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.

1. Introduction

Functional connectivity MRI (fcMRI) examines the synchrony of slow-wave fluctuations in BOLD signal (<0.08 Hz) between geographical regions in the brain, and infers common functional relationships when correlated BOLD time series exist between regions of interest (ROIs)(Biswal et al., 1995; Biswal et al., 2010; Fox and Raichle, 2007; Zhang and Raichle, 2010). Similarly, antagonistic relationships between regions are inferred from anticorrelations in their respective BOLD signal fluctuations over time(Fox et al., 2005; Fransson, 2005), when not induced by postprocessing strategies(Anderson et al., 2011b; Fox et al., 2009; Murphy et al., 2009).

fcMRI methods were first used to map the sensorimotor cortex of the resting human brain(Biswal et al., 1995), with subsequent investigations demonstrating consistent intrinsic connectivity networks detectable by fcMRI during wakeful rest (Cordes et al., 2000; Damoiseaux et al., 2006; Greicius et al., 2003; He et al., 2009; Kiviniemi et al., 2009; Lowe et al., 1998; Tomasi and Volkow, 2011). Recent anatomical work within resting state functional connectivity analysis has benefitted from large datasets from multiple centers. In conjunction with the Neuroimaging Informatics Tools and Resources Clearinghouse (NITRC) and the International Neuroimaging Data-sharing Initiative (INDI), a large scale resting state fMRI dataset has already been made openly accessible through the 1000 Functional Connectomes Project(Biswal et al., 2010) containing resting state fMRI data obtained from over 1400 subjects by 28 international laboratories.

Among the most robust distributed connectivity networks is the default mode network(Raichle et al., 2001; Raichle and Snyder, 2007), comprised from brain regions that are consistently more active during wakeful rest than during numerous cognitive tasks(Gusnard and Raichle, 2001). These observations led to the hypothesis that this network of regions might be supporting default activity of the human brain (Greicius et al., 2003; Gusnard et al., 2001; Raichle et al., 2001), such as attending to internal stimuli, self-reflection, or internal narrative(Cavanna and Trimble, 2006; Gusnard et al., 2001; Northoff et al., 2006).

Despite such extensive work on clarifying the functional network anatomy of the brain, there are yet relatively few reports attempting to extend static connectivity measurements to whole brain dynamical models(Deco et al., 2010). One approach used known structural relationships in the macaque brain from anatomical tracing studies to simulate interactions of neural oscillators in each region using weak coupling coefficients(Honey et al., 2007). Another approach using macaque connectivity demonstrated ultraslow coherent network fluctuations in a model using anatomic connectivity, time delays, and noise(Ghosh et al., 2008). The relationship of anatomic topology, coupling strength, time delay between regions, and noise to temporal dynamics was further explored in a report using a network of Wilson-Cowan modulators to simulate slow coherent fluctuations(Deco et al., 2009).

One limitation of such approaches to network modeling is that precise information from anatomical tracer studies is not available for the more complex human brain, and precise path lengths and anatomic topology are difficult to measure. Also, the complexity of connectivity between regions becomes computationally intractable with the number of nodes studied in a dynamical network. As an alternate approach, we investigate a dynamical model that treats as computational units regions of gray matter on the scale of several millimeters. Rather than build neural networks from high temporal-resolution oscillations, we use an iterative approach at discrete time points to evaluate dynamical relationships between large-scale distributed networks. We report that using only functional connectivity measurements between a lattice of brain regions covering the gray matter, the brain’s default mode network emerges in such simulations as a dynamically stable state, with other described intrinsic connectivity networks demonstrating reproducible metastability across a wide range of initial conditions.

