Estimation of direct nonlinear effective connectivity using information theory and multilayer perceptron

Highlights

• The direct nonlinear effective connectivity of high-dimensional datasets is estimated.

• A combination of regressor selection, MLP modeling and Granger Causality is proposed.

• βmRMR-MLP-GC can deal with highly nonlinear, high-dimensional datasets.

• In simulations, βmRMR-MLP-GC yields both high sensitivity and specificity.

• βmRMR-MLP-GC detects Back-to-Front alpha information flows in resting brain.

Abstract

Background

Despite the variety of effective connectivity measures, few methods can quantify direct nonlinear causal couplings and most of them are not applicable to high-dimensional datasets.

New method

In this paper, a novel approach (called βmRMR-MLP-GC) is proposed to estimate direct nonlinear effective connectivity of high-dimensional datasets. βmRMR is used to select a suitable subset of candidate regressors for approximating each neural (here EEG) signal. The multilayer perceptron (MLP) is used for multivariate characterization of EEG signals while the optimum MLP structure is selected using an iterative cross-validation scheme. Finally a causality measure is defined based on Granger Causality (GC) concept to quantify the casual relations among EEG channels.

Results

Applying βmRMR-MLP-GC to high-dimensional simulated datasets with different linear and nonlinear structures yields sensitivity and specificity values higher than 95%. Also, applying it to eyes-closed resting state EEG of six normal subjects in the alpha frequency band yields significant net activity propagations from the posterior to anterior brain regions. This is in accordance with the most previous studies in this field.

Comparison with existing method(s)

βmRMR-MLP-GC is compared with Granger Causality Index, Conditional Granger Causality Index, and Transfer Entropy. It outperforms these methods in terms of sensitivity and specificity in simulated datasets. Also, βmRMR-MLP-GC detects the most number of significant and reproducible Back-to-Front net information flows among the specified brain regions and highlights the posterior brain regions as dominant source of alpha activity propagation.

Conclusions

βmRMR-MLP-GC provides a novel tool to estimate the direct nonlinear causal networks of high-dimensional datasets.

Introduction

The human brain is definitely one of the most complex natural systems in the world. Despite lots of guided studies for discovering its functional organization, there are still many unknowns about it. Founded on the “brain specialization concept” most studies aimed at exploring the brain regions specialized for particular brain tasks in the past years (Jirsa and McIntosh, 2007). The emergence of “brain integration concept” directed many researches in recent years toward the brain connectivity (Jirsa and McIntosh, 2007). Brain connectivity is a broad concept that can be generally divided into three categories: Structural, Functional and Effective connectivity. The structural connectivity refers to the structural connections of brain regions via nerve fibers. The functional connectivity deals with the temporal interdependencies among the activity of brain regions. The effective connectivity characterizes the causal (directed) effects among brain regions. Reviews of the most commonly used functional and effective connectivity measures can be found in Greenblatt et al. (2012), Sakkalis (2011), Pereda et al. (2005), and Muskulus et al. (2009).

Exploring the effective connectivity networks helps neurologists to investigate the changes of brain causal networks due to the brain disorders such as autism (Wicker et al., 2008), Alzheimer (Liu et al., 2012), schizophrenia (Diaconescu et al., 2011), epilepsy (Amini et al., 2010a, Amini et al., 2010b). This helps the physicians to find effective treatments for the brain disorders.

Among the functional brain imaging modalities, EEG and MEG are capable of capturing the temporal dynamics of cortical connectivity owing to their high temporal resolution (He et al., 2011). Consequently, they are popular modalities for functional/effective connectivity estimation, despite their limitations in terms of spatial resolution and volume conduction effects (Schoffelen and Gross, 2009).

According to Wiener Causality concept (Wiener, 1956), if adding the past and present information of (system) X to the past and present information of (system) Y improves predicting the future of (system) Y, X is the cause of Y. Granger (1969) limited the very general definition of Wiener Causality (Wiener, 1956) to linear bivariate autoregressive models and proposed a mathematical formulation to infer the Wiener Causality quantitatively (Granger, 1969). The terms Wiener causality (WC) and Granger Causality (GC) are usually used interchangeably. Geweke (1982) proposed the most practical GC-based effective connectivity measure, which is usually known as Granger Causality Index (GCI) (Geweke, 1982). To distinguish between direct and indirect causal links, a variety of multivariate linear GC-based measures have been developed, for example Conditional Granger Causality Index (CGCI) in the time domain (Ding et al., 2007), and DTF, PDC, dDTF and ffDTF in the frequency domain (Kaminski and Liang, 2005, Wu et al., 2011). They are all based on linear Multivariate Auto-Regressive (MVAR) models.

