Elsevier

NeuroImage

Volume 58, Issue 2, 15 September 2011, Pages 323-329
NeuroImage

Comments and Controversies
Wiener–Granger Causality: A well established methodology

https://doi.org/10.1016/j.neuroimage.2010.02.059Get rights and content

Abstract

For decades, the main ways to study the effect of one part of the nervous system upon another have been either to stimulate or lesion the first part and investigate the outcome in the second. This article describes a fundamentally different approach to identifying causal connectivity in neuroscience: a focus on the predictability of ongoing activity in one part from that in another. This approach was made possible by a new method that comes from the pioneering work of Wiener (1956) and Granger (1969). The Wiener–Granger method, unlike stimulation and ablation, does not require direct intervention in the nervous system. Rather, it relies on the estimation of causal statistical influences between simultaneously recorded neural time series data, either in the absence of identifiable behavioral events or in the context of task performance. Causality in the Wiener–Granger sense is based on the statistical predictability of one time series that derives from knowledge of one or more others. This article defines Wiener–Granger Causality, discusses its merits and limitations in neuroscience, and outlines recent developments in its implementation.

Introduction

For most of its history, neuroscience has primarily been concerned with examining the physiological correlates of experimentally delivered stimuli and overt behavioral responses. More recently, there has been growing interest in studying the effect that one part of the nervous system has on another, either in the absence of identifiable behavioral events or in the context of task performance. Such effects are typically examined by stimulating or lesioning the first part and investigating the outcome in the second. In peripheral and spinal pathways, the interventional techniques of stimulation and ablation have proven to be powerful methods for inferring causal influences from one neuron or neuronal population to another. For the study of causal relations within the brain, interventional techniques also have utility, although that utility is diminished by the high levels of convergence and divergence in brain pathways, as well as the highly nested reciprocity of projections. This article deals with a different approach to the problem of causal influence in the brain, called Time Series Inference (TSI). This approach, although relatively recent in neuroscience, is showing promise as a valuable adjunct to more traditional interventional approaches.

TSI, unlike stimulation or ablation, does not require intervention in the nervous system. It is based on temporal relations existing between time series recordings of neural activity, which may be obtained noninvasively as electroencephalographic (EEG), magnetoencephalographic (MEG), or functional Magnetic Resonance (fMRI) data. Of course, the time series may also be obtained from invasive single-unit, multi-unit, local field potential, or electrocorticographic recordings. TSI depends on the statistical predictability of one time series by another time series. If the two time series represent neural activity from different neurons or neuronal populations, then inference about causal relations between those neurons or neuronal populations is possible. Another advantage of TSI methods is that they naturally accommodate stochastic processes, and thus are well suited to the ubiquitous variability that is found in neural time series data. Furthermore, under proper conditions, TSI methods can be effectively used to relate neural activity to cognitive function.

In this article, we will describe Wiener–Granger Causality (WGC) as the type of TSI most commonly employed in neuroscientific studies. TSI methods, in one form or another, are well established and widely used in many different fields of study. The use of TSI methods in neuroscience is relatively new, but already has appeared in different implementations. For example, the popular Dynamic Causal Modeling (DCM) (Friston et al., 2003) approach is based on TSI, although this fact is not commonly appreciated. We will focus here on WGC techniques, which, as described below, are based on a relatively small set of straightforward assumptions. DCM, which we will not discuss at length, involves TSI with additional and more complicated assumptions (see accompanying articles in this issue).

In summary, our purpose here is to detail the implementation of WGC via AutoRegressive (AR) modelling, discuss the assumptions required to apply WGC to neural time series data, describe some of the limitations of its use, and provide some examples of applications to neural time series data. Finally, we will describe some potentially important extensions of currently employed techniques.

Section snippets

Time-domain WGC

The problem of defining ‘causality’ is non-trivial for complex systems, where, unlike simple systems observed in the every-day world, an intuitive understanding of cause and effect is lacking. In 1956 Norbert Wiener introduced the notion that one variable (or time series) could be called ‘causal’ to another if the ability to predict the second variable is improved by incorporating information about the first (Wiener, 1956). Wiener however lacked a practical implementation of his idea. Such an

Assumptions and challenges

In this section we identify some key challenges for WGC analysis, both in general and with regard to the commonly employed neuroimaging methods of fMRI, EEG, and MEG. In each case we outline recent work that may address, at least in part, these challenges.

Causal inference in neural systems

Any discussion of causal inference should be prefaced by one fundamental question: What do we expect from a measure of causality? One useful perspective is that causal measures in neuroscience should reflect effective connectivity, namely the directed influences that neuronal populations in one brain area exert on those in another (Friston, 1994). As applied to neuroimaging data (e.g., by way of DCM), effective connectivity analysis aims at identifying, from recorded neural time series data,

Acknowledgments

AKS is supported by EPSRC Leadership Fellowship EP/G007543/1 and by a donation from the Dr. Mortimer and Theresa Sackler Foundation. The authors thank Ms. Wei Tang, Dr. Lionel Barnett, Dr. Adam Barrett, Dr. Alard Roebroeck, Dr. Abraham Snyder, and our reviewers for helpful comments.

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