Dynamical stability of intrinsic connectivity networks

Abstract

Functional connectivity MRI (fcMRI) has become a widely used technique in recent years for measuring the static correlation of activity between cortical regions. Using a publicly available resting state dataset (n = 961 subjects), we obtained high spatial-resolution maps of functional connectivity between a lattice of 7266 regions covering the gray matter. Average whole brain functional correlations were calculated, with high reproducibility within the dataset and across sites. Since correlation measures not only represent pairwise connectivity information, but also shared inputs from other brain regions, we approximate pairwise connection strength by representing each region as a linear combination of the others by performing a Cholesky decomposition of the pairwise correlation matrix. We then used this weighted connection strength between regions to iterate relative brain activity in discrete temporal steps, beginning both with random initial conditions, and with initial conditions reflecting intrinsic connectivity networks using each region as a seed. In whole brain simulations based on weighted connectivity from healthy adult subjects (mean age 27.3), there was consistent convergence to one of two inverted states, one representing high activity in the default mode network, the other representing low relative activity in the default mode network. Metastable intermediate states in our simulation corresponded to combinations of characterized functional networks. Convergence to a final state was slowest for initial conditions on the borders of the default mode network.

Figure 1

Simulating correlated BOLD data using Cholesky decomposition. A Intrinsic noise time series were independently generated for 7266 ROIs (1000 time points per ROI), such that each time series showed a 1/f distribution, with 0 mean and 1 standard deviation (matrix A from [1]). B Comparison of Fisher-transformed correlation for each connection from actual measured data with simulated correlation. Simulated correlation was obtained by the Pearson correlation coefficient between each row of TA, where T is the matrix obtained from the Cholesky decomposition. Y-axis shows the mean correlation values of 1000 trials, averaged after Fisher transformation. Red line shows y=x.

Figure 2

Reproducibility of functional correlation measurements. A Distributions of the difference in correlation between randomly selected subsamples of subjects across all 26.3 million connections. Subsets of 50, 100, 240, or 480 subjects were compared. Each histogram shows for two unique subsamples of the population the distribution of difference in mean correlation across all connections between the two groups. B Standard deviation of difference in correlation across all connections as a function of the number of subjects averaged. The y-axis represents the standard deviation of difference in mean correlation across connections for two subsamples of the population. The standard deviation for each connection across all subjects was averaged across connections and was 0.2828. The red fitted curve is 0.2828/sqrt(number of subjects). C Comparison of mean Fisher transformed correlation values from 2 unique subsamples of 480 and 481 subjects. Red line shows y=x. D Pseudocolor plot showing mean Fisher transformed correlation values for connections between each ROI. Color range was limited to −0.2 to 0.6 to optimize image contrast.

Figure 3

Effect of Cholesky decomposition on intrinsic connectivity networks. To the left are shown Fisher-transformed full correlation values of each ROI to 3 seed ROIs in the posterior cingulate, left intraparietal sulcus, and left primary auditory cortex. Corresponding values of the weighted connectivity matrix (Cholesky decomposition) are shown in the center column for the same seeds. Partial correlation values are shown in the right column for the same seeds. Images were normalized by subtracting the mean and dividing by the standard deviation across ROIs, with color showing standard deviations across ROIs for better comparison of image contrast in the three techniques.

Figure 4

Density maps comparing distribution of full correlation, Cholesky decomposition, and partial correlation techniques. A Distribution of Cholesky decomposition vs. partial correlation. Color scale shows filled contour plots of the logarithm of the number of connections in each bin. Bin size was 0.01 in each axis. B Density of Cholesky decomposition vs. partial correlation showing only connections between ROIs greater than 6 cm apart in Euclidean distance. C As above, comparing Cholesky decomposition with full correlation. D Cholesky decomposition vs. full correlation for connections between ROIs greater than 6 cm apart.

Figure 5

Convergence to the default mode network. A Difference between steps for each of 100 simulations from random initial conditions, measured as the sum of absolute value of differences between normalized intensity values at each ROI between the two steps. Only a subset of the simulations is shown to better allow visualization of traces. B Pseudocolor plot showing difference between steps for the same 100 simulations. C Final convergence state for one of the simulations. Colors represent normalized activity across ROIs. All of the final convergence states from these simulations were qualitatively identical or an additive inverse of the image shown (negative values where positive values are shown) although in a minority of simulations (<1%) the final convergence state was instead the visual network. D Final convergence state obtained only from data from subjects in the Beijing (left) and Cambridge (right) datasets.

Figure 6

Convergence states for a single subject. A Each row represents the final convergence state from data obtained from 50 minutes of BOLD imaging while the subject was watching cartoons during an independent imaging session. B Each row represents final convergence state from data obtained from 50 minutes of BOLD imaging in a resting state, eyes open. C For each 50-minute session, the final convergence state was measured as a vector of activity across 7266 ROIs. The plot shows correlationcoefficients between the activity vector for each pair of sessions. Pairs of unique sessions were more similar for when subjects were watching cartoons in both sessions (r = 0.71) or resting in both sessions (r = 0.68), than when one session was watching cartoons and the other was in the resting state ( r= 0.43). A two-tailed t-test comparing correlation coefficients for different tasks vs the same task was significant at p = 1.9 e - 17).

Figure 7

Clustering of metastable states. To the left is a dendrogram showing clustering of 949 simulations producing metastable states where a local minimum was seen during convergence. The images to the right show averages of the metastable states for each cluster, obtained at the time point where a local minimum was seen in the difference between steps of the simulation.

Figure 8

Steps to convergence, starting with the correlation network for each ROI. A Color represents the iteration at which the simulation converged to within 0.05% of the final convergence state. Initial conditions for each ROI consisted of the normalized correlation across ROIs to the seed ROI. B Initial conditions for which the simulation required 10 or more steps to converge, superimposed on the activity from the final default mode convergence state. C Brain regions are shown in blue for which initial conditions with high activity only in this region resulted in convergence to a state with high default mode network activity. Initial conditions with high activity in regions in red converged to a state with low default mode network activity.