Approaches to the Detection of Direct Directed Interactions in Neuronal Networks
In this chapter, we address the challenge of detecting interactions among neuronal processes by means of bivariate and multivariate linear analysis techniques. For linear systems, both undirected and directed measures exist. Coherence is a commonly used undirected bivariate measure to detect the interaction between two nodes of a network, while multivariate measures like the partial coherence distinguish direct and indirect connections. The partial directed coherence additionally features the direction influences between nodes. We introduce the theoretical framework of these analysis techniques, discuss their estimation, and present their application to simulated and real-world data.
KeywordsCoupling Scheme Multivariate Time Series Multivariate System Coherence Analysis Nonlinear Stochastic System
Special thanks goes to Bernhard Hellwig, Florian Amtage, and Professor Carl Hermann Lücking who provided us not only with the tremor data but also with knowledge about the neurophysiology of Parkinsonian tremor. This work was supported by the German Science Foundation (Ti315/2-1) and by the German Federal Ministry of Education and Research (BMBF grant 01GQ0420).
- P. Bloomfield. Fourier Analysis of Time Series: An Introduction. John Wiley & Sons, New York, 1976.Google Scholar
- D. R. Brillinger. Time Series: Data Analysis and Theory. Holden-Day, San Francisco, 1981.Google Scholar
- P. J. Brockwell and R. A. Davis. Time Series: Theory and Methods. Springer, New York, 1998.Google Scholar
- R. Dahlhaus and M. Eichler. Causality and graphical models for time series. In P. Green, N. Hjort, and S. Richardson, editors, Highly Structured Stochastic Systems, pp. 115–137. Oxford University Press, Oxford, 2003.Google Scholar
- M. Eichler. Graphical modeling of dynamic relationships in multivariate time series. In B. Schelter, M. Winterhalder, and J. Timmer, editors, Handbook of Time Series Analysis, Chapter 14, pp. 335–372. Wiley-Vch, Weinheim, 2006.Google Scholar
- E. L. Lehmann. Testing Statistical Hypotheses. Springer, New York, 1997.Google Scholar
- M. B. Priestley. Spectral Analysis and Time Series. Academic Press, London, 1989.Google Scholar
- B. Schelter, M. Winterhalder, R. Dahlhaus, J. Kurths, and J. Timmer. Partial phase synchronization for multivariate synchronizing systems. Phys. Rev. Lett., 96:208103, 2006a.Google Scholar
- J. Timmer, M. Lauk, S. Häuβler, V. Radt, B. Köster, B. Hellwig, B. Guschlbauer, C. H. Lücking, M. Eichler, and G. Deuschl. Cross-spectral analysis of tremor time series. Int. J. Bif. Chaos, 10:2595–2610, 2000.Google Scholar