Abstract
A challenging problem in cognitive neuroscience is to relate the structural connectivity (SC) to the functional connectivity (FC) to better understand how largescale network dynamics underlying human cognition emerges from the relatively fixed SC architecture. Recent modeling attempts point to the possibility of a single diffusion kernel giving a good estimate of the FC. We highlight the shortcomings of the singlediffusionkernel model (SDK) and propose a multiscale diffusion scheme. Our multiscale model is formulated as a reactiondiffusion system giving rise to spatiotemporal patterns on a fixed topology. We hypothesize the presence of interregional coactivations (latent parameters) that combine diffusion kernels at multiple scales to characterize how FC could arise from SC. We formulated a multiple kernel learning (MKL) scheme to estimate the latent parameters from training data. Our model is analytically tractable and complex enough to capture the details of the underlying biological phenomena. The parameters learned by the MKL model lead to highly accurate predictions of subjectspecific FCs from test datasets at a rate of 71%, surpassing the performance of the existing linear and nonlinear models. We provide an example of how these latent parameters could be used to characterize agespecific reorganization in the brain structure and function.
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Introduction
Over the last two decades, investigation of the slow correlated fluctuations in the resting state functional magnetic resonance images (rsfMRI) of the brain have yielded valuable insights about the spontaneous functional organization of the brain. The structural connectivity (SC) derived from diffusion tensor imaging (DTI) reveals the whitematter fiber connections between regions of interest (ROIs)^{1,2}. On the other hand, functional connectivity (FC) estimates the linear statistical dependency between ROIs based on blood oxygen level dependent (BOLD) activation^{3}. It has been observed that BOLD signals are nonstationary in nature^{4,5,6}. This nonstationarity is understood in terms of states (measured in small time windows) and their transitions and are expressed as functional connectivity dynamics (FCD)^{7}. On the other hand, static FC enables investigation of the analytical properties of network behavior^{8}.
Even in resting conditions BOLD activations of ROIs have been observed to form selforganizing patterns called resting state networks (RSNs). FC can be interpreted as a superposition of all these RSNs. Relating structural connectivity (SC) of the brain to its functional connectivity (FC) is a fundamental goal in neuroscience because it enables our understanding of how the relatively fixed SC architecture underlies human cognition and diverse behaviors^{9}. It has been observed that regions with direct structural link are also functionally connected. However the converse is not necessarily true^{10}. The question of how SC shapes FC has been the object of computational modeling but remains an open question^{11}. In the recent years, connectivity analysis using whole brain computational models and graph theoretic techniques have given unprecedented insights about brainwide correlations during rest and task conditions^{12,13,14}. Computational models are designed to expand our understanding and explaining the functioning of the brain. The more biologically real the model is, the more computationally expensive it is. Hence, gaining analytical insights becomes increasingly difficult with complex models.
In the realm of noiseinduced correlated deviations, there are linear and nonlinear mean field models that attempt to answer this open question incorporating various kinds of dynamics and biological details^{15,16,17,18}. A biophysical attempt to relate SC to FC is a linear model based on graph diffusion of brain dynamics is outlined in^{18}. This linear diffusion model considers brain dynamics, the diffusing quantity, firing rate of the neuronal population, undergoing random walk on the SC graph. This linear diffusion model considers that the mean regional activity diffuses over the anatomical fibers governed by a deterministic linear differential equation^{18}. The analytically tractable solution becomes the graph diffusion kernel which is hypothesized to resemble the FC. This model fixes one global parameter across all subjects. Another model proposed by Saggio et al.^{19} considers a linear autoregressive model with additive Gaussian white noise, coupling matrix being SC. This model becomes a linear system of coupled stochastic first order differential equations in which the BOLD activities diffuse on the anatomical constraints, i.e. SC. This model computes covariance between regional activities whose analytical expression works out to be a function of SC. Such a model would find it difficult to account for intersubject variability in the functional expression. Extending the idea of linearity to supercritical bifurcations and multistability, a series of nonlinear stochastic models have been proposed that explain the underlying biological behavior efficiently^{20,21,22}. These models differ in their representation scheme for the ROIs^{23}. Whereas Kuramoto oscillator model^{24} abstracts out the biophysical details, Deco and Jirsa’s meanfieldmodels^{20,21} consider dynamics of specific biological analogues such as mean firing rate and mean activity of the regions. These neural and mesoscopic models can be seen as variants of reactiondiffusion system at the heart of which lie the WilsonCowan equations^{25,26}. WilsonCowan equations, a variant of reactiondiffusion systems, provide a coarsegrained description of the largescale neuronal network in terms of oscillatory selforganizing patterns. New experimental evidence supports these equations^{27}.
