Structure-function coupling in the human connectome: A machine learning approach

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Introduction
The function of most biological systems is strongly constrained by their underlying structure and anatomy. A major goal in neuroscience is to understand structure-function relationships in nervous systems across different scales and species (1). The structural substrate of a nervous system comprises neurons and neuronal populations, interconnected by a complex anatomical network of axons and axonal fiber bundles, collectively referred to as the connectome (2,3). The connectome shapes and constrains neural activity to conform to a canonical repertoire of co-activation patterns, commonly known as functional networks (4,5), which in turn support cognitive, perceptual and other mental functions (6,7).
Structure-function relationships in nervous systems are increasingly conceptualized as associations between structural and functional brain networks. In the human brain, these relationships can be studied with magnetic resonance imaging (MRI).
Tractography performed on diffusion-weighted MRI can be used to infer structural connectivity (8,9), while functional connectivity can be derived from resting-state or task-based functional MRI (10). Numerous studies suggest that functional connectivity is strongest among regions that are structurally connected, but this correspondence is far from perfect (11), and many functional connections are formed in the absence of a direct structural link (12). The correspondence is even weaker at the macroscale, where canonical brain networks, such as the default-mode network, are generally not evident within structural networks (13,14).
Structure-function coupling is therefore likely to be more complex than a one-to-one linear correspondence. High-order interactions among neural elements, involving polysynaptic communication across multiple structural connections, may give rise to complex mappings between structure and function (15). Indeed, by accounting for such high-order interactions, biophysical models (16)(17)(18), graph harmonics (19), network communication schemes (20,21) and multivariate statistical models (22,23) have furnished evidence of tighter structure-function coupling in brain networks than suggested by earlier univariate correlational models.
The most well studied of these approaches are dynamic biophysical models. They can be used to simulate neural activity within structurally interconnected neuronal populations (24)(25)(26)(27)(28)(29). Functional connectivity can then be computed between the neuronal populations based on the simulated activity and compared to empirical functional connectivity estimated from functional MRI (30)(31)(32)(33). Despite these modeling advances, structure remains a relatively modest predictor of function in brain networks. Structural connectivity rarely explains more than -of the variance in functional connectivity across the connections comprising mesoscale brain networks, and the agreement between predicted (i.e. simulated) and empirical (i.e. true) functional connectivity is comparably modest.
The imperfect coupling between structure and function raises the possibility that structure and function are indeed truly decoupled to a certain extent in nervous systems, meaning that function cannot be entirely predicted based on structure alone. In many biological systems, some degree of decoupling between structure and function is important to isolate low-level noise fluctuations from functional-level dynamics.
Decoupling can also facilitate regional and individual specialization of behavior (19,34).
Furthermore, structure-function coupling is dynamic and changes throughout the lifespan, particularly during adolescence (35), suggesting that perfect structure-function coupling might not be expected.
An alternative possibility is that imperfect structure-function coupling in brain networks reflects limitations of current biophysical and statistical brain network models, which could potentially overlook key neurophysiological processes and high-order interactions. Correlating empirical estimates of functional and structural connectivity overlooks the contribution of polysynaptic anatomical pathways; while using biophysical models to predict function from structure requires careful tuning of numerous physiological parameters, such as gating variables, synaptic coupling and kinetic dynamics for synaptic activity. Parameter tuning geared toward maximizing structure-function coupling has often been performed on the same empirical data used to evaluate model performance, potentially leading to exaggerated coupling estimates, due to overfitting. Moreover, the precise neurophysiological processes governing structure-function coupling remain poorly understood, and thus it is unclear which of the many available mean field and neural mass models are best suited to investigating these phenomena (28). A biophysical model should be chosen that characterizes the relevant mechanisms and dynamics that give rise to structure-function coupling, yet it is challenging to determine a priori which, if any, of the vast catalogue of available models adequately captures the relevant neurophysiological processes.
Hence, it remains unclear whether the relatively modest associations between structure and function in the connectome represent a fundamental property of the brain, or whether this reflects inherent limitations of current models for assessing structurefunction coupling. Here, we aim to shed light on this question with a fundamentally different approach to assessing structure-function coupling in the connectome. Rather than using a model that is predicated on specific assumptions and neurophysiological processes, we instead develop a deep learning framework to learn structure-function relationships in the absence of any underlying biophysical models, network communication theories and/or assumptions on high-order interactions. Although deep learning does not provide mechanistic insight into the physiological processes that govern structure-function coupling, the accuracy with which functional connectivity can be predicted from structural connectivity within a deep learning framework may be greater compared to current biophysical models. In this way, we aim to determine the margin for improvement in current state-of-the-art biophysical models and evaluate whether some decoupling of structure and function is a fundamental property of the human connectome.
In the present study, we predicted functional connectivity from structural connectivity in each of healthy adults using a novel deep learning framework. Whereas most previous studies consider prediction of group-averaged functional connectivity (18,30,36), we generated a personalized prediction for each individual, thereby preserving inter-individual variation in predicted functional connectivity and enabling the investigation of whether this variation associated with measures of behavior and cognition. We benchmarked each individual's predicted functional network to empirical functional connectivity derived from their resting-state function MRI data. We hypothesized that deep learning would yield significantly improved individual and group-level predictions of functional connectivity compared to biophysical models. This study provides new evidence suggesting that structure-function coupling in the human connectome is stronger than previously implied and demonstrates that predictions of function from structure are sufficiently accurate to explain inter-individual variation in some behavioral measures. Figure 1a provides a schematic overview of the study.

