On Structural Controllability of Symmetric (Brain) Networks

The question of controllability of natural and man-made network systems has recently received considerable attention. In the context of the human brain, the study of controllability may not only shed light into the organization and function of different neural circuits, but also inform the design and implementation of minimally invasive yet effective intervention protocols to treat neurological disorders. While the characterization of brain controllability is still in its infancy, some results have recently appeared and given rise to scientific debate. Among these, [1] has numerically shown that a class of brain networks constructed from DSI/DTI imaging data are controllable from one brain region. That is, a single brain region is theoretically capable of moving the whole brain network towards any desired target state. In this note we provide evidence supporting controllability of brain networks from a single region as discussed in [1], thus contradicting the main conclusion and methods developed in [2]. We consider brain networks modeled by a weighted graph G = (V, E), where V = {1, . . . , n} and E ⊆ V × V are the vertex and edge sets, respectively. Let A = [aij ] be the weighted adjacency matrix of G, where aij = 0 if (i, j) 6∈ E and aij ∈ R≥0 if (i, j) ∈ E . We assume that A is symmetric and that the graph G has no self loops, forcing the diagonal entries of A to zero. These assumptions are dictated by the use of DSI/DTI scans to reconstruct brain networks [1]. Let x : N→ R be the vector containing the state of the brain regions over time. The network dynamics with control region i ∈ V read as x(t+ 1) = Ax(t) + biu(t), (1) where u : N→ R is the control input injected into the i-th brain region, and the input vector bi satisfies bj = 0 if j 6= i and bi = 1. The network (1) is controllable if and only if the controllability matrix C(A, bi) is invertible [3], where C(A, bi) = [ bi Abi · · · An−1bi ] . (2) Assessing controllability of network systems is numerically difficult because the controllability matrix typically becomes ill-conditioned as the network cardinality increases; e.g., see [4], [5]. Because different controllability tests suffer similar numerical difficulties, a convenient tool to study controllability of networks is to resort to the theory of structural systems. To formalize this discussion, notice that the determinant det(C(A, bi)) = φ(aij) is a polynomial function of the nonzero entries of the adjacency matrix. The network (1) is uncontrollable when the weights are chosen so that C(A, bi) is not invertible or, equivalently, when φ(aij) = 0. Let S contain the choices of weights that render the network (1) uncontrollable, that is, S = {aij : (i, j) ∈ E and φ(aij) = 0}. (3) Notice that each element of S can be represented as a point in R, where d = |E| is the number of nonzero entries of A. Formally, the set S defines an algebraic variety of R [6]. This implies that controllability of (1) is a generic property because it fails to hold on an algebraic variety of the parameter space [7]. Thus, when assessing controllability of (1) as a function of the network weights, only two mutually exclusive cases are possible: (i) either there is no choice of weights aij , with (i, j) ∈ E , rendering the network (1) controllable, or (ii) the network (1) is controllable for all choices of weights aij except, possibly, those lying in a proper algebraic variety of the parameter space R (see Example 1 below). Loosely speaking the above discussion implies that, if one can find a choice of weights aij such that (1) is controllable, then almost all choices of weights aij yield a controllable network. In this case, the network is said to be structurally controllable [6], [8], [9]. In what follows we show that brain networks are structurally controllable from one single region, thus providing theoretically-validated and numerically-reliable support to the result in [1]. This further shows that the result in [2] is likely incorrect and misleading. In fact, even in an unfortunate choice of weights that prevents controllability, that is, a choice of weights that lies in a proper algebrain variety, a random and arbitrarily small deviation of network weights due to perturbation or uncertainty in estimating neural connections would guarantee controllability. Classic results on structural controllability cannot be directly applied to symmetric (brain) networks. In fact, these results assume that the network weights can be selected arbitrarily and independently from one another, a condition that cannot be satisfied when the weights need to be symmetric. To overcome this limitation we proceed as follows: first we show that network controllability remains a generic property when the weights are symmetric; then, we find a choice of symmetric weights that guarantees controllability. This ensures that brain networks are structurally controllable from one single node, even with symmetric weights, and that almost all choices of symmetric edge weights yield controllable networks. Tommaso Menara and Fabio Pasqualetti are with the Mechanical Engineering Department, University of California at Riverside, {tomenara, fabiopas}@engr.ucr.edu. Shi Gu and Danielle S. Bassett are with the Department of Bioengineering and the Department of Electrical and Systems Engineering, University of Pennsylvania, {gus,[email protected]}. ar X iv :1 70 6. 05 12 0v 1 [ m at h. O C ] 1 6 Ju n 20 17