A novel mathematical construction for identifying attractors from task-driven fMRI data

Functional brain imaging data can provide a window into the task-driven network states that shape brain function (or dysfunction). Conventionally, these network states can be represented as bivariate correlation matrices (which are formed from fMRI time series from multiple brain regions/nodes within any task window). Here, we treat these conventional connectivity matrices as connectivity terrains in order to recover local structure. In principle, any such terrain can be traversed node by node, where from any node, one can move towards its nearest functional neighbor (i.e., its maximally correlated node). In terrains with meaningful structure, such traversals across multiple nodes should converge to attractor nodes; here, the nodes that flow into a shared attractor form an attractor basin, which effectively is a sub-network within the system. Extant methods (e.g., degree distribution and characteristic path length) can summarize global network properties but cannot identify attractor nodes and basins. Here, we construct a new relation, called transitive maximal correlation (TMC) that can recover attractors and attractor basins in connectivity terrains. Node A is said to be transitively maximally correlated to node B if and only if B is an attractor into which A flows. We first develop the mathematical basis for deriving a TMC matrix TMC(M) from a bivariate correlation matrix M (before explaining this with hypothetical data). We next apply the TMC relation to connectivity terrains derived from real fMRI time series data, where these data were acquired in two distinct task-domains (that varied in their extent of cross-cerebral demand): i) associative learning and ii) visually guided motor control. We show that TMC is remarkably sensitive to inter-hemispheric structure in the connectivity terrain; here, attractor pairs that were inter-hemispheric homologues were more likely to be observed for the cross-cerebral learning task, than the more circumscribed motor-control data. We confirm the condition-specific sensitivity of TMC showing that observed attractor basins differed significantly across conditions of the learning task. Finally, we demonstrate that TMC complements graph theoretic constructions like path length and betweenness centrality. We suggest that TMC is a mathematically sound and novel method for capturing functional properties of brain networks.