Fig. 1. Schema of study design and methodology (a)
Whole-brain structural and functional connectivity matrices were mapped for each of 1000 healthy adults using diffusionweighted and resting-state functional MRI data, respectively. Using 10-fold cross-validation, a deep neural network was trained to predict each individual's functional connectivity (FC) matrix based on their structural connectivity (SC) matrix. Functional connectivity inferred from the functional MRI data is referred to as empirical FC (eFC), whereas functional connectivity predicted by the neural network is referred to as predicted FC (pFC). The correlation between eFC and pFC was computed for each individual comprising the test set of each fold and used to evaluate prediction performance and the strength of structure-function coupling. (b) The architecture of the proposed feed-forward fully connected neural network used in this study. Each layer (represented by a blue rectangular box) is a combination of two sub-layers (represented by concatenated purple and gray boxes). The input to the neural network is the upper triangle of the SC matrix, which comprises the connectivity strengths between all pairs of regions comprising an established cortical parcellation atlas. The output is the upper triangle of the pFC matrix. (c) Schematic of inter-and intra-subject correlation coefficients computed between SC, eFC and pFC to assess prediction performance and the strength of structure-function coupling. Correlations were computed between pairs of distinct individuals (inter-subject) or between distinct pairs (modalities) of connectivity matrices within the same individual (intra-subject). SC: structural connectivity matrix. eFC: empirical functional connectivity matrix inferred from resting-state function MRI. pFC: functional connectivity matrix predicted from SC. N: total number of participants.

Datasets
Two related datasets were used in this study. The primary dataset comprised healthy adults participating in the Human Connectome Project (HCP) (37). Data pre-processing and connectivity mapping for the diffusion-weighted and resting-state functional MRI data comprising the primary dataset is described below.
Additionally, a supplementary dataset comprising healthy adults was provided by Wang and colleagues (18) and used for validation and replication experiments. Data preprocessing and connectivity mapping for the supplementary dataset is described in detail elsewhere (18). While some individuals were common to the primary and supplementary datasets, the pre-processing pipelines, tractography algorithms and connectivity mapping methods differed substantially between the two datasets. For the supplementary dataset, only one member from each family was included to ensure independence of structural and functional connectivity measures between individuals.

Structural connectivity mapping
We obtained minimally pre-processed diffusion-weighted MRI data for each individual from the HCP, where the MRI acquisition protocols and minimal pre-processing pipelines are described in detail elsewhere (37). In brief, 90 gradient directions were acquired at each of three b-values ( , and s/mm 2 ) using spin-echo planar imaging. Pre-processing included correction of eddy-current, head motion and gradientnonlinearity distortions, followed by transformation of the corrected volumes to native structural space and rotation of gradients vectors.
Whole-brain tractography was performed on the pre-processed diffusion-weighted MRI data. This involved estimating multiple fiber orientations for each white matter voxel using constrained spherical deconvolution (CSD) (38) and then using deterministic fiber tracking to propagate streamlines throughout the white matter volume. We opted for CSD-based deterministic tractography based on recent recommendations (39).
Streamlines were uniformly seeded from a white matter mask derived from an automated structural segmentation. The boundaries of the white matter mask were dilated by one voxel to fill potential gaps between grey and white matter boundaries. Structural connectivity strengths were found to exhibit orders of magnitude of variation between pairs of cortical regions. To alleviate potential numerical scaling issues and stabilize the training of the neural networks, we applied a Gaussian resampling to the structural connectivity matrices (12). Finally, each resampled structural connectivity matrix was rescaled to a mean of and a standard deviation of . This is a rankreserving normalization. Normalization is essential for data samples with non-uniform scales. Without normalization, data points residing many deviations from the mean can unduly impact the weights of the neural network, resulting in instability and poor performance.

