Temporal Information of Directed Causal Connectivity in Multi-Trial ERP Data using Partial Granger Causality

Abstract

Partial Granger causality (PGC) has been applied to analyse causal functional neural connectivity after effectively mitigating confounding influences caused by endogenous latent variables and exogenous environmental inputs. However, it is not known how this connectivity obtained from PGC evolves over time. Furthermore, PGC has yet to be tested on realistic nonlinear neural circuit models and multi-trial event-related potentials (ERPs) data. In this work, we first applied a time-domain PGC technique to evaluate simulated neural circuit models, and demonstrated that the PGC measure is more accurate and robust in detecting connectivity patterns as compared to conditional Granger causality and partial directed coherence, especially when the circuit is intrinsically nonlinear. Moreover, the connectivity in PGC settles faster into a stable and correct configuration over time. After method verification, we applied PGC to reveal the causal connections of ERP trials of a mismatch negativity auditory oddball paradigm. The PGC analysis revealed a significant bilateral but asymmetrical localised activity in the temporal lobe close to the auditory cortex, and causal influences in the frontal, parietal and cingulate cortical areas, consistent with previous studies. Interestingly, the time to reach a stable connectivity configuration (~250–300 ms) coincides with the deviation of ensemble ERPs of oddball from standard tones. Finally, using a sliding time window, we showed higher resolution dynamics of causal connectivity within an ERP trial. In summary, time-domain PGC is promising in deciphering directed functional connectivity in nonlinear and ERP trials accurately, and at a sufficiently early stage. This data-driven approach can reduce computational time, and determine the key architecture for neural circuit modelling.

Appendices

Appendix A. Theory of Time-Domain Granger Causality Approaches

The estimation in GC methods requires multivariate autoregressive (MVAR) modelling. The MVAR is an extension of autoregression (AR) model for arbitrary number of channels and works based on stochastic process theory and stationarity assumption. It has been considered as a mathematical model of EEG (Wang et al. 2008). By assuming that the present measurement (X _t ) at a given source is a linear combination of prior (p) measurements of all sources, plus a residual error term, the MVAR model can be defined as:

$$ {X}_t={\displaystyle \sum_{k=1}^p{A}_k{X}_{t-k}}+{\varepsilon}_t $$

(A. 1)

where X _t  = [X _1t , X _2t , …, X _Nt ] is a vector of signal measurements across N-nodes at time t. The variable p is the model order and corresponds to the number of lags; A _k  = {a _i,j (k)} is an N × N dimensional coefficient matrix contains autoregressive coefficients identifying the effect of source i on source j at a delay of k time samples; ${\varepsilon}_t=\left[{\upvarepsilon}_{1t},{\upvarepsilon}_{2t},\dots, {\upvarepsilon}_{Nt}\right]\sim \mathcal{N}\left(0,\Sigma \right)$ is the zero mean uncorrelated noise vector, and Σ is its covariance matrix. The coefficients A _k are usually determined by the recursive entropy-based LWR algorithm (Morf et al. 1978) that is computationally more robust than classical moment-based method (Yule-Walker) (Ding et al. 2006). The autocovariance matrix can then be elicited from AR coefficients. The mathematical formalism of the three GC types for a two-dimensional stationary joint time series is schematically shown in Fig. 12.

Fig. 12

The schematic autoregressive representation of the three time-domain CG-analysis approaches described by Guo et al., in (Guo et al. 2008a). a). BGC-, b). CGC- and c). PGC- analysis of two jointly distributed, covariance-stationary time series influencing by exogenous inputs and latent variables.

The BGC is applied in pairwise situations. Figure 12a schematically shows two stationary time series (X _t and Y _t ), which are jointly interactive. If the prediction error of X _t + 1 using p previous values of X _t and Y _t (full model, Eq. A. 3) is less than the prediction error using only the series X _t (restricted model, Eq. A. 2), then Y _t is said to have a causal influence (Granger-cause) on X _t (F1 _Y → X , Eq. A. 4). Similarly, if the prediction error of Y _t in the full model is less than that in the restricted, then X _t is said to Granger-cause Y _t (F1 _X → Y ) (Barrett et al. 2010):

$$ \left\{\begin{array}{c}\hfill {X}_t={\displaystyle \sum_{k=1}^p{a}_{1k}{X}_{t-k}}+{\varepsilon}_{1t}\hfill \\ {}\hfill {Y}_t={\displaystyle \sum_{k=1}^p{b}_{1k}{X}_{t-k}}+{\varepsilon}_{2t}\hfill \end{array}\right. $$

(A. 2)

$$ \left\{\begin{array}{c}\hfill {X}_t={\displaystyle \sum_{k=1}^p{a}_{2k}{X}_{t-k}}+{\displaystyle \sum_{k=1}^p{c}_{2k}{Y}_{t-k}}+{\varepsilon}_{3t}\hfill \\ {}\hfill {Y}_t={\displaystyle \sum_{k=1}^p{b}_{2k}{Y}_{t-k}}+{\displaystyle \sum_{k=1}^p{d}_{2k}{X}_{t-k}}+{\varepsilon}_{4t}\hfill \end{array}\right. $$