Recently, a new paradigm of understanding the oscillatory patterns of corticocortical activity is proposed that utilizes spectral analysis of the connectome or structural connectivity (SC)^{28}. It has been observed that these connectomespecific harmonics predict oscillatory functional networks of the human brain possibly through interplay of excitation; for instance mediated by the glutamatergic principal cells, and inhibition; for instance mediated by the GABAergic interneurons. The pushandpull between diffusing excitatory cells and suppressing inhibitory cells can result in selforganizing pattern formation. The emergent harmonics or the standing waves are the allowed spatial frequencies, or the eigenfunctions of the graph Laplacian operator on the anatomically constrained SC largely determined by the selection of the diffusion parameters of excitation and inhibition. Surampudi et al.^{29} observed that physical diffusion on largescale graphs, i.e. SC, at multiple diffusion scales exhibits scaledependent relationships among various regions of interest (ROIs). These multiscale diffusion kernels are similarly motivated to capture reactiondiffusion systems operating on a fixed underlying connectome (SC) and hence can be interpreted as components of FC at different diffusion scales (see Supplementary Fig. S1 for the motivation of multiscale approach). However, our investigations revealed that a combination of multiple diffusion kernels was not sufficient to explain the selforganizing resting state patterns found in FC and hence necessitates the need of additional explanatory parameters.
The extant whole brain computational models can be characterized along two dimensions–interpretability and complexity, where the linear and nonlinear models lie at the opposite ends of the spectrum. The former are analytical models with few parameters that can be interpreted and tuned easily, whereas the latter are fairly complex models with richer dynamics but tend to be analytically intractable. The proposed model possesses the analytical beauty of linear models and yet is complex enough to capture the biological details. We hypothesize that the presence of regional multiscale coactivations that initiate diffusion would be necessary to bridge the gap between structurally confined diffusion phenomenon and empirically observed FC and that these coactivations would be common across the cohort. We further provide a plausible mathematical reasoning for the existence of these coactivations along with diffusion kernels by linearizing a variant of reactiondiffusion model and extending it to generate FC. Moreover, we also describe a succinct multiple kernel learning (MKL) procedure to retrieve these coactivations by formulating it as an optimization formulation. MKL techniques are well explored in the machine learning community^{30,31}. Our proposed model while retaining the parsimony of a simple linear approach, proposes a novel learning scheme for optimizing the bestfitting kernels for SCtoFC mapping. Our detailed empirical results demonstrate the validity of the proposed model on a larger dataset.
Results
Model description
Our model is based on the hypothesis that there are interregional multiscale coactivations (among ROIs) for initiating diffusion. These coactivations that are shared across all subjects enable our model to generate the observed FC patterns from the structurally confined diffusion kernels. This is achieved by first postmultiplying the coactivation parameters (denoted henceforth as π_{ i }′s) with the respective diffusion kernel matrices derived from subjectspecific structural matrix (SC) and then taking a linear combination of these for approximating/predicting the functional connectivity (FC). We propose to learn these coactivation parameters in a multiple kernel learning (MKL) framework, thus retaining the parsimony of a simple linear approach. In the experiments reported in this paper, 16 diffusion kernels were used and their respective interregional coactivation parameters (16 corresponding π_{ i }′s) were learned (see Supplementary Figs S2–S4 along with their corresponding sections for scale selection methodology). To evaluate our model, we mainly used the data scanned at the Berlin Centre for Advanced Imaging, Charité University, Berlin, Germany that has SCFC pairs of total 47 subjects (see Supplementary section 8 for data preprocessing methodology).
Model performance
We compared performance of the proposed model (henceforth designated as MKL model) with two previously proposed approaches: singlediffusionkernel (SDK) model of Abdelnour et al.^{18} and the nonlinear dynamicmeanfield (DMF) model described in Deco et al.^{32} (see Methods for implementation details of these models). To remain consistent with the previous studies, we used Pearson correlation coefficient between empirical and predicted functional connectivities (FC) as the measure of model performance. To obtain a benchmark, we computed the Pearson correlation between empirical SCFC pairs for all subjects and found mean value for these correlations to be 0.3 with a standard deviation of 0.02. These values are taken as baseline correlation values henceforth.
