An efficient method for effective connectivity of brain regions

Abstract

Recent attention has been focused on detecting interregional connectivity in a resting state of the brain, in general, described in terms of functional connectivity based on functional magnetic resonance imaging (fMRI) data. The fMRI functional data are given in the form of multivariate time-series. The authors have proposed a model for the effective connectivity of brain regions based on multivariate autoregressive (MAR) model. MAR modeling allows for the identification of effective connectivity by combining graphical modeling methods with the concept of Granger causality. In our current model, multivariate time-series methods of the brain regions were performed only when the length of the time-series T is sufficiently large. This is opposite of the mechanism used in functional imaging that measures relatively short time-series over thousands of voxels of the brain. As a method of coping with this situation and also in case of sufficiently large T or T ≤ d (regions), the authors present a novel and highly efficient modeling approach to detect effective connectivity of the brain regions. This proceeds in two steps: (i) accurate estimation of MAR coefficients (paths) using an analytic ridge regression approach, and (ii) network model selection by testing the associated partial correlations. The usefulness of the proposed method is confirmed by the analysis result of simulated and real fMRI experiments, and performance is shown to be high. © 2012 Wiley Periodicals, Inc. Concepts Magn Reson Part A 40: 14–24, 2012.

INTRODUCTION

Functional magnetic resonance imaging (fMRI) has been used to study human brain activity for almost two decades (1–9). Currently, interest has been shifted away from mapping of activation toward identifying the connectivity that weaves them together into dynamical systems (5, 6). The study of functional integration with fMRI data has been pioneered by several different research groups (10–18). It is believed that processing of a functional task only can be performed through interaction of brain regions within a network (19–21). Various methods have been proposed to extract interaction information from fMRI data, most of which is either functional or effective connectivity (16, 22, 23). Varela et al. identified anatomical and functional networks to be the basis of the brain's computations (24) and their work has led to a rapid increase in investigations of functional integration with fMRI for various cognitive tasks. More importantly, the availability of functional neuroimaging techniques, such as fMRI, optical images, and EEG/MEG are analyzed through the methods of statistical time-series analysis.

A model-free approach or technique to study functional integration is to investigate the correlation between measured time courses of different brain areas, referred to as functional connectivity (13). The use of techniques such as multivariate autoregressive (MAR) modeling allows the identification of effective connectivity by combining graphical modeling methods. An example of the use of graphical modeling comes from Bach and Jordan (1) with applications to fMRI connectivity provided from Salvador (25). The effective connectivity is also closely related to the concept of Granger causality (26–30). The causality analysis (31–34) with multi-time-series theory has originated graphical time-series modeling as exemplified in various reports (3, 30, 35). A different line of work is represented by Pearl (31, 32, 36) and Spirtes (33, 34, 37), among others, who studied graphs with directed edges that represent causal relations between variables. Structural equation modeling (SEM) is another statistical technique for testing and estimating causal relations using a combination of statistical data and qualitative causal assumptions (38). This technique has been adopted for testing causal relations among brain regions based on covariance matrix ∑ structure (4, 6, 10, 18, 39). In the standard SEM technique (2), a dynamical system Y = AY + ω subject to some input ω, where the authors presume the system input is unknown and Gaussian distributed. To generate from the model, we sample ω run the dynamical system to its fixed point, and use that fixed point as a sample of Y. This fixed point is given by Y = (I -- A)^--1ω, which produces the standard SEM covariance matrix ∑ structure for Y. One limitation of SEM is that the use is restricted to structural models of relatively low complexity since models with reciprocal connections and loops often become nonidentifiable (2). Contrary to SEM, MAR models explicitly address the temporal aspect of causality in the time-series data. These MAR models take into account causal dependence of the present on the past time. The MAR models extend this approach to the brain regions of interest (ROI).

fMRI provides a unique opportunity to observe simultaneous recordings of activity throughout the brain evoked by cognitive and sensorimotor challenges. Each voxel of the brain is represented by a time-series of neurophysiologic activity that underlies the measured blood oxygenation level dependence (BOLD) response. Given these multivariate voxel-based time-series, the authors can infer large-scale network behavior among functionally specialized regions based on MAR(p) models, which provides accurate estimation of path parameters and the most significant network, by ridge regression and partial correlation test, respectively, and the performance of the method is acceptable when compared with the conventional network analysis methods such as Granger causality mapping (GCM), DCM, and SEM (15, 18, 38). The authors have presented statistical methods for testing hypotheses concerning partial correlations. The procedures reviewed in the article involve applications of general statistical theory to fMRI data analysis concerning partial correlations.

