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Conceived and designed the experiments: WT SLB CMS GLS MC. Performed the experiments: WT CMS. Analyzed the data: WT SLB CMS. Contributed reagents/materials/analysis tools: WT. Wrote the paper: WT SLB CMS GLS MC.

The authors have declared that no competing interests exist.

Functional brain network studies using the Blood Oxygen-Level Dependent (BOLD) signal from functional Magnetic Resonance Imaging (fMRI) are becoming increasingly prevalent in research on the neural basis of human cognition. An important problem in functional brain network analysis is to understand directed functional interactions between brain regions during cognitive performance. This problem has important implications for understanding top-down influences from frontal and parietal control regions to visual occipital cortex in visuospatial attention, the goal motivating the present study. A common approach to measuring directed functional interactions between two brain regions is to first create nodal signals by averaging the BOLD signals of all the voxels in each region, and to then measure directed functional interactions between the nodal signals. Another approach, that avoids averaging, is to measure directed functional interactions between all pairwise combinations of voxels in the two regions. Here we employ an alternative approach that avoids the drawbacks of both averaging and pairwise voxel measures. In this approach, we first use the Least Absolute Shrinkage Selection Operator (LASSO) to pre-select voxels for analysis, then compute a Multivariate Vector AutoRegressive (MVAR) model from the time series of the selected voxels, and finally compute summary Granger Causality (GC) statistics from the model to represent directed interregional interactions. We demonstrate the effectiveness of this approach on both simulated and empirical fMRI data. We also show that averaging regional BOLD activity to create a nodal signal may lead to biased GC estimation of directed interregional interactions. The approach presented here makes it feasible to compute GC between brain regions without the need for averaging. Our results suggest that in the analysis of functional brain networks, careful consideration must be given to the way that network nodes and edges are defined because those definitions may have important implications for the validity of the analysis.

Modern cognitive neuroscience views cognition in terms of brain network function. A network is a physical system of nodes connected to each other by edges. From the network perspective, cognitive function depends on activity patterns involving the nodes and edges of functional brain networks. It is important then, to appropriately define the nodes and edges of functional brain networks in order to understand cognition. In this study we consider the nodes of functional brain networks to be brain regions, and demonstrate a method that effectively measures the edge pattern between regions with a technique called Granger Causality. Our method is made possible by the utilization of recent advances from the field of statistics. Our approach is generally applicable to functional brain network analysis and contributes to the understanding of network properties of the brain.

The modern understanding of human cognition relies heavily on the concept of large-scale functional brain networks, and large-scale functional network analysis of Blood-Oxygenation-Level-Dependent (BOLD) signals from functional Magnetic Resonance Imaging (fMRI) is playing an increasingly important role in cognitive neuroscience

A node is typically represented in brain network studies of fMRI BOLD activity as a lumped Region Of Interest (ROI), formed by averaging the BOLD signals of all the ROI's voxels

Here we present a novel procedure for the analysis of directed interregional functional interactions that is based on the BOLD activity of the individual voxels of ROIs and the Granger Causality (GC) measure of directed interaction between voxels. GC tests whether the prediction of the present value of one time series by its own past values can be significantly improved by including past values of another time series in the prediction. If so, the second time series is said to Granger cause the first, and the degree of significance of the improvement may be taken as the strength of GC

Previous evidence from GC analysis of fMRI BOLD data argues against the assumption of homogeneous interregional functional interactions, and thus suggests that averaging BOLD signals prior to edge measurement may not be appropriate. Bressler et al.

An approach to the problem of heterogeneous functional interaction between ROIs is to compute the distribution of GC values using a bivariate AR model for each pairwise combination of voxels in the ROIs. This pairwise-GC approach, followed by Bressler et al.

A) Sequential driving pattern, where voxel

Our approach to the problem of spurious GC significance rests on the concept of conditional GC

To make use of the MVAR model for ROI-level GC analysis necessitates overcoming one further problem that often occurs in model estimation: the number of available observations (data points) limits the number of parameters (model coefficients) that can accurately be estimated. This problem commonly arises in neurobehavioral studies where the number of data points that can realistically be acquired limits the size of the MVAR model that can be estimated. This limitation can be mitigated, however, if it is assumed that the voxel-voxel functional interactions between ROIs are sparse (i.e., have a low connectivity density)

This paper presents a novel application of the MVAR model to study voxel-based region-to-region interactions in the brain, particularly long-range, top-down interregional interactions in visuospatial attention. We demonstrate that the LASSO algorithm can be effectively used to pre-select model variables, thereby enabling estimation of the coefficients of a voxel-based MVAR model of two predefined ROIs. The originality of our methods derives from: (1) estimation of the MVAR model for fMRI voxel-level BOLD time series from two ROIs; (2) use of the LASSO algorithm for variable pre-selection prior to MVAR model estimation; (3) use of the General Cross-Validation criterion for determining optimal predictors in the MVAR model from the LASSO algorithm; and (4) creation of two types of summary statistics at the ROI level that represent the separate measurement of density and strength of GC between ROIs.