Functional connectivity mapping
Minimally pre-processed resting-state functional MRI (rs-fMRI) data were sourced for each individual from the HCP (37). Functional MRI was acquired using a multiband gradient-echo planar imaging period of minutes ( ms, multiband factor ), with eyes open and relaxed fixation on a projected bright cross-hair on a dark background. A session with two runs, with right-to-left and left-to-right phase encodings, were acquired. Minimal pre-processing included data cleaning with ICA-FIX, and nonlinear transformation to Montreal Neurological Institute (MNI) standard space.
The two runs were temporally demeaned and then concatenated, effectively yielding a total of volumes for each individual.
A functional connectivity matrix was computed for each individual using the Pearson correlation coefficient to infer functional connectivity strength. In particular, voxelspecific time series were first spatially averaged over all voxels comprising a region to yield regionally averaged time series. Element ( ) of an individual's functional connectivity matrix stored the Pearson correlation coefficient between the regionally averaged time series of regions and . Given that these matrices were inferred from empirically acquired data, we refer to them as representing empirical functional connectivity (eFC), establishing a distinction from predicted functional connectivity (pFC).

Deep learning network
A feed-forward fully connected neural network was trained to predict functional connectivity from structural connectivity. The network architecture is shown in Fig. 1b. The neural network was trained with the Adam optimizer, with momentum parameters and .
The overall aim of the proposed deep learning framework was to learn the complex mapping from SC to eFC, with the goal of predicting functional connectivity (pFC) for unseen SC matrices comprising a test set (Fig. 1a). To achieve this, the objective function of the neural network was designed to satisfy two requirements: 1) maximize the similarity between eFC and pFC, and 2) preserve inter-individual differences in functional connectivity (Fig. 1c). The first requirement was realized by minimizing the prediction error between eFC and pFC. For the second requirement, we introduced a regularization term to ensure that the correlation in pFC between different individuals (inter-pFC) was comparable to the correlation in eFC between different individuals (inter-eFC). This ensured that the neural network learned the mapping between SC and eFC while preserving inter-individual differences, rather than predicting a groupaveraged representation of functional connectivity for all individuals.
Given these two requirements, the objective function for the proposed neural network was given by where denotes parameters of the neural network (weights and bias) to be learned,

( ) is the loss function and is a regularization constant for the regularization function
given by ( ( ) ). Mean square error was used as the loss function ( ) in this study, as given by where is the number of subjects in a training batch, indexes a subject, is the actual output (eFC) and is the predicted output (pFC). Mean square error ensured that the spatial dependence between SC and FC was preserved during the training. As discussed, the objective of the neural network was to predict pFC while maintaining inter-eFC difference. To achieve this, the Pearson correlation coefficient was introduced in the where represents the inter-eFC correlation of the training dataset, which remained constant throughout the training process. It should be noted that varied for different datasets. In the above formulation, represents the inter-pFC correlation, which was expected to be less than or equal to inter-eFC. In this way, the regularization function ( ( ) ) ensured that the inter-individual differences in pFC remained comparable to inter-individual differences in eFC. The regularization function is similar to the L1norm.
Multiple neural networks were trained using varying values of that ranged between and . Other hyper-parameters were optimized (Table 1) based on an exhaustive grid search performed only in the training data (learning rate= to , dropout -, -, batch size -).

