Evaluation of different measures of functional connectivity using a neural mass model

Abstract

We use a neural mass model to address some important issues in characterising functional integration among remote cortical areas using magnetoencephalography or electroencephalography (MEG or EEG). In a previous paper [Neuroimage (in press)], we showed how the coupling among cortical areas can modulate the MEG or EEG spectrum and synchronise oscillatory dynamics. In this work, we exploit the model further by evaluating different measures of statistical dependencies (i.e., functional connectivity) among MEG or EEG signals that are mediated by neuronal coupling. We have examined linear and nonlinear methods, including phase synchronisation. Our results show that each method can detect coupling but with different sensitivity profiles that depended on (i) the frequency specificity of the interaction (broad vs. narrow band) and (ii) the nature of the coupling (linear vs. nonlinear).

Our analyses suggest that methods based on the concept of generalised synchronisation are the most sensitive when interactions encompass different frequencies (broadband analyses). In the context of narrow-band analyses, mutual information was found to be the most sensitive way to disclose frequency-specific couplings. Measures based on generalised synchronisation and phase synchronisation are the most sensitive to nonlinear coupling. These different sensitivity profiles mean that the choice of coupling measures can have dramatic effects on the cortical networks identified. We illustrate this using a single-subject MEG study of binocular rivalry and highlight the greater recovery of statistical dependencies among cortical areas in the beta band when mutual information is used.

Introduction

The complicated nature of brain connectivity, composed of local networks interconnected by long range pathways, is well known and is the principal determinant of its functional organisation Abeles, 1991, Sporns et al., 2000. The ensuing architecture means that the brain can be understood as a dynamical system, comprising a huge number of state variables generating many distinct and metastable states. This metastability endows the brain with a characteristic lability, that is, sensitivity to change, which arises from a capacity to self-organise through modulating its internal structure (Friston, 2000). This modulation is context dependent, occurs over different time scales and can be transient (e.g., changes of connectivity due to attention modulation) or enduring (e.g., somatotopic reorganisation due to limb amputation).

Integration or functional connectivity is usually inferred by statistical dependencies among signals in coupled neuronal systems. It is possible to analyse these dependencies using noninvasive macroscopic measures such as magnetoencephalography (MEG) (Hamalainen et al., 1993) and electroencephalography (EEG) (Nunez, 1981). However, the variability of MEG or EEG signals and our incomplete understanding of the neuronal processes generating them make their analysis particularly difficult. Indeed characterising dynamic patterns is tricky due to the intrinsic complexity and nonlinearity of the underlying neuronal processes (Abarbanel and Rabinovich, 2001).

There is an uncertain consensus about the best method to characterise neuronal couplings with MEG or EEG. Interactions can be synchronous or asynchronous, linear or nonlinear and so forth, engendering many different types of synchronisation, that is, a common behaviour of at least one property of MEG or EEG generators Boccaletti et al., 2002, Pikovsky et al., 2001. This may be why different dependency measures appear in the MEG or EEG literature: these include linear (cross-correlation, coherence; Clifford Carter, 1987) or nonlinear (mutual information; Roulston, 1999), nonlinear correlation (Pijn et al., 1992), mutual dimension (Buzuk et al., 1994), nonlinear interdependencies or generalised synchronisation Arnhold et al., 1999, Schiff et al., 1996, Stam and van Dijk, 2002, Stam et al., 2003, neural complexity (Tononi et al., 1994)) or are based upon the quantification of phase locking of oscillations (phase synchronisation; Lachaux et al., 1999, Tass et al., 1998). This battery of measures has been assembled during the last decade, making research in this area exciting but quite confusing. Recently, an attempt has been made to evaluate the performance of some of these measures in a case study of EEG signals (Quian Quiroga et al., 2002). The authors conclude that, with the exception of mutual information, all these measures give qualitatively equivalent results, with nonlinear measures being more sensitive.

The aim of this paper was to compare a set of commonly employed measures to assess their relative efficacy for detecting neuronal coupling. Our strategy was to use simulated neuronal processes in which we could manipulate the coupling between two cortical areas. Using surrogate or null data, we determined the null distribution of each measure to enable its sensitivity to be evaluated. This represents an assessment in terms of each measure's ability to detect statistical dependencies among remote neurophysiological indices. These dependencies define, operationally, “functional connectivity”. In this paper, we focus on measuring functional connectivity in continuous neuronal time series. Note that this does not constitute an analysis of “effective connectivity” (the influence one neuronal system exerts over another): We are not interested, here, in the mechanism of the coupling or indeed how the dependencies are expressed (e.g., linear vs. nonlinear). We simply want to establish whether dependencies exist and find their most sensitive measure. In a later paper, we will deal with similar issues in the context of transient responses and induced oscillations.

The measures considered in this work were linear (cross-correlation function or first-order cumulant) and nonlinear. The nonlinear measures represented static (mutual information) and dynamic measures. The latter were phase synchronisation and one based on the concept of generalised synchronisation. We anticipated that measures encompassing nonlinear and dynamic dependencies would supervene over linear and static metrics, given the way neuronal interactions are mediated. However, each estimator has its own bias and variance that may be important when dealing with finite time series.

In this paper, we use a model of MEG or EEG signals in which we changed the coupling strength between two cortical areas. This allowed us to assess the sensitivity of various interaction measures using oscillatory signals that simulate normal, ongoing MEG or EEG activity. The principal characteristics of the model have been presented in a companion paper (David and Friston, in press) and are described only briefly here. Our model is a generalisation, to multiple neuronal populations, of the Jansen model (Jansen and Rit, 1995) operating in its oscillatory regime. The Jansen model was based on previous work in the 1970s Freeman, 1978, Lopes da Silva et al., 1974 modelling EEG and visual-evoked potentials. Recently, it has been modified to generate epileptic activity (Wendling et al., 2000). A specific characteristic of our model is that it exhibits oscillatory behaviours that are driven by stochastic noise. Our model generates more complicated activity than the Jansen model because different neuronal populations are endowed with different kinetics. Our generalised model was designed to assess the consequences of coupling changes in the context of oscillatory dynamics. This complements the modelling initiative in (Wendling et al., 2001) for epileptic activity.

Modelled cortical areas are composed of interacting inhibitory and excitatory populations with slow and fast kinetics. Two operations are used to integrate the model: a spike density-to-membrane potential conversion, which depends linearly upon neuronal kinetics; and the inverse conversion, which is considered instantaneous and nonlinear (a sigmoid function). Different areas are coupled by excitatory connections whose strength is set by a coupling parameter. We have shown, qualitatively, that coupling areas results in a phase locking of their activities. The purpose of using this model here is to demonstrate that the sensitivity profile of each measure differs and that a complete description of interactions calls for the use of several approaches.

In the first section, we introduce the different measures of interdependencies among MEG or EEG signals that are considered in subsequent sections. Then we use our neural mass model to evaluate their sensitivity to coupling in the context of nonlinear interactions. Finally, we show how findings obtained from simulations translate using real MEG data from a binocular rivalry study.