Graph-theoretical analysis for energy landscape reveals the organization of state transitions in the resting-state human cerebral cortex

The resting-state brain is often considered a nonlinear dynamic system transitioning among multiple coexisting stable states. Despite the increasing number of studies on the multistability of the brain system, the processes of state transitions have rarely been systematically explored. Thus, we investigated the state transition processes of the human cerebral cortex system at rest by introducing a graph-theoretical analysis of the state transition network. The energy landscape analysis of brain state occurrences, estimated using the pairwise maximum entropy model for resting-state fMRI data, identified multiple local minima, some of which mediate multi-step transitions toward the global minimum. The state transition among local minima is clustered into two groups according to state transition rates and most inter-group state transitions were mediated by a hub transition state. The distance to the hub transition state determined the path length of the inter-group transition. The cortical system appeared to have redundancy in inter-group transitions when the hub transition state was removed. Such a hub-like organization of transition processes disappeared when the connectivity of the cortical system was altered from the resting-state configuration. In the state transition, the default mode network acts as a transition hub, while coactivation of the prefrontal cortex and default mode network is captured as the global minimum. In summary, the resting-state cerebral cortex has a well-organized architecture of state transitions among stable states, when evaluated by a graph-theoretical analysis of the nonlinear state transition network of the brain.

and S1C). The pairwise interactions, Jij, were distributed among positive and negative values, ranging from -0.4807 to 1.1950 ( Figure S1B and S1D). The baseline sensitivities of the inferior parietal lobe, superior frontal gyrus, caudal middle frontal gyrus, and pars-triangularis (HIP, HSF, HcMF, and HTr) were more negative than those of other regions. Four pairwise interaction parameters, JOp-Tr, JcMF-IP, JIP-PC, and JIC-PC, were relatively larger than others. Thus, each ROI of the system would be inactivated without pairwise interactions. Estimated strong positive and negative pairwise interactions reflect how two nodes were easily co-activated, and the activation patterns of the local minima (Fig. S1E). The superior parietal (SP), isthmus cingulate (IC) and precuneus (PC) were positively and negatively connected with other ROIs.  The state transition network that relate to transition process to the lowest local minimum is presented. All states which appeared in the state transition process were assigned nodes. Orange, light blue, and green color represent transition, transient, and local minima states, respectively. (C) Both of TS2 and ~TS2 systems did not show any correlation between the energy barrier and effective path lengths.

B. State transition network analysis of the resting state in right-hemisphere
In the main text, cortical regions in the left-hemisphere were mainly investigated. To confirm if brain dynamics in the right-hemisphere contain similar properties, we further constructed the maximum entropy model (MEM) for the right-hemisphere. The activation patterns of rs-fMRI data were reproduced with a high accuracy of fit (rD = 85.5 %) and reliability (ER = 99.9 %) ( Figure S5A). Baseline sensitivity parameters Hi and pairwise interaction, Jij, are displayed in Figure S5B. Strong positive correlation was observed between the estimated MEM parameters of the right-hemisphere and the left-hemisphere (r=0.982, p=7.932 × 10 -137 ). Although most of MEM parameters of the right-hemisphere were similar to those of the left-hemisphere, several MEM parameters were different; e.g., HSF, HrMF, and JIP-rMF ( Figure S5C).
Analysis of energy landscape identified 18 local minima (having lower energy than their neighbor states) of the right-hemisphere cerebral cortex system at rest. From analysis of the state transition network among full states (STN-FS) and state transition processes (STN-GM) from local minima (LM) toward the global local minimum (LM15), we confirmed that similar properties of state transitions such as existence of hub nodes and multistep process were conserved in the resting-state cerebral cortex system of the right-hemisphere ( Figure S6). Activation patterns of local minima and transition rates were similar between the right and left cerebral cortex systems, and clustered (well organized) state transition processes was also identified in the right-hemisphere cerebral cortex system ( Figure S7). More specifically, we identified three groups, and similar to the left-hemisphere cerebral cortex system at rest, we found TS1 which appears to mediate transition between two large groups. When we exclude this hub state, its complementary state ~TS1 appeared to serve as a detour for inter-group transitions with similar transition rates (99 %). Thus, "redundant" pathways in inter-group transition processes existed in both the left and right cerebral cortex systems.

C. Energy landscape of randomized system
To show the characteristic of the resting state brain network, we generated and evaluated 500 randomized MEMs (sets of MEM parameters) in two ways.
First, we generated 500 MEM parameter sets by choosing random values within a range of [-2.2, 0.1] for Hi and a range of [-0.5, 1.2] for Jij. These ranges were determined by the minimum and maximum values of the currently estimated MEM parameters from the resting state fMRI. By using energy landscape analysis, we found that the total numbers of local minima from randomly generated MEMs were less than three (supporting information Figure  S8A), which are extremely smaller than that of the resting state brain (n = 14).
Second, we generated another 500 MEMs, by permutating Hi and permutating Jij parameters (similar to a study in Watanabe et al (2013)). As a result, we found that most systems have smaller numbers of local minima (the mean was around 5) than that of resting state fMRI. We did not find an organized structure in the state transition network of energy landscapes from those permuted MEMs, as shown in the perturbation analysis exampled in Figure 5 (C) and (D) with a = 0.8 and a = 1.2. Even in some systems with total numbers of local minima similar to that of experimental data, their energy landscapes were simpler than that of resting state fMRI (see below supporting information Figures S8B and S9). We presented a simulated system which contains five local minima as a representative example of random MEMs. The system has a simple transition network structure (no clustered or organized structure similar to that of Figure 5) (supporting information Figure S9).