Controllability Analysis of Functional Brain Networks
CControllability Analysis of Functional Brain Networks
Shikuang Deng a and Shi Gu a,b,*a School of Computer Science and Engineering, University of Electronic Science andTechnology of China, Chengdu, China * Corresponding Author.
Abstract
Network control theory has recently emerged as a promising approach for understanding brainfunction and dynamics. By operationalizing notions of control theory for brain networks, it offersa fundamental explanation for how brain dynamics may be regulated by structural connectivity.While powerful, the approach does not currently consider other non-structural explanations of braindynamics. Here we extend the analysis of network controllability by formalizing the evolution ofneural signals as a function of effective inter-regional coupling and pairwise signal covariance. Wefind that functional controllability characterizes a region’s impact on the capacity for the wholesystem to shift between states, and significantly predicts individual difference in performance oncognitively demanding tasks including those task working memory, language, and emotional intelli-gence. When comparing measurements from functional and structural controllability, we observedconsistent relations between average and modal controllability, supporting prior work. In the samecomparison, we also observed distinct relations between controllability and synchronizability, reflect-ing the additional information obtained from functional signals. Our work suggests that networkcontrol theory can serve as a systematic analysis tool to understand the energetics of brain statetransitions, associated cognitive processes, and subsequent behaviors.1 a r X i v : . [ q - b i o . Q M ] M a r ntroduction Large-scale noninvasive neuroimaging provides an accessible window into the rich, complex neu-rophysiological dynamics of the human brain. Such dynamics are supported by a relatively fixedbackbone of white matter fiber bundles spanning cortical and subcortical structures in an intricatenetwork characterized by highly nontrivial topology [1, 2]. The relationship between underlyingwhite matter network architecture and large-scale functional dynamics has been the focus of muchseminal work, with methods ranging from statistical analyses to biophysical modeling [3, 4]. Yet,across these diverse studies, simple intuitions regarding the mechanisms by which neural activity ispropagated along white matter tracts to enable spatially distributed changes in neurophysiologicaldynamics have been difficult to attain [5–7]. Such difficulties are in part due to the fact that thearchitecture of the white matter network and the rich dynamics of functional neuroimaging arefrequently studied in isolation.Network control theory is a particularly promising mathematical framework to address thesedifficulties [8,9]. Informed by both a precise empirical estimate of white matter network architectureand a model of the dynamics that such an architecture can support, network control theory offersstatistics, models, and analytical insights to map and predict the effects of regional activationon time-evolving whole-brain states [10, 11]. Originally developed in the physics and engineeringliterature [12], the approach is flexible to applications across scales and species, including cellularmodels [13],
C. elegans [14], fly [11], mouse [11], macaque [10], and human [10], and has beenextended to study cognitive function [15], development [16], heritability [17], disease [18, 19], andthe effects of stimulation [20–24]. More recently, methods for optimal control have been appliedto better understand the mechanisms by which the human brain might switch between diversecognitive states [25–28]. Despite its broadening utility, current work utilizing notions of networkcontrol are somewhat limited by the assumption that all effective relations between regions aretime-invariant and encapsulated in the underlying white matter network architecture. Such anassumption leaves the approach agnostic to the distinct ways in which structure can be utilized forinter-regional communication [29], both to support diverse states in health [27] and in disease [30].Sensitivity to the connectivity elicited by a given state can be partially attained by using meth-2ds for the estimation of effective connectivity [31]. Common examples of such methods includedynamic causal modeling [32], structural equation modeling [33], Granger causality, and transferentropy [34], and can be extended to account for physiological state [35] as well as unknown driversmodulating activity even in quiet resting periods [36]. A limitation of the majority of methods thatestimate effective connectivity is that they do not also estimate the dynamics that occur atop thatactivity. If one wishes to estimate both connectivity and dynamics at once, one naturally turns tothe engineering approach of systems identification [37], a methodology for building mathematicalmodels of dynamic systems using measurements of the system’s input and output signals. Systemidentification has been successfully applied to both micro- [38] and macro-circuits [39], as well asto human functional magnetic resonance imaging (fMRI) [40, 41]. While little work has been donein this area, one could naturally consider exploiting system identification [42] combined with net-work controllability [43] to investigate the control properties of the brain reflected in functionalneuroimaging data.Here we took exactly this tack using fMRI data from the Human Connectome Project YoungAdult 900s release [44,45]. Using system identification, we fit the 0- and 1-shift correlation matricesto estimate the system’s stochastic linear dynamics [42]. Using these fitted dynamics, we built alinear control system, setting the transition matrix as the effective connectivity and the controlmatrix as the canonical form multiplied by the covariance of noise. With this formulation, we askedwhether the model dynamics were globally controllable [10,46], and whether the energy required forsuch control was small or large [47]. Next, by examining the distribution of minimal control sets andcalculating statistics probing two distinct control strategies, we sought to better understand the roleof various cognitive systems in regulating whole-brain dynamics. In terms of the additional insightto the structural controllability, we first hypothesize that functional controllability will vary acrossdistinct task states and rest, potentially supporting online adaptation to task demands. We furtherhypothesize that individual differences in functional controllability can predict task performance.Finally, we examined relations between measurements of functional controllability and of structuralcontrollability to directly assess the value added by the former. Broadly, our study extends currentwork in network control theory by coupling it with systems identification to better understand therole of effective connectivity in shaping whole-brain dynamics.3 esults
We used the minimally preprocessed data in the HCP Young Adults 900 released subjects [44],which provided × (Region × TR) time series for each of the 758 subjects (Fig. 1 A). Weconstructed a control dynamics model and estimated the effective connectivity (Fig. 1 B,C). Fromthe effective connectivity, we estimated three control-related statistics: the average controllability,modal controllability, and global synchronizability (Fig. 1 D). Finally, we examined these statisticsduring the resting state and during the performance of cognitively demanding tasks.
