenge to correct probabilistic tracking results for the impact of fiber distance and further work is needed to address this methodological limitation. Our model improved with an additional added weight of homotopic connections, which is supporting the data by Messé et al. [24]. This finding points to a related limitation of the probabilistic tracking algorithms to correctly assess long distance and lateral transcallosal fibers. In agreement with previous studies, we show that this limitation can be addressed by adding a preprocessing step to the structural connectome reconstruction. Lastly, we want to point out that the parcellation scheme, especially the spatial resolution, has a strong effect on the SC and FC, as shown in previous studies [81–83]. We did not include other parcellation schemes as alternatives in this work because a different parcellation effects all steps in the processing pipeline at the same time. Most importantly, a different parcellation also changes the predefined space in which the model prediction is evaluated, so that the resulting correlation values are not directly comparable to the results of our presented reference procedure. We chose a parcellation scheme which has been used in several previous studies [16,22,40,44,84] and implemented in Freesurfer. The effect of parcellation schemes on structure-function relationships is a very important topic that is currently under investigation. We show that our SAR model already explains much of the variance in the empirical EEG data. Our results indicate that the Kuramoto model moderately improved results compared to the reference model. The SAR model has a small number of parameters allowing a fast exploration of the parameter space [49] and the SAR model served several studies in which complexity and information-theoretical measures characterizing FC were explored [49,85,86]. As a downside, the SAR model has a smaller number of parameters and therefore lacks the modeling capacity to further optimize the dynamics to better fit to the empirical data. Furthermore, the SAR model cannot model individual frequencies and their interactions, making the Kuramoto model a viable alternative. It has been shown that the Kuramoto model features complex synchronization dynamics which can be related to the explanation of oscillatory phenomena in the human cortex, such as fluctuating beta oscillations [48] or metastable synchronization states [21]. A more detailed analysis of the synchronization properties of the Kuramoto model in the human connectome was done by Villegas et al. [87], where frustration and the transition between synchronous and asynchronous phases were analyzed [88]. The Kuramoto model was also used to study the effects of lesions on cortical dynamics and binding by synchrony [69,89]. However, it has been shown that more complex models with more parameters are usually not better in explaining fMRI functional connectivity from structural data [24–26]. Highly parameterized models which require the numerical integration of differential equations take several orders of magnitude more computational time to obtain a reliable estimate of FC than the simple model used here. For certain neurophysiological questions however, the wider parameter space of complex models can be used to explore neural processing properties. The relative benefit of a dynamical model has to counterbalance the higher computational demand. Therefore, the choice of model depends on the investigated scientific question [26,45]. In this study we used the simpler SAR model as a reference because the focus was to investigate alternatives also in many other stages of the processing pipeline and a more complex simulation model would impede identifying the best alternative in the other stages of the processing pipeline, due to the high dimensional parameter space. The inverse problem is ill-posed since the higher number of possible active neuronal sources is higher than the number of recording channels. Thus, the ground truth of brain activity patterns generating the measured signal is impossible to infer. A variety of alternative methodological approaches have been developed regarding source imaging. Particular caution should be exercised concerning the influence of different inverse solutions on the resulting data [90]. Here, we presented a comparison of the performance of three commonly used inverse methods regarding the global correlation between empirical and simulated FC in our technical framework (Fig 7). All source reconstruction algorithms perform in a similar range with resembling r-values between 0. 674 and 0. 728. Although the algorithms differ regarding the assumptions made about the source signal, the high correspondence in performance of the three source reconstruction techniques mutually validates their respective inverse solutions. Next, we aimed to investigate the best approach for comparing empirical and simulated FC particularly in sensor and source space, see Fig 1. In the sensor space scenario, the simulated signal, as the mean field source activity generated by the SAR model, was projected into sensor space to generate a simulated EEG signal by applying the leadfield (i. e. forward model). For this approach we found slightly higher correlations between simulated and empirical data (Fig 6A). However, we also found that the high correspondence between empirical and simulated EEG sensor space FC was independent of the underlying SC: Shuffling SC before the simulation did not abolish the correlation between the empirical and simulated FC as was the case when the comparison was done at the source space level. This lack of specificity of the simulated FC regarding the anatomical skeleton strongly suggests that the sensor level connectivity matrix is shaped mainly by the leadfield (Fig 6). In fact, the leadfield can already explain most of the variance (81. 9%) in the empirical FC of the sensor space. In contrast, the inverse solution in the source reconstruction procedure removes much of these volume conduction correlations so that the comparison of coherence in source space appears reasonable. We conclude that the volume conduction model of the head is mixing the source time series such that the coherence in sensor space reflects to a high degree the structure within this mixing matrix and the sensor space is a suboptimal stage for investigating structure-function relationships by large-scale modeling approaches. Thus, one should refrain from such a comparison in sensor space with metrics that do not exclude zero-lag interactions. In order to assess the accuracy of simulated global network characteristics, the comparative spatial level should be at source space in order to avoid signal mixing by the leadfield matrix and allow to include zero-lag interactions. The results offer an important ground for modeling studies using source connectivity analyses for MEG/EEG data. One of the main differences between fMRI/PET and MEG/EEG connectivity studies is that for MEG/EEG a multitude of different metrics to quantify FC are currently available and no single metric is predominantly employed or has emerged as being superior [91,92]. This issue hampers comparability between studies and physiologic interpretation. It was our aim to use our theoretic framework for a systematic comparison of different functional