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-\title{Dynamic graph signal processing metrics to predict human behaviour}
+\title{Dynamics of brain activity captured by graph signal processing of neuroimaging data to predict human behaviour}
 
 \begin{document}
 
 \name{Thomas A.W. Bolton$^{\star \dagger}$ \qquad Dimitri Van De Ville$^{\star \dagger}$}
 \address{$^{\star}$ Institute of Bioengineering, \'{E}cole Polytechnique F\'{e}d\'{e}rale de Lausanne, Switzerland \\
      $^{\dagger}$ Department of Radiology and Medical Informatics, University of Geneva, Switzerland}
 
 \maketitle
 
 \begin{abstract}
-Multimodal imaging approaches in which structural and functional characterisations of the brain are considered jointly have started to show potential in elucidating the underpinnings of human cognition. In graph signal processing (GSP), functional activation time courses are viewed as signals living on a graph summarising anatomical connectivity. The fraction of the functional signals that is in line with structure (termed \textit{alignment}), and the one that is not (\textit{liberality}), have been linked to behaviour when computed as a temporal average. Here, we examine whether there is also information of interest at the level of temporal fluctuations of alignment and liberality. We consider the prediction of an array of behavioural scores, and show that in many cases, a dynamic characterisation yields additional significant insight.
+Joint structural and functional modeling of the brain based on multimodal imaging increasingly show potential in elucidating the underpinnings of human cognition. In the graph signal processing (GSP) approach for neuroimaging, brain activity patterns are viewed as graph signals expressed on the structural brain graph built from anatomical connectivity. The energy fraction between functional signals that are in line with structure (termed \textit{alignment}) and those that are not (\textit{liberality}), has been linked to behaviour. Here, we examine whether there is also information of interest at the level of temporal fluctuations of alignment and liberality. We consider the prediction of an array of behavioural scores, and show that in many cases, a dynamic characterisation yields additional significant insight.
 \end{abstract}
 
 \begin{keywords}
 Graph signal processing, alignment, liberality, dynamic functional connectivity, behaviour
 \end{keywords}
 
 \section{Introduction}
 \label{sec:intro}
 
 Magnetic resonance imaging (MRI)-based approaches have enabled the quantification of physical wiring between remote brain areas, and the resulting structural connectome is governed by precise organisational rules, in which regions are clustered into modules linked by a group of densely interconnected hubs~\cite{Fornito2015}. This structural scaffold, combined with local nonlinear processing, generates spatiotemporally complex functional activation patterns that can be captured by functional MRI during the resting-state~\cite{Yeo2011}. 
 
 A recently introduced analytical avenue has been to study brain structure and function in conjunction: this way, confounding factors that impede the functional signals, such as physiology- or motion-driven signal changes~\cite{CaballeroGaudes2017}, may be damped out by a structure-informed analysis. Meanwhile, functional brain features of truly neural relevance would be associated to a structural counterpart. 
 
 Recently, whole-brain functional connectivity maps and grey matter density were analysed jointly at the voxel level through deep learning to establish their joint predicting potential in the context of schizophrenia~\cite{Plis2018}. At the coarser scale of a regional brain parcellation, structural connectivity information was also used to inform the total variation-based deconvolution of fMRI data~\cite{Bolton2019d}.
 