Biophysical model
We used an established relaxed mean-field biophysical model (rMFM (18)), which represents the modified variant of an established dynamic mean-field model (MFM (31)). The MFM is described in more detail by Wang and colleagues (18). Briefly, the MFM described the mean neural activity for each brain region using the following set of non-linear stochastic differential equations ∑ where , ( ) and denote the total input current, the population firing rate, and the average synaptic gating variable at the cortical region, respectively. is determined by the recurrent connection strength , the excitatory subcortical input , and interregion information flow, is a scaling constant and nA represents synaptic coupling. Inter-region information flow is controlled by , which we set to the empirically measured structural connectivity strength between region and , inferred from the diffusion-weighted MRI data for each individual. Parameter values for ( ) were set to be n/C, Hz and s and the kinetic parameters were set to and s, where ( ) is uncorrelated Gaussian noise whose amplitude is controlled by .
The recurrent connection strength and excitatory subcortical input were assumed to be the same across regions in the MFM, whereas this constraint is relaxed in rMFM.
Therefore, each region had its own recurrent connection strength and excitatory external (e.g., subcortical) input . The rMFM parameters corresponding to , , and were optimized by maximizing the similarity between simulated and empirical FC.
Further details of the model and values of the parameters are described elsewhere (18).
Finally, the Balloon-Windkessel hemodynamic model (41) was applied to the simulated neural activity for each region and the Pearson correlation coefficient was used to compute a functional connectivity matrix for the simulated data.

Behavioral measures
As described in detail elsewhere (42), independent component analysis (ICA) was used to parse 109 behavioral and cognitive assessments into five independent summary measures characterizing inter-individual variation in: i) cognition, ii) illicit substance use, iii) tobacco use, iv) personality and emotional traits, and iv) mental health. The 109 individual assessments were sourced from the HCP (37) and tapped alertness, cognition, emotion, sensory-motor function, personality, psychiatric symptoms, substance use, and life function. In-scanner task performance in emotional processing, language processing, social recognition, reward/decision making, working memory and motor execution were also included among the 109 items.
Briefly, under ICA, the matrix of residuals Y (behaviors individuals) was factorized into , where represents the estimated independent sources (components individuals) and represents the de-mixing matrix (behaviors components).
Bootstrapping was used to sample individuals with replacement and the FastICA algorithm was performed independently on each bootstrapped sample to estimate and , where consensus among the bootstrapped samples was then derived using clustering.
The ICA was repeated for 10 trails, where similarity of the resulting estimates was further evaluated and then averaged to ensure stability and reliability of the estimated independent components. The ICA procedure was repeated for candidate models ranging from three to thirty independent components and the optimal model of five components was selected.
Lasso regression in a 10-fold cross-validation was used to predict each of the fiveindependent behavioral/cognitive measures based on a feature space comprising functional connectivity between all pairs of cortical regions. The lasso regularization parameter was optimized using an internal cross-validation on the training data of each fold of the outer cross-validation. Prediction accuracies were quantified as the Pearson correlation coefficient between the predicted and actual measures in the test data.

Results
We mapped whole-brain structural and functional brain networks for each of healthy adults (primary dataset) participating in the Human Connectome Project (HCP) (37). Structural connectivity (SC) and functional connectivity (FC) were estimated between all 68 regions comprising the Desikan-Killiany cortical parcellation atlas (43) using diffusion-weighted and resting-state functional MRI data, respectively. Using 10fold cross-validation, deep neural networks were then trained to predict each individual's functional network using only their structural network (Fig. 1a,b).
Replication experiments were performed using a supplementary dataset (n=452) for which structural and functional connectivity were mapped using alternative pipelines and cortical parcellation atlases (Materials and Methods).
We refer to functional connectivity inferred from resting-state functional MRI as empirical FC (eFC), whereas functional connectivity predicted by the neural networks is referred to as predicted FC (pFC). To assess prediction accuracy and quantify the overall strength of structure-function coupling, we computed the correlation coefficient between eFC and pFC across all pairs of regions. Denoted as eFC-pFC, this correlation was computed separately for each individual. To ensure that the predictions preserved inter-individual variation in functional connectivity, we also computed the correlation in pFC and eFC between distinct pairs of individuals, referred to as inter-pFC and inter-eFC, respectively (Fig. 1c). We aimed to ensure that the distribution of inter-pFC across all pairs of individuals showed comparable variation to the distribution of inter-eFC. As a secondary measure of structure-function coupling, for each individual, we also computed the correlation between SC inferred from the diffusion-weighted MRI data and eFC and pFC, referred to as SC-eFC and SC-pFC, respectively (Fig. 1c). Pearson correlation is the most common measure of structure-function coupling and requires minimal assumptions (3,12,17,36).
To benchmark the performance of the neural network, a state-of-the-art biophysical model (31) was trained to perform the same prediction tasks. Parameters of the biophysical model were fitted as part of a cross-validation framework, following procedures described by Wang and colleagues (18). We use the nomenclature pFCNN and pFCBM to differentiate between predictions of functional connectivity from the neural network and biophysical model, respectively. In cases where the distinction is selfevident, we simply use pFC to denote predicted function connectivity. Further details can be found in Materials and Methods.

