Control theory illustrates the energy efficiency in the dynamic reconfiguration of functional connectivity

Abstract

The brain's functional connectivity fluctuates over time instead of remaining steady in a stationary mode even during the resting state. This fluctuation establishes the dynamical functional connectivity that transitions in a non-random order between multiple modes. Yet it remains unexplored how the transition facilitates the entire brain network as a dynamical system and what utility this mechanism for dynamic reconfiguration can bring over the widely used graph theoretical measurements. To address these questions, we propose to conduct an energetic analysis of functional brain networks using resting-state fMRI and behavioral measurements from the Human Connectome Project. Through comparing the state transition energy under distinct adjacent matrices, we justify that dynamic functional connectivity leads to 60% less energy cost to support the resting state dynamics than static connectivity when driving the transition through default mode network. Moreover, we demonstrate that combining graph theoretical measurements and our energy-based control measurements as the feature vector can provide complementary prediction power for the behavioral scores (Combination vs. Control: t = 9.41, p = 1.64e-13; Combination vs. Graph: t = 4.92, p = 3.81e-6). Our approach integrates statistical inference and dynamical system inspection towards understanding brain networks.

Fig. 1. Conceptual schematic.

a We begin with the preprocessed BOLD time series from 400 cortical (Schaefer atlas) and 19 subcortical regions. The static functional connectivity matrix is calculated as the Pearson’s correlation across the full-time sequence. Two types of dynamic functional connectivity are estimated by the autoregression approach and the sliding-window approach^,, respectively. b For a pair of given initial and target states, different types of interaction matrices would result in distinct trajectories thus varied energetic cost.

Fig. 2. Spatial distribution of functional controllability.

We show the distribution of mean (a) average and (b) modal controllability across the different functional brain networks. The mean average control values (Mean ac-Value) and mean modal control values (Mean mc-Value) are calculated across all subjects. The system mean is calculated across the within-system regions for all subjects. We adopt the eight brain networks from Yeo’s partition to compare differences in controllability for different networks^,.

Fig. 3. Energy efficiency explains the dynamic reconfiguration of functional connectivity.

Each point in (a–h) represents a transition between a pair of initial and target states. The E_s is the energy cost when the control dynamics are induced by the sFC-Correlation. The $E_d$ is the energy cost when the control dynamics is induced by the dFC-Slidingwindow. The E_r is the energy cost when the control dynamics is induced by the same matrices of the dFC-Slidingwindow but in a randomly shuffled order. The y-axis of (f)–(h) represents the log of the relative energy inflation when changing the connectivity matrices from the observed order to the random order. a We successively set Default Mode network (DM), the combination (CN) of Default Mode, Dorsal Attention (DA), Salience (SA), and Fronto-parietal networks (FP), or the whole network (All) as the control set. The T-tests measure their significance vs. zero. b When the network is controlled through DMN with varied control intervals $\left(\tau \right)$, the improvement of energetic cost (i.e., the vertical difference between blue and red dots) is higher on short control intervals (i.e., smaller $\mathrm{\tau }$) than the long control intervals. Similar patterns exist when c the control set is the combination of DMN, DA, SA, and FP but with less significance. d The difference is less visually significant when controlling on the whole network. e When we permute the order of sequential matrices in the temporal control paradigm, we also observe an increase in the energy cost. f–h The relative increase of log energy is negatively correlated to the log energy corresponding to the observed order of reconfiguration for dynamic functional connectivity.

Fig. 4. Prediction of scores in the behavioral tasks.

The z-value is the Fisher z-value of the correlation between the observed and predicted scores. a The control theoretical measurements of the dorsal attention network contribute most in the prediction model. The z-value measures the decline of the model performance when the measurement values on the corresponding area are removed from the input feature vector. b The graph-theoretical measurement in the visual system contributes most in the prediction model. c The control theoretical measurements of the dorsal attention and default mode networks contribute positively high in the prediction model. d The graph-theoretical measurement of the dorsal attention, default mode, and fronto-parietal networks contributes positively high in the prediction model.

Fig. 5. Complementary power of graph and control theoretical measurements in predicting behavioral scores.

We demonstrate the gain of predictive power by using a pair of features combining both graph and control theoretical measurements over using features chosen from either graphical or controllability measurements. a The improvement is positive for 41 of 58 tasks over the input features selected from either control or graph-theoretical measurements. In (b–d), we further examine the prediction improvement separately on three groups. We exhibit the distribution of prediction improvement on the task scores b that can be significantly predicted by control measurements, c that can be significantly predicted by graph measurements, and d that can be significantly predicted by the combination of the control and graph measurements. We say the score can be significantly predicted when the correlation of the predicated and observed scores is significant with $p<0.05$. The prediction improvement in (b–d) is calculated as the z-values of the difference of correlations between the predicted and observed scores.