Fig. 1. Conceptual Schematic. (A)
We begin with the preprocessed BOLD time series from84 cortical regions. (B)
We then build a linear stochastic model to represent the BOLD dynamicsby estimating the effective connectivity between regions, as well as the covariance of the intrinsicnoise. (C)
We let the intrinsic noise covariance be the control matrix, and we let the effectiveconnectivity be the state transition matrix. This control model preserves the regional activity co-variance pattern estimated from the stochastic model. (D)
Finally, we detect control sets, quantifycontrollability statistics, and examine their relations with connectivity.4 lobal Controllability of the System
First, we sought to address the question of whether the functional brain network is as globallycontrollable as the structural brain network. Global controllability refers to the capacity to drivea system to any desired state by injecting input into a single node [10], and can be examinedby calculating the smallest eigenvalue of the system’s controllability Gramian. We calculated thecontrollability Gramian of the effective connectivity matrix with perturbations to single-nodes, andfound that the smallest eigenvalues ranged from − to − . Consistent with observations instructural networks [46], we concluded that the fitted functional control system is controllable fromevery region, although the energy required may be large and biological infeasible. Distribution of Minimal Control Sets
While global controllability is of theoretical interest, a more practical concern is to identify a setof nodes that can drive desired state transitions with little energy cost. We therefore examined the α -minimal control set (Eqn. 5) where the weighted adjacency matrix A was set to be the estimatedeffective connectivity matrix from Eqn. 1. Here, we set α = 1 and identified the 1-minimum controlset for each subject; sensitivity and robustness analyses for different values of α can be found inthe SI (see Fig. S1).We first identified the minimal control set for each subject and then calculated the frequency withwhich each node was found in the control sets of all subjects. By stipulating the null hypothesisthat the control set was randomly chosen among all nodes, we calculated the z -values of thesefrequencies with a permutation test. In Fig. 2A, we showed that the areas most consistentlyidentified as members of the control set were distributed broadly across the brain, including thelateral orbital gyrus, insula, inferior parietal lobule, and middle temporal gyrus.Next we sought to determine whether the areas consistently identified in the control set tendedto be hubs of either functional or effective connectivity networks. We found that the probability ofappearing in the minimal control set was negatively correlated with the node strength calculatedfrom either the effective or functional connectivity matrix (Fig.2 C). The finding was consistentacross both resting and task conditions. Importantly, this negative relation is intuitive; when5ontrolling a network following the minimal control strategy, control nodes are likely to be weaklyconnected areas because connections to these nodes could be difficult to cover if control nodes werelocated far away. Because the control set depends on the lower bound of connectivity, functionalbrain networks – which often have many edge weights close to zero – may be very difficult to control. Fig. 2. The Spatial Distribution of Minimal Control Sets. (A)
Across subjects, the brainregions contained in the 1-minimum control set are spatially distributed, both at rest and duringthe working memory task condition. (B)
In the fronto-parietal and auditory systems, we observeda moderately high concentration of control nodes, as measured by the probability of appearance inthe 1-minimum control set across subjects. We did not observe significant differences between restand task conditions in the assignment of control nodes to cognitive systems. C We observed thatcontrol nodes tend to be located in brain regions that are weakly connected to the rest of the brain,as measured by the z-value of node strength calculated from the functional or effective connectivitymatrices.
Regional Distribution of the Average and Modal Controllability
After examining the distribution of the nodes in minimal control sets, we further asked how con-trollable the system was from a given node. For the model setting, we multiplied the square rootof the noise covariance matrix to the typical control matrix B (Equation. 4) to take care of thebetween-region interaction. Next we computed the average controllability, which was defined asthe H -norm of the system, and the modal controllability, which was defined as the inverse- H ∞ -norm of the system. We found that the regions with high average controllability were located in6recentral & postcentral, cuneus, temporal suprior, and frontal inferior opercular areas, which weremostly local executive hubs. The regions with high modal controllability were located in olfactorygyrus, and orbitofrontal medial area that were typically thought involved in solving complex tasks.Further, we examined the relationship between controllability and connectivity. We discovered thatthe average controllability displayed a negative correlation with the effective connectivity strength( r = − . , p = 1 . × − ) and no significant correlation with functional connectivity strength( r = 0 . , p = 7 . × − ) while the modal controllability showed a negative correlation withthe functional connectivity strength ( r = − . , p = 2 . × − ) and no significant correlationwith the effective connectivity strength ( r = − . , p = 2 . × − ). This supports the claimthat the average and modal controllability characterize distinct aspects of the functional dynamics. Controllability Variation from Resting to Task State
Controllability was defined through the interaction among regions and related to the energy costassociated with state transitions. From the resting state to the task state, the cognitive controlmechanism of the brain altered the dynamics to adapt to the task execution. Here we hypothe-sized that the average controllability decreased while the model controllability increased from theresting state to task state. Testing with the 2-sample t-test, we found that the mean of aver-age controllability in the memory task was significantly lower than that of the resting state with t = − . , p = 2 . × − , and the mean of modal controllability on the memory task wassignificantly higher than that of the resting state, with t = 52 . , p = 1 . × − . Compared tothe structural controllability that focuses on the static connectivity defined through the diffusion,functional controllability allowed for the dependence on observed functional imaging sequences thusbetter characterized the overall preference of control strategy regarding the underlined states. Relationship between Controllability and Cognitive Task Performance
In the previous section, we showed that the functional controllability characterized the transitionfrom resting to task states. Here we further asked whether the individual differences in cognitiveperformance can be predicted from the functional controllability. To answer this question, wetrained a linear model on of the data to predict the score in working memory test with7 ig. 3. Comparison of Controllability Measurements between the Resting State andthe Working Memory Task State.