connectivity metrics. We compared six commonly used metrics that differ regarding their sensitivity towards zero-phase lag coupling and amplitude variations. The definition of PLV, PLI, WPLI and LPC theoretically renders those metrics insensitive to amplitude variations. We found no major difference in performance between COH and PLV and no major difference between ICOH, PLI, WPLI, and LPC. This result is easily understood on the basis that the SAR model presents the steady state solution including a small noise component only. An important finding is the high correspondence in model performance between coherence and PLV. Coherence is the cross-spectrum between two sensors normalized with the auto-spectra whereas PLV quantifies the consistency of a phase difference between two signals across time. Both measures are high if there is a consistent phase difference regardless of whether the latter is near zero, 180° or inbetween. Similar results between coherence and PLV have been found in previous studies [93–95]. The similarity of both measures in our study suggests that amplitude variations between areas are of less weight than phase variations. Another main finding is the drop in model performance with the metrics ICOH, PLI, WPLI and LPC which are by design less sensitive to zero-phase coupling. Regarding the latter, a major concern exists whether such coupling in scalp recordings would be contaminated by volume conduction artifacts. Obviously, synchrony at sensor level could result from two channels picking up activity from a common source since the activity of the source signal passes through the layers of cerebrospinal fluid, dura, scalp and skull acting as a spatial filter. This effect leads to the detection of spurious synchrony, even if the underlying sources are independent [96]. Based on the assumption that the quasi-static approximation holds true for EEG, volume conduction would occur with zero-phase lag [97]. Thus, the most commonly used approach to deal with the problem of volume conduction is to neglect interactions that have no phase delay. This is, however, a potentially overly conservative approach. To address the question of how these biased measures of interactions are suited for comparing modeled and empirical connectomes, we compared global model performance based on connectivity metrics that are sensitive and robust to zero-phase lags in this study. ICOH, PLI, WPLI and LPC all showed a significantly lower match between simulated and empirical FC (around r = 0. 18) compared to coherence and PLV (Fig 8). For all six metrics, the global correlation was essentially abolished if the underlying SC was shuffled prior to simulation (yellow bars in (Fig 8) ). Also, the overall model performance for ICOH, PLI, WPLI and LPC was considerably smaller than the mere correlation between SC and empirical FC (middle row in Fig 5A). What are the possible reasons for this performance drop with ICOH, PLI, WPLI and LPC? One reason could lie in the fact that the reference model SAR does not include delays, thus the simulated FC mainly consists of instantaneous interactions and a comparison with an empirical FC in which those interactions have largely been removed would be futile. However, the results were very similar using the Kuramoto model. The large-scale connectomes derived from all of the four biased metrics did not much reflect the coupling that emerged from our model of fast dynamics based on structural connectivity. Presumably, a considerable amount of functionally relevant synchrony takes place with near zero or zero-phase lag which is not detected using the biased scores. In fact, zero-phase lag synchronization has been detected between cortical regions in a visuomotor integration task in cats [98]. More recently, a study of spike train recordings showed how paths among somatosensory areas were dominated by instantaneous interactions [99]. But synchrony across areas incorporating delays can also lead to high coherence [100]. A recent modeling study investigated the detection rates of synchrony by different EEG phase synchronization measures (PLV, ICOH, WPLI) in a network of neural mass models. They found that no single phase synchronization measure performed substantially better than all the others, and PLV was the only metric able to detect phase interactions near ±0° or ±180° [91]. This study challenged the supposed superiority of biased metrics in practical applications, because they are biased against zero-phase interactions that do truly occur in the brain. Taken together we argue that by using biased metrics to detect neural synchrony a major portion of relevant coupling is neglected. However, as the relevant stage for comparisons is the source space, the undesired influence of volume conduction effects on the estimated connectivity is partly reduced [101]. Since effects of field spread can never be completely abolished also in the source space, we cannot rule out that volume conduction artifacts have influenced the high correlation in our model. The empirical functional connectome was constructed based on band-pass filtered EEG in the alpha frequency range. Since different FC maps have been detected for different frequency bands [9], it is conceivable that biased vs. unbiased FC metrics might vary in their performance depending on the frequency. In summary, our framework demonstrates how technical alternatives and choices along the modeling path impact on the performance of a structurally informed computational model of global functional connectivity. We show that determining the resting-state alpha rhythm functional connectome, the anatomical skeleton has a major influence and that simulations of global network characteristics can further close the gap between brain network structure and function.
Brain imaging techniques are broadly divided into the two categories of structural and functional imaging. Structural imaging provides information about the static physical connectivity within the brain, while functional imaging provides data about the dynamic ongoing activation of brain areas. Computational models allow to bridge the gap between these two modalities and allow to gain new insights. Specifically, in this study, we use structural data from diffusion tractography recordings to model functional brain connectivity obtained from fast EEG dynamics occurring at the alpha frequency. First, we present a simple reference procedure which consists of several steps to link the structural to the functional empirical data. Second, we systematically compare several alternative methods along the modeling path in order to assess their impact on the overall fit between simulations and empirical data. We explore preprocessing steps of the structural connectivity and different levels of complexity of the computational model. We highlight the importance of source reconstruction and compare commonly used source reconstruction algorithms and metrics to assess functional connectivity. Our results serve as an important orienting frame for the emerging field of brain network modeling.
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Modeling of Large-Scale Functional Brain Networks Based on Structural Connectivity from DTI: Comparison with EEG Derived Phase Coupling Networks and Evaluation of Alternative Methods along the Modeling Path