Trade-off between prediction accuracy and inter-individual heterogeneity
We trained neural networks to maximize the correlation between eFC and pFC, subject to ensuring that inter-individual variability was preserved. We hypothesized that neural networks can, at least in principle, learn complex mappings between SC and FC, without making any assumptions about specific neurophysiological mechanisms (20). The intraindividual similarity between eFC and pFC was maximized by minimizing the squared error between these two matrices when training the neural networks, whereas interindividual heterogeneity in pFC was controlled by a tuning parameter, denoted here with (Materials and Methods).
The choice of was found to parameterize a trade-off between intra-individual eFC-pFC correlation strengths (i.e. prediction accuracy) and inter-individual variability in pFC (i.e. prediction heterogeneity). Increasing the value of resulted in less interindividual variability in pFC (Fig. 2b), but higher prediction accuracies (Fig. 2c). To arbitrate this trade-off, we mapped inter-individual variability in eFC (Fig. 2a) and selected the value of ( ) that resulted in the best match between the mean inter-individual variability in pFC and eFC. This ensured that interindividual correlation in pFC was similar to eFC.

Prediction of group-averaged functional connectivity
Structure-function coupling in the human connectome is typically investigated using group-averaged measures of connectivity (4,12,18,21,36). Therefore, before seeking to predict FC from SC for individuals, we first aimed to evaluate structure-function coupling at the group level using both the biophysical model and neural network. Using

Prediction of individual functional connectivity
Having found that group-level structure-function coupling is substantially stronger than suggested by prior studies, we next considered the more challenging task of predicting function from structure in individual participants. Using 10-fold cross-validation, FC matrices were predicted for each individual comprising the test set of each fold using either the neural network or biophysical model. The parameters of the biophysical model were estimated from the group-averaged connectivity matrices for the training set of each fold and the estimated parameters were then used to predict FC for each individual comprising the test set.
Prior tuning of the neural network ( Fig. 2) ensured that inter-individual heterogeneity was well matched between the predicted and empirical FC matrices, as indexed by the correlation between matrices from pairs of distinct individuals (Fig. 3e). This provided assurance that the neural network was not simply predicting a group-average representation of FC for each individual. The biophysical model also preserved interindividual heterogeneity quite well (Fig. 3e). We found that the neural network yielded significantly more accurate predictions of individual eFC matrices compared to the biophysical model (eFC-pFCNN: , eFC-pFCBM: , Fig. 3f, leftmost boxplots). However, prediction accuracies varied considerably between individuals and were lower in accuracy compared to predictions of group-averaged FC.
Interestingly, the FC matrix predicted by the biophysical model for each individual (pFCBM) was more strongly correlated with the individual's SC matrix than their eFC matrix (Fig. 3f, rightmost boxplot), suggesting that the biophysical model recapitulated SC to a greater extent. Indeed, the coupling between SC and pFCBM was substantially greater than the coupling between SC and eFC (Fig. 3f). In contrast, the coupling between SC and pFCNN was comparable to the coupling between SC and eFC ( ).
Therefore, the neural networks not only accurately predicted function in the connectome, the predicted FC matrices (pFCNN) also preserved the linear relationship between structure and function, as indexed by the correlation coefficient. These findings were replicated in the supplementary dataset ( Supplementary Figs. S1 and S2).

