title: Assessing the strength of directed influences among neural signals: an approach to noisy data.
abstract: BACKGROUND
  Measurements in the neurosciences are afflicted with observational noise. Granger-causality inference typically does not take this effect into account. We demonstrate that this leads to false positives conclusions and spurious causalities.
  
  
  METHODS
  State space modelling provides a convenient framework to obtain reliable estimates for Granger-causality. Despite its previous application in several studies, the analytical derivation of the statistics for parameter estimation in the state space model was missing. This prevented a rigorous evaluation of the results.
  
  
  RESULTS
  In this manuscript we derive the statistics for parameter estimation in the state space model. We demonstrate in an extensive simulation study that our novel approach outperforms standard approaches and avoids false positive conclusions about Granger-causality.
  
  
  METHODS
  In comparison with the naive application of Granger-causality inference, we demonstrate the superiority of our novel approach. The wide-spread applicability of our procedure provides a statistical framework for future studies. The application to mice electroencephalogram data demonstrates the immediate applicability of our approach.
  
  
  CONCLUSIONS
  The analytical derivation of the statistics presented in this manuscript enables a rigorous evaluation of the results of Granger causal network inference. It is noteworthy that the statistics can be readily applied to various measures for Granger causality and other approaches that are based on vector autoregressive models.
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, King's College, Old Aberdeen AB24 3UE, United Kingdom; Institute for Pure and Applied Mathematics, University of Aberdeen, King's College, Old Aberdeen AB24 3UE, United Kingdom; Freiburg Center for Data Analysis and Modeling (FDM), University of Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany; Institute for Physics, University of Freiburg, Hermann-Herder-Strasse 3a, 79104 Freiburg, Germany; Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, Albertstrasse 19, 79104 Freiburg, Germany. Electronic address: l.sommerlade@abdn.ac.uk.
authors.forename: Linda
authors.initials: L
authors.lastname: Sommerlade
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, King's College, Old Aberdeen AB24 3UE, United Kingdom; Institute for Pure and Applied Mathematics, University of Aberdeen, King's College, Old Aberdeen AB24 3UE, United Kingdom. Electronic address: m.thiel@abdn.ac.uk.
authors.forename: Marco
authors.initials: M
authors.lastname: Thiel
authors.affiliation: Freiburg Center for Data Analysis and Modeling (FDM), University of Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany; Institute for Physics, University of Freiburg, Hermann-Herder-Strasse 3a, 79104 Freiburg, Germany; Department of Neuropediatrics and Muscular Disease, University Medical Center of Freiburg, Mathildenstrasse 1, 79106 Freiburg, Germany. Electronic address: Malenka.Mader@fdm.uni-freiburg.de.
authors.forename: Malenka
authors.initials: M
authors.lastname: Mader
authors.affiliation: Freiburg Center for Data Analysis and Modeling (FDM), University of Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany; Institute for Physics, University of Freiburg, Hermann-Herder-Strasse 3a, 79104 Freiburg, Germany. Electronic address: Wolfgang.Mader@fdm.uni-freiburg.de.
authors.forename: Wolfgang
authors.initials: W
authors.lastname: Mader
authors.affiliation: Freiburg Center for Data Analysis and Modeling (FDM), University of Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany; Institute for Physics, University of Freiburg, Hermann-Herder-Strasse 3a, 79104 Freiburg, Germany; BIOSS Centre for Biological Signalling Studies, University of Freiburg, Schänzlestr. 18, 79104 Freiburg, Germany. Electronic address: JeTi@fdm.uni-freiburg.de.
authors.forename: Jens
authors.initials: J
authors.lastname: Timmer
authors.affiliation: Institute of Medical Sciences, University of Aberdeen, Foresterhill, Aberdeen AB25 2ZD, United Kingdom. Electronic address: b.platt@abdn.ac.uk.
authors.forename: Bettina
authors.initials: B
authors.lastname: Platt
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, King's College, Old Aberdeen AB24 3UE, United Kingdom; Institute for Pure and Applied Mathematics, University of Aberdeen, King's College, Old Aberdeen AB24 3UE, United Kingdom; Freiburg Center for Data Analysis and Modeling (FDM), University of Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany. Electