Effective Connectivity Modeling for fMRI: Six Issues and Possible Solutions Using Linear Dynamic Systems

Abstract

Analysis of directionally specific or causal interactions between regions in functional magnetic resonance imaging (fMRI) data has proliferated. Here we identify six issues with existing effective connectivity methods that need to be addressed. The issues are discussed within the framework of linear dynamic systems for fMRI (LDSf). The first concerns the use of deterministic models to identify inter-regional effective connectivity. We show that deterministic dynamics are incapable of identifying the trial-to-trial variability typically investigated as the marker of connectivity while stochastic models can capture this variability. The second concerns the simplistic (constant) connectivity modeled by most methods. Connectivity parameters of the LDSf model can vary at the same timescale as the input data. Further, extending LDSf to mixtures of multiple models provides more robust connectivity variation. The third concerns the correct identification of the network itself including the number and anatomical origin of the network nodes. Augmentation of the LDSf state space can identify additional nodes of a network. The fourth concerns the locus of the signal used as a "node" in a network. A novel extension LDSf incorporating sparse canonical correlations can select most relevant voxels from an anatomically defined region based on connectivity. The fifth concerns connection interpretation. Individual parameter differences have received most attention. We present alternative network descriptors of connectivity changes which consider the whole network. The sixth concerns the temporal resolution of fMRI data relative to the timescale of the inter-regional interactions in the brain. LDSf includes an "instantaneous" connection term to capture connectivity occurring at timescales faster than the data resolution. The LDS framework can also be extended to statistically combine fMRI and EEG data. The LDSf framework is a promising foundation for effective connectivity analysis.

Keywords: dynamic systems; effective connectivity; fMRI.

Figure 1

Neural responses over a single block (~30 s.) in three regions during the Attention to Visual Motion task as estimated via deterministic (A) and stochastic (B) models. (A) Connectivity in the deterministic (DCM) model is seen in the slope of the block onset. The region V1 (blue) increases first and most rapidly and V5 (green) increases with a similar slope. The region SPC (red) increases more slowly; its slope is less similar to V5 than V5 is to V1, thus the connectivity from V5 to SPC is less than that from V1 to V5. This slope magnitude and slope similarity is the nature of the deterministic connectivity. (B) In contrast, the three regions respond similarly in the onset of the block for the stochastic model. The connectivity is in the covariance within the block. V1 and V5 are more connected than V5 and SPC because their estimated responses are more correlated within the block.

Figure 2

Actual connectivity parameter values from simulation (smooth lines) versus LDSf parameter estimates using the joint model (jagged lines). True parameter values in (A,B) vary according to sine waves while true parameter values in (C,D) are constant.

Figure 3

Parameter values in switching linear dynamic system. Shown are the parameter values for three regions (A) and the probability of each cognitive regime (B). Parameters are mixtures of the stationary models from each regime weighted by the probability of being in each regime. The connectivity parameter values are similar but not identical for repetitions of the same condition and the parameters vary within a block of the same condition as well. However the main source of variance in the parameters is the change from condition to condition.

Figure 4

The eigenvalues (A) and eigenvalue spacing (B) identified by SSA for the Lorenz system at varying window sizes. Eigenvalues are correctly identified in three pairs, each pair representing a single variable from the system. The three pairs clearly rise above the zero floor.

Figure 5

Model used for CCA-LDSf simulations. Model connectivity is shown in (A). The model contains three anatomical regions (blue circles). Regions 1 and 2 contain a single signal each while region three contains three signals (red circles). The signals form two distinct task related networks [1,2,3] and [3,4,5]. (B) For anatomical region 3, 100 voxels were created by mixing signals 3, 4, and 5 by known weights that varied by voxel. The weights were always non-zero and positive.

Figure 6

Results from the CCA-LDSf model. CCA-LDSf was used to identify a small number of “best” voxels from the 100 voxels in region 3 that interacted with regions 1 and 2 (see Figure 5). Signal three contains the optimal signal. Voxels with high weights on signal three (red weights) should be selected while voxels with low weights on signal three should be avoided. Values of weights for each voxel as estimated by the CCA-LDSf model are shown in bars. A voxel with near maximal true weight on signal 3 was chosen as the optimal voxel by the CCA-LDSf.

Figure 7

Network impulse response functions for left primary motor and premotor cortex during right hand tapping. Probability density (0.025–0.975) estimated via bootstrap is color-coded for each response function. A 1 SD impulse to the premotor cortex causes a small but sustained increase in the motor cortex while a 1 SD impulse to the primary motor cortex causes a smaller, less sustained decrease in the premotor cortex.