It is widely assumed that the interactions among neuronal populations are nonlinear (Marinazzo et al., 2011, Pereda et al., 2005, Ioannides and Mitsis, 2010, Stam, 2005; Gourévitch et al., 2006). Consequently, using the linear connectivity measures may oversimplify the functional organization of brain and even lead to incorrect estimation of causal relations. A few multivariate and nonlinear effective connectivity measures have been proposed based on GC in the literature. Some of them are parametric and model-based like the kernel-based nonlinear Granger Causality measures (Marinazzo et al., 2008, Guo et al., 2008) and locally linear MVAR models on the reconstructed attractor space (Chen et al., 2004) whereas the others are nonparametric. Some of those nonparametric methods are partial entropy with non-uniform embedding (Faes et al., 2011), and Partial Transfer Entropy (PTE) (Gomez-Herrero, 2010, Vakorin et al., 2009) which is an extension of well-known Transfer Entropy (TE) measure (Schreiber, 2000). Some other nonparametric methods are based on the Correlation Integral (CI) (Zhidong et al., 2010, Gourévitch et al., 2006).

An ideal effective connectivity measure has three characteristics: (1) It uses no restrictive model, therefore it can explore both linear and nonlinear causal connections. (2) It is multivariate in order to distinguish the direct connections from indirect ones. It has been shown that bivariate measures may yield misleading results on multivariate data (Kus et al., 2004). (3) It must be practically applicable to high-dimensional datasets (e.g. EEG/MEG with high number of channels), and to systems with long memory or large coupling delays.

Despite the existence of many effective connectivity measures, very few measures have all three features above. Due to the lack of a systematic dimensionality reduction stage, none of the multivariate nonlinear measures mentioned above are applicable to high-dimensional datasets except the method proposed by Marinazzo et al. (2008) and may be the one proposed by Faes et al. (2011). Both these measures use some kinds of dimensionality reduction techniques. The nonparametric nonlinear measures (in fact, information theoretic-based and CI-based measures) need the estimation of multivariate probability density functions. Consequently, they are not applicable to high-dimensional datasets since their required data samples exponentially grow with the number of variables. As a result, PTE (Gomez-Herrero, 2010, Vakorin et al., 2009) and CI-based methods (Zhidong et al., 2010, Gourévitch et al., 2006) are not applicable to high-dimensional datasets. Even the method proposed by Faes et al., 2011 may produce inaccurate results if the number of selected partial terms in the partial entropy becomes slightly big. Also in the methods proposed by Chen et al. (2004) and Guo et al. (2008) the number of modeling parameters is proportional to the square of number of dimensions so they do not have the third desired feature of an ideal effective connectivity measure, too.

In this paper, we propose a new approach to estimate the total (linear and nonlinear) effective connectivity network of high-dimensional datasets (e.g. EEG/MEG datasets with large number of channels). We use multilayer perceptron (MLP) to explore linear and nonlinear causal dependencies among neural signals. MLP is a member of the artificial neural networks (ANNs) family, which has the “universal approximation” property (Bishop, 1995, Ripley, 1996). MLP has been used successfully for modeling mean profile of resting state EEG signals (Kawano et al., 2003, Nagashino et al., 2002). As a novel work, in this paper we use MLP for effective connectivity estimation. The performance of MLP like any other regression method may be deteriorated if the number of its input regressors (and so the number of its parameters) increases. In such a case, an appropriate input selection method should be used before MLP modeling. May et al. (2011) reviewed the practical input selection approaches for ANNs.

In this paper, to keep the advantage of detecting nonlinear structure of high-dimensional datasets, an information theoretic-based approach is used for input regressor selection. This approach is called βmRMR (β minimal-Redundancy-Maximal-Relevance) (Hejazi and Cai, 2009). βmRMR is a modified version of the well-known mRMR (Peng et al., 2005). It iteratively selects the input regressors that add maximum amount of new information about the output to the previously selected set of inputs. Since βmRMR is based on information theory, it can keep the advantage of MLPs for exploration of highly nonlinear structures. Moreover due to its careful design, it can yield reliable results even for very large set of candidate regressors. Then we combine βmRMR and MLP modeling to approximate time series of channels. Finally we combine βmRMR-MLP with GC concept to construct a new method called βmRMR-MLP-GC for estimating the direct total (linear and nonlinear) effective connectivity of high-dimensional datasets. We also use Time-Shifted surrogate data to evaluate the significance of the effective connectivity estimates. We applied the proposed method on both simulated datasets and experimental EEG data and compared its performance with three of the most widely used effective connectivity measures: GCI (Geweke, 1982); CGCI (Ding et al., 2007); and TE (Schreiber, 2000).

This paper is organized as follows: in Section 2, we review MLP, βmRMR input regressor selection, and Granger Causality Index. Then we combine these three tools to propose our method called βmRMR-MLP-GC. Then the simulation designs for evaluating the performance of our proposed method are described and our EEG data are introduced. In Section 3, we report the results of applying βmRMR-MLP-GC to the simulated datasets and EEG data. In Section 4, we discuss some results, approach some issues related to βmRMR-MLP-GC, and propose some future works. Finally, in Section 5 we conclude the paper.