(A. 3)

$$ \left\{\begin{array}{c}\hfill F{1}_{Y\to X}=Ln\left(\frac{\operatorname{var}\left({\varepsilon}_{1t}\right)}{\operatorname{var}\left({\varepsilon}_{3t}\right)}\right)\hfill \\ {}\hfill F{1}_{X\to Y}=Ln\left(\frac{\operatorname{var}\left({\varepsilon}_{2t}\right)}{\operatorname{var}\left({\varepsilon}_{4t}\right)}\right)\hfill \end{array}\right. $$

(A. 4)

The second approach, conditional-GC, an extension of BGC, can be used for more than two time series, namely, when two time series are mediated by a third source. Thus, in the presence of such intervening influence, CGC compares the prediction error between the restricted model (Eq. A. 5) and full model (Eq. A. 6) when it is conditioned by a third, measurable series Z _t . As shown in Fig. 12b, the causal influence of Y _t on X _t given Z _t (F2 _{Y → X|Z} ) is defined by the log-ratio of residual variance matrices (Eq. A. 7) of the restricted and full model (Ln(S ₁₁/∑₁₁)) in Eq. A. 8). One variable causally influencing a second variable if the prediction error variance of the first is reduced after including the second variable in the model, with all other variables included in both cases.

$$ \left\{\begin{array}{c}\hfill {X}_t={\displaystyle \sum_{k=1}^p{a}_{1k}{X}_{t-k}}+{\displaystyle \sum_{k=1}^p{c}_{1k}{Z}_{t-k}}+{\varepsilon}_{1t}\hfill \\ {}\hfill {Z}_t={\displaystyle \sum_{k=1}^p{b}_{1k}{Z}_{t-k}}+{\displaystyle \sum_{k=1}^p{d}_{1k}{X}_{t-k}}+{\varepsilon}_{2t}\hfill \end{array}\right. $$

(A. 5)

$$ \left\{\begin{array}{c}\hfill {X}_t={\displaystyle \sum_{k=1}^p{a}_{2k}{X}_{t-k}}+{\displaystyle \sum_{k=1}^p{b}_{2k}{Y}_{t-k}}+{\displaystyle \sum_{k=1}^p{c}_{2k}{Z}_{t-k}}+{\varepsilon}_{3t}\hfill \\ {}\hfill \begin{array}{l}{Y}_t={\displaystyle \sum_{k=1}^p{d}_{2k}{X}_{t-k}}+{\displaystyle \sum_{k=1}^p{e}_{2k}{Y}_{t-k}}+{\displaystyle \sum_{k=1}^p{f}_{2k}{Z}_{t-k}}+{\varepsilon}_{4t}\\ {}{Z}_t={\displaystyle \sum_{k=1}^p{g}_{2k}{X}_{t-k}}+{\displaystyle \sum_{k=1}^p{h}_{2k}{Y}_{t-k}}+{\displaystyle \sum_{k=1}^p{k}_{2k}{Z}_{t-k}}+{\varepsilon}_{5t}\end{array}\hfill \end{array}\right. $$

(A. 6)

$$ \begin{array}{l}S=\left[\begin{array}{cc}\hfill \operatorname{var}\left({\varepsilon}_{1t}\right)\hfill & \hfill \operatorname{cov}\left({\varepsilon}_{1t},{\varepsilon}_{2t}\right)\hfill \\ {}\hfill \operatorname{cov}\left({\varepsilon}_{2t},{\varepsilon}_{1t}\right)\hfill & \hfill \operatorname{var}\left({\varepsilon}_{2t}\right)\hfill \end{array}\right]\hfill \\ {}\varSigma =\left[\begin{array}{ccc}\hfill \operatorname{var}\left({\varepsilon}_{3t}\right)\hfill & \hfill \operatorname{cov}\left({\varepsilon}_{3t},{\varepsilon}_{4t}\right)\hfill & \hfill \operatorname{cov}\left({\varepsilon}_{3t},{\varepsilon}_{5t}\right)\hfill \\ {}\hfill \operatorname{cov}\left({\varepsilon}_{4t},{\varepsilon}_{3t}\right)\hfill & \hfill \operatorname{var}\left({\varepsilon}_{4t}\right)\hfill & \hfill \operatorname{cov}\left({\varepsilon}_{4t},{\varepsilon}_{5t}\right)\hfill \\ {}\hfill \operatorname{cov}\left({\varepsilon}_{5t},{\varepsilon}_{3t}\right)\hfill & \hfill \operatorname{cov}\left({\varepsilon}_{5t},{\varepsilon}_{3t}\right)\hfill & \hfill \operatorname{var}\left({\varepsilon}_{5t}\right)\hfill \end{array}\right]\hfill \end{array} $$

(A. 7)

$$ F{2}_{Y\to X\Big|Z}=Ln\left(\frac{S_{11}}{\varSigma_{11}}\right) $$

(A. 8)