We report the results of both MVAR model simulations and the application of MVAR model estimation to an empirical fMRI BOLD dataset obtained during a visuospatial attention task

We conclude that LASSO variable pre-selection and MVAR model estimation can be effectively used to measure Granger Causality between cortical regions from voxelwise fMRI BOLD signals. Through the MVAR model, it is beneficial to analyze all the voxels in an ROI, instead of taking an average over the ROI. In this way, directed interregional functional interactions are captured with less distortion of the information carried in the BOLD time series.

Simulation MVAR models were created based on Equation 3 (see _{xx}, B_{yx}, B_{xy} and B_{yy}) were constructed separately. For each submatrix, some coefficients (_{ij}_{xx} and B_{yy}, 0.2 for B_{yx}, and 0.1 for B_{xy}). For each simulation, 200-point-long time series for each pseudo-voxel were created by model iteration. A total of 56 simulation models were created. The density of model connectivity was systematically increased with increasing model identification number by augmenting the number of voxel pairs connected by non-zero

Sim. ID | B_{xx} |
B_{yy} |
B_{yx} |
B_{xy} |
Sim. ID | B_{xx} |
B_{yy} |
B_{yx} |
B_{xy} |

1 | 0.0656 | 0.0592 | 0.0493 | 0.0420 | 29 | 0.1944 | 0.1768 | 0.1593 | 0.1580 |

2 | 0.0656 | 0.0592 | 0.0473 | 0.0373 | 30 | 0.1944 | 0.1768 | 0.1413 | 0.1473 |

3 | 0.0656 | 0.0592 | 0.0453 | 0.0427 | 31 | 0.1944 | 0.1768 | 0.1373 | 0.1627 |

4 | 0.0656 | 0.0592 | 0.0400 | 0.0433 | 32 | 0.1944 | 0.1768 | 0.1240 | 0.1647 |

5 | 0.0656 | 0.0592 | 0.0427 | 0.0453 | 33 | 0.1944 | 0.1964 | 0.1853 | 0.1500 |

6 | 0.0656 | 0.0592 | 0.0413 | 0.0520 | 34 | 0.1944 | 0.1964 | 0.1707 | 0.1460 |

7 | 0.0656 | 0.0592 | 0.0320 | 0.0580 | 35 | 0.2267 | 0.2160 | 0.1807 | 0.1620 |

8 | 0.0656 | 0.0592 | 0.0340 | 0.0507 | 36 | 0.2267 | 0.2160 | 0.2000 | 0.1920 |

9 | 0.0978 | 0.0984 | 0.0973 | 0.0553 | 37 | 0.2267 | 0.2160 | 0.1800 | 0.1767 |

10 | 0.0978 | 0.0984 | 0.0860 | 0.0727 | 38 | 0.2267 | 0.2160 | 0.1727 | 0.2193 |

11 | 0.0978 | 0.0984 | 0.0827 | 0.0760 | 39 | 0.1944 | 0.1964 | 0.1367 | 0.1860 |

12 | 0.0978 | 0.0984 | 0.0860 | 0.0607 | 40 | 0.2267 | 0.2160 | 0.1687 | 0.2160 |

13 | 0.0978 | 0.0984 | 0.0907 | 0.0680 | 41 | 0.2589 | 0.2552 | 0.2380 | 0.2200 |

14 | 0.0978 | 0.0984 | 0.0713 | 0.0847 | 42 | 0.2267 | 0.2356 | 0.2113 | 0.1947 |

15 | 0.0978 | 0.0984 | 0.0787 | 0.0800 | 43 | 0.2267 | 0.2356 | 0.2213 | 0.1927 |

16 | 0.0978 | 0.0984 | 0.0713 | 0.0907 | 44 | 0.2267 | 0.2356 | 0.2113 | 0.2180 |

17 | 0.1300 | 0.1376 | 0.1233 | 0.0933 | 45 | 0.2589 | 0.2552 | 0.2220 | 0.2360 |

18 | 0.1300 | 0.1376 | 0.1220 | 0.1073 | 46 | 0.2267 | 0.2356 | 0.2027 | 0.2113 |

19 | 0.1300 | 0.1376 | 0.1187 | 0.1107 | 47 | 0.2267 | 0.2356 | 0.1740 | 0.2013 |

20 | 0.1300 | 0.1376 | 0.1300 | 0.0993 | 48 | 0.2589 | 0.2552 | 0.1720 | 0.2613 |