Figure 1 shows the performance comparison of the proposed method with other two models in three different setups. In the first setup, a randomly chosen set of half of the subjects was used for training (23 pairs) and the other half (23 pairs) for testing. Figure 1(a) shows the model performance for all the test participants for the three models. Since SDK and DMF models do not incorporate learning in their formulation, we gleaned the optimal values based on the training subjects. The optimal parameter settings were taken as the values at the mode of the performance distribution histogram for the training set and the same were used for estimating FC for test subjects. We took the best fitting scale on the training subjects for the SDK model and similarly selected the optimal global coupling parameter, G, for the DMF model. Optimal scale for SDK model worked out to be 0.8 and similarly optimal value of G for DMF was 2.85. As can be seen from Fig. 1(a) and (b), the MKL model performs consistently better for each test subject when compared to the other two models. In the remaining two setups, in order to crosscheck whether MKL model suffers from overfitting, we computed leaveoneout (Fig. 1b) and 5fold crossvalidation (Fig. 1(c)) results. These results clearly show the consistency in the performance of MKL model and indicate that the performance is not due to overfitting nor it is due to any particular optimistic traintest split.
In all the experiments and for all the three models in order to compare group statistics, we compute the predicted FC for each test subject and then find the Pearson correlation coefficient with the corresponding empirical FC, followed by taking the mean of all these correlation coefficients. We designate the resulting mean correlation as mean FC in the rest of the paper.
EdgeConnectivity Analysis
Mean FCs are visualized primarily in two modes, via the edgeconnectivity pattern analysis and using the seedbased connectivity analysis. To understand the edge and node distribution across the communities, we rendered the mean predicted FCs on brain surface. The visualization of edgeconnectivity patterns of four mean FCs is shown in Fig. 2 (see Supplementary Fig. S6 for the estimated FCs). In the figure, the colors demarcate the communities for a particular model on the corresponding brain surface. It can be seen that the community structure of the mean FC predicted by the MKL model (shown in Fig. 2(b)) best resembles that of the mean empirical FC (shown in Fig. 2(a)). The other two models predict either a dense FC network (as shown in Fig. 2(c) for the DMF model) or a sparse FC network (as shown in Fig. 2(d) for the SDK model), where both the scenarios are far from the empirically observed network. Additionally, the predicted mean FC by MKL model and the empirical mean FC seem similar in terms of community assignment and interhemispheric connections.
Further, to see elementwise variance in the mean predicted FCs, we also drew scatter plots between the predicted and empirical FCs in Supplementary Fig. S7, where only the nondiagonal lower triangular matrix entries were displayed. These plots suggest that MKL model preserves the global structure of the empirical FC as well as the elementwise connectivity patterns significantly better than the other models.
Seedbased Connectivity Analysis
To further validate the nature of reconstruction of the connectivity patterns for various ROIs, we performed a seedbased correlation analysis using the mean FC matrices predicted from the three models. We chose the left posterior cingulate cortex (PCC) as a seed region since it has been known to have an important functional role as a hub region of the default mode network^{33}. Figure 3 plots the correlation values between left PCC and all other regions on the brain surface reconstructed from the DesikanKilliany atlas^{34}. Cool (hot) colors suggest low (high) connectivity (correlation) of that particular region with the left PCC (shown as dark red color in the mean empirical FC). The MKL model could reconstruct the connectivity pattern with higher precision than the other two models. It appears that due to very high correlation of left PCC with all other regions, DMF model could not as clearly distinguish the boundary between regions. SDK model could not possibly distinguish them due to very sparse connectivity between left PCC and all other regions.
Effect of Thresholding
The richclub organization of the structural connectivity (SC) matrix is previously demonstrated to be the backbone for generating the functional connectivity patterns^{35,36,37}. Therefore we set out to investigate the impact of using thresholded SCs for predicting FCs. We pruned the SCs of all subjects by keeping only top T% of the connections (see Supplementary Section 1.4 for details). Each of these sparse matrices was passed as input to the learned MKL model. For each sparse SC, corresponding FC was predicted. Prelearned π_{ i }′s were used for predicting FCs in the MKL model. Similarly, fixed diffusion scale and G parameters were used for SDK and DMF models for comparative evaluation. Figure 4 shows the mean correlation between empirical FC and predicted FCs for each of the sparse SC matrices. As can be seen, DMF and SDK models attain their respective optimum performance even when only few (as low as 10%) strongest edges in SC remained. On the other hand, MKL model requires both strong edges and few local edges, and hence its performance starts increasing from T = 15% and is significantly superior at all thresholds above this value. This result suggests that functional patterns may be primarily decided by the initial coactivations captured by π_{ i }′s and that the structural constraints of individual SCs provide paths for these activities to diffuse, giving support to our hypothesis. Nevertheless, stable performance with sparsification as high as with T = 20% indicates that all the models obey the basic richclub principle.
