A practical guide to methodological considerations in the controllability of structural brain networks

Abstract

Predicting how the brain can be driven to specific states by means of internal or external control requires a fundamental understanding of the relationship between neural connectivity and activity. Network control theory is a powerful tool from the physical and engineering sciences that can provide insights regarding that relationship; it formalizes the study of how the dynamics of a complex system can arise from its underlying structure of interconnected units. Given the recent use of network control theory in neuroscience, it is now timely to offer a practical guide to methodological considerations in the controllability of structural brain networks. Here we provide a systematic overview of the framework, examine the impact of modeling choices on frequently studied control metrics, and suggest potentially useful theoretical extensions. We ground our discussions, numerical demonstrations, and theoretical advances in a dataset of high-resolution diffusion imaging with 730 diffusion directions acquired over approximately 1 hour of scanning from ten healthy young adults. Following a didactic introduction of the theory, we probe how a selection of modeling choices affects four common statistics: average controllability, modal controllability, minimum control energy, and optimal control energy. Next, we extend the current state of the art in two ways: first, by developing an alternative measure of structural connectivity that accounts for radial propagation of activity through abutting tissue, and second, by defining a complementary metric quantifying the complexity of the energy landscape of a system. We close with specific modeling recommendations and a discussion of methodological constraints. Our hope is that this accessible account will inspire the neuroimaging community to more fully exploit the potential of network control theory in tackling pressing questions in cognitive, developmental, and clinical neuroscience.Recent advances in network control theory offer a formal means to study how the temporal dynamics of a complex system emerges from its underlying network structure [9][10][11]. Applying this theory to the brain requires that one first builds a network model in which brain regions (nodes) are anatomically connected to one another (edges) [12,13]. The state of the brain network system is then reflected in the pattern of neurophysiological activity across network nodes, and state trajectories represent the temporal sequence of brain states that the system traverses [14,15]. With definitions of the network and its state in hand, we can consider the problem of network controllability, which in essence amounts to asking how the system can be driven to specific target states by means of internal or external control input [16]. In the context of the brain, such input can intuitively take the form of electrical stimulation [17-21], task modulation [22][23][24], or other perturbations from the world or from different portions of the body [25,26]. Practically, network control theory and its associated toolkit e...

Structure-informed functional connectivity driven by identifiable and state-specific control regions

“…the coactivations among brain regions, can be explained by the correlations between the activities of these regions resulting from a linear dynamics spreading through the structure of the brain. This model, termed structure-informed FC, happens to be mathematically linked to the Gramian matrix used in controllability studies [28,18,30]. This provides a novel interpretation of FC in which we can leverage control theory to explain state-specific FC configurations arising from a fixed anatomical architecture.…”

“…The solution to Equation (2) is known as the controllability Gramian. Here, in contrast to previous studies where Σ is used to derive quantitative control properties of individual nodes in the network [28,18,30], we interpret the Gramian as the state-covariance matrix obtained by stochastic excitation of the system through a set of control nodes. This allows us to relate it to the concept of functional connectivity.…”

“…While most studies on brain communication dynamics focused on the mechanisms governing the information flow in the absence of external intervention [16,38,39,32,40], it has been pointed out that the connectome is a controllable network capable of integrating context-dependent signals [20]. Earlier work on the controllability of the connectome typically fixed the transition matrix A (Equations 1-2) as the adjacency matrix of the connectome [28,18,30]. This is a common approach, but we should stress that it is possible to define multiple flow dynamics upon a fixed network structure [41].…”

“…As in former connectomic studies [15,28,18,30], our approach relies on linear time-invariant modelling (Equation 1). Despite the known non-linearities of neural dynamics [42], first-order approximations have been proved useful in capturing various aspects of brain functioning at different spatio-temporal scales [43,44,45].…”

“…While its value can be chosen arbitrarily, we choose here to fix it a priori to β = 0.72s in order to match the sampling rate of the empirical fMRI data used in this work [29]. In order to stabilize System (6), as done in several studies [18,71] (see the detailed discussion in [30]), the entries of matrix A were further divided by 1 + λ max (A), where λ max (A) is the largest eigenvalue of A. In our case, the smallest eigenvalue of L is always 0 since the Laplacian of an unsigned graph is positive semi-definite, and always possesses a zero eigenvalue [72].…”