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Construct Validation of a DCM for Resting State fMRI

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Construct Validation of a DCM for Resting State fMRI

Adeel Razi et al. Neuroimage.

Abstract

Recently, there has been a lot of interest in characterising the connectivity of resting state brain networks. Most of the literature uses functional connectivity to examine these intrinsic brain networks. Functional connectivity has well documented limitations because of its inherent inability to identify causal interactions. Dynamic causal modelling (DCM) is a framework that allows for the identification of the causal (directed) connections among neuronal systems--known as effective connectivity. This technical note addresses the validity of a recently proposed DCM for resting state fMRI--as measured in terms of their complex cross spectral density--referred to as spectral DCM. Spectral DCM differs from (the alternative) stochastic DCM by parameterising neuronal fluctuations using scale free (i.e., power law) forms, rendering the stochastic model of neuronal activity deterministic. Spectral DCM not only furnishes an efficient estimation of model parameters but also enables the detection of group differences in effective connectivity, the form and amplitude of the neuronal fluctuations or both. We compare and contrast spectral and stochastic DCM models with endogenous fluctuations or state noise on hidden states. We used simulated data to first establish the face validity of both schemes and show that they can recover the model (and its parameters) that generated the data. We then used Monte Carlo simulations to assess the accuracy of both schemes in terms of their root mean square error. We also simulated group differences and compared the ability of spectral and stochastic DCMs to identify these differences. We show that spectral DCM was not only more accurate but also more sensitive to group differences. Finally, we performed a comparative evaluation using real resting state fMRI data (from an open access resource) to study the functional integration within default mode network using spectral and stochastic DCMs.

Keywords: Bayesian; Dynamic causal modelling; Effective connectivity; Functional connectivity; Graph; Network discovery; Resting state; fMRI.

Figures

Fig. 1
Fig. 1
This figure summarises the results of simulating fMRI responses to endogenous fluctuations over 512 time points (scans) with a TR of 2 s — here we only show initial 256 time bins. The simulation was based upon a simple four-region hierarchical network or graph, shown on the lower right, with positive effective connectivity (black) in the forward or ascending direction (and lateral direction) and negative (red) in the backward or descending direction. The four regions were driven by endogenous fluctuations (upper right panel) generated from an AR(1) process with autoregressive coefficient of one half (and scaled to a standard deviation of one quarter). These fluctuations caused distributed perturbations in neuronal states and consequent changes in haemodynamic states (shown as cyan) in the upper right panel, which produce the final fMRI response in the lower left panel.
Fig. 2
Fig. 2
This figure reports the results of Bayesian model inversion using data shown in the previous figure. This inversion produced predictions (solid lines) of sample cross spectra (dashed lines) and cross covariance functions, shown in the upper two panels for spectral DCM. The real values are shown on the left and the imaginary values on the right. Imaginary values are produced only by extrinsic (between regions) connections. The first half of these responses and predictions correspond to the cross spectra between all pairs of regions, whilst the second half are the equivalent cross covariance functions — note that cross covariance only has real part. The lower panel shows the predicted response, in time, for the four regions and the associated error between the predictions and the observed responses.
Fig. 3
Fig. 3
This figure shows the posterior estimates that result from the Bayesian inversion of the simulated time series. The posterior means (grey bars) and 90% confidence intervals (pink bars) are shown with the true values (black bars) in the left column. The upper and lower panel reports spectral and stochastic DCMs respectively. The grey bars depict the posterior expectations of connections, where intrinsic (within region) or self-connections are parameterised in terms of their log scaling (such that a value of zero corresponds to a scaling of one). The extrinsic (between regions) connections are measured in Hz in the usual way. It can be seen that, largely, the true values fall within the Bayesian confidence intervals for spectral DCM but not for stochastic DCM. The right panel shows the same results but plotting the estimated connection strengths against their true values. For spectral DCM (resp. stochastic DCM), the blue (resp. cyan) circles correspond to extrinsic connections and the red (resp. magenta) circles to intrinsic connectivity.
Fig. 4
Fig. 4
This figure reports the results of Monte Carlo simulations assessing the accuracy of posterior estimates in terms of root mean square error (RMS) from the true value. Both panels show the results of 32 simulations (red diamonds) for different run or session lengths. For the upper panel – that reports the results for spectral DCM – the average root mean square error (black bars) decreases with increasing run length to reach acceptable (less than 0.1 Hz) levels after about 300 scans. In the lower panel – that reports the results for the stochastic DCM – we see same trend of average root square error decreasing with increasing run lengths but it never attains the (heuristic) threshold of 0.1 Hz.
Fig. 5
Fig. 5
This figure reports the Bayesian parameter averages of the effective connection strengths using the same format as in Fig. 3. Because we have pooled over 32 simulated subjects, the confidence intervals are much smaller (and also note the characteristic shrinkage one obtains with Bayesian estimators). The right column (resp. left column) shows the results for spectral DCM (resp. stochastic DCM) revealing the similarity between the Bayesian parameter averages from long runs (upper panel) and shorter runs (lower panel), of 1024 and 256 scans, respectively.
Fig. 6
Fig. 6
This figure reports the results of a simulated group comparison study of two groups of 24 participants (with 512 scans per participant). The upper row shows the Bayesian parameter averages of the differences using the same format as previous figures. For the spectral DCM (left panel) it can be seen that increases in the extrinsic forward connections from the second to the third region (seventh parameter) has been estimated accurately. Similarly, the decrease in the backward connection from the fourth to the third region is also estimated accurately. For the stochastic DCM (right panel), the estimation of the differences in the two parameter sets is not as accurate — although the direction is detected correctly. The equivalent classical inference — based upon the t-statistic is shown on lower row. Here the posterior means from each of 48 subjects were used as summary statistics and entered into a series of univariate t-tests to assess differences in group means. The red lines correspond to significance thresholds at a nominal false-positive rate of p = 0.05 corrected (solid lines) and uncorrected (broken lines). Clearly the connections with differences survive the corrected threshold for spectral DCM (left panel) whereas for the stochastic DCM (right panel) few other connections are also above threshold.
Fig. 7
Fig. 7
This figure reports the results of changing the priors on measurement noise when characterising group differences for both spectral and stochastic DCMs. The left column shows the Bayesian parameter averages of the differences for spectral DCM and the right column for stochastic DCM — using the same format as in the previous figures. For these results, we kept the prior covariance of the (log) precision parameters constant whilst varying the prior expectation of (log) precision parameters within the range of 2 and 10 with a step size of 2 (except the value of 6 for which the results are already reported in Fig. 6).
Fig. 8
Fig. 8
This figure reports the sensitivity of both schemes based on simulations of the sort reported in the previous figure. For these results, we varied the connection from node 4 to node 3 in the range of − 0.4 to 0.2 with a step size of 0.1 such that group differences were in the range of 0.3 to − 0.3. The left panel shows the Bayesian parameter averages of the differences for spectral DCM using the same format as previous figures — and the right panel shows the results for stochastic DCM.
Fig. 9
Fig. 9
Summary of empirical time series used for the illustrative analysis. The time series (right-hand panels) from four regions are the principal eigenvariates of regions identified using seed connectivity analyses (upper left insert) for a typical subject. These time series we used to invert the DCMs (both spectral and stochastic) with the (fully-connected) architecture shown in the lower left panel.
Fig. 10
Fig. 10
This figure summarises the results of model inversion using the model and data of previous figure. We only show the results for connections with non-trivial connection strengths greater than 0.5 Hz (and omit self-connections for simplicity). The upper row shows the results for the spectral DCM in same format in previous figures for simulated data. The leftmost panel shows the Bayesian parametric averages over 22 subjects. The middle panel shows the results of classical t-tests reporting t-statistics for each connection, whereas the right panel shows only those edges on the graph that survive the corrected threshold in the middle panel. The lower row reports the results for stochastic DCM in the same format as for the spectral DCM.
Fig. 11
Fig. 11
This figure plots the distribution of the posterior expectations of the two schemes over subjects for the strongest connection from left IPC to right IPC (see Fig. 10). The posterior expectations were ranked in descending order. The upper left panel shows the posterior expectations (light grey bars) for the spectral DCM with superimposed confidence interval (pink bars). A similar plot for stochastic DCM is shown in the lower left panel. We also show scatter plot of the posterior expectations over subjects for the two schemes.

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