Interestingly, MKL model captures the differences in sparsity levels better when compared to the other two models, especially when SC was pruned to keep the strongest edges between 10–20%. This behavior suggest that pretrained π_{ i }′s in MKL model do not compensate for major loss of information in sparsified SCs, thereby indicating avoidance of overfitting. Please see Supplementary Section 1.4 for the exact threshold values for this procedure.
Robustness of the MKL Model
The proposed model learns a latent representation, Π that maps the relationship between SC and FC. This being the crucial difference between MKL and other models, we performed extensive robustness tests to verify the usefulness of learning the π_{ i }′s. To ascertain that the model’s representation learns important features and does not capture the SCFC mapping by chance, we conducted the following four randomization experiments. In the first one we randomize the input to the model (i.e., SCs) (see Fig. 5) and in the second the learning itself is conducted based on perturbed SCs (see Supplementary Fig. S5). In the third experiment we disturb the scalespecific relation between the learned π_{ i }′s and H_{ i }′s (see Eq. 21) and finally in the fourth experiment the constituent rows of π_{ i }′s are randomly permuted (see Fig. 6).
Perturbing the model input
To verify whether the model learns the SCFC relationship correctly or predicts the average FC independent of SC, we provided the MKL model with perturbed SCs in two possible scenarios: first, while testing, and second, while training.
Each subjectwise SC was perturbed N = 250 times, hence forming 250 sets of subjectwise perturbed SCempirical FC pairs (please see Supplementary section 1.5 for perturbation procedure). In the first perturbation analysis, we trained the MKL model with the original subjectwise training SCFC pairs, and tested the model with these 250 perturbation sets. These same sets were used for evaluating the other two models. We calculated the mean correlation values between predicted and empirical FCs, thus obtaining 250 mean correlation values for every model. Figure 5(a)–(c) show the histograms of these mean correlation values for MKL, DMF and SDK models, respectively. As expected, all the three models have significant drop in their performance indicating their sensitivity towards meaningful SC matrices while arriving at predictions.
In the second perturbation analysis, we trained N = 250 MKL models using the 250 purtubed sets and evaluated them using the subjectwise empirical SCFC pairs. We did not have to perform this analysis for the other two models as this analysis is the same as that of the above for these models. Supplementary Fig. S5 shows the histogram of the 250 mean correlation values that is distributed across a wide range of correlation values instead of peaking at a particular value, thus indicating a garbagein, garbageout phenomenon from a machine learning perspective! This result, along with the results in Fig. 4, demonstrate that MKL model is not learning just a transformation from a subgraph of SC to an average FC but that the learning is holistic.
Altering the model parameters
After confirming that the model does not learn a random mapping between SCFC pairs, we alter the learned mapping to further confirm model’s robustness. We considered two ways of altering the model parameters (π_{ i }′s). These parameters are mathematically represented as a set of m matrices (π_{ i }′s) corresponding to m diffusion scales (m here is set to 16, also see Eq. 21). We sought to experimentally verify that Π can be interpreted as holding complementary information of a cohort of SCs. Hence it is likely that any perturbation of Π would disturb the synergistic correspondence to empirical SCs and cause performance degradation. In order to experimentally validate this intuition, we ran two types of permutation tests.
Firstly, we sought to estimate the importance of the arrangement of π_{ i }′s, i.e., the ordering of the scalespecific matrices constituting Π. For this we swapped every matrix π_{ i } (1 ≤ i ≤ m) one at a time with π_{ m } (corresponding to the lowest scale, i.e., π_{16}). Figure 6(a) shows the mean correlation while performing swapping. Pearson correlations are plotted against the swapped indices. Because of noswap the last correlation (corresponding to i = 16) depicts optimal performance. This plot suggests that indeed matrices have positional significance (in other words, scalespecificity), so they cannot be reorganized to predict FC. This is a property that is also subtly captured in Eq. (21) in the sense that these matrices have a strict correspondence to their scales, consequently they embed scalespecific diffusion kernels to enable correct prediction of FC.
Secondly, we sought to estimate regional importance of the entries of π_{ i } matrices across scales. We concatenate all m π_{ i }′s into a single matrix (Π) of size mn × n. We permute the rows of this matrix and test the model performance. A row of Π captures regional coactivations at that scale between the region corresponding to that row and all other regions. We permute the rows of this large matrix 250 (N). Each newly generated Π is used for testing the model performance. Figure 6(b) shows the histogram of the mean correlations of all the N permutations. Clearly the plot shows that permuted Π significantly deteriorates the model performance. This figure underlines the importance of maintaining the structure of coactivation between pairs of regions.
