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Within the last few decades, attempts have been made to characterize the underlying mechanisms of brain activity by analyzing neural signals recorded, directly or indirectly, from the human brain. Accordingly, inference of functional connectivity among neural signals has become an indispensable research tool in modern neuroscience studies aiming to explore how different brain areas are interacting with each other. Indeed, remarkable advances in computational sciences and applied mathematics even allow the estimation of causal interactions among multichannel neural signals. Here, we introduce the brief mathematical background of the use of causality inference in neuroscience and discuss the relevant mathematical issues, with the ultimate goal of providing applied mathematicians with the current state-of-the-art knowledge on this promising multidisciplinary topic.

Traditional functional neuroimaging studies have focused on the functional specification of brain areas. However, only a limited amount of information regarding the underlying neuronal mechanisms can be obtained when such spatial specification is studied. Recently, research interests have shifted toward describing how different brain areas interact with each other, with the hope of better understanding the functional organization of the cortical network [

Over the past few decades, a number of measures for “directional” coupling between neural activities have been developed [

Here, we introduce several mathematical signal analysis methods for estimating directional coupling between neural activities, all of which have been widely used in basic and applied neuroscience. Additionally, this paper attempts to illustrate the important mathematical issues that need to be addressed to improve the conventional causality estimators, with the aim to stimulate interest in this imperative multidisciplinary research topic among applied mathematicians.

Recently, a number of causality estimation techniques have been developed to infer causality among multiple neural signal generators. The MVAR model—a linear multivariate time series model with a long history of application in econometrics [

Granger causality [_{0} equals RSS_{1}. Conversely, if the null hypothesis is rejected, that is,

To test this hypothesis, a traditional

However, the MVAR model is problematic when estimating the appropriate model order

Directed transfer function (DTF) is a widely used tool in identifying information flow between multichannel neural signals. Even though both Granger causality and DTF are based on MVAR modeling, the DTF procedure differs slightly from Granger causality. As described above, Granger causality uses the variance of prediction errors to estimate the causal interaction, while DTF uses a matrix transfer function derived from MVAR model coefficients [

The DTF can then be defined in terms of the elements of the transfer matrix

Partial directed coherence (

Although the MVAR-based causality estimators described above have been shown to be useful in many neuroscience problems, they are not applicable for all types of neural signals. For example, signals which have severely unbalanced model residual variances are not appropriate for

The generalized partial directed coherence (gPDC) was first proposed by Baccalá et al. to circumvent the numerical problem associated with time series scaling [

The Geweke’s Granger causality is derived from Geweke’s formulation [

The direct directed transfer function (dDTF) was proposed by Korzeniewska [

The dDTF method was proposed to circumvent some problems associated with DTF, specifically its inability to differentiate between the direct and indirect connections [

To date, Granger causality has been extensively applied to the analysis of neural signals [

The DTF algorithm has also been extensively applied to various aspects within neuroscience, particularly to the analyses of electrophysiological signals such as EEG, MEG, and iEEG, because frequency-domain analysis is generally required in these modalities. Franaszczuk et al. first applied the DTF algorithm to the localization of ictal onset zones in temporal lobe epilepsy patients [

As with DTF, PDC analyses have recently been applied to a variety of practical applications: Sun et al. demonstrated that PDC is a useful tool for evaluating changes in cortical interdependences in the context of different psychotic or mental states and can also be used to diagnose affective disorders [

Information theoretic measures have widely been utilized to quantify mutual dependence between time series. Although standard time-delayed mutual information can estimate mutual dependence between neural signals, it is not able to distinguish information flow [

Many researchers now apply the TE algorithm to the field of neuroscience [

To robustly estimate the direction of information flow in multivariate time series, Nolte proposed a new causality estimator called phase slope index (PSI) [

To estimate causal interactions between the nonlinear bivariate neural signals, nonlinear Granger causality (NGC) was developed [

Recently, Gourévitch et al. proposed a new method for estimating nonlinear causal interactions [

Although PNGC showed promising results when applied to complex systems, it is still dependent on model order and scale [

The most popular causality estimators—GC [

Moreover, MAVR-based causality estimators guarantee accurate causality inference only when datasets (signals) satisfy stationary conditions [

Another critical limitation affecting the reliability of causality estimators is the linear modeling of neural signals [

Generally, MVAR-based causality estimators require the appropriate selection of signal sample number. In one study, Schlögl assessed the dependency of several MVAR algorithms on the number of time samples, demonstrating that sufficient numbers of samples are required to obtain a reliable estimate of causal interactions among neural signals [

While most non-MVAR-based causality estimators, such as PNGC, nonlinear Granger causality, TE, and PSI, were introduced to circumvent the well-described problems of MVAR-based causality estimators, many can only be applied to causality inferences of bivariate neural signals. As such, further research is required to extend bivariate causality inferences to include multivariate (more than three) causality inference. Furthermore, a method for determining the proper model order in PNGC remains an ongoing problem [

Here, we summarized the mathematical techniques used in causality estimation, all of which have been extensively applied to infer causal relationships among multichannel neural signals. We also described the limitations of current methods and presented several ongoing problems, some of which may be of interest to applied mathematicians. We hope that this paper will serve as a useful guide for researchers in the field of applied mathematics and helps raise awareness of this important research topic.

Y.-J. Jung and K.-H. Kim are co-first authors and contributed equally to this work.

This work was supported by the Original Technology Research Program for Brain Science through the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Education, Science, and Technology (no. 2010-0018840).