The average and modal controllability display oppositepreference of spatial distribution and maintain a similar pattern for both (A)(B) resting stateand working memory task state (C)(D) . When transitioning from the resting state to the workingmemory state, (E) the average controllability significantly decreases (
F W ER < − ) while (F) the modal controllability significantly increases ( F W ER < − ). The largest differences appearat olfactory cortex and the middle frontal gyrus.controllability measurement and calculated the correlation between the predicted score and observedscore on the left of the data. In Fig. 4, we first showed that although both the controllabilitycalculated on the resting and working memory task states were able to predict the scores in theworking memory task, the correlations were more significant when predicted from the task data( r = 0 . , p = 3 . × − for average controllability and r = 0 . , p = 1 . × − formodal controllability) than the resting data ( r = 0 . , p = 3 . × − for average controllabilityand r = 0 . , p = 1 . × − for modal controllability). The predictability of cognitive testperformance from controllability measurement holds for the language and emotional intelligencetasks as well. Indeed, the correlation between the predicted and real language task scores were r = . , p = 0 . from average controllability and r = 0 . , p = 0 . from modal controllability,and the correlation between the predicted and real emotional intelligence scores were r = 0 . , p =0 . from average controllability and r = 0 . , p = 0 . from modal controllability. Next, toexamine whether the correlations were significantly different for the resting and testing states, wecomputed the Fish-z values for comparing two correlations correspondingly and found that both theaverage controllability ( z = 1 . , p = 0 . ) and the modal controllability ( z = 3 . , p = 4 × − )maintained a more significant prediction with the controllability measurement on the task datathan the resting data. In addition, the correlation between observed and predicted accuracy were r = 0 . , p = 6 . × − and r = 0 . , p = 4 . × − when predicted with weighted nodalstrength calculated from effective connectivity and the functional connectivity correspondingly (seeSI Fig. S2), lower than those predicted with controllability measurement as well. Specially, themedial frontal gyrus, dorsolateral prefrontal cortex, the supramarginal gyrus and the occipital cortexrelated to the execution of working memory task appeared as the most positively sensitive areas onthe perspective of modal controllability. The temporal pole hippocampus, insula and suproparietalturned out to be the most positively sensitive area on the average controllability. These resultssupport the validity of applying the functional controllability to characterize the state transitionfrom the resting to task states. Relationship among Controllability Measurements
In the previous section, we investigated the functional controllability and utilized it to investigatethe transition from resting to task state. Here we further ask how these measurements were relatedto each other and how they were different from the established measurements in structural control-lability [10]. To answer these questions, we computed the global average and modal controllabilityby setting the whole brain controlled and correlated them with the global synchronizability. Thissetting of controlling all nodes could be viewed as an average effect of the controllability acrossregion where the interaction and signal fluctuation happened throughout the whole system. Forthe calculation of structural controllability adapted from [10], we normalized the effective connec-tivity matrix by its largest singular value and used it as the state transition matrix, together withsetting identity matrix as the control matrix. From Fig. 5A, we can see that when all regions were9 ig. 4. Prediction of Performance in the Working Memory Task from Controllability.