Discussion
The holy grail in cognitive neuroscience is understanding how the static brain structure gives rise to dynamic function both during rest and task conditions. Several models have been proposed to characterize the structurefunction relationship^{38}. Simple linear diffusion models^{18,19} as well as complex nonlinear, wholebrain computational models^{32} have been proposed. Linear graph models^{18} admit closed form deterministic and testable solution to macroscopic interactions of brain activity without requiring any details of neural coding or their biophysical substrate. On the other hand nonlinear complex driftdiffusion models based on excitatory and inhibitory neuronal populations, though not analytically tractable, give rise to rich dynamics^{32}.
Abdelnour et al.^{18} conceived a model of functional connectivity (FC) with only one diffusion kernel defined at an optimal scale. This optimal kernel operates on an identity matrix, meaning that the amount of activity reaching other regions from a single source is representative of the statistical dependence between those regions. This statistical dependence resembles activity heat maps which exhibit interindividual variations. However, Surampudi et al.^{29} showed that single kernel models do not generalize to a larger cohort and demonstrated that FC can be decomposed into multiple diffusion kernels with subject nonspecific combination coefficients.
In this work, we proposed a multiple kernel learning (MKL) method that learns interregional coactivations (denoted as π_{ i }′s) and reshapes the structurally confined diffusion kernels to give rise to functional connectivity estimates. MKL model is a generalization of the SDK diffusion model (see Supplementary and Fig. S1 for the strengths and limitations of SDK models). Resting state functional connectivity could be considered as a signal on a brain graph expressed at multiple different spatiotemporal scales. Our approach essentially finds a way to unfold these solutions on the brain graph combining multiple scales to accurately estimate the empirical FC. One way to interpret the proposed multiscale diffusion model is to treat it as a variant of a reactiondiffusion system on the graph determined by the underlying structural connectivity (SC) matrix.
In this work, we adopt the representation of the graph signal in terms of eigenvectors of the graph Laplacian similar to what has been recently proposed^{28}. The proposed MKL framework devises a scheme for learning the hidden parameters (π_{ i }′s) to estimate FC. The initial regional activity u_{0} in the reactiondiffusion type model is a vector, hence the matrix \({u}_{0}{u}_{0}^{{\rm T}}\) is a rank 1 matrix. As it is a positive semidefinite (PSD) matrix, it will only have one nonzero eigenvalue. Eigendecomposition in Eq. (11) suggests a possible physical interpretation, that the initial mean activity distribution, an eigenvector of the graph Laplacian, resembles standing wave patterns on the graph. Total number of such standing waves is equal to the number of nodes of the graph. Hence our hypothesis is that the initial regional coactivations (π_{ i }′s) correspond to one of the standing waves present at some time k_{ i }τ significantly changing the pattern at that reaction instance (please refer to section titled Methods/Proposed MKL model for notations). Functional connectivity can then be articulated as a superposition of such standing wave patterns and their regional coactivations.
In order to predict FC from the proposed diffusion model, we estimated Π by solving a LASSO optimization formulation. We hypothesized that these hidden parameters are learnable from the training data and remain fixed at the time of testing. Consequently different FC matrices for the test subjects would be arising by virtue of the underlying differences in the respective structural connectivity matrices (SCs). This would mean that the parameters Π are not merely a derivative of SCs but instead they complement the missing aspects by capturing the statistical dependence between two regions that are modulated by some intermediate region that may not be in physical proximity and that too operating at multiple resolutions or scales. Thus by incorporating the inter and intrahemispheric functional connectivity terms for a brain region, the learned optimal Π parameters enable more accurate matching of the structurefunction correlation. All the computational models can be visualized to lie on the spectrum spanned by biological interpretability and analytical ease. Whereas linear models enjoy simplicity of solution of their models, nonlinear models tend to explain the complex biological reality. MKL model seems to find a sweet spot and enjoys best of both by analytically providing the solution and explaining the patterns in terms of largescale excitatoryinhibitory interactions. Since LASSO optimization is the most expensive computational step, the computational complexity of the proposed MKL model would be dominated by the cost of LASSO optimization.