Does connectome topology inform the prediction of functional connectivity?
Having found that neural networks can accurately predict the function of both groupaveraged and individual structural connectomes, we next sought to investigate the extent to which unique topological structures in the SC matrices were utilized to inform the predictions of FC. To this end, we used the Maslov-Sneppen rewiring algorithm (44,45) to randomly rewire each individual's SC matrix. This resulted in the randomization of high-order topological structure in SC, but not basic properties such as node degree and connection density. We generated randomized SC matrices for each individual and provided the randomized SC matrices to the neural networks that were trained on the true (non-randomized) data. We did not re-train the neural networks using the randomized SC matrices, but rather sought to test the extent to which removal of topological structure would decrease prediction accuracies. The FC matrix predicted for each of the randomized SC matrices for each individual was correlated with the corresponding individual's eFC matrix, thereby generating a null distribution for the correlation between empirical and predicted FC.
Using this approach, the null hypothesis of high-order topological structure in SC not informing the prediction of function could be rejected for (n=820) of individuals (p ). Fig. 4 shows SC (Panel a), eFC (b) and pFCNN (c) matrices for four representative individuals, as well as scatter plots showing the correlation between eFC and pFCNN (d) and the null distribution for this correlation (e). Randomizing the SC matrices resulted in a to reduction in prediction accuracies, on average ( Supplementary Fig. S3a). Furthermore, the higher inter-individual correlation in the pFCNN matrices predicted from the randomized SC matrices (Supplementary Fig. S3b) suggests that the topology of the SC matrices is also important for preserving interindividual heterogeneity. We conclude that the tight coupling between structure and function predicted by the neural networks cannot be attributed to trivial topological properties for most individuals, although node degree and other basic properties preserved by Maslov-Sneppen rewiring are clearly informative for a minority of individuals (see Discussion).
Control analyses were undertaken in which the group-averaged eFC matrix for the training set of each fold was correlated with the group-averaged eFC matrix for the test set ( Supplementary Fig. S3a). This correlation was significantly higher than eFC-pFCNN Finally, in supplementary analyses, the same neural network architecture was trained using randomized structural connectomes to predict actual (non-randomized) FC matrices. We found that training with randomized SC yielded significantly lower prediction accuracies compared to training with the true SC matrices (p<0.005), as shown in Supplementary Fig. S8. This suggests that the architecture of the individual structural connectomes significantly contributed to prediction performance.

Cognition associates with predicted functional connectivity
The ability to accurately predict an individual's functional network from their structural connectome raises the prospect of using predicted functional connectivity to explain inter-individual variation in behavior. We aimed to assess whether predicted functional connectivity could explain inter-individual variation in five measures of behavior and cognition with comparable accuracy to empirical functional connectivity inferred from resting-state functional MRI.
We compared empirical (eFC) and predicted functional connectivity (pFC) with respect to accuracy in predicting inter-individual variation in five behavioral measures characterizing: i) cognition, ii) illicit substance use, iii) tobacco use, iv) personality and emotional traits, and iv) mental health. These five measures were parsed from behavioral items using independent component analysis (ICA), as described in detail elsewhere (42). Individuals with missing values for some of the behavioral items were excluded, leaving individuals in the primary dataset and in the supplementary dataset. We used lasso regression in a 10-fold cross-validation to predict interindividual variation (46), where functional connectivity between ( ) pairs of regions was included in the feature space. Given that SC was correlated with pFCBM ( Fig. 3f), we regressed SC from both the empirical (eFC) and predicted functional connectivity (pFCBM and pFCNN) matrices and the residuals were used in the lasso regression.
After correcting for multiple comparisons across the five behavioral measures, eFC, pFCNN and pFCBM were each found to predict significant inter-individual variation in cognitive performance (p ), but not the other four behavioral measures. A comparison of predicted and actual cognition prediction for 2 bootstraps is shown in Fig. 5(a-c), whereas an average across 200 bootstrapped samples is presented in Fig. 5d.
The correlation between predicted and actual cognitive performance was highest for eFC ( , Fig. 5a), followed by pFCNN ( , Fig. 5b) and pFCBM ( , Fig. 5c). This ranking was replicated in the supplementary dataset ( Supplementary Fig. S9). Therefore, while empirically acquired functional MRI data explained the most inter-individual in cognitive performance, if a reduction in explanatory power is tolerable, neural networks could potentially be used to predict functional connectivity without acquiring any empirical functional MRI data. The relation between individual differences in SC-FC coupling and cognitive performance was also investigated, but no significant association was found for eFC, pFCNN and pFCBM (all ).
To further compare eFC and pFC, we identified subnetworks of connections that were associated with cognitive performance. To this end, we utilized the network-based