2 Materials and Methods

2.1 fMRI Data Sources

fMRI data was extracted from the open-access ‘1000 Functional Connectomes Project’ (http://fcon_1000.projects.nitrc.org/) in which resting-state fMRI scans have been aggregated from 28 sites. (Biswal et al., 2010) For inclusion we required whole-brain coverage from MNI coordinates z=−35 to z=70. Any subject for whom postprocessed data did not cover all 7266 ROIs was discarded prior to analysis. Although postprocessing steps were performed using an automated batch, the results of normalization, segmentation, and realignment steps were manually inspected for all subjects, and any subject for whom the normalized and segmented images were not in close alignment with the MNI template on visual inspection were discarded. The Dallas sample was not included because of ambiguity about left/right orientation at the time of analysis. From 1051 subjects for whom batch postprocessing was initiated, 961 subjects from 23 sites were included in the analysis sample. The datasets from which these subjects were obtained are listed in Table 1. Mean age of the subjects was 27.3 +/− 11.7 s.d. years (range 13–79). 525 subjects were male, 394 female, and gender of 42 subjects was unknown.

Table 1.

Number of subjects used in the analysis from each center from the 1000 Functional Connectomes Project.

Site	Number of subjects	Site	Number of subjects	Site	Number of subjects
AnnArbor	18	Leiden	31	Oulu	100
Baltimore	23	Leipzig	37	Oxford	14
Bangor	1	Milwaukee	59	PaloAlto	14
Beijing	188	Munchen	12	Pittsburgh	4
Berlin	26	NewYork	65	Queensland	17
Cambridge	194	Newark	17	SaintLouis	31
Cleveland	30	Ontario	6	Taipei	19
ICBM	37	Orangeburg	18	Total	961

Additional data from a single subject was also analyzed. Data from this subject have been previously published(Anderson et al., 2011d), although analyses presented herein are unique to this report. For this subject, (male, 39 years old), eleven scan sessions were obtained. In six of the sessions, the subject was watching cartoons (Looney Tunes Golden Collection, Volume 1, Warner Brothers) for ten 5-minute BOLD scans per session (50 minutes BOLD data per session). In the other 5 sessions, 50 minutes of resting BOLD data was obtained (10 5-minute scans, eyes open).

2.2 fMRI Post-processing

The following sequence was used for image post-processing of all BOLD image datasets. Using SPM8 toolbox (Wellcome Trust, London), BOLD images were realigned (realign, estimate and write), coregistered to MPRAGE image (coregister, estimate and write), and normalized to MNI template (normalize, estimate and write, T1.nii template). Gray matter, white matter and CSF were segmented from MPRAGE image using SPM8 segment function (modulated, normalized, thorough clean). Images were bandpass filtered between 0.001 and 0.1 Hz and a linear detrend was performed at each voxel in the brain. Because each site used slightly different TR, we note that this detrend step may introduce some heterogeneity of filtering between sites associated with the detrend operation. Time series were averaged from 2 ROIs in the white matter (bilateral centrum semiovale, CSF (lateral ventricles), soft tissues of the head and face, and 6 rigid motion correction parameters from realignment step as previously described(Anderson et al., 2011b; Anderson et al., 2010) and for each voxel, a general linear model was used to find a best fit for white matter, CSF, soft tissues, and motion parameter time series, which were subtracted from the voxel’s time series. No regression was performed of the global signal or gray matter.

2.3 Methods for Calculating Connectivity

Typically, functional connectivity has been measured by calculating the correlation coefficient between BOLD time series in different brain regions or networks(Biswal et al., 1995; Fox and Raichle, 2007). Alternately, approaches in the frequency domain have used coherence to similarly identify a relationship between the two time series(Sun et al., 2004). Nonlinear metrics of synchrony between the time series, such as mutual information, have also been described(Salvador et al., 2007). Yet while these approaches can establish a relationship between two regions that presumably incorporates information about underlying structural connectivity between the regions, it also may include information about shared connections with a third region. For our purposes, we would like to improve the quantitative relationships of connectivity between brain regions by attempting to account for such indirect connections. One method that has been used for approximating direct connections only is to regress out the effects of all other brain regions using partial correlation(Marrelec et al., 2006; Smith et al., 2011). We used both full correlation, partial correlation, and a novel method using the Cholesky decomposition of the correlation matrix between brain regions to estimate direct connections between brain regions. Details for these methods are given below.