The aim of our study is to develop an accurate framework to detect effective connectivity over brain regions using the MAR approach of fMRI time-series based on the idea of “Granger causality” (26). The “Granger causality” is a term for a specific notion of causality in time-series analysis. A time-series X is said to Granger-cause Y if it can be shown, usually through a series of t-tests and F-tests on lagged values of X (and with lagged values of Y also included). Those X values provide statistically significant information about future values of Y. This technique has been adapted to ® (26, 28, 31, 32).

Assuming that y_t(1) and y_t(2) are the measured time courses of two brain regions (or voxels), causality quantifies the usefulness of unique information in one of the previous time-series in predicting values of the other. Thus, temporal precedence based on a linear combination of its own past values and the past values of other regions. Finally, our method quantifies the causality among ROI of the brain using accurate estimation of MAR coefficients (paths) and the network model selection involves testing the associated partial correlations.

BOLD

blood oxygenation level dependence

fMRI

functional magnetic resonance imaging

GCM

Granger causality mapping

ML

maximum likelihood

MSE

mean square error

MAR

multivariate autoregressive

ROI

regions of interest

SEM

structural equation modeling.

MATERIALS AND METHODS

The Proposed MAR (p) Model

A d-dimensional MAR (p) models of d regions as shown in Fig. 1(a) and the time-series allows for the computation of the network weights and must be understood as an estimation of effective connectivity. These models take into account the causation of past activations on current dynamics, addressing temporal aspects of causality in time-series. Thus, the activation at time instant t, derived from the BOLD signal in fMRI, is modeled as a linear combination of the vectors that have the signals during p previous time points:

• 

The generalized form of above model is, . The fMRI activations for the d regions at time instant, t = p + 1,…,T are collected in y_t (d × 1) and the white noise input ε_t ∼ N(0,∑) is introduced for modeling spatial correlation between regions. The matrices of MAR coefficients A(i)_{d × d} = {β}_1≤j,k≤d are for the different time lags. In general, a row (column) A(i) of corresponds to the influence field of ingoing (outgoing) connections from (to) the other regions in Fig. 1(a); this can be represented as corresponding rows (columns) in matrices.

Figure 1. (a) d-dimensional regions on the brain. (b,c) Colored area shows brain activities of four subjects at resting state on the right and left hemispheres of the lateral surface of the brain. (d) BA 17, BA38, and BA10 are regions of interest in Brodmann's map.

Setting:

• 

The authors can recast the dynamics of the network of regions as a multivariate regression model.

• (1)

where Y is (T -- p) × d, X is (T -- p) × (pd), β is (pd) × p, and E is (T -- p) × d. In model [1], each row Y corresponds to a typical scan of fMRI data; columns represent the time-series for each region. Least-squares solution of model [1] is obtained by minimization of function S(β) = (Y -- Xβ)^--1 (Y -- Xβ), and can be written as:

• (2)

This is also the maximum likelihood estimate assuming normal distribution. Solution [2] assumes that X is of full column rank, therefore X^TX is invertible. This is not a feasible solution in case of ill-conditioned fMRI data where X^TX is singular, or in the case of d ≥ T. MAR models for a large value of d are also a current challenge in statistics. It is common to estimate effective connectivity based on ROIs; estimation of by least squares can lead to over estimation or other bias. To avoid these situations, it is convenient to make assumptions on the shape of the coefficients. These can be imposed by applying constraints on (40). The simplest case of imposing constraint is via the ridge regression method producing the solution as:

• (3)

Ridge regression is frequently used to obtain stable estimators. In this study, the authors argue that the estimation of by ridge regression controls the overestimation or bias or singularity and deals also in the case of huge dimensionalities. The mean square error (MSE) of the ridge estimator (9) is:

• (4)