In summary, on the model continuum, the proposed MKL model lies somewhere between simple linear diffusion models^{18,19} and complex nonlinear drift diffusion models^{32}. Consequently, we compared our simulation results predicting BOLD functional connectivity using the proposed model with models at either end of the complexity spectrum. The experimental results showed that the correlation structure of BOLD functional resting state brain networks is significantly well captured by our model. Prediction accuracy of the MKL model for the 23 test subjects is close to 0.70 whereas that of the nonlinear model comes second best at 0.52 and that of the SDK model around 0.37. We conducted a series of tests that perturbed the inputs to the model as well as permuted the learned parameters Π. The test results attest to the robustness of the proposed model. Interestingly the model not only captures the variability of scales across participants but also demonstrates a possible application in characterizing agerelated differences in learning optimal parameters for the accurate estimation of FC (refer to Supplementary section 4 and Figs S8–S10). Even in the face of considerable amount of variability present in the data, the proposed MKL model is still able to predict subjectspecific FCs with high accuracy. Beyond this, functional connectivity subsumes the influence of different regions across scales and age groups providing a viability of Π being a useful parameter for classification purposes for other domains of application in health and disease. Overall, our method might be considered the missing link in the estimation and improvement of predicting subjectspecific restingstate functional connectivity that remained elusive so far for complex nonlinear and linear models. Given the strength of the analytical approach and tractability, the proposed model could be a suitable method for predicting taskbased functional connectivity across different age groups.
One major limitation of our work is that it is not so straightforward like the linear diffusion model to invert the FC to recover the SC matrix. Currently, in the MKL model the procedure to predict SC from FC would rely on a given Π. One way of finding SC is by estimating the diffusion kernels for individual subjects by solving the same system of linear equations used to find FC. Laplacian of a graph could then be estimated. Carefully recovering multiple diffusion kernels might turn out to cause numerical instability to the Laplacian (see Supplementary section 5 for details of the proposed inversion) and this issue needs to be resolved in the future. While in the current formulation we are empirically determining the number of scales (m) and their spacing, optimization formulation could be modified to estimate these automatically. Moreover, the current model does not consider the nonstationary nature of functional connectivity, the so called functional connectivity dynamics (FCD). Future studies can look at optimization procedures of MKL for modeling the dynamic functional connectivity which is more realistic than modeling stationary FC.
Methods
Notations
This section introduces the notations used here as well as in the Supplementary material. Please refer to Table 1 for all the notations.
Proposed Multiple Kernel Learning (MKL) Model
In this section we describe the proposed model as a variant of Reactiondiffusion (RD) systems^{39} wherein the regional mean activities diffuse on the graph determined by anatomical pathways (SC). RD systems explain the formation of complex selforganizing patterns naturally occurring in nature^{40}. RD systems have been employed to model interaction among populations of neurons and the emerging patterns of functional connectivity among neural ensembles^{41,42,43,44}. Just as statistical thermodynamics relates brownian motion of fluid particles to mean motion of a whole fluid, WilsonCowan equations characterize the macroscopic statistical behavior of mean fields of the resulting neural activities^{26,45}. Atasoy et al.^{28} embed anatomical constraints in terms of the graph Laplacian matrix of the SC matrix in the WilsonCowan equations to explain the macroscale excitatory and/or inhibitory interactions of the regional activities. These excitatory and/or inhibitory interactions result in the formation of complex functional patterns such as RSNs. We extend our model from Atasoy et al.^{28} and explain the formation of FC through RSNs. We hypothesize that the cumulative mean activities of all the regions is generated by intraregional microscale dynamics which diffuses interregionally on the structural connectome. We propose a physical model that implicitly captures the pairwise functional interactions between ROIs by explicitly associating them with their extent of influence through the diffusion kernels on the SC (see Supplementary section 3 for details on graph Laplacian and diffusion kernels).
The derivation of the expression for FC consists of five major stages. We consider that FC matrix encompasses effects of diffusion from multiple reactions. In the first stage, we formulate the differential equation for the time evolution of regional activities (Eqs 1–3). In second stage, we characterize the time evolution of the regional activities in an arbitrarily small time interval (Eqs 5 and 6). In the third stage, we integrate the diffusion process over all the connectome harmonics (Eqs 7 and 8). In the fourth stage we accumulate the diffusions happening in various time intervals to generate the complete expression for FC (Eqs 9–21). This FC assumes the form of a combination of diffusion kernels weighted by scalespecific parameters (Eq. 21). In the final stage, we propose an optimization framework for estimating these global parameters (Eqs 22–27).
Let the cumulative mean activities for all regions be denoted by u(t)_{n×1} at time t. We assume that these activities belong to either excitatory and/or inhibitory interactions. The temporal evolution of regional activities are modeled as the following linear variant of WilsonCowan equations:
where \({\mathscr{D}}\) is the spatial propagation operator. τ is a characteristic time scale that speeds up or slows down the evolution of the system. Mean activity of a region i, u_{ i }(t), can be abstracted out from biological details as a onedimensional (1D) time varying signal. A vector of these 1D signals indexed by the nodes of the graph represents a graph signal. We represent the graph signal in terms of its Fourier components using graph Fourier transform^{46}:
where Ψ is the eigenvector matrix of graph Laplacian (see Supplementary section 3) and β (t) is its Fourier representation at time t. With this decomposition temporal dynamics is explicitly represented using spatial basis functions. Further we conceptualize the spatial operator \({\mathscr{D}}\) in the form of a diffusion kernel defined at scale σ^{2}/2 on the structural brain graph Laplacian (Λ) corresponding to the time interval τ between two consecutive reaction instances.
Substituting Equations (2) and (3) and combining the fact that Ψ is invertible, differential Equation (1) can be solved for β (t) which represents the signal evolution in the time interval between two reaction instances, as follows:
where β_{0} represents the initial mean activity. Equation (6) depicts how the mean activity (β_{0}) of every region diffuses on the graph. Finally the graph signal between two reaction times can be expressed in a closed form (substituting Equation (6) in (2)) as:
where, u_{0} = Ψβ_{0} captures initial activity just after reaction, or at the start of diffusion. u_{0} depends on the magnitude of reaction phenomenon, hence may change after every reaction instance. Given the temporal evolution of graph signal, we will next derive how this leads to the evolution of functional connectivity and RSNs. RSNs have unique correspondence with graphharmonics/eigenvectors of the structural graph Laplacian^{28}. We develop the model for a single graphharmonic, i.e., for all RSNs corresponding to that graphharmonic. Finally, we superpose all the patterns of the resting state networks and explain the formation of FC.
The graph signal u_{0} may not change significantly in every reaction. Equation (8) represents the diffusive phenomenon of the graph signal over the characteristic time τ. Let u_{0} change significantly at scalar multiples of τ, i.e., t + k_{0}τ, t + k_{0}τ + k_{1}τ, … with corresponding amplitudes a_{0}, a_{1}, …, respectively. For now we consider generating the functional connectivity \(({{\bf{C}}}_{f}{}_{{k}_{0}})\) for the time interval between two consecutive reactions; at times t + k_{0}τ and t + k_{0}τ + k_{1}τ.
As u_{0} is also a signal on graph, we can express the positive semidefinite (PSD) matrix \({{\bf{u}}}_{0}{{\bf{u}}}_{0}^{{\rm{{\rm T}}}}\) in terms of its eigendecomposition. And as it is only a single harmonic, Δ is a diagonal matrix with only one nonzero entry.
Hence, \({{\bf{C}}}_{f}{}_{{k}_{0}}\) takes the following form:
We can denote the initial activity matrix as \({a}_{0}^{2}{\theta }_{{k}_{0}}\). As reaction instances are not usually far apart in time, instead of double exponentiation we utilize the first order Taylor approximation for the exponent of the integrand; i.e. \({{\bf{I}}}_{n}{e}^{{\rm{\Lambda }}{\sigma }^{2}/2}\approx {\rm{\Lambda }}{\sigma }^{2}/2\). Hence, \({{\bf{C}}}_{f}{}_{{k}_{0}}\) becomes
We call the matrix independent of time as
Now with multiple reactions happening at multiples of τ, we sum over all the reaction instances to get the functional connectivity matrix:
where,
Observing the structure of the FC matrix, FC is conceptualized as being represented by diffusion kernels and their corresponding interregional mean activities. So, the larger the value of k_{ i }, the lesser is its contribution to FC. This means that summation on a finite number of diffusion scales is sufficient for reproducing FC (in this work we considered 16 diffusion scales based on pilot simulations). Now after combining the functional patterns of all the graphharmonics, we approximate empirical FC with m number of diffusion scales γ_{ i }′s. The model thus takes the form as follows:
where, H_{ i } denotes the diffusion kernel at scale γ_{ i }. Further the model in Equation (21) suggests that the scale of diffusion is determined by a characteristic time constant (τ), spatial diffusion variance (σ^{2}) and the time interval between consecutive reaction instances. Matrix π_{ i } represents the scalespecific initial relationships in the mean regional activities.
Proposed model represents the functional connectivity in terms of diffusion kernels operating on scalespecific matrices. It can be inferred that Adelnour et al.^{18} envisage FC comprising only one diffusion kernel defined at an optimal scale. The optimal kernel operates on an identity matrix; meaning only the concerned region has nonzero mean activity independent of other regions, i.e., the amount of activity reaching other regions from the single source is representative of the statistical dependence between those two regions. Surampudi et al.^{29} demonstrated that FC can be decomposed into multiple diffusion kernels whose combination coefficients are unique to the cohort. In addition to the multiple scales, proposed model provides interregional relationships instead of individually active regions. The proposed model generalizes both the aforementioned models as statistical dependence between two regions may be modulated by some intermediate regions without physical proximity that too at multiple resolutions or scales. Moreover, the model provides a biological interpretation of the diffusion scales and has an organic relationship to the reactiondiffusion system.
Optimization formulation
We hypothesize that the global parameters π_{ i }′s are estimated from the training subjects (indexed by s and varies from 1 to p) and remain fixed for all the test subjects. In order to estimate π_{ i }′s we utilize an optimization formulation that minimizes an objective function J comprising the mean squared error between empirical and predicted FCs.
To keep the number of reacting regions less, we employ L_{1} norm on Π^{j}′s.
We apply sparsity on each column and compute every column of Π^{j} separately. The objective function takes the form well known in regression analysis as least absolute shrinkage and selection operator (LASSO) that performs both variable selection and regularization. We passed the respective matrices into lasso solver to get the solution. We arrived at the model parameters experimentally, for example, the number of scales m is empirically chosen.
DMF Model
We used the reduced dynamic mean field model as the nonlinear model for comparative analysis^{32}. This approach considers models with synaptic gating variable with passive decay differential equation along with Gaussian fluctuations. Firing rate was approximated based on inputoutput sigmoid function of the synaptic gating variable. The whole dynamics of each local network of excitatory and inhibitory populations of spiking neurons interconnected via excitatory synapses can be expressed by a single onedimensional equation. The global brain dynamics of the network of interconnected local networks can be described by the following set of coupled nonlinear stochastic differential equations^{32}:
Here S_{ i } is synaptic gating variable of area i. x_{ i } is population mean firing rate for region i. J_{ N } is the excitatory synaptic coupling. ν_{ i } in (28) is uncorrelated standard Gaussian noise with noise amplitude σ = 0.001 nA. I_{0} is the external input current. C_{ ij } represents entries of the SC matrix which captures the structural connectivity between regions i and j. Parameter values were selected as in Deco et al.^{32}. A forward BOLD model was used that converts the local synaptic activity of a given cortical area into an observable BOLD signal. The simulated BOLD signal was downsampled at 2 secs to have the same temporal resolution as in the empirically measured BOLD signal. Simulation length for computing the model FC was equivalent to 8 minutes. The coupling parameter G (see Equation (30)) is varied between 0 to 3. We use individual empirical SC  FC matrices for exploration of subjectwise parameters for optimal fit. The optimal G value varied among the subjects from 0.5 to 3. The mode of the distribution of the parameters obtained for training subjects was taken as the optimal Gs for the training cohort and was found to be 2.85. The same value was used to estimate predicted FCs for all the test subjects.
SDK model
We used a linear diffusion model described in Abdelnour et al.^{18}. This model considers SC matrix of a participant as the weighted adjacency matrix and computes graph Laplacian (described in Supplementary section 3). Then it assumes a set of scales and defines diffusion kernels at each scale. Iteratively for each diffusion kernel, the estimated FC is compared with empirical FC of that subject in terms of Pearson correlation. The scale at which Pearson correlation was found to be maximum is considered subjectspecific optimum scale and its corresponding diffusion kernel is hypothesized as FC. Amongst the training subjects, we found that the mode of the optimal diffusion scale was 0.8, and this was maintained as a fixed parameter for all the test subjects.
Data availability statement
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. The codes for the MKL model are available at https://github.com/govindasurampudi/MKL.
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Acknowledgements
DR is supported by the Ramalingaswami Fellowship (BT/RLF/Reentry/07/2014) from Department of Biotechnology (DBT), Ministry of Science & Technology, Government of India. We acknowledge the efforts of an honors student, Joyneel Misra, at IIITHyderabad who assisted the authors in the mathematical development of the model, improving figures and their captions, and proofreading the manuscript.
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S.G.S., A.S. and D.R. designed the study. S.G.S. and S.N. collected data. S.G.S. and S.N. ran the analyses and generated figures. S.G.S., A.S., R.B.S., D.R. and V.J. analyzed the results as well as the mathematical formulation. All authors have contributed in writing the manuscript.
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Surampudi, S.G., Naik, S., Surampudi, R.B. et al. Multiple Kernel Learning Model for Relating Structural and Functional Connectivity in the Brain. Sci Rep 8, 3265 (2018). https://doi.org/10.1038/s41598018214560
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DOI: https://doi.org/10.1038/s41598018214560
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