Stimulation-Based Control of Dynamic Brain Networks

The ability to modulate brain states using targeted stimulation is increasingly being employed to treat neurological disorders and to enhance human performance. Despite the growing interest in brain stimulation as a form of neuromodulation, much remains unknown about the network-level impact of these focal perturbations. To study the system wide impact of regional stimulation, we employ a data-driven computational model of nonlinear brain dynamics to systematically explore the effects of targeted stimulation. Validating predictions from network control theory, we uncover the relationship between regional controllability and the focal versus global impact of stimulation, and we relate these findings to differences in the underlying network architecture. Finally, by mapping brain regions to cognitive systems, we observe that the default mode system imparts large global change despite being highly constrained by structural connectivity. This work forms an important step towards the development of personalized stimulation protocols for medical treatment or performance enhancement.


Normalization of structural matrices across subjects
When doing controllability calculations, we must ensure that the structural adjacency matrix describing the tractography connectivity is Schur stable. In Gu et al. [1], this was obtained by normalizing each structural matrix by one plus its largest singular value. Because each matrix effectively received a slightly different normalization factor, regional controllability results were ranked in order to perform averaging across scans and subjects. To avoid the loss of information that results from ranking the data, we instead employed a different form of normalization. We first calculated the maximum eigenvalue for each structural matrix in the data set. From this pool of maximum eigenvalues, we then selected the maximum value, and divided all structural matrixes by two times this quantity. This ensures Schur stability and allows us to compare directly compare regional controllability values obtained from different structural matrices. The controllability results presented in this paper therefore represent the resultant numerical values of regional controllability calculations, as opposed to the ranked values presented in Gu et al [1].

Average controllability and the steady state response
In linear network control theory, the controllability of a network refers to the possibility of altering the configuration of the network nodes via external stimuli and in a predictable way. To quantify the degree of controllability of a network, we first model the dynamical interaction among network nodes by means of a discrete-time, linear, time-invariant system: In the equation above, x is a vector containing the states of the network nodes, A = A T is a (stable) weighted adjacency matrix of the network, u is the external control signal, and B K identifies the control nodes; see also [1,2].
In our simulations, we use a constant input current to stimulate brain regions, and therefore are interested in the steady state response of the network. With a constant control input, the network steady state is where u constant is the value of the constant input. Thus, the steady state effect of a constant input to the i-th region is characterized by the i-th column of the matrix (I − A) −1 . The largest value of the i-th column gives steady state response of the region maximally effected by the input, while the average of the i-th column gives the average steady state effect over all regions.
The use of a constant input as regional stimulation is a special case of the more general paradigm of a time-varying input normally used to define network control statistics such as those used in the main manuscript (average/modal controllability). We therefore would like to relate this steady state response to the average controllability which describes the more general paradigm. The degree of controllability of a network can be quantified in different ways [2], but in this paper, we use the classical definition of the Controllability Gramian, that is, and measure the average degree of controllability as Trace(W K ), which has a specific system theoretic interpretation [1][2][3]. Moreover, notice that In other words, the controllability degree with control nodes K equals the sum of the diagonal entries of (I − A 2 ) −1 indexed by K. Because of the normalization of the adjacency matrix adopted in this work, it can be verified that so that the average controllability information can be reconstructed from the steady state response matrix (I − A) −1 . Specifically, for stimulation of a single region, the largest entry of the i-th column of the steady state matrix will be approximately equal to the average controllability. As seen in S1 Fig (a), when we plot the functional effect of stimulation as a function of the largest steady state value, we do indeed reproduce the results of Fig. 5a. Additionally, we see a similar result when plotting the functional effect of stimulation for the average steady state value (S1 Fig (b)). Therefore, in the main manuscript, we present our findings in terms of the more general regional controllability values instead of the steady state response, which also allows for comparison with previous work using these measures to study the properties of structural brain networks [1]. However, it should be noted that the average/modal controllability may not always be an adequate approximation of the steady state response of a network, and therefore care should be taken in using these statistics in other studies without first demonstrating their consistency with the steady state response statistics.

Mapping regions to cognitive systems
Similar to the assignment of brain regions in Gu et al. [1] and inspired by Power et al. [4], we initially assigned each of the 83 brain regions to one of 9 cognitive systems. Our only divergence from Gu et al. was the creation of a "ventral temporal association" category to bin perceptual regions associated with invariant object representations and multisensory activation. For further analysis, this assignment was coarse grained into 4 cognitive systems [5]. The placement of each region into each cognitive system is summarized in Table A Table A. Assignment of brain regions to cognitive systems.