Controllability provides a perspective of understanding the progression of signal and energy acrossthe brain network. Here we want to show that functional controllability can predict the performancein the working memory task. For the resting state, (A) the correlation between the true testaccuracy and the predicted accuracy from average controllability is r = 0 . with p = 3 . × − and (B) the correlation when predicting scores from modal controllability is r = 0 . with p =1 . × − . For the working memory task state, (C) the correlation between the true test accuracyand the predicted accuracy from average controllability is r = 0 . with p = 3 . × − and (D) the correlation of modal controllability is r = 0 . with p = 1 . × − . The functionalcontrollability on working memory related areas including the medial frontal gyrus, dorsolateralprefrontal cortex, the supramarginal gyrus and the occipital cortex contribute most positively forthe predicted accuracy.controlled, the average controllability was negatively correlated with the modal controllability forboth the functional controllability ( r = − . , p = 3 . × − ) and structural controllability( r = − . , p = 4 . × − ). The synchronizability, which measured the network’s ability ofpersisting in a synchronous state, displayed a negative correlation with the average controllabilityin both the functional controllability (Fig. 5B, r = − . , p = 4 . × . − ) and the structuralcontrollability (Fig. 5E r = − . , p = 3 . × − ). However, the trends were different when10orrelating synchronizability with modal controllability where the correlation was negative for thefunctional controllability ( r = − . , p = 4 . × − ) and positive for the structural control-lability ( r = 0 . , p = 1 . × − ). This could be caused by the fact that in the calculationof structural controllability, only the effective connectivity was adopted with normalization, mak-ing the between-controllability measurement somewhat driven by the asymptotic property of themeasurement [10] that resulted in higher correlations. While in the functional controllability, boththe effective connectivity and noise covariance matrices were involved without extra normalization,which drove the correlation between the two controllability measurements away from the asymp-totic behavior thus lower and less significant. One thing to notice was that these relationshipswere not fully consistent with those for the controllability on structural brain networks through thedevelopment [16] (see SI Fig. S3 for the replication on structural controllability and SI Fig. S4-6 forthe relationship between nodal connectivity strength and controllability), suggesting a mechanicaldifference between the functional and structural controllability analyses. Discussion
Brain is a complex dynamical system that enables various behaviors through moving itself amongmultiple cognitive states. Although the trajectories of these state transition are biologically con-strained by the white matter microstructure and partially explained by the network control theorybased on streamlines [16], the fitness to the observed dynamics is not satisfying probably due tothe intrinsic complexity of modelling the evolving manner of functional signals from the structuralconnectivity [25, 26, 48, 49]. In practice, it is more feasible yet still critically important to build acontrol framework from the functional time series, e.g. BOLD, EEG, MEG, and etc with the con-straints from structural connectivity constructed from diffusion imaging. In this work, we proposeda novel framework of analyzing brain’s functional dynamics from the control perspective wherecontrollability measurements were defined through the system norms and investigated on both theresting and task states.The controllability on structural brain networks predicts the the ability of alternating large-scaleneural circuits based on the assumption that the transition of brain states can be modeled by the11 ig. 5. Relationship among Controllability Measurements.
We demonstrate the relation-ship among the average controllability, modal controllability, and syncronizability when we set thecontrol set as the whole brain. For the functional controllability, the average controllability displayssignificant (A) negative correlations with modal controllability ( r = − . , p = 7 . × − and (B) positive correlation with synchronizability( r = 0 . , p = 4 . × − ). In addition, (C) the modal controllability shows significant negative correlation with synchronizability ( r = − . , p = 4 . × − . For the structural controllability, (D) the negative correlation betweenaverage controllability and modal controllability is much stronger ( r = − . , p = 4 . × − . (E) A similarly stronger negative correlation between average controllability and synchronizability( r = − . , p = 3 . × − ) exists as well. (F) However, different from the case of func-tional controllability, the modal controllability and synchronizability displays a positive correlation( r = 0 . , p = 1 . × − .structural connectivity [10]. Then, why do we still need the functional controllability? From theopinion of [31], this transition, which quantifies the impact of one region on another in movingthe brain states, is indeed one type of effective connectivity. Thus, in addition to obtaining a12egenerated model from structural connectivity that dismisses the difference over functional states,it is critically important to model and validate the control theoretical analysis based on effectiveconnectivity. Further, in what space should we model these transitions? The current work offunctional controllability examines this problem of defining a set of nodes’ role in moving the brainstates from a data-driven approach, inferring the transition and co-varying patterns in the signalspace. Biologically, the neural stimulus spreads across neurons along the synapse, reflected in thefluctuation of metabolic consumption [50]. Although rigorously the control input should be on thesource space [51], here we focus on the control mechanism and assume that the principle appliesin the signal space as well. This assumption is widely supported by the researches that the signalspace inference are able to reveal the intrinsic interactions between regions [48, 52, 53]. Thus it ismeaningful to consider the functional controllability in the signal space.The change of functional controllability from the resting to task state provides a system-controlperspective of understanding how the system shifts the association among regions for the adaptionto the executive task. We showed that the whole brain’s average controllability decreases and themodal controllability increases from the resting state to the task state. This suggests that the restingstate is potential ‘ground state’ with better maintain of averaged energy cost. Relatively, for thetask state or pluripotent ‘excited state’, more energy would be consumed in order to facilitate thecognitive processes with improved controllability on each mode. This complements the previousreasoning on the regional preference of control strategies [10]. In addition, it unveils how