Where λ₁, λ₂,…,λ_q are the eigenvalues of X^TX. The first term on the right-hand side of [4] is the sum of variances of the parameter in and the second term is the square of the bias. The bias in increases with k > 0 and the variance decreases as k increases. For the estimation of ridge coefficients [3], we choose a good value of k, i.e., k^* such that the reduction in variance term [4] is greater than the increase in the squared bias. We used the method of generalized cross-validation (GCV) (41) for choosing a good value of k^* for k. We estimated the GCV score for a good k^* by GeneNet package of R. The GeneNet is a package for analyzing gene expression (time-series) data and focuses on the inference of gene networks. In particular, GeneNet implements the methods of different groups (42–44) for investigating large-scale gene association networks in which the estimation procedure is based on GCV score. Finally, the optimal choice of provided the MSE of ridge estimator minimum as compared to the least-squares estimator (45).

MAR (p) Network Model Selection

The d-dimensional Model [1] representing directed causal influences is given by the non-zero entries in the matrix of MAR (p) coefficients. The coefficients are an estimate; it is unlikely that any of its components are exactly zero. Therefore, the authors need to test statistically whether the entries of are vanishing. However, instead of inspecting MAR (p) coefficients or paths directly, it is preferable to test the corresponding partial correlation coefficients. A partial correlation coefficient measures direct interaction strength between regions (46, 47). The authors estimated a correlation coefficient r_ij.m between the time courses of two connections or nodes, i and j from the total nodes (v = d × p). This value is called the sample partial correlation between nodes i and j with other nodes (v = d × p -- 2) partially out; it designated r_ij.m where m is a set of secondary subscripts such that v is the number of subscripts in m.

To reconstruct the selection of connectivity from given d-dimensional fMRI data, the following procedure is typically used. First, an estimate of the partial correlation coefficients matrix R_R = {r_ij.m}_1≤d,j≤d is obtained, via the unbiased sample covariance matrix C_R. Each entry of matrix R_R is a measure of the interaction between the time courses of two connections or nodes i and j where v other variables are held constant, In principle, given an appropriate choice of k^* Equation [3] allows the authors to compute following covariance matrix C_R for the estimation of partial correlation coefficients matrix R_R.

• (5)

where N = T -- p is the sample size. If the authors set π = (π_ij) = C, the inverse covariance matrix of C_R, then partial correlation coefficients of order v can be estimated of all connections or variables in d regions through the following relationship:

• (6)

where r_ij.m is the estimated value of partial correlation between two time course variables, i and j when m is a set of secondary subscripts such that v is the number of subscripts in m. Second, to address the statistical testing of nonzero partial correlation, the authors consider the following hypotheses:

• (7)

Third, the following statistical tests are used for each path or to determine which entries r_ij.m in the estimated partial correlation matrix R_R are significantly different from zero.

• (8)

The distribution of a partial correlation coefficient of order v variables based upon N observations is noted in older sources (48–51). Finally, the inferred correlation structure is visualized by a Fig. 4(a), with edges of values corresponding to nonzero partial correlation coefficients. However, the key advantage of our MAR (p) network selection is that it is specifically designed to meet huge dimensional requirements and accuracy, using estimators of ridge regression and network selection by testing the partial correlation coefficients.

ANALYSIS OF SIMULATED DATA

The purpose of the simulations was to generate the network of effective connectivity with the best choice of p based on time-series from which the network connection of the idealized cortical network could be estimated. For the evaluation of diagnostic methods, a number of examples were calculated to evaluate the performance of different samples and regions. In a comparative simulation study, the authors have investigated the power of diverse approaches to recover the true path of the model MAR (p) network with a “best choice,” i.e., p = 2. The authors randomly generated simulated time-series data (52) with R (a programming language and software environment for statistical computing and graphics) of different sample sizes, with T varying between 5 and 200, d varying between 2 and 9, p varying between 2 and 6. In addition to the MAR (p) model, the authors estimated regression coefficients by ridge Regression [3]. These simulation results of ridge regression coefficients of the model MAR (p) in case of d = 3 and p = 6 are summarized in Table 1. All these regression coefficients were tested with the above MAR model selection criteria based on the partial correlation test. The nonzero or significant ridge regression coefficients of simulation data of the model MAR (6) as shown in Table 1 and the network of effective connectivity among three regions in Fig. 2(a). Similarly, Figs. 2(b–f) shows different examples of the effective connectivity network resulting from simulations run with sample size T varying between 5 and 200, p = 2,3,4 and d = 2,3,9.