-Another well-suited analytical approach is graph signal processing (GSP)~\cite{Shuman2013}, in which a structural graph $\mathcal{G}$ that describes anatomical connectivity between brain regions is constructed, and the functional activation pattern at a given time point $\mathbf{x_t}$ is treated as a signal on $\mathcal{G}$. In the context of an attention switching task,~\cite{Medaglia2016} showed an association between switch cost and \textit{liberality} of the functional signals with respect to the underlying anatomical scaffold. In a subsequent work,~\cite{Preti2019} also observed that the balance between functional \textit{alignment} (the fraction of the functional signals that is in line with structure) and liberality follows a gradient along the cortical surface that also distinguishes lower-order sensory functions from higher-order cognitive ones.
+Another well-suited analytical approach is graph signal processing (GSP)~\cite{Shuman2013}, in which a structural graph $\mathcal{G}$ that describes anatomical connectivity between brain regions is constructed, and the functional activation pattern $\mathbf{x}_t$ at a given time point $t$ is treated as a signal on $\mathcal{G}$. In the context of an attention switching task,~\cite{Medaglia2016} showed an association between switch cost and \textit{liberality} of the functional signals with respect to the underlying anatomical scaffold. In a subsequent work,~\cite{Preti2019} also observed that the balance between functional \textit{alignment} (the fraction of the functional signals that is in line with structure) and liberality follows a gradient along the cortical surface that also distinguishes lower-order sensory functions from higher-order cognitive ones.
 
 Since alignment and liberality have so far been quantified as temporal averages over a full scanning session, the now well acknowledged dynamics of functional brain reconfigurations (see~\cite{Preti2017} for a comprehensive review) is not captured. Here, we explore whether a dynamic GSP characterisation provides additional information to predict an array of behaviourally relevant scores.
 
 \section{Materials and Methods}
 \label{sec:format}
 
 \subsection{Data preprocessing}
 
 We considered structural, functional and behavioural data acquired within the scope of the \textit{Human Connectome Project} (HCP)~\cite{VanEssen2013}. Diffusion-weighted imaging data from $56$ unrelated subjects was processed with the \textit{MRtrix} toolbox~\cite{Tournier2012} (multi-shell multi-tissue response function estimation, spherical deconvolution, tractogram generation with $10^7$ output streamlines between $R = 360$ regions from the Glasser atlas~\cite{Glasser2016}), and averaged to yield a common graph representation $\mathcal{G}=(\mathcal{V},\mathbf{A})$ of the data, with $\mathcal{V}$ the set of nodes (brain regions) and $\mathbf{A}\in\mathbb{R}^{RxR}$ the adjacency matrix summarising average physical wiring between them across subjects.  
 
 We analysed minimally preprocessed resting-state fMRI data of $181$ subjects (LR acquisition session, 15 min of recordings at a TR of 0.72 s), which had already underwent realignment, co-registration and warping to Montreal Neurological Institute space. We discarded the first ten volumes, and performed linear detrending of voxel-wise time courses, regression of a Discrete Cosine Transform basis (frequency cutoff: 0.01 Hz), atlasing~\cite{Glasser2016}, and scrubbing at a framewise displacement threshold of 0.3 mm with cubic spline interpolation to re-estimate excised volumes. 
 
 We initially considered the behavioural scores of $951$ subjects for whom less than 5\% of entries were missing, and used probabilistic PCA to fill in missing values while deriving $62$ summarising measures across anthropometric, cognitive and psychometric domains. We performed z-scoring for each domain measure, and sampled the scores linked to the $181$ subjects for which functional data was preprocessed.
 
 \subsection{Graph signal processing analyses}
 
 Let the adjacency matrix $\mathbf{A}$ and the subject-specific regional activation time courses $\mathbf{X}\in\mathbb{R}^{R\times T}$, with $R=360$ regions and $T=1190$ time points. The symmetric adjacency matrix can undergo an eigendecomposition as $\mathbf{A} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}^{\top}$, with $\mathbf{V}=[\mathbf{v}_1|\mathbf{v}_2|...\mathbf{v}_R]$ containing the \textit{eigenmodes} (spatial frequencies on the graph), arranged in descending eigenvalue order. Because $\lambda_k = \mathbf{v}_k^{\top} \mathbf{A} \mathbf{v}_k = \sum_{i\neq j} A_{i,j} [\mathbf{v}_k]_i [\mathbf{v}_k]_j$, larger eigenvalues are associated to lower frequencies on the graph.\todo{Don't include bold on indexes...}
 
 Each functional signal time point $\mathbf{x_t}$ lives on $\mathcal{G}$, and can be expressed as a linear combination of the graph's constituting eigenmodes~\cite{Shuman2013}:
 \begin{equation}
 \mathbf{x_t} = \mathbf{V} \mathbf{\tilde{x}_t},
 \end{equation}
 where $\mathbf{\tilde{x}_t}$ is a vector of size $R$ containing the graph frequency coefficients. 
 
 We can then define the diagonal filtering matrix $\mathbf{H}$, and fill it in order to perform filtering on the graph. For lowpass filtering, $H_{i,i} = 0$ if $\lambda_i < \lambda_{LP}$ and 1 otherwise, while for highpass filtering, $H_{i,i} = 0$ if $\lambda_i > \lambda_{HP}$ and 1 otherwise. The filtered regional time courses are then retrieved as:
 \begin{equation}
 \mathbf{X_{filt}} = \mathbf{V} \mathbf{H} \mathbf{V}^{\top} \mathbf{X}.
 \end{equation}
 
 Lowpass and highpass filtering respectively yield the \textit{aligned} and \textit{liberal} subparts of the original functional signals~\cite{Medaglia2016}, and depend on the selection of the filtering parameter, as more stringent filtering will retain a narrower graph frequency band. Here, we compute the average power spectral density (PSD) of the signals across subjects as a function of graph frequency, and deem aligned/liberal the larger/lower eigenvalue components that tally to $P_{PSD}$\% of the PSD.
 
 \subsection{Brain/behaviour relationships}
 
 From the processed time courses $\mathbf{X_{Al}}$ and $\mathbf{X_{Lib}}$, we considered three metrics to characterise each region $r\in[1,2,...,R]$:
 \begin{itemize}
 	\item The average alignment or liberality strength along time, akin to~\cite{Medaglia2016} and~\cite{Preti2019}
 	\item The standard deviation of these same quantities along time
 	\item Their instantaneous change, which we quantified as $\frac{1}{T-1}\sum_{t=1}^{T-1} \frac{|X(r,t+1)-X(r,t)|}{2}$.
 \end{itemize}
 
 The last two metrics are inspired from the dynamic functional connectivity work of~\cite{Liu2018b}. We used the Scikit-learn toolbox~\cite{Pedregosa2011} to implement a nested cross-validation scheme in which an elastic net regression strategy was deployed. We used 3-fold cross-validation for the inner loop, and 2-fold cross-validation for the outer loop, and probed parameter ranges $\alpha = [0.25,0.5,0.75,1,1.5,2]$ and $R_{\ell} = [0.01,0.2,0.4,0.6,0.8,1]$. $\alpha$ controls the extent of regularisation ($\alpha=1$ indicates balanced data fitting and regularisation terms), while $R_\ell \in [0,1]$ controls the ratio between $\ell_1$ and $\ell_2$ regularisation terms (a value of 0 amounts to $\ell_2$ regularisation). The $R^2$ coefficient of determination was used as a quality metric.
 
 For prediction accuracies with $R^{2} > 0$, we went back to the full subject population, and fitted an elastic net model using optimal parameter values. We compared the resulting $R^2$ value to a null distribution computed non-parametrically. Because we conducted $96$ such tests, we assessed significance at $\alpha=0.0005$. 
 
 \section{Results}
 
 Prediction quality for the assessed metrics and range of $P_{PSD}$ values is displayed in Figure~\ref{ISBI_F1}. In total, only 10.13\% of $R^2$ values exceeded 0, which indicates that in the majority of cases, prediction accuracy is very unsatisfying (worse than when using the mean of all data points).
 
 %%%%%%%%%%%%%%%%%%%
 \begin{figure*}[h!]
 \centering
 \includegraphics[width=0.82\textwidth]{ISBI2020_FIG1.eps}
 \caption{\textbf{Significance of brain/behaviour relationships.} Across all 62 behavioural domains and investigated $P_{PSD}$ values, $R^2$ coefficient obtained upon nested cross-validation.}
 \label{ISBI_F1}
 \end{figure*}
 %%%%%%%%%%%%%%%%%%%
 
 Some behavioural domains could be predicted well regardless of the stringency in defining the aligned/liberal contributions (\textit{i.e.}, for any $P_{PSD}$ value), such as \textit{Strength} (with standard deviation of liberality); in other cases, a low or a high $P_{PSD}$ value was required (respective examples include \textit{Depression}---instantaneous change of alignment---and \textit{WM (Bad)}---standard deviation of liberality).
 
 On the best case across $P_{PSD}$ values for each behaviour and metric, significance was reached for 7/17/13 behavioural scores for mean/standard deviation/instantaneous change of alignment, and 15/14/9 for liberality. In Figure~\ref{ISBI_F2}, we plot the best nested cross-validation $R^2$ values for each behaviour, as well as matching $R^2$ coefficients recomputed on the whole subject population. We also display the type of metric that yielded this best case. It can be seen that alignment and liberality each contribute to some of the optimal predictions, while standard deviation dominates over mean and instantaneous change. 
 
 %%%%%%%%%%%%%%%%%%%
 \begin{figure*}[h!]
 \centering
 \includegraphics[width=0.85\textwidth]{ISBI_FIG2.eps}
 \caption{\textbf{Comparison of metrics predictive potential.} \textbf{(A)} $R^2$ coefficients for all 62 behavioural domains in the best case scenario, when computed in nested cross-validation fashion (top) or back to the full dataset (bottom). \textbf{(B)} For each behavioural domain, the type of metric (alignment versus liberality, and mean versus standard deviation versus instantaneous change) that yielded the best prediction accuracy is colour coded.}
 \label{ISBI_F2}
 \end{figure*}
 %%%%%%%%%%%%%%%%%%%
 
 \section{Discussion and conclusion}
 
 In this work, we compared a set of GSP metrics in terms of predicting a broad range of behavioural variables. We observed that successful prediction could be achieved across diverse functional domains, such as sensory perception, motor abilities, cognitive skills or arousal. Alignment and liberality had already been examined in previous work~\cite{Medaglia2016} when quantified as temporal averages; the present analyses confirm the relevance of the two metrics.
 
 Our major finding is that beyond the use of a static measure, dynamic GSP metrics (whose use had been suggested in~\cite{Huang2018} without being practically applied) also contain relevant information: in several cases, the standard deviation or instantaneous change of alignment or liberality actually enabled significant prediction accuracies when it was not the case using static measures (see for example the \textit{Strength} score in Figure~\ref{ISBI_F1}).
 
 There is an increasing understanding that the brain transits between moments of extensive cross-network interactions (when functional connectivity strongly differs from the underlying structure), and moments of more modular and structurally-conform activity~\cite{Betzel2016,Fukushima2018}. Dynamic GSP metrics likely capture such transitions as temporal reconfigurations of alignment and/or liberality.
 
 An essential aspect is the definition of what an aligned or a liberal component is: so far, either a fixed number of eigenmodes was considered~\cite{Medaglia2016}, or a median split of the PSD was used~\cite{Preti2019}. Our results show that in some cases, predictive potential is insensitive to this parameter choice, whereas in others, it is necessary to include a sufficient amount of eigenmodes (that is, use larger $P_{PSD}$ values), probably because the meaningful structural interplays for the behaviours at hand are then embedded in more intermediate frequency eigenmodes. It is possible that the rarer occurrence of the opposite (better prediction as less eigenmodes are considered) was partly due to the ability of the elastic net approach to suppress irrelevant features by $\ell_1$ regularisation; further examination using other regression strategies will help clarify this point.
 
 An interesting perspective for future work will be to assess to what extent the observed effects exist at the whole-brain level, or at a spatially finer mesoscale. Practically speaking, one could for instance envision the derivation of \textit{Slepian vectors}, which are linearly recombined eigenmodes with maximised concentration in a user-defined subset of nodes~\cite{Petrovic2019}. It would then become possible to investigate alignment and liberality at the level of individual networks, and thus further refine our understanding of human cognition. 
 
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