2.4 Calculation of Full and Partial Correlation Matrices

An MNI template for gray matter (SPM8, grey.nii, intensity >0.5) was parcellated into 7,266 regions of interest (ROIs) by removing voxels from the image that were less than 5 mm from other retained voxels. Then all gray matter voxels were assigned to the closest remaining voxel’s ROI. The ROIs ranged from 2 to 12 voxels in extent (mean 4.9 +/− 1.3 voxels at isotropic 3 mm resolution). Time series data for each ROI were generated from the averaged time series of all voxels pertaining to the respective ROI. Postprocessed time series data from every ROI were compared to the time series from all other ROIs using Pearson correlation coefficients. The resultant 7,266 by 7,266 matrix of correlation coefficients constituted a whole-brain functional correlation matrix for an individual subject. The whole-brain correlation matrices for each subject were Fisher transformed by evaluating hyperbolic arctangent to improve normality(Fox et al., 2005) and averaged across all subjects to produce a mean full correlation matrix.

The mean full correlation matrix was converted to correlation values by reverse Fisher transformation, and partial correlation values were obtained using method of Marralec et al.(Marrelec et al., 2006; Smith et al., 2011). We inverted the full correlation matrix C_ij to obtain Y_ij = C⁻¹. Partial correlation coefficients P_ij were obtained by the relationship:

$${\mathrm{P}}_{\text{ij}}=-{\mathrm{Y}}_{\text{ij}}/\text{sqrt}({\mathrm{Y}}_{\text{ii}}{\mathrm{Y}}_{\text{jj}})$$ [1]

2.5 Decomposition of Functional Correlation Matrix

Correlation coefficients can measure synchrony between two time series, but are only an indirect estimate of anatomic connectivity(Honey et al., 2009). One reason for differences is that shared inputs to two regions contribute to correlation values. For example, if two regions x and y had no direct relationship, but both exhibited positive correlation with a third region z, it would be expected that x and y would nevertheless show significant positive correlation.

We approached this problem by considering a linear model in which we start with an intrinsic noise time series A_it = [a_i(t)] for region i and time point t and a transition matrix T=[t_ij] of coefficients representing connectivity between region i and region j. A is constructed to have mean 0 and standard deviation 1 for each row by subtracting the mean and dividing by the standard deviation. For m regions and n time points, A will represent an m × n matrix, and T will represent an m × m matrix such that:

$$TA=B$$ [2]

where B will be an m × n matrix where row i represents a time series for region i, simulating a BOLD time series. The correlation between rows b_i and b_j should approximate c_ij, the measured correlation value from actual BOLD time series between region i and region j. Then we assign matrix C = [c_ij] as the dot product of the rows of B:

$$TAA’T’=nC$$ [3]

But A represents an intrinsic noise signal for each node which should be independent for each row. For sufficiently long time series A of length n, since the rows a_i(t) are independent with uniform standard deviation, then A A’ will approximate the identity matrix times the length of the time series n and we must find a matrix T such that

$$TT’=C$$ [4]

But this equation is known to have a unique solution T that is upper triangular if and only if C is positive definite(Trefethen and Bau III, 1997), and this solution can be obtained from the Cholesky decomposition of C. We designate t_ij for i>j as the approximated connectivity between region i and region j, and constructed the weighted connectivity matrix used in the analysis by setting t_ji = t_ij for i>j and normalizing each row of the matrix by subtracting the mean and dividing by the standard deviation. Cholesky decompositions were performed using the standard Matlab function chol.m.

