Journal of Neuroscience Methods 183 (2009) 31–41 Contents lists available at ScienceDirect Journal of Neuroscience Methods journal homepage: www.elsevier.com/locate/jneumeth Functional similarities and distance properties Michael Muskulus a,∗ , Sanne Houweling b , Sjoerd Verduyn-Lunel a , Andreas Daffertshofer b a Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands b Research Institute MOVE, Faculty of Human Movement Sciences, VU University Amsterdam, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands article info Article history: Received 15 May 2009 Received in revised form 23 June 2009 Accepted 27 June 2009 PACS: 05.45.-a 87.19.L- 89.75.-k Keywords: Functional connectivity Multidimensional scaling Phase locking Granger causality MEG abstract The analysis of functional and effective brain connectivity forms an important tool for unraveling structure–function relationships from neurophysiological data. It has clinical applications, supports the formulation of hypotheses regarding the role and localization of functional processes, and is often an ini- tial step in modeling. However, only a few of the commonly applied connectivity measures respect metric properties: reflexivity, symmetry, and the triangle inequality. This may hamper interpretation of findings and subsequent analysis. Connectivity indices obtained by metric measures can be seen as functional dis- tances, and may be represented in Euclidean space by the methods of multidimensional scaling. We sketch some classes of measures that do allow for such a reconstruction, in particular the class of Wasserstein distances, and discuss their merits for interpreting cortical activity assessed by magnetoencephalography. In an application to magnetoencephalographic recordings during the execution of a bimanual task, the Wasserstein distances between relative circular variances indicated cortico-muscular synchrony as well as cross-talk between bilateral primary motor areas in the ˇ-band. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Functional connectivity can be defined as the occurrence of a significant statistical interdependency between activities of distant neurons or neural populations. In combination with the con- straining anatomy, this definition forms a proper starting point for unraveling the relationship between structural and functional features of the brain. Down to the present day, the quest for a comprehensive understanding of structure–function interaction has attracted a lot of attention (Stephan et al., 2008; Lee et al., 2003). Structure can be rather complicated but is typically considered material and fixed, whereas function reflects statisti- cal similarity between dynamical processes in the brain. Related concepts are anatomical and effective connectivity, respectively, where the latter refers to causal relationships between signals (Friston et al., 1993b; Ramnani et al., 2004). A functional con- nectivity analysis often precedes the formulation of a causal (or directed) model, yielding numerous applications. Apart from its fundamental role in determining functionally important neuronal processes, it has important clinical applications (Stam, 2005). The neurophysiological underpinnings, however, are still under ∗ Corresponding author. Tel.: +31 71 527 7058; fax: +31 71 527 6985. E-mail addresses: muskulus@math.leidenuniv.nl (M. Muskulus), s.houweling@fbw.vu.nl (S. Houweling), verduyn@math.leidenuniv.nl (S. Verduyn-Lunel), marlow@fbw.vu.nl (A. Daffertshofer). debate, partly because of the huge variety of connectivity mea- sures employed, rendering methods inscrutable (Pereda et al., 2005) and questioning the possible contribution of functional connectivity to an integrative understanding of brain function- ing (Horwitz, 2003; Fingelkurts et al., 2005). To dispel doubts, we outline general properties of functional connectivity mea- sures and their implications for analysis. We aim for facilitating the selection of proper measures and, by this, distill convincing arguments for their relevance for an understanding of brain dynam- ics. In a nutshell, the large majority of commonly implemented con- nectivity measures do not respect fundamental metric properties. Here, we focus on three important properties: (i) reflexivity, (ii) symmetry, and (iii) the triangle inequality. If a connectivity measure disregards one or more of these three properties, its interpreta- tion may be ambivalent when multiple signals are assessed. Put differently, such measures can be very successful for a pair-wise comparison of signals, i.e., in the bivariate case, but they may lead to spurious results in multivariate settings (Kus et al., 2004). On the contrary, if a connectivity measure does respect all properties (i)–(iii), then it describes a proper functional distance. We argue that this is a necessary condition for a truly integrative analysis. Of course, genuine multivariate statistical methods suffice for this pur- pose, but implementation can be cumbersome and results might be difficult to interpret. More important, commonly used multivariate methods require explicit assumptions about the data, e.g., signals ought to be normally distributed to apply principal or independent 0165-0270/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jneumeth.2009.06.035