theadjustment of regional activation contributes to the systematic alternation to executing tasks. Forthe cognitive tasks considered here, we observed larger increase in the task related area and smallerdecrease in the default mode area. This indicates that the compensation on the task state is lowercompared to other regions, backing the previous literature stating that the default mode is probablyoptimized for the baseline state. Further, this result enriches the reasoning on the relationshipbetween weakly connected area and modal controllability. The modal controllability quantifies aregion’s ability in controlling the amplitude of signal amplifying thus it is higher on the weaklyconnected area because the amplifying effect was transitioned through the connectivity, makingthe less connected region result in higher controllability of amplifying amplitude. When comparedthrough the resting and task states, the increase of modal controllability on the corresponding13egions improves the controllability on its associated modes thus the execution of task as well [54].Controllability not only characterizes the shift of states from resting to task states but alsopredicts the individual difference in the scores in the working memory tasks. Why? On the onehand, as shown in previous literature [55, 56], the functional connectome encodes the informationof individual difference. On the other hand, from the perspective of system control, higher modalcontrollability indicates more efficient utility in executing the mode thus the coefficients on the task-related area would contribute positively in the predictive model [22, 57]. The connectivity-basedmodels, although informative about the spatial location of the regions related to the task perfor-mance, lack a mechanical explanation of predictability, which is complemented by the proposedcontrollability framework.The controllability of functional brain networks is closely related to that of structural networks.Both frameworks relied on the time-invariant linear model, which could be not real consideringthe nonlinearity of brain system yet still provides a fair estimation both from the perspective ofbehavior and control theory. Second, the average controllability display strong positive correlationswith nodal connectivity strength in the structural networks and with effective connectivity strengthin the functional networks. This is supported by the results revealed in [48] that the structuralconnectivity can act as a predictor of the effective connectivity, which also provides the consistencybetween the current framework and the controllability analysis on structural brain networks [10].In addition, this positive correlation suggested that although not exactly overlapped, the structuraland functional hubs maintain efficient roles in driving the brain to many easily-reachable states,providing an explanation of the cognitive association for both structural and functional hubs.However, the differences also exist between the two proposed networks. On the modeling per-spectives, as the functional control model fits the BOLD time series directly while the structuralcontrol model studies the induced dynamics, the two frameworks are generally only applicable totheir own modalities. The newly proposed one does have some convenience. For example, the con-trollability frameworks on the structural networks requires the normalization of transition matrix toensure the Schur stability of the system while the proposed framework on functional networks satis-fied the constraints automatically from the model fitting. This avoids the potential bias introducedby the normalization when the controllability statistics need to be averaged across subjects [58].14n addition, the definition of controllability on structural brain networks was derived from the con-nectivity [9, 10] and regionally defined while the currently proposed framework is defined via thesystem norms that explicitly links the connectivity to a formulated energy and naturally extendableto control sets consisting of multiple and even all nodes. On the application scenario, structuralcontrollability framework provides a mechanical explanation of how the underlined structure sup-ports the executive function [57] and neural development [16], as well as the evolution of dynamictrajectories associated with the state transition [26]. Yet it remains to explain how the biomarkersdefined through the activation of and the statistical association among regions are related to thesystem’s controllability from a mechanical view. For example, different brain activation and con-nectivity patterns act as biomarkers to unveil the representative phenotype of psychiatric diseaselike depression [59, 60] and schizophrenia [61, 62]. Yet it is unclear how one or multiple regions, i.e.the control set, drive the whole neural circuit to move across states and result in the abnormalityin brain functions. Our functional controllability model potentially bridges the gap by analyzingthe time-series directly rather than inferring from the structure [19]. It allows future possibility ofapplication to intervene the neural circuits via certain nodes for psychiatric medication [63].Methodologically, it is worth pointing that the current effort still shares certain limitations be-fore. First, by changing the dynamics into a stochastic one, the nonlinear effect still remained tobe solved in future. Secondly, due to the constrains of unpredictable noise in the measurement,the approximation to the observed trajectories are still unsatisfactory especially for the real values.Finally, the controllability measurements are highly related to the effective and functional nodestrength. Although the linear dynamics could predict the trend as we show in the article, quanti-tatively, the amount of modeled dynamics is still around the limit point of the linear system thusmay not be able to quantify the long-range high level dependence on connectivity. Materials and Methods
The theory in this work consists of two parts. The first part is the inference on dynamics, whichassumes that the dynamical patterns of BOLD signals are driven by the intrinsic fluctuation ofnoise and can be reflected on its covariance structure. The second part is how a region imposes its15mpact on others, i.e. the definition of controllability. We denote the state at time t for a brain as x t , which is an N × vector with N as the number of regions. Usually, the evolutionary dynamicsof the states is formulated as describing the state’s time derivative dx/dt with the state variable x and other parameterized related terms, e.g. the noise. In this work, we attempt to fit the dynamicsof x t , followed by investigating it from the control perspectives where we examine both the spatialdistribution of minimal control sets and control measurements. Preprocessing of fMRI Data
We used the minimally preprocessed fMRI data conducted using HCP Functional Pipeline v2.0 [45].Subjects with incomplete resting state or two task data were excluded. Then we used DPARSF [64]and SPM12 [65] to process these minimally preprocessed data. First, we removed the constant,linear and quadratic trend from these functional images. Next, several nuisance signals includingcerebrospinal fluid signal, white matter signal, and motion effect were regressed from the timecourse of each voxel using multiple linear regression and Friston’s 24 head motion parameters.Then 3D spatial smoothing was applied to each volume of the fMRI data using a Gaussian kernelwith Full-width at Half Maximum (FWHM) equaling to 4 mm. Finally, ALFF and fALFF(0.01-0.1 Hz) was used to inhibit the energy of physiologically meaningless brain regions, and temporalband-pass filtering (0.01-0.1 Hz) was applied to reduce the influence of low-frequency drift and thehigh-frequency physiological noise. For resting state fMRI time series, the first and last 50 volumeswere discarded to suppress equilibration effects. For task data, the break time in the task wasdeleted to remove the effects of resting state.The AAL2 atlas [ ? ] was used for the parcellation ofbrain cortex into region. We kept 84 cortical regions only excluding non-cortical regions includingamygdala, caudate, putamen, pallidum, thalamus, vermis and cerebellum areas. Finally there are758 subjects used in the current analyses, including 422 females and 336 males aged from 22 to 37years old. Construction of the Control Dynamics
We start from the linear stochastic model, where the changing rate of the state is determined by thecurrent state and the random diffusion following Gilson’ s steps [42]. Mathematically, the dynamic16odel is given by d x = ( − τ x x + Cx ) dt + d W t , (1)where τ x is the constant of state decay over time, C is the effective connectivity matrix and d W is awiener process with covariance Σ . To fit the dynamics, we estimate three unknown parameters, τ x , C and Σ by minimizing the loss between model-derived and empirical covariances. First, assumingthe stationarity of this system, we can derive the relationship between autocorrelation Q andcovariance Σ with the Ito’s formula [66] which implies JQ + Q J † + Σ = 0 , (2)where J = − τ x I + C is the Jacobian of equation 1. Further, the theoretical formula of the τ -delayautocorrelation can be computed as Q τ = Q exp( J † τ ) via similar derivation. On the other hand,we can define the empirical estimation of autocorrelation ˆ Q k with k-shift. We hope the fittedautocorrelations are as close to the empirical ones as possible. Thus the loss function of fitting thedynamics is given by the weighted sum of the distance between each pair of estimated and empiricalauto-covariance matrices, i.e. L ( J , Σ) = K (cid:88) k =1 λ k l ( Q k ( J , Σ) , ˆ Q k ) , (3)where l ( · ) is the loss function between two covariance matrices, K is the number of shifts we wantto use for the estimation and λ is a scalar to weight among the losses for these Q k ’s.Using the gradient descent, ˆ J, ˆΣ can be recursively updated.Analogous to the classical control representation ˙ x ( t ) = Ax ( t ) + Bu ( t ) , the state transitionmatrix A can be modeled with the Jacobian J and the control input matrix B is reformulated as ˆΣ · B K , which simultaneously selects the control sets with N × K matrix B K and preserves theco-varying pattern estimated from the stochastic modeling with ˆΣ . Consequently, we built up thelinear time-invariant dynamic model for brain’s functional signals as d x dt = ˆ Jx + ˆΣ · B K u ( t ) , (4)where u ( t ) is the input vector to be determined. In this manuscript, for the ease of notation, weuse A = ˆ J and B = ˆΣ · B K when there is no ambiguity. When we say functional connectivity, werefer to the Pearson’s correlation for the time series of each pair of regions.17 dentification of Minimal Control Sets Theoretically, if the transition matrix A for the linear system is non-degenerate, the system isalmost surely controllable from a single node [67]. But the control energy could be so high thatresults in unreasonable trajectories in practice. Here we adapt the minimum dominant set algorithmin [68] and define the α -minimum control set ( α -MCS) as the solution of the following optimizationproblem: min K (cid:88) k ∈K β k , s.t. (cid:88) k ∈K ,k (cid:54) = i β k a ik ≥ α · (1 − β i ) · max ( A ) , (5)where A = { a ij } is the transition matrix, β i takes 1 if node i is chosen as a control node and 0otherwise, and K is the control set. When the network is binary and α = 1 , it reduces to the regularproblem of identifying the minimal control set. When A is weighted, the optimization problem findsthe minimal set such that every nodes is either in the control set or connected to the control setwith overall strength above a threshold α scaled by the maximum weight in the weighted adjacencymatrix. Average Controllability
The average controllability of the linear stable system refers to its H -norm, which intuitivelyquantifies the average distance the system can reach in the state space with unit input energy.Mathematically H norm is the energy of the output of the system ˙ x = Ax + (cid:88) i B i ω i (6)where ω i = δ i ( t ) is the δ -function and B i is the i-th column control matrix in Eqn[4]. The averagecontrollability is then defined as a c = H = (cid:115) trace (cid:20) B T (cid:18)(cid:90) + ∞ exp ( A t + A T t ) dt (cid:19) B (cid:21) (7)where B is the control matrix. If the average controllability is high, it means that the brain is moreefficient in moving into many easily reachable states.18 odal Controllability The modal controllability of the linear stable system is defined as the inverse of H ∞ -norm, whichquantifies the inverse of maximal possible vector amplification with sin( · ) input. Mathematically,it is defined as g c = ( H ∞ ) − = (cid:18) sup ω ∈ R σ { G ( jω ) } (cid:19) − (8)where j is the virtual unit with j = − , G ( s ) = ( s I − A ) − B , and σ denotes the largest singularvalue. A higher modal controllability then corresponds to a easier control of the dynamics in thedirection of highest energy cost. Global Synchronizability
The global syncronizability refers to the inversed spread of the Laplacian eigenvalues, which intu-itively measures the ability of the network’s dynamics to persist in a synchronous state where allnodes have the same magnitude of activity [16]. Mathematically, it is defined as s c = (cid:115) d ( N − (cid:80) N − i =1 | λ i − λ | (9)where λ i is the positive eigenvalues of the Laplacian matrix L with L ij = δ ij (cid:80) k A ik − A ij and d = (cid:80) i (cid:80) j (cid:54) = i A ij /N is the average strength of each node. Prediction with Linear Model
To examine the effectiveness of nodal measurement to predict the performance in memory task,we build a linear model with the nodal measurement z = { z i } (e.g. controllability measurement orweighted nodal strength) as input and the task score t as the output. Mathematically, we write themodel as t ∼ η · z , where η = { η i } represents the contribution of each region’s nodal measurementin predicting the score in cognitive tasks. Acknowledgments
This work is primarily supported by NSFC 61876032.19 uthor contributions
S.G. designed the research. S.G. and D.S.K performed the research, contributed to new analyticaltools and wrote the draft.
Competing Financial Interests
The authors declare no competing financial interests.20 eferences [1] Hagmann, P. et al.
Mapping the structural core of human cerebral cortex.
PLoS Biol , e159(2008).[2] Misic, B. et al. Cooperative and competitive spreading dynamics on the human connectome.
Neuron , 1518–1529 (2015).[3] Deco, G., Tononi, G., Boly, M. & Kringelbach, M. L. Rethinking segregation and integration:contributions of whole-brain modelling. Nature Reviews Neuroscience , 430 (2015).[4] Breakspear, M. Dynamic models of large-scale brain activity. Nat Neurosci , 340–352 (2017).[5] Rosenthal, G. et al. Mapping higher-order relations between brain structure and function withembedded vector representations of connectomes.
Nat Commun , 2178 (2018).[6] Becker, C. O. et al. Spectral mapping of brain functional connectivity from diffusion imaging.
Sci Rep , 1411 (2018).[7] Medaglia, J. D. et al. Functional alignment with anatomical networks is associated withcognitive flexibility.
Nat Hum Behav , 156–164 (2018).[8] Liu, Y.-Y., Slotine, J.-J. & Barabási, A.-L. Controllability of complex networks. Nature ,167 (2011).[9] Pasqualetti, F., Zampieri, S. & Bullo, F. Controllability metrics, limitations and algorithmsfor complex networks.
IEEE Transactions on Control of Network Systems , 40–52 (2014).[10] Gu, S. et al. Controllability of structural brain networks.
Nature communications , 8414(2015).[11] Kim, J. Z. et al. Role of graph architecture in controlling dynamical networks with applicationsto neural systems.
Nature physics , 91 (2018).[12] Kailath, T. Linear Systems (Prentice-Hall, 1980).2113] Wiles, L. et al.
Autaptic connections shift network excitability and bursting.
Sci Rep , 44006(2017).[14] Yan, G. et al. Network control principles predict neuron function in the caenorhabditis elegansconnectome.
Nature , 519 (2017).[15] Cornblath, E. J. et al.
Sex differences in network controllability as a predictor of executivefunction in youth.
Neuroimage , 122–134 (2018).[16] Tang, E. et al.
Developmental increases in white matter network controllability support agrowing diversity of brain dynamics.
Nature Communications , 1252 (2017).[17] Lee, W. H., Rodrigue, A., Glahn, D. C., Bassett, D. S. & Frangou, S. Heritability and cognitiverelevance of structural brain controllability. Cereb Cortex bhz293 (2019).[18] Jeganathan, J. et al.
Fronto-limbic dysconnectivity leads to impaired brain network controlla-bility in young people with bipolar disorder and those at high genetic risk.
Neuroimage Clin , 71–81 (2018).[19] Bernhardt, B. C. et al. Temporal lobe epilepsy: Hippocampal pathology modulates connectometopology and controllability.
Neurology , e2209–e2220 (2019).[20] Taylor, P. N. et al. Optimal control based seizure abatement using patient derived connectivity.
Front Neurosci , 202 (2015).[21] Muldoon, S. F. et al. Stimulation-based control of dynamic brain networks.
PLoS computationalbiology , e1005076 (2016).[22] Medaglia, J. D. et al. Network controllability in the inferior frontal gyrus relates to controlledlanguage variability and susceptibility to tms.
Journal of Neuroscience , 6399–6410 (2018).[23] Stiso, J. et al. White matter network architecture guides direct electrical stimulation throughoptimal state transitions.
Cell Rep , 2554–2566.e7 (2019).[24] Khambhati, A. N. et al. Functional control of electrophysiological network architecture usingdirect neurostimulation in humans.
Netw Neurosci , 848–877 (2019).2225] Betzel, R. F., Gu, S., Medaglia, J. D., Pasqualetti, F. & Bassett, D. S. Optimally controllingthe human connectome: the role of network topology. Scientific reports , 30770 (2016).[26] Gu, S. et al. Optimal trajectories of brain state transitions.
Neuroimage , 305–317 (2017).[27] Cornblath, E. J. et al.
Context-dependent architecture of brain state dynamics is explained bywhite matter connectivity and theories of network control. arXiv , 02849.[28] Shine, J. M. et al.
Human cognition involves the dynamic integration of neural activity andneuromodulatory systems.
Nat Neurosci , 289–296 (2019).[29] Palmigiano, A., Geisel, T., Wolf, F. & Battaglia, D. Flexible information routing by transientsynchrony. Nature neuroscience , 1014 (2017).[30] Shah, P. et al. Local structural connectivity directs seizure spread in focal epilepsy. bioRxiv .[31] Friston, K. J. Functional and effective connectivity: a review.
Brain connectivity , 13–36(2011).[32] Friston, K. J., Li, B., Daunizeau, J. & Stephan, K. E. Network discovery with DCM. Neu-roimage , 1202–1221 (2011).[33] McIntosh, A. R. Tracing the route to path analysis in neuroimaging. Neuroimage , 887–890(2012).[34] Barnett, L., Barrett, A. B. & Seth, A. K. Granger causality and transfer entropy are equivalentfor gaussian variables. Phys Rev Lett , 238701 (2009).[35] Havlicek, M. et al.