Discussion
Biophysical models characterize the relationship between synaptic connectivity and neural dynamics, enabling detailed investigation of structure-function coupling in nervous systems across different species and scales (16,28). These models suggest that structure is at best moderately coupled to function (18). Network communication theory (20,21) and statistical approaches (22,23) provide comparable estimates of structurefunction coupling in the human connectome. In this study, we found that structure and function are significantly more tightly coupled in the human connectome than previously suggested by state-of-the-art biophysical and network communication models. Analogous to most biological systems, our study suggests that the structure and function of mesoscale brain networks is indeed inextricably linked, and there is a significant margin to improve existing models. Mechanistic insight into the physiological and network principles underlying structure-function coupling in nervous systems will likely be advanced with refined biophysical models and graph-theoretic analyses. Recent studies demonstrate that incorporating the hierarchical organization of the cerebral cortex in the biophysical model can improve the accuracy of FC prediction (29) and graph theory can potentially increase our understanding of the association between indirect SC connections and FC (48).
Previous studies seeking to predict functional brain networks from the structural connectome have typically considered prediction of group-averaged networks. Using a biophysical model, Honey and colleagues (12) achieved a correlation between empirical and simulated functional connectivity of . More recently, studies have sought to predict brain function from structural connectomes using network communication models (49), connectome embeddings (50) and statistical approaches (22,23). These studies report comparable or somewhat lower accuracies than Honey and colleagues.
However, comparing measures of structure-function coupling between studies is challenging due to critical differences in experimental design, such as Importantly, we were able to predict brain function for individual participants based on their structural connectomes, yielding personalized predictions. The ability to predict personalized functional connectivity opens the possibility of investigating associations between brain function and inter-individual variation in cognitive performance and behavior, without using empirical functional MRI data. That is, functional connectivity predicted from an individual's structural connectome can be used in lieu of empirically measured functional connectivity. Given that functional connectivity explains greater inter-individual variation in behavior than structural connectivity (49), but measuring functional connectivity empirically is challenging due to head motion, physiological confounds and other methodological issues, it may be possible to investigate behavioral associations using functional connectivity predicted from an individual's structural connectome. This may be particularly advantageous when empirically acquired functional MRI data is low quality or contains artifacts. However, the high-quality functional MRI data used here explained 10-15% more inter-individual variation in cognition than functional connectivity predicted from structural connectomes.
Functional MRI also enables measurement of dynamics in functional MRI, whereas we only considered the prediction of static functional connectivity in this study. Therefore, while neural networks may enable brain function to be inferred from anatomy in circumstances where empirical functional MRI data is not acquired or contaminated with artifacts, functional connectivity should be measured empirically whenever possible.
The prediction of group-averaged and individual functional connectivity was replicated in a supplementary dataset in which the structural connectomes were mapped using an alternative tractography pipeline (Supplementary Figs. S1-S2). This ensured that the predictions were robust to connectome construction choices. Also, the participants selected for the supplementary dataset were not genetically related, which demonstrates that the high accuracy of the proposed method for the primary dataset was not influenced by inherited similarities in structural and functional connectivity.
Supplementary analyses were also undertaken to ensure prediction accuracies were not contingent on the choice of brain parcellation atlas ( Supplementary Figs. S4-S7). Mean square error was used as a supplementary metric to evaluate the performance of the models (Supplementary Fig. S11). Mean square error was lower for pFCNN compared to pFCBM, particularly for the primary dataset. While accurate predictions of individual functional connectivity were achieved ( ), further improvement may be possible by using a larger number of individuals to train the neural network or stratifying the training process according to demographic features, such as age and sex.
The objective function used to assess prediction accuracy is an important consideration.
In this study, we aimed to maximize the correlation between empirical and predicted functional (eFC-pFC), subject to preservation of inter-individual differences in functional connectivity. The latter was achieved with a hyperparameter that was carefully tuned to ensure that inter-individual correlation was on average matched between empirical (eFC) and predicted (pFC) functional connectivity. As a supplementary measure of prediction performance, we also investigated the correlation between structural and functional connectivity, a classical measure of structure-function coupling. Interestingly, functional connectivity predicted by the neural network inherently preserved the extent of empirically measured structure-function coupling (SC-pFCNN: r ; SC-eFC: r ). This was not the case for the biophysical model, which tended to inflate the extent of empirically measured structure-function coupling (SC-pFCBM: r ), suggesting that the predicted functional connectivity largely recapitulated structural connectivity, rather than capturing high-order interactions in function.
However, the performance of the biophysical model improved in this regard for groupaveraged prediction (SC-pFCBM: r ; SC-eFC: r ). Hence, while biophysical models can characterize group-averaged brain function, neural networks may be able to better unveil more subtle individual differences that underpin the complex mapping from structure to function in the connectome.
Mean square error was used as a loss function for training the proposed neural network, which ensured that the spatial embedding of the SC and FC matrices were preserved during the training process. We confirmed that the neural network learned nontrivial topological features of the structural connectome; that is, predictions were not driven by global effects and/or potential artifacts. Providing the trained neural network with topologically randomized structural connectomes significantly reduced prediction accuracies for most individuals. It is important to remark that the randomization procedure preserved the degree of each region and the overall connection density of the structural connectome. These basic network features could have been leveraged by the neural network and may explain why randomization did not lead to a decrease in prediction accuracy for some individuals. It might be argued that the neural network simply learned a prototypical functional connectivity architecture that was representative of the training group, without learning inter-individual differences in the mapping from structure to function. To refute this argument, we showed that predicted functional connectivity explained significant inter-individual variation in cognitive performance. Furthermore, we found that group-averaged empirical functional connectivity computed in the training data was on average a better predictor, compared to the individual predictions yielded by the neural network ( Supplementary Fig. 3a).
Hence, if preservation of inter-individual differences is not important, group-averaged measures of connectivity should be considered.
Several limitations require consideration. First, unlike biophysical and network communication models, deep learning does not provide insight into the putative neurophysiological mechanisms that underpin structure-function coupling in nervous systems. However, the advantage of deep learning is that complex relations between brain structure and function can be uncovered, without preconditioning on any specific mechanisms. Also, the proposed neural network does not guarantee that the predicted functional connectivity is a covariance matrix. However, sophisticated neural networks, e.g. transformers (51) that take into account the dependencies between data points could be used in the future to satisfy the covariance constraints of functional connectivity. Second, it is inevitable that the structural connectomes comprised a proportion of spurious connections due to tractography limitations (52). Given that systematic errors in connectome reconstruction are most likely not informative of brain function, it is improbable that the neural networks learned to utilize spurious connections to predict functional connectivity, although this possibility cannot be discounted. Finally, the performance of the neural network is dependent on the training dataset, including the specific brain parcellation scheme, connectome reconstruction pipeline, measures of structural and functional connectivity and data pre-processing methodology. The neural network is unlikely to generalize to connectomes that are constructed using alternative pipelines and we advise against applying the trained network to connectomes that were not constructed according to the pipeline used in this study. To improve generalizability, future work should consider diversifying the training dataset with the inclusion of connectomes from a variety of reconstruction pipelines and diffusion MRI data of varying quality. Multiple methods for pre-processing rs-fMRI data and estimating connectivity matrices are available. Global signal regression can alleviate head motion and respiration confounds, which may improve the estimation of eFC (53).
Moreover, eFC can also be computed using mutual information or dynamic time warping to better account for the non-linearities in the signal (54). Similarly for structural connectivity, streamline counts and lengths can be normalized to control for distance biases and variation in region sizes (8). These alternatives can potentially impact the estimation of structure-function coupling and should be investigated in future work. The FC prediction models evaluated in this study did not take the distance between nodes into account explicitly. Modelling the effects of geometry on structure-function relationships can potentially improve the associations reported here.
In conclusion, we found that structure-function coupling in the human connectome is much tighter than suggested by previous studies. While we were unable to uncover mechanisms underpinning the tight coupling between brain structure and function, our work provides insight into the margin by which biophysical and network communication models of structure-function coupling can be improved. The advantage of deep learning is that the mapping from an individual's structural connectivity matrix to functional connectivity can be learned in the absence of any explicit model. We were able to generate personalized predictions of functional connectivity for specific individuals based on their structural connectome and demonstrated that these predictions explained significant inter-individual variance in cognitive performance. In the future, deep learning could potentially be used to infer brain function for individuals without empirically acquired functional MRI data.
Code and data availability: The code for the proposed framework is publicly available at https://github.com/sarwart/mapping_SC_FC. The HCP dMRI, rs-fMRI and behavioral data are publicly available (https://www.humanconnectome.org/).