Root-Cause Analysis of Activation Cascade Differences in Brain Networks

Abstract

Diffusion MRI imaging and tractography algorithms have enabled the mapping of the macro-scale connectome of the entire brain. At the functional level, probably the simplest way to study the dynamics of macro-scale brain activity is to compute the “activation cascade” that follows the artificial stimulation of a source region. Such cascades can be computed using the Linear Threshold model on a weighted graph representation of the connectome. The question we focus on is: if we are given such activation cascades for two groups, say A and B (e.g., controls versus a mental disorder), what is the smallest set of brain connectivity (graph edge weight) changes that are sufficient to explain the observed differences in the activation cascades between the two groups? We have developed and computationally validated an efficient algorithm, TRACED, to solve the previous problem. We argue that this approach to compare the connectomes of two groups, based on activation cascades, is more insightful than simply identifying “static” network differences (such as edges with large weight or centrality differences). We have also applied the proposed method in the comparison between a Major Depressive Disorder (MDD) group versus healthy controls and briefly report the resulting set of connections that cause most of the observed cascade differences.

A Appendix

1.1 A.1 Optimization of TRACED

A key observation is that if adding a single edge (x, y) into a solution set does not change the activation status of node y, we will inevitably need to add additional edges pointing to y to build a final solution. Otherwise, for a solution C with (x, y), we can always find a better solution $C' = C -\{(x, y)\}$ with $U'_C(s) = U'_{C'}(s)$.

Therefore, we can improve the original TRACED algorithm, by adding a collection of edges in each iteration, so that $U'_C(s)$ changes when we create a new partial solution. This way we can reduce the number of partial solutions that we create during the search for the optimal solution. How do we find the collection of edges that can cause the change in $U_C(s)$? We know that we focus on change of activation status of nodes in $U'_C(s) \triangle U(s)$, and so we can discuss the case of nodes $U(s) \setminus U'_C(s)$ and $U'_C(s) \setminus U(s)$ separately.

• 1. For each node v in $U(s) \setminus U'_C(s)$, we can check if there is an ensemble of edges from $U(s) \cap U'_C(s)$ pointing to this node, so that if we include the ensemble into the solution, v would be active in the updated $U'_C(s)$. It is guaranteed that we can find at least one such collection of edges. Otherwise, we cannot explain why this v could be active in U(s).

• 2. For nodes in $U'_C(s) \setminus U(s)$, we can further find its subset $T_C(s)$ so that for each node $v \in T_C(s)$, $\sum _{u \in U(s) \cap U'_C(s)} w(u, v) \ge \theta$. We can prove that $U'_C(s) \setminus U(s)$ will no longer be in $U'_C(s)$ if and only if we add an ensemble of edges for each node in $T_C(s)$ into C. If for a node v in $T_C(s)$ we do not add edges connecting to v into C, v will remain active and present in $U'_C(s)$. If we add edges connecting to v for every node v in $T_C(s)$, none of the nodes in $U'_C(s) \setminus U(s)$ receive an activation more than $\theta$, so that they will no longer be active.

With this modification, each partial solution C corresponds to a state $U'_C(s)$, and it is guaranteed that there are no edges that can be removed from C without changing that state. Therefore, all partial solutions corresponding to one state are equivalent, in terms of the edges that need to be added to the solution to reach another state. Therefore, we can construct a graph of solutions, where each node x corresponds to a state, and each edge $(x, y, \{e_1, \dots \})$ corresponds to an ensemble of edges $\{e_1, \dots \}$ needed to be added into the partial solutions corresponding to state x so that the new solution leads to state y. Such an edge is also weighted, with a weight that is equal to the number of edges in the collection. Notice that there can be multiple edges between two nodes, each corresponding to one collection of edges and may have a different weight different than other edges.

With such a graph of solutions, our goal is equivalent to finding the weighted shortest path between the initial state $U'(s)$ and the final state $U = U'_{\hat{C}}(s)$ in the graph. This is because the sum of the weights of edges along a path in the graph of solutions would be the number of actual edges we include in the final solution. We can find the shortest path using Dijkstra’s algorithm since we have only positive weights. The major benefit of having this graph of solutions is that we can deal with the case of multiple optimal solutions more explicitly. They will be represented as multiple shortest paths from the initial state to the final state.