Figure 2. (a) A network of of MAR(6) model when d = 3, based on simulation data. (b–f) Different examples of the network of effective connectivity resulting from simulations run with sample size T varying between 5 and 200, p = 2,3,4 and d = 2,3,9.

Table 1. Estimates of of MAR(6) When d = 3 with the Use of Model (3) based on Simulation Data

p
1	0.519 (0.5)	0.213 (0.2)	0.021 (0)	0.034 (0)	−0.192 (−0.2)	−0.009 (0)	−0.014 (0)	0.022(0)	0.252 (0.3)
2	−0.318 (−0.3)	0.101 (0)	−0.014 (0)	−0.002 (0)	−0.246 (−0.2)	0.003 (0)	0.702 (0.7)	0.007(0)	−0.643 (−0.6)
3	0.116 (0.1)	0.004 (0)	0.029 (0)	0.010 (0)	−0.416 (−0.4)	−0.802 (−0.8)	0.012 (0)	0.052(0)	0.385 (0.4)
4	−0.032 (0)	−0.001 (0)	0.046 (0)	0.027 (0)	0.113 (0)	0.005 (0)	−0.030 (0)	0.058(0)	−0.042 (0)
5	0.024 (0)	0.104 (0)	−0.010 (0)	−0.048 (0)	−0.221 (0)	−0.103 (0)	0.010 (0)	−0.011(0)	0.036 (0)
6	−0.007 (0)	−0.012 (0)	0.012 (0)	0.034 (0)	0.016 (0)	−0.016 (0)	−0.019 (0)	−0.003(0)	−0.003 (0)

There exists various ways to select the order, i.e., p in the model MAR (p) such as Bayesian technique (42). The selection of order of the model MAR (p) and Bayesian gives inferences about connections they made on the basis of the estimated posterior distribution; but our proposed criteria allows for the identification of a network connections on the basis of order 2 and the distribution of a partial correlation coefficient of order v. On the other hand, a partial correlation test provides a way to circumvent the issue of network model selection that SEM or DCM has to face. These connections describe most significant and accurate causal relations, in the sense of Granger (26).

RESULTS AND ANALYSIS OF REAL DATA

The authors also tested our proposed model for the effective connectivity network of the brain regions based on real cognitive neuroscience fMRI experiments of four subjects or participants.

Participants

Participants in the fMRI experiments were four healthy, right-handed male volunteers (age range 21–31, mean 26) from the KAIST and SNU student communities arranged by the Yonsei College of Medicine, Severance Hospital. All subjects of the fMRI experiments underwent a one-run scan, particularly a resting state. During the resting state, subjects were instructed to keep their eyes closed, relax their mind, and remain as motionless as possible. They did not have a history of psychiatric disorder, significant physical illness, head injury, neurological disorder, or alcohol or drug dependence. After being given a complete explanation of the intended study, written informed consent was obtained from all subjects, and the proposed study was approved by KAIST and SNU institutional review boards.

Image acquisition

Images were acquired with a 3.0-T MRI. Functional images of 165 volumes were acquired with T2-weighted gradient echo planar imaging sequences sensitive to the BOLD contrast. Each volume consisted of 31 slices with a thickness of 4.5 mm to cover the majority of the whole brain (TE, 30 ms; TR, 2 s; 80 × 80). The scans lasted for 370 s.