To show with simulated data how the Cholesky decomposition may be related to individual time courses from brain regions, we generated 7266 intrinsic noise time series of length 1000 time points (matrix A, 7266 × 1000). To better simulate BOLD data(Anderson, 2008; Cordes et al., 2001), these noise time series were generated with each row having mean 0 and standard deviation 1, with 1/f frequency distribution (pink noise) using the method of Little et al. (www.eng/ox/ac.uk/samp/software/powernoise/powernoise.m)(Little et al., 2007). The 7266 × 7266 correlation matrix was averaged across all 961 subjects after Fisher transformation, and the mean was converted back to correlation values by evaluating the hyperbolic tangent. Cholesky decomposition matrix T was obtained and simulated BOLD time series (matrix B) was generated from the matrix product in [2]. Pearson correlation coefficients were obtained between each pair of rows in B and a 7266 × 7266 matrix of correlation values was obtained. This process was repeated 1000 times, with the resulting correlation matrix averaged across trials after Fisher transformation with results shown in Figure 1. Compared to the actual measured BOLD correlation values, there is close agreement to the time series B, representing a linear combination of independent noise vectors as specified by the transformation matrix T generated by the Cholesky decomposition. Thus, a biophysical model in which one considers each brain region to have intrinsic fluctuations (noise) that combine with the intrinsic activity of other brain regions according to a weighted connectivity matrix T will generate time series with correlation matrix C satisfying Equation 4.

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.

2.6 Whole Brain Functional Simulations

To model brain activity over time, each of the 7,266 ROIs was assigned a pseudorandom value from a normal distribution with mean 0 and standard deviation 1. These starting conditions were considered ‘step 0’ of the simulation. For step 1 of the simulation, the intensity values at every ROI of the brain were assigned by considering every other ROI’s intensity value from the previous step, multiplying that intensity value by its respective weighted connectivity coefficient, then summing the products of intensity and connectivity values in order to determine the new intensity value at each ROI. The simulation was repeated for 5,000 sets of randomized initial conditions, with 40 steps in each simulation.

2.7 Identifying and Categorizing Metastable States

The rate of change in neural activity between steps in the whole brain simulation was determined by finding the sum of the absolute values of intensity differences between corresponding ROIs for successive steps. Metastable patterns of neural activity were identified by local minima in the rate of change. Each of the metastable states across all 5,000 iterations were clustered using the dendrogram.m function with city block p-distribution and average linkage using the MatLab statistical toolbox (R2010b). The characteristic networks for each cluster was determined by calculating the mean for all metastable states within the same dendrogram cluster.

2.8 Determining Convergence

In addition to the 5,000 whole brain simulations in which randomized starting conditions were assigned to each of the 7,266 ROIs, we performed an additional 7,266 simulations using parameters obtained from resting state functional correlation measurements as the starting conditions. In this paradigm, the whole brain initial conditions for each iteration were set to the correlation values corresponding to the seed from one of the respective 7,266 ROIs. For each set of 7,266 initial conditions corresponding to a different ROI seed, we measured the number of steps in the simulation required for the system to converge to within a tolerance of less than 0.05% change in the mean absolute value of the intensity across the ROIs. The scalar corresponding to the number of steps for convergence for each seed ROI was then assigned to the respective ROI, and mapped onto a gray matter whole brain image.

3 Results

To determine a standard map of functional correlation, we averaged pairwise functional correlation measurements between 7266 brain regions of interest (ROI) covering the gray matter for 961 healthy adult control subjects available through the open access 1000 Functional Connectome Project resting state database. Regions were selected by parcellating an image of brain gray matter (SPM8 toolbox, Wellcome Trust, London, grey.nii) into regions such that each region’s center was at least 5 mm distant from every other region, effectively yielding ROIs of 5 mm diameter(Anderson et al., 2011c; Anderson et al., 2011e). Pairwise correlation measurements between these regions comprised 26.3 million connections for each subject.