21 | 0.1300 | 0.1376 | 0.1193 | 0.1147 | 49 | 0.2911 | 0.2944 | 0.2847 | 0.2467 |

22 | 0.1300 | 0.1376 | 0.0993 | 0.1213 | 50 | 0.2589 | 0.2552 | 0.2387 | 0.1773 |

23 | 0.1300 | 0.1376 | 0.0980 | 0.1313 | 51 | 0.2911 | 0.2944 | 0.2540 | 0.2680 |

24 | 0.1300 | 0.1376 | 0.0880 | 0.1327 | 52 | 0.2589 | 0.2552 | 0.2373 | 0.2120 |

25 | 0.1944 | 0.1768 | 0.1800 | 0.0920 | 53 | 0.2589 | 0.2552 | 0.1880 | 0.2280 |

26 | 0.1944 | 0.1768 | 0.1727 | 0.0933 | 54 | 0.2911 | 0.2944 | 0.2367 | 0.2947 |

27 | 0.1944 | 0.1768 | 0.1587 | 0.1473 | 55 | 0.2911 | 0.2944 | 0.2053 | 0.2680 |

28 | 0.1944 | 0.1768 | 0.1560 | 0.1500 | 56 | 0.2911 | 0.2944 | 0.1827 | 0.2907 |

To verify model validity, we determined that the correlations of the model residuals were low for all 56 models. A representative residuals correlation matrix from one of the simulations is displayed in the

We then considered the effect of averaging the BOLD activity of all voxels in an ROI on the measurement of interregional GC. The GC between two ROIs, each of which is represented by an averaged time series, was measured by a single

GC was computed from averaged voxel time series as _{yx} and _{xy}, and then normalized to

We next examined how well voxel-based methods recovered the actual GC patterns of the four submatrices across the simulation models shown in

The X voxels in A-C are represented by green dots and Y voxels by red dots. All the

To determine how typical were the results seen in

The fraction of significant

The

The GC strength (W summary statistic) in each submatrix, computed directly from the simulation model, is compared with the

The comparison of LASSO-GC versus pairwise-GC across 56 runs can be considered as 56 repeated tests of the two methods for their efficiency in estimating model parameters. The fact that LASSO-GC yielded more accurate estimations than pairwise-GC over a range of different parameter settings demonstrates LASSO-GC's robustness. To further validate this conclusion, we repeated each 56-run test on 20 separate iterations, each iteration using an independently generated dataset with the parameters from

An fMRI BOLD dataset from a slow event-related visuospatial attention task paradigm was analyzed with the LASSO-GC method. Details about the experimental design and the fMRI recording are available in

As with the simulation results, the correlations of the MVAR model residuals were found to be low, indicating that the models were valid. Because of the large data dimension, not all ROI pairs could be examined. Instead, 10 ROI pairs were randomly selected from each subject for examination: the residuals correlation matrix for one representative ROI pair is displayed in

The results of functional connectivity analysis between the VOC and DAN are presented in _{VP−>VP}_{FEF−>VP}_{VP−>FEF}_{FEF−>FEF}

Patterns are shown for one exemplary ROI pair from one subject. The

Connectivity based on the correlation measure is also considered. For correlations measured directly on the fMRI BOLD time series, a larger fraction of connections is significant at _{VP−VP}_{VP−FEF}_{FEF−FEF}

To extend the functional connectivity analysis to the full fMRI BOLD dataset, we applied the LASSO-GC method to all 60 VOC-DAN ROI pairs in each of the 6 subjects. The

The

Paired-sample t-tests with subjects as repeated measures (df = 5 for all comparisons) were performed on both

We have shown that Granger Causality (GC) computed from voxel-level BOLD signals better reflects the pattern of directed functional interaction between ROIs than that computed from voxel-averaged signals. We conclude that brain regions are not unitary elements, that network structure exists at the voxel level, and that ROI-level GC connectivity is best measured by summary scores computed over voxel-level connectivity patterns.

We emphasize that our conclusions apply specifically to GC between pre-defined ROIs, and do not necessarily extend to the computation of maps showing GC between a “seed” signal, averaged over the voxels in one cortical region, and voxels throughout the rest of the cortex

In addition to mapping, another common analytic method in the literature examines region-to-region correlations based on averaged signals and identifies topological properties from large-scale networks that involve hundreds of ROIs

We have demonstrated that the LASSO-GC method can better identify GC connectivity between ROIs in simulated fMRI BOLD data than the pairwise-GC method by more accurately estimating the connectivity density and strength. The pairwise-GC method can yield spuriously significant coefficients if correlated predictors are present in the MVAR model. The close fit of the LASSO-GC results to the actual results from the simulation models demonstrates that the LASSO-GC method is better able to avoid false positives, and also shows the sensitivity of this method in detecting model changes. By contrast, GC values computed from averaged data do not systematically follow changes in simulated ROI models, suggesting that summary statistics computed from voxel-to-voxel GCs are better able to represent ROI-level connectivity than single region-to-region GCs computed after averaging over ROI voxels.

