Figure 3: Conceptual illustration of the whole-brain dynamic landscape, bifurcation, and phase transition.

A multistable dynamic landscape (a) contains multiple attractors, shown as troughs occupied by purple balls. Each attractor corresponds to a distinct pattern of activation over the whole brain (i-iv). Influenced by external input or intrinsic noise, the model brain may transition from its current state (attractor i, bright purple ball) to a different one (ii, iii, or iv, dim purple balls), indicated by black arrows. Structural features of a model brain can alter the shape of the landscape, causing some attractors to appear or disappear through a process mathematically named a bifurcation (a→b, a→c, or the reverse). By modifying the repertoire of attractors, bifurcation alters the set of possible transitions and the coordination between regions during transitions. For example, in landscape (a), the left and right hemisphere can be co-activated during a transition (i→iii), or activated separately through other transitions (i→ii, or i→iv). In contrast, in landscape (b), the left and right hemisphere can only be co-activated, and in (c), only activated separately. Numerically, a repertoire of attractors can be represented as a matrix, where each row represents an attractor and each column represents a brain region (repertoire matrix a, b, c, with entries shown as blue/red color blocks). The overall inter-regional coordination across attractors can be estimated by the rank correlation between the columns of the repertoire matrix. The resulted square coordination matrix summarizes how brain regions transition together over the entire landscape, serving as a signature of the landscape (coordination matrix a, b, c, shown to the right of each repertoire). In more complex landscapes (not shown), there are many more attractors, and they correspond to subtler patterns of activation (d,e; see also Figure 4). The coordination between brain regions during a transition is correspondingly more complex (f=e-d), with some regions co-activated (red) while others co-deactivated (blue).