Analysis

Data analysis was performed with R and statistical parametric mapping software package (SPM: http://www.fil.ion.ucl.ac.uk/spm/). The data of the functional images were preprocessed and analyzed using SPM version, i.e., SPM8b. Subjects scans were corrected for head movement. After being spatially normalized with the standard template “Montreal Neurological Institute,” the data were spatially smoothed by a 8-mm FWHM Gaussian kernel to decrease spatial noise. For the resting state data, a SPM8b standard low-pass frequency filter was applied to remove physiological high-frequency noise, e.g., respiratory and cardiac. For the resting condition, statistical analysis was performed on each subject data by generalized linear model, group analysis of four subject by t-test and SPMs map was obtained as shown in Figs. 1(b,c). The resulting map was used to determine the effective connectivity of brain regions among specific primary visual area or Brodmann Area 17 (BA 17), thinking & management (BA 38), and memory & emotion area (BA 10) as shown in Fig. 1(d). For measuring the effective connectivity among three mentioned Brodmann regions (17, 38, 10), we estimated the all ridge regression coefficients in our Model [3] with 0 ≤ k ≤ 100 as shown in Fig. 3. The optimal estimated value corresponding to appropriate choice of k^* where estimated error score of GCV for good k , i.e., k^* = 5 shown as in Fig. 3(d), then we computed covariance matrix C_R [5] and partial correlation coefficients matrix R_R with the use of relation [6]. Finally, for nonzero optimal parameter , the authors applied Eqs. [7] and [8] to get the significant paths shown in Fig. 4(a).

Figure 3. (a–c) Graphs showing estimated values of ridge regression coefficients with 0 ≤ k ≤ 100 of Brodmann areas BA17, BA38, and BA10, respectively, by the proposed Model (3) in which each line represents all paths (coefficients) from one region to the other two regions at different k values. (d) Graph showing estimated error score of GCV for good k , i.e., minimum at k^*.

Figure 4. (a) Slice view network among three brain regions at time t, t -- 1, and t -- 2 with the optimal values of ridge regression coefficients of the proposed model. (b) Connectivity of real data among three Brodmann regions with the use of proposed model.

Results

In this study, the authors used a network model based on graph theory to describe functional connectivity. Thus, in Fig. 4(a) the nodes denote the brain regions and the links denote the connections or information flow among them. Fig. 4(b) the connections values are optimal ridge regression coefficients of three brain region in the resting state across all subjects. Fig. 4(b) arrows shows that there exists significant functional connectivity between the brain regions; thus, it is considered as an important path in the network. The starting region of thinking & management area processing in the model and information flows via the memory & emotion area to the primary visual area. In Fig. 4(b), thicker arrows represent strong connectivity among regions or most significant path where a region (BA 38) is caused by other region (BA 10) at two level, i.e., t -- 1 and t -- 2.

DISCUSSION

Effective connectivity aims at examining the influence that regions exert on each other (11). The authors have proposed a method for identifying large scale effective connectivity patterns at a resting state of fMRI data. The method is based on estimating MAR (p) models by a two-stage process that first applies ridge regression and then partial correlation t-test for testing the effective connectivity. The method is demonstrated to perform well in identifying patterns of network connectivity by means of simulations and real fMRI on an idealized cortical network. The simulation and real data networks also show that, the simplest of the methods, ridge regression coefficients (paths), analyzing the partial correlation coefficients, and performs as well as more sophisticated when compared with the mixture of penalized regression techniques (29) or conventional SEM (18, 38).

In this article, the authors propose to use partial correlation tests of functional dependency using a covariance matrix C_R related to effective connectivity. Methods using partial correlation were used to eliminate the effect of the experimental design (53–55) but in our approach, its use to causal dependencies between the regions instead of subtract and remove mutual dependencies on common influences among brain regions. The causal dependencies between regions, the ensuing functional connectivity (i.e., partial correlation) reflects interactions between the regions. Therefore, the use of partial correlation, allowing access to a quantity that relates to direct interaction, takes the analysis of functional connectivity closer to the characterization of functional interactions in terms of effective connectivity. In this framework, the authors measured the interaction strengths among the interested regions of the brain. Once the regions and the corresponding time courses were selected, the partial correlation test developed a procedure for the measured path ( ) that led investigation of dependencies to effective connectivity. This technique provided relevant insight into the causal relationships between regions. The networks of causal connections are found by testing each path with the hypotheses; partial correlation is significantly different from zero. This partial correlation test (51, 52) for the network has the interesting features of providing a convenient summary of dependencies in regions related to the direct functional interactions (i.e. effective connectivity) of the brain regions.

