Uncovering Interactions in the Frequency Domain

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Abstract
Oscillatory activity plays a critical role in regulating biological processes at levels ranging from subcellular, cellular, and network to the whole organism, and often involves a large number of interacting elements. We shed light on this issue by introducing a novel approach called partial Granger causality to reliably reveal interaction patterns in multivariate data with exogenous inputs and latent variables in the frequency domain. The method is extensively tested with toy models, and successfully applied to experimental datasets, including (1) gene microarray data of HeLa cell cycle; (2) in vivo multi-electrode array (MEA) local field potentials (LFPs) recorded from the inferotemporal cortex of a sheep; and (3) in vivo LFPs recorded from distributed sites in the right hemisphere of a macaque monkey. When predicting the structure of a network (a gene network, a protein network, a metabolic network or a neuronal network) based upon simultaneously recorded multi-variable temporal data, a major tool is either the Bayesian network or the Granger causality. We focused on the Granger causality, and it has become increasingly important in recent years because of the huge body of temporal data available in, for example, molecular biology (microarray gene data) and physiology (multi-electrode array recordings of multi-neurons). However, all methods of estimating the Granger causality tend to ignore latent variables, which are ubiquitous in experimental data. Here, we have developed a method that can eliminate the influence of latent variables in predicting the network structure. The method is then extended to the frequency domain. The ability of the method to eliminate the influence of latent variables is extensively verified in toy models and then applied to a gene circuit, a neuronal network, and a network of brain areas. Both in the time and frequency domains, our approach can be used to detect a network structure when multi-dimensional temporal data are available.