Frequency-dependent functional connectivity analysis of fMRI data in fourier and wavelet domains

Abstract

The analysis of physiological relationships between multiple brain regions has been spurred by the emergence of functional magnetic resonance imaging (fMRI), which provides high spatial resolution of dynamic processes in the human brain with a time resolution in the order of seconds. Many different conceptual approaches and algorithmic solutions have been proposed for connectivity analysis in the last 10 years or so (see some examples in Bullmore et al 1996, McIntosh 1999, Horwitz 2003, Ramnani et al 2004). These can be broadly sub-divided into analyses of functional connectivity - usually defined as a statistical association between spatially remote time series - or effective connectivity - the causal influence that one time series exerts over another (Espinosa & Gerstein 1988, Friston et al. 1997). While functional connectivity between brain regions is frequently depicted through undirected graphs, effective connectivity networks are more naturally portrayed by directed graphs. In addition to this well-rehearsed distinction, we can also categorise available methods according to the mathematical domain in which they are implemented. While the great majority of methods for both functional and effective connectivity analysis of fMRI data have been implemented in the time domain, considerably fewer methods to date have been implemented in the Fourier domain (Cordes et al. 2001, Sun et al. 2004, Salvador et al. 2005a, Yamashita et al. 2005); and the development of approaches in the wavelet domain has been very recent (Achard et al. 2006). In comparable analyses of brain functional networks based on electromagnetic (EEG or MEG) data, the adoption of tools in the Fourier and wavelet domains has been more widespread (see reviews by Samar et al. 1995 and Koenig et al. 2005). The main motivation for further consideration of Fourier and wavelet methods for functional connectivity analysis of fMRI data is that associations between brain regions may not be equally subtended by all frequencies; rather, some frequency bands may be of special importance in mediating functional connectivity. There is, for example, abundant prior evidence that functional connectivity measured with subjects lying quietly in the scanner at "rest" is subtended predominantly by very low frequencies, ≤0.1Hz, for many pairs of connected regions (Biswal et al. 1995, Lowe et al. 2000, Cordes et al. 2001, Robouts et al. 2003). This presumably reflects at a bivariate or multivariate level of analysis the well-replicated but still incompletely understood phenomenon of low frequency endogenous oscillations in resting fMRI (Maxim et al. 2005, Salvador et al. 2005b), optical imaging (Mayhew et al. 1996) and EEG time series (Leopold & Logothetis 2003). Functional connectivity analysis using tools that naturally support the frequency decomposition of physiological associations between regions may therefore be of interest in "denoising" the analysis (by restricting attention to a frequency band of special relevance); in supporting multimodal analysis of brain connectivity combining fMRI and EEG/MEG data; and in exploring changes in frequency-dependent connectivity related to drug treatments or pathological states. Here we aim simply to provide a technical introduction to methods in the Fourier and wavelet domains that are appropriate to frequency-dependent analysis of functional connectivity in fMRI, and to illustrate them through examples.