Metastable Resting State Brain Dynamics

Abstract

Metastability refers to the fact that the state of a dynamical system spends a large amount of time in a restricted region of its available phase space before a transition takes place, bringing the system into another state from where it might recur into the previous one. beim Graben and Hutt (2013) suggested to use the recurrence plot (RP) technique introduced by Eckmann et al. (1987) for the segmentation of system's trajectories into metastable states using recurrence grammars. Here, we apply this recurrence structure analysis (RSA) for the first time to resting-state brain dynamics obtained from functional magnetic resonance imaging (fMRI). Brain regions are defined according to the brain hierarchical atlas (BHA) developed by Diez et al. (2015), and as a consequence, regions present high-connectivity in both structure (obtained from diffusion tensor imaging) and function (from the blood-level dependent-oxygenation-BOLD-signal). Remarkably, regions observed by Diez et al. were completely time-invariant. Here, in order to compare this static picture with the metastable systems dynamics obtained from the RSA segmentation, we determine the number of metastable states as a measure of complexity for all subjects and for region numbers varying from 3 to 100. We find RSA convergence toward an optimal segmentation of 40 metastable states for normalized BOLD signals, averaged over BHA modules. Next, we build a bistable dynamics at population level by pooling 30 subjects after Hausdorff clustering. In link with this finding, we reflect on the different modeling frameworks that can allow for such scenarios: heteroclinic dynamics, dynamics with riddled basins of attraction, multiple-timescale dynamics. Finally, we characterize the metastable states both functionally and structurally, using templates for resting state networks (RSNs) and the automated anatomical labeling (AAL) atlas, respectively.

Keywords: BOLD fMRI; brain hierarchical atlas; diffusion tensor imaging; metastability; recurrence structure analysis; resting state.

Figure 1

Functional image preprocessing pipeline. Dual acquisition is needed, high-resolution anatomical images (T1) and functional images at rest. Following state-of-the-art pipeline of neuroimaging preprocessing, time series of the blood oxygenation level dependent (BOLD) signal were obtained for each region of interest (ROI), defined by a functional atlas of 2,514 ROIs. Finally, the time series were averaged using different partitions of the brain hierarchical atlas (BHA). The partition with maximal metastability was the one with M = 40 modules (see section 2.4 for details).

Figure 2

Segmentation complexities measured as number of metastable states against number of structure-function modules for normalized data using the cosine distance.

Figure 3

Selection of segmented rsfMRI time series. Upper panels: rsfMRI time series averaged over 40 modules, lower panels: symbolic sequences s′ resulting from optimal partitions into metastable states. Selected subjects: (A) #1, (B) #7, (C) #18, (D) #30.

Figure 4

Markov utility functions from averaged rsfMRI time series over 40 modules. Subjects: (A) #1, (B) #7, (C) #18, (D) #30.

Figure 5

Bistable population resting state dynamics as resulting from Hausdorff clustering.

Figure 6

Spatial projections of metastable brain resting states. (A) Projection onto AAL regions. Notice that just for illustration purposes, although the spatial projection has been performed over all the 45 AAL brain regions, some of the labels have been removed from the x-axis. Brain areas corresponding to maximum projections (written in text) were lingual, calcarine, precuneus, occipital and cingulate cortices. (B) Projection onto RSN regions. Maximum projections occurred for the auditory and medial visual networks. (C) Brain localization of blue attractor with regard to AAL and RSN partitions. Notice that projection strength might have positive or negative values depending on the direction of the basis vectors.

Figure 7

Anatomical AAL projections of the two metastable states obtained from optimizing RSA over 40 modules. Subjects: (A) #1, (B) #7, (C) #18, (D) #30.

Figure 8

Resting state network (RSN) projections of the two metastable states obtained from optimizing RSA over 40 modules. Subjects: (A) #1, (B) #7, (C) #18, (D) #30.

Figure 9

Three possible mathematical mechanisms to model switching dynamics: (A1–A3) slow-fast systems with a hysteresis loop; (B1–B3) systems with intermingled basins of attraction; (C1–C3) systems with a robust heteroclinic cycle. In all parts (left, central, right), the top panel shows a sketch of the phase space; the middle panel shows a time series of the minimal model representing one of the frameworks displaying alternating switches between two metastable states s₁ and s₂; the bottom panel shows minimal ODEs for a given framework. All equations are phenomenological but the resulting dynamics can be found in biophysical models of brain activity.