Physiologically informed dynamic causal modeling of fMRI data.
Neuroim-age , 355–372 (2015).[36] Park, H. J., Friston, K. J., Pae, C., Park, B. & Razi, A. Dynamic effective connectivity inresting state fMRI.
Neuroimage , 594–608 (2018).[37] Zipser, D. Identification models of the nervous system.
Neuroscience , 853–862 (1992).2338] Mitra, A. & Manitius, A. A systems identification approach to estimating the connectivity ina neuronal population model. Conf Proc IEEE Eng Med Biol Soc , 4860–4863 (2014).[39] Murphy, J. W., Kelly, S. P., Foxe, J. J. & Lalor, E. C. Isolating early cortical generators ofvisual-evoked activity: a systems identification approach.
Exp Brain Res , 191–199 (2012).[40] Tauchmanova, J. & Hromcik, M. Subspace identification methods and fMRI analysis.
ConfProc IEEE Eng Med Biol Soc
J Neural Eng , 066016 (2018).[42] Gilson, M., Moreno-Bote, R., Ponce-Alvarez, A., Ritter, P. & Deco, G. Estimation of di-rected effective connectivity from fmri functional connectivity hints at asymmetries of corticalconnectome. PLoS computational biology , e1004762 (2016).[43] Zhou, K., Doyle, J. C., Glover, K. et al. Robust and optimal control , vol. 40 (Prentice hall NewJersey, 1996).[44] Van Essen, D. C. et al. The wu-minn human connectome project: an overview.
Neuroimage , 62–79 (2013).[45] Glasser, M. F. et al. The minimal preprocessing pipelines for the human connectome project.
Neuroimage , 105–124 (2013).[46] Menara, T., Katewa, V., Bassett, D. S. & Pasqualetti, F. The structured controllability radiusof symmetric (brain) networks. In , 2802–2807 (IEEE, 2018).[47] Klamka, J. Controllability of linear dynamical systems (1963).[48] Deco, G., Jirsa, V. K. & McIntosh, A. R. Emerging concepts for the dynamical organizationof resting-state activity in the brain. Nature Reviews Neuroscience , 43 (2011).[49] Gu, S. et al. The energy landscape of neurophysiological activity implicit in brain networkstructure.
Scientific reports , 2507 (2018).2450] Kandel, E. R. et al. Principles of neural science , vol. 4 (McGraw-hill New York, 2000).[51] Valdes-Sosa, P. A., Roebroeck, A., Daunizeau, J. & Friston, K. Effective connectivity: influ-ence, causality and biophysical modeling. Neuroimage , 339–361 (2011).[52] Cohen, J. D. et al. Computational approaches to fmri analysis.
Nature neuroscience , 304(2017).[53] Fox, M. D. et al. The human brain is intrinsically organized into dynamic, anticorrelatedfunctional networks.
Proceedings of the National Academy of Sciences , 9673–9678 (2005).[54] Cui, Z. et al.
Optimization of energy state transition trajectory supports the development ofexecutive function during youth. bioRxiv et al.
Functional connectome fingerprinting: identifying individuals using patternsof brain connectivity.
Nature neuroscience , 1664 (2015).[56] Shen, X. et al. Using connectome-based predictive modeling to predict individual behaviorfrom brain connectivity. nature protocols , 506 (2017).[57] Cornblath, E. J. et al. Sex differences in network controllability as a predictor of executivefunction in youth.
NeuroImage , 122–134 (2019).[58] Wu-Yan, E. et al.
Benchmarking measures of network controllability on canonical graph models.
Journal of Nonlinear Science et al.
Activity and connectivity of brain mood regulating circuit in depression: afunctional magnetic resonance study.
Biological psychiatry , 1079–1088 (2005).[60] Veer, I. M. et al. Whole brain resting-state analysis reveals decreased functional connectivityin major depression.
Frontiers in systems neuroscience , 41 (2010).[61] Lynall, M.-E. et al. Functional connectivity and brain networks in schizophrenia.
Journal ofNeuroscience , 9477–9487 (2010).[62] Fitzsimmons, J., Kubicki, M. & Shenton, M. E. Review of functional and anatomical brainconnectivity findings in schizophrenia. Current opinion in psychiatry , 172–187 (2013).2563] Sitaram, R. et al. Closed-loop brain training: the science of neurofeedback.
Nature ReviewsNeuroscience , 86 (2017).[64] Yan, C.-G., Wang, X.-D., Zuo, X.-N. & Zang, Y.-F. Dpabi: data processing & analysis for(resting-state) brain imaging. Neuroinformatics , 339–351 (2016).[65] Ashburner, J. et al. Spm12 manual.
Wellcome Trust Centre for Neuroimaging, London, UK (2014).[66] Itô, K. Stochastic integration. In
Vector and Operator Valued Measures and Applications ,141–148 (Elsevier, 1973).[67] Menara, T., Bianchin, G., Innocenti, M. & Pasqualetti, F. On the number of strongly struc-turally controllable networks. In
American Control Conference (ACC), 2017 , 340–345 (IEEE,2017).[68] Sun, P. G. & Ma, X. Understanding the controllability of complex networks from the micro-cosmic to the macrocosmic.
New Journal of Physics19