3.1 Reproducibility of functional correlation measurements

Before attempting dynamical modeling, we characterized the reproducibility of the functional correlation measurements by comparing mean correlation values for a randomly selected subsample of the total control population (Figure 2A) compared to a different unique subsample of the same size. Subject subsamples of all sizes showed a normal distribution of connectivity differences between groups, with standard deviation of the error inversely proportional to the square root of the number of subjects averaged (Figure 2B). When we divided the sample into 2 groups of 480 and 481 subjects, we observed close agreement of correlation values for all 26.3 million connections (Figure 2C), indicating consistent reproducibility of functional correlation outcomes for large population samples from the 1000 Functional Connectomes database. The mean pairwise functional correlation matrix is shown in Figure 2D. By extending the relationship seen in Figure 2B, we estimated that each individual measurement between 2 ROIs had an accuracy of less than 0.01 units of Fisher-transformed correlation compared to what would be expected for a similarly constructed population of 961 subjects.

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.

3.2 Weighted connectivity calculations

Correlation measurements, however, are only an approximation of the expected connectivity strength between two regions, and may be systematically misleading by incorporating shared input from other regions in a pairwise correlation measurement. We attempted to adjust for this relationship by approximating relationships between regions using a Cholesky decomposition of the pairwise correlation matrix as described in the Methods section, wherein weighted connectivity between regions more closely approximates what would be expected if each region were expressed as a linear combination of the other brain regions. We subsequently refer to this pairwise association matrix after Cholesky decomposition as the weighted connectivity matrix between regions.

To compare the results of the Cholesky decomposition solution with the original measured functional correlation data as well as with partial correlation analysis, we usedseed voxels from three major networks of interest (default mode, attention control, and primary auditory networks) to illustrate the effect of the method on known intrinsic connectivity networks. For each of the seed voxels, the weighted connectivity measurements from the Cholesky decomposition demonstrated higher specificity of interregional relationships than was shown by full correlation of time series data (Figure 3).

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.

When compared to partial correlation measurements, the Cholesky decomposition results show greater similarity to partial correlation than to full correlation (Figure 4), with many values close to zero in both partial correlation that had larger positive or negative values in full correlation analysis. This presumably reflects connections that have shared correlation with other brain regions but weak or absent direct connection. Cholesky decomposition differs from partial correlation in our data in that partial correlation shows near complete absence of long-distance connections, while such long-distance connections are largely preserved in the Cholesky method.

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.

3.3 Whole brain simulation and DMN convergence

We then used the weighted connectivity matrix derived from Cholesky decomposition to create a simulation of the brain using a two-step iterative process. First, the “activity” of an ROI at any step in the simulation is the weighted sum of the activity values of all other ROIs from the previous step multiplied by their weighted connectivity coefficient with the ROI. Second, the resulting values were normalized across the brain by subtracting the mean activity across all brain regions and dividing by the standard deviation of activity across brain regions. The normalization step prevents any one brain region from achieving unrealistically high or low activity. The simulation models what might be expected where brain activity in one step is determined by relative brain activity in the previous step in addition to the information in the weighted connectivity matrix. We repeated the simulation using 5000 randomized initial conditions in which initial brain activity was selected from a normal probability density function with mean 0 and standard deviation of 1. Each of the 5000 iterations showed stable convergence by 40 steps in the simulation (Figure 5A & 5B) with robust convergence to the default mode network in 4955 of the simulations (Figure 5C). Forty-five of the 5000 simulations converged to a different stable state characterized by high activity in the visual network consisting of the occipital lobe and posterior medial parietal lobe. The simulations that converged to the default mode network were distributed roughly equally between states where the default mode regions converged to positive activity and states where the default mode regions converged to negative activity, since a simulation converging to negative activity is identical to a simulation converging to positive activity with the sign of all initial conditions inverted. The convergence to the default mode network during weighted connectivity-based simulations was robust across resting state data subsets from acquisition sites, and not a unique characteristic of the whole set average. The weighted connectivity matrices specific to the two largest data subsets (Beijing n=188, Cambridge n=194) both demonstrated robust convergence to the default mode network during whole brain simulations based on their respective weighted connectivity matrices with only slight differences in final convergence state (Figure 5D).

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.

3.4 Single subject reproducibility, and task-specific influence on whole brain simulations

In order to assess the scalability of weighted connectivity-based whole brain simulations to the single subject level, we acquired resting state data from a healthy control subject for five, one-hour blocks. The BOLD time series from each of the five, one-hour blocks was processed as previously described in order to create weighted connectivity matrices for each of the five, one-hour resting state acquisitions. The convergences to steady-state brain activity demonstrate robust, reproducible default mode network configurations for each of the five, one-hour datasets on the same subject (Figure 6B). Further, when the same subject was instructed to watch cartoons for six additional, one-hour periods, the steady-state convergence patterns from each of the weighted connectivity-based whole brain simulations also showed reliable reproducibility at the single subject level (Figure 6A). The convergence state for resting scans was reproducibly different from when the subject was watching cartoons, as shown in Figure 6C. One of the resting state scans (Session 11) showed slightly less similarity to the other resting state scans in final convergence state, but was still uniformly more similar to other resting state convergence states to those obtained from when the subject was watching cartoons. Moreover, the convergence state was reproducibly different for the individual subject than for the population. For example, the posterior cingulate node wasmore anterior for the individual than for the population, and the relative intensity of nodes reproducibly differed between the two task conditions as well as between the individual and population.

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).

3.5 Metastable intermediate networks

In addition to final convergence to the default mode network, many of the simulations (n=949/5000) produced a metastable intermediate state characterized by a local minimum during iterations of the difference between steps, with eventual convergence to a different stable state. We used hierarchical clustering to categorize similarities between intermediate states, and observed six major clusters in the metastable configurations (Figure 7). The averaged outcomes for the metastable intermediates from each cluster exhibited spatial distributions with features of well characterized functional networks, including the visual (A,C), sensorimotor (D,E), dorsal attention (D,E), and salience-detection networks (F). In some cases, (D,E) intermediate states were a hybrid of two or more functional networks (sensorimotor, dorsal attention) of opposite polarity. In other cases, intermediate states resembled a portion of the default mode network (B,C) in which one or more of the core nodes of the default mode network were absent.

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.

In order to assess the effect of activity in any one brain region on convergence dynamics, we performed 7266 additional simulations, using activity corresponding to the functional correlation of each region in the brain with a seed region, and repeating for each region as the seed. We then measured the number of iterations needed to converge to the final convergence state within a tolerance of 0.05%. Seed ROIs thatresulted in delayed convergence to the default mode network were located almost exclusively at the margins of the default mode network (Figure 8A & B). Sixteen of the 7266 simulations converged to the visual network. The seeds for which initial conditions converged to the visual network did not show any clustering or clear pattern in spatial distribution, but were scattered throughout the infratentorial and supratentorial brain.

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.

In converging to default mode network patterns of steady-state activity, dynamic iterations ultimately converge to one of two inverted mirror image states, one representing high activity in the default mode network, the other representing low relative activity in the default mode network. The determining factor for whether a steady-state outcome will represent high default mode versus low default mode activity is predicted by whether the initial starting conditions correspond more closely to correlation patterns for regions within the default mode network, or to correlation patterns for regions outside of the default mode network (Figure 8C).

4 Discussion

4.1 Standard map of functional connectivity

A population standard map of functional connectivity for the human brain underlies large scale imaging initiatives such as the NIH’s Human Connectome Project and the INDI/NITRC 1000 Functional Connectome database. Our results confirm the reliability of datasets obtained from multiple sites converging to a reproducible whole brain functional connectivity map. By expressing the functional correlation matrix for each region as a linear combination of other regions, and iterating normalized brain activity over discrete time steps, we are able to obtain a convergent solution for the default mode network from a wide range of initial conditions. In greater than 99% of our simulations, the system converged to the default mode network (DMN), and in a large number of trials (n=949) passed through an intermediate metastable state before proceeding to its final convergence state.

4.2 Connectivity hubs in the human brain

Regions of the posterior cingulate cortex, precuneus, medial prefrontal cortex, and medial temporal lobes have been classified as ‘hubs’ of structural and functional connectivity in anatomical studies of the human brain(Buckner et al., 2009; Cole et al., 2009; Hagmann et al., 2008; Tomasi and Volkow, 2011). It is not surprising, then, that a dynamical system would converge toward a state in which the major hubs of connectivity were preferentially represented. This result is also consistent with the behavioral observation that during the resting state, in which the BOLD data used for our simulations were obtained, the default mode network is consistently more active than during other cognitive tasks(Raichle et al., 2001).

4.3 Convergence outcomes in whole brain model

In <1% of cases, the simulated brain converged to the visual network rather than the DMN, indicating that the functional connectivity relationships from the population likely exhibit not just metastability of multiple states, but actual multistability. Given the enriched density of high-participant hubs in the occipital lobe described by Hagman, et al(Hagmann et al., 2008), this is also not surprising that the visual system would act as an attractor in some starting conditions. A possible implication of this finding is that the human brain is a multistable system driven toward one of several stability outcomes by a combination of environmental stimuli and previous state conditions. Future work may investigate the various conditions under which the system is driven toward DMN versus non-DMN convergence points.

The number of steps required for a given region’s functional connectivity profile to converge to the DMN was prolonged for voxels at the margin of the default mode network and the attention control network. It has been previously shown that resting state functional connectivity can exhibit sharp transitions at areal boundaries, and that these transitions may serve to define functional domains(Cohen et al., 2008). Our results indicate that convergence to a network is delayed more by activity at such boundaries than by activity in functionally opponent networks, such as activity within the attention control network.

For different sets of initial conditions simulating relative brain activity, dynamic iterations ultimately converge to one of two inverted mirror image states, one representing high activity in the default mode network, the other representing low relative activity in the default mode network(Fox et al., 2005; Fransson, 2005). The default mode network is balanced by relative activity in other brain regions and activity in any of those regions will lead to a state of decreased default mode activity when activity is iterated over time using connectivity-based evolution of brain activity. Among regions that converge to low default mode network activity, even regions that themselves are not directly anticorrelated with or functionally opponent with the default mode network are preferentially connected to regions that are, resulting in patterns of brain activity that over time result in stable default mode network suppression. In contrast, activity in regions associated with the default mode network will converge to a dynamically stable state with higher relative activity within the default mode network.

4.4 Representing intrinsic connectivity by Cholesky decomposition

To obtain convergent results that showed the architecture of intrinsic connectivity networks, we had to express the functional correlations between brain regions as a linear combination of other regions rather than simple correlation coefficients between regions. Applying correlation coefficients without transformation to the same iteration led to convergence states that were a gradient from the front to back of the brain that appeared to overemphasize local connectivity. The Cholesky decomposition method we propose may allow greater specificity in determining actual connections between brain regions, which may be useful for generating more accurate graph-theoretical representations of interregional connectivity(Hagmann et al., 2008; Sporns et al., 2004).

4.5 Comparison of Cholesky decomposition to partial correlation analysis

The Cholesky decomposition is a different approach to weighting brain connections than correlation-based methods. Instead of correlation or partial correlation, which express how synchronized brain regions are, the Cholesky approach directly evaluates how activity in one region may arise from a linear combination of other brain regions. Like partial correlation, it allows disentanglement of direct connections from connections to a shared input. But unlike partial correlation, instead of breaking down and losing long distance connections in a dense connectome with many closely related or overlapping nodes, the Cholesky decomposition preserves long-distance connections but with much greater specificity than seen using full correlation. Moreover, whereas partial correlation may be easily calculated when the correlation matrix is invertible, the Cholesky decomposition requires only a positive definite correlation matrix.

4.6 Model limitations

Our results are drawn from averaging across a broad age range included in the subject sample. It is likely that different subject demographics may influence dynamical stability of intrinsic connectivity networks given known differences in functional connectivity across development(Dosenbach et al., 2010; Fair et al., 2008). It is also likely that the types and frequencies of metastable and multistable convergence points will vary across the lifespan according to the natural development of neural connectivity with age.

We also note that our model iterates normalized rather than absolute brain activity to constrain activity values to physiologically plausible levels. Exploring other types of normalization may better represent physiological mechanisms such as adaptation or other nonlinearities known to exist in the brain. Nevertheless, there is evidence that normalization mechanisms occur in the brain, wherein neural conductances are reduced by pooled neural activity from a population of neurons(Reynolds and Heeger, 2009). Some form of normalization is certainly required given that energy constraints on the brain prohibit indefinitely additive circuitry, and blood flow to the brain is relatively constant over time, from which BOLD measurements are derived.

An additional limitation of our approach is that discrete steps in the simulation do not reflect explicit timing of interactions brain regions, and do not allow us to conjecture about transition times between initial conditions and convergence states, or temporal duration of metastable intermediate configurations. Undirected correlation measurements are also a simplified paradigm for connectivity. In order to create more realistic whole brain simulations, improved methods are needed for modeling the asymmetries of effective connectivity between regions, including circuits such as corticostriatal projections known to contain one-way (non-reciprocal) connectivity. Further, a solution to the Cholesky decomposition requires a positive definite correlation matrix. Although the mean functional correlation matrix for the population we studied meets this requirement, it is possible that other study populations will not result in positive definite matrices. In such cases, other methods or extensions would be necessary to analyze the functional correlation matrix using alternative decomposition strategies, such as decomposition to a “best fit” triangular matrix.

4.7 Future applications of dynamical whole brain modeling

In spite of these limitations, there are numerous possible applications for this approach to whole brain modeling. It is conceivable that various disease states may be typified by their variations in the multistability and metastability of large scale networks, given known differences in functional connectivity between patient populations(Anderson et al., 2011a; Calhoun et al., 2008; Greicius et al., 2004; Just et al., 2004; Rombouts et al., 2009). Methods for modeling stability of distributed brain networks may allow identification of relatively small subsets of nodes for which perturbation can affect brain network stability through control theory or other dynamical systems methods(Liu et al., 2011), and allow informed design of therapeutic strategies such as transcranial magnetic stimulation or deep brain stimulator placement. Changes in functional connectivity are known to exist with cognitive tasks(Fransson, 2006), and examination of changes in convergence states with data obtained during specific tasks may allow characterization of how stimulation may prime the brain for greater stability in different distributed networks. Lastly, there is potential to examine the normal course of brain development by examining changes in dynamical stability of brain networks with age and brain maturation. In short, methods that move beyond static functional correlation to examine dynamical network properties may provide additional characterization of brain networks and states relevant to behavioral and pathophysiological mechanisms.

5 Conclusions

By mathematically decomposing the functional correlations across 7,266 ROIs in the human brain, we are able to approximate the underlying weighted connectivity between gray matter regions. This approach is superior to existing analytical methods such as partial correlation analysis, in the sense that it preserves contributions from long range connectivity better than partial correlation analysis when large numbers of nodes are included in the model. The weighted connectivity can then be used in a whole brain dynamical model, demonstrating multistable convergence outcomes, with the default mode network showing greatest stability. This method is reproducible at a single subject level of analysis, and is sensitive to changes in functional connectivity affected by task-specific dynamics. We also demonstrate metastable qualities of the human connectome, with intermediate configurations resembling well-characterized functional networks. The default mode network steady-state convergence outcomes are either reflective of high relative DMN activity or low relative DMN activity, contingent upon whether starting conditions for neural activity are correspondent to in-network or out-network connectivity patterns for the default mode network. Finally, we demonstrate the borders between default mode network and attention control network are the slowest to converge to the default mode network.