Finally, our approach taking into account multidimensional connectivity using cross-connectivity measures of Models [3] and [5], the authors have showed the existence of a large organized functional connectivity network among Brodmann area (17, 38, 10) in the resting brain of four subjects with fMRI data. More importantly, the authors have found that such a network can be modulated from a conscious resting state to thinking, management, memory, emotion, and visual of voluntary movement state, exhibited by significant changes of functional connectivity of some brain regions with different brain activity.

CONCLUSION

This article presents novel statistical methods for accurate estimation and inference of brain connectivity from the fMRI technique based on MAR model and partial correlation analyses. The authors have found a large organized functional connectivity network related to thinking, visual, and decision areas in the resting brain with fMRI of human. More notably, the authors have shown that such a network can be modulated during the resting state. In addition, the authors have provided evidence in the results section for the existence of a functional connectivity network among the brain regions in the resting state. Thus, our study can be considered as fully assessing joint interaction among multiple brain regions with different states of brain activity, which may be helpful to understand basic neurophysiologic mechanisms that operate in the resting state. Of course, exact mental process supported by the connectivity network during rest due to its uncontrolled nature is essentially difficult and needs to be studied in the future. Finally, our method is a powerful method to characterize neural interactions among brain regions either some particular tasks or resting state using a functional imaging data.

Acknowledgements

This research funded by Brain Korea 21 (BK21), NAP of Korea Research Council of Fundamental Science &Technology P90015, and National Research Foundation of Korea 2010-0012185, 2010-0018837.

Biographical Information

Fayyaz Ahmad received his Ph.D. degree in Mathematical sciences from the Korea Advanced Institute of Science and Technology (KAIST) in 2010. Currently, he is working as a Research Scientist in the department of Electrical Engineering at KAIST. His research interests include modeling and analyzing fMRI data, time series analysis, and multivariate statistical analysis for effective connectivity in the brain.

Dr. Muhammad Maqbool has obtained his Ph.D. degree in experimental condensed matter physics from Ohio University, USA, in 2005 and his Master of Science degree in Medical and Radiation Physics from the University of Birmingham, UK, in 1998. Currently, he works as an Assistant Professor of Physics and Medical Physics at Ball State University, USA. His area of research is Medical and Health Physics, Photonics, Biophotonics, and Surface Physics. He has published 32 articles in peer reviewed journals and 4 conference publications. He has also invented a titanium infrared microlaser on optical fibers. Currently, he works on the light emission from erbium oxide nanopowders for internal body visualization and optical devices applications.

Eunwoo Kim received B.S. and M.S. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST) in 2008 and 2010, respectively. Currently, he is a Ph.D. candidate in Department of Electrical Engineering at KAIST. His research interests include Modeling and Analyzing fMRI data.

Biographical Information

HyunWook Park is a professor of the Department of Electrical Engineering at KAIST. He served as the Department Head of Electrical Engineering at KAIST from December 2005 to January 2011. He received the B.S. Degree in Electrical Engineering from Seoul National University, Seoul, Korea in 1981 and the M.S. and Ph.D. degrees in Electrical Engineering from Korea Advanced Institute of Science and Technology (KAIST), Seoul, Korea in 1983 and 1988, respectively. He has been a professor of electrical engineering department since 1993 and an adjunct professor of bio and brain engineering department since 2003, KAIST, Korea. He was a Research Associate at the University of Washington from 1989 to 1992 and was a Senior Executive Researcher at the Samsung Electronics from 1992 to 1993. He is a senior member of IEEE. He has served as Associate Editor for International Journal of Imaging Systems and Technology. His current research interests include image Computing System, Image Compression, Medical Imaging, and Multimedia System.

Dae-Shik Kim studied Psychology and Computer Science at the Darmstadt University of Technology. He subsequently performed Masters and Ph.D. research works at the Max-Planck-Institute for Brain Research in Frankfurt, Germany. Following postdoctoral works in optical imaging (M.I.T., Cambridge) and computational neurosciences (RIKEN, Japan), he conducted research in high-field MRI (Assistant Professor at the Center for Magnetic Resonance Research, University of Minnesota) and Diffusion Tensor Imaging (Associate Professor at Boston University). In 2009, he joined the Department of Electrical Engineering at Korea Advanced Institute of Science and Technology (KAIST) as a full professor. His research interest includes Systems, Developmental, and Computational Neurosciences, and functional connectivity mapping of the human brain.