The estimated

Directional asymmetry in GC connectivity between the Dorsal Attention Network (DAN) and Visual Occipital Cortex (VOC) was reported in our previous work

The problem of bias actually has multiple facets. It is known from theory that the LASSO method may be biased if predictors are highly correlated. There are two main problems caused by correlated predictors. First, some predictors in a system may not be included in the model of the system. This is the case when estimation of multiple bivariate AR models is employed in place of MVAR model estimation: the estimation may be biased by undetected influences from the excluded predictors. The use of LASSO helps to mitigate this problem by allowing estimation of a full MVAR model. Second, even when all the predictors are taken into account, correlation among predictors may still bias model estimation, a situation often referred to as the collinearity problem for multiple regressions. An example of such bias would be the case of a group of strongly correlated predictors, where LASSO tended to select only one predictor from the group. Extensions of LASSO have been proposed to mitigate this problem by selecting the entire group instead of a single predictor. Such extensions include fused LASSO

Although the MVAR models used in this paper were implemented with order one, models having higher order (

In conclusion, our work suggests that the LASSO algorithm can be effectively employed for pre-selection of voxels that are then used in an MVAR model to measure functional connectivity between ROIs, using voxel-based fMRI BOLD signals. It indicates that the

We first consider an fMRI BOLD dataset from

The relationship between X and Y can be expressed in the form of a Multivariate Vector AutoRegressive (MVAR) model. A general matrix representation of the model is:_{t}_{t-k}_{k}_{t}

When expanded, the product term in Equation 3 becomes:

Each element of the Z_{t-k}th^{k}_{ij}_{k}_{t}^{k}_{ij}_{t}

The model order (

We also employed pairwise-GC analysis for comparison with MVAR analysis. In the pairwise-GC approach, coefficients are estimated (and the significance of GC determined) by constructing a separate bivariate model for each pair of voxels, one in X and one in Y:

In pairwise-GC analysis, the assumption is made that the predictors are independent of one another. Under this assumption, the GC between X and Y can be assessed solely from the bivariate models in Equation 5, and it is not necessary to estimate the coefficients representing GC within X or Y. In fact, however, the predictors may be correlated for BOLD time series, making the pairwise-GC approach problematic. If the predictors are correlated, estimation by separate bivariate (or partial) models may be biased, and all of the coefficients in the B matrix should be estimated simultaneously

The Least Absolute Shrinkage and Selection Operator (LASSO) technique is a method that makes model estimation feasible when only a limited number of observations is available. Under the assumption that the B matrix is sparse (i.e., many coefficients are zero), the LASSO algorithm effectively determines which

In the MVAR model, pre-selection is carried out in a row-wise manner. LASSO adds a constraint on each row equation of Equation 3 that restricts the total absolute values of the coefficients. The constraint is expressed as:

Regression of the

Finding a least-squares solution of Equation 7 requires a subset of the

The next step in model estimation is to tune the parameter

A subsequent Ordinary Least Squares (OLS) procedure is then applied to the new row equation with the selected predictors. To avoid using the same data to estimate the LASSO and OLS models, we randomly sort the data trials into two sets. One set is used to estimate the LASSO coefficients for model selection, and the other is then used to estimate OLS coefficients for the new row equation. In the second step, if the model order is one, as in our application, there is only one coefficient for each predictor. Either an

The full B matrix may be estimated by following the above procedures for every row equation in Equation 3. It consists of four submatrices (B_{xy}, B_{yx}, B_{xx}, and B_{yy}), where the first subscripted index represents the predictor and the second represents the dependent variable. Thus, B_{xy} represents connectivity from X to Y, B_{yx} represents connectivity from Y to X, and B_{xx} and B_{yy} represent connectivity within X and within Y, respectively. In order to measure GC from one ROI to another (i.e., X→Y or Y→X), one or more statistics are needed to summarize the voxel-to-voxel GCs represented by significant coefficients in B_{xy} or B_{yx}.

The first summary statistic that we used was the fraction (

Red dots represent the voxels of ROI Y, green dots the voxels of ROI X, and arrows the significant

The second summary statistic used was the average strength of significant GC from voxels in one ROI to voxels in another (_{yx}. For any given voxel _{xy}. Although not the focus of this paper, _{xx}, or within ROI Y using B_{yy}.

Simulation models were constructed using the R statistical computing package. For the purpose of comparing the GC strength of a simulation model with its estimated values, we computed the simulation

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_{1}penalized regression: A review.