In PGC, It is assumed all confounding influences are reflected in correlations among the residuals, can be detected and partly be factored out analogous to partial correlation (Guo et al. 2008a). Partial correlation removes the influence of known unwanted variables via correlation measure while PGC tries to mitigate the influence of unknown variables indirectly and via their influence on prediction error (Bressler and Seth 2011). Figure 12c shows the vector autoregressive representation of PGC of two assumed independent sources, X _t and Y _t, are influenced by latent variables (L ₁ and L ₂), exogenous inputs (E ₁ and E ₂ ) and the modulatory factor (driven by series Z _t). The corresponding residual error of two confounding factors are ε ^E _t (E: exogenous) and B(L)ε ^L _t (L: latent) along with residual error ε _t can be incorporated into multivariate autoregressive representation (Eqs. A. 9 and A. 10). B (L) is the polynomial matrix of appropriate lag operator L related to latent input depends on its history (Guo et al. 2010). Therefore, the causal influence (e.g. F3 _{Y → X|Z} ) in PGC can be defined by the log-ratio of the variance of corresponding residual errors (Eq. A. 11) considering the covariance term to control the indirect interactions (Eq. A. 12).

(A. 9)

(A. 10)

$$ \begin{array}{l}S=\left[\begin{array}{cc}\hfill \operatorname{var}\left({u}_{1t}\right)\hfill & \hfill \operatorname{cov}\left({u}_{1t},{u}_{2t}\right)\hfill \\ {}\hfill \operatorname{cov}\left({u}_{2t},{u}_{1t}\right)\hfill & \hfill \operatorname{var}\left({u}_{2t}\right)\hfill \end{array}\right]\hfill \\ {}\varSigma =\left[\begin{array}{ccc}\hfill \operatorname{var}\left({u}_{3t}\right)\hfill & \hfill \operatorname{cov}\left({u}_{3t},{u}_{4t}\right)\hfill & \hfill \operatorname{cov}\left({u}_{3t},{u}_{5t}\right)\hfill \\ {}\hfill \operatorname{cov}\left({u}_{4t},{u}_{3t}\right)\hfill & \hfill \operatorname{var}\left({u}_{4t}\right)\hfill & \hfill \operatorname{cov}\left({u}_{4t},{u}_{5t}\right)\hfill \\ {}\hfill \operatorname{cov}\left({u}_{5t},{u}_{3t}\right)\hfill & \hfill \operatorname{cov}\left({u}_{5t},{u}_{3t}\right)\hfill & \hfill \operatorname{var}\left({u}_{5t}\right)\hfill \end{array}\right]\hfill \end{array} $$

(A. 11)

$$ F{3}_{Y\to X\Big|Z}=Ln\left(\frac{S_{11}-{S}_{12}{S}_{22}^{-1}{S}_{21}}{\varSigma_{11}-{\varSigma}_{13}{\varSigma}_{33}^{-1}{\varSigma}_{31}}\right) $$

(A. 12)

Appendix B. Nonlinear Toy Model

By means of three GC measures, we compared our proposed biologically plausible nonlinear model whose kernel function is sigmoidal (Section 2.2.2) with the nonlinear model whose kernel function is Gaussian-like from the original paper (Guo et al. 2008a). We tested the models in two different conditions. First, we set all coefficient related to confounding influences to zero (a _i  = b _i  = c _i  = 0) in both models, i.e. noiseless condition. Surprisingly, the measures applied to the nonlinear model of the original paper showed a false positive from source 4 to 5, while the sigmoidal model correctly revealed all expected effects (Fig. 13). The accuracies for all the three measures are clearly poorer under the noisy condition (a _i  = b _i  = c _i  = 5), but the expected interactions could still be detected by PGC method.

Fig. 13

The comparison between sigmoidal nonlinear model (Section 2.2.2) and Gaussian nonlinear model introduced by Guo et al. (2008a) in two conditions. The models tested using three GC measures in two conditions: with and without noise (confounding effects).

Appendix C. Standard versus Oddball Responses

Figure 14 shows a comparison made between the results of standard and oddball responses at 300 ms post-stimulus using the PGC measure. According to the results, the causal influences of oddball responses are significant in bilateral temporal, right frontal and right parietal regions, while the causal interactions of standard responses only appear in the bilateral areas. Moreover, in the results of standard responses, no causal influence emerges in cingulate and prefrontal cortices and between STG and IFG area.

Fig. 14

PGC analysis performed in 19-channel ERP trials of five subjects, which elicited from standard and oddball responses. The causal effects of oddball responses (up) are lateralised to the temporal and right frontal areas while the causal interactions of standard responses (down) show a weak pattern of connectivity in bilateral temporal areas.

Information Sharing Statement

The dataset for the above-mentioned auditory oddball task is based on the published work by Garrido et al. (2007a) upon approval by the authors in the paper, it is now available at https://github.com/vyoussofzadeh/MMN_ERP. Video clip and source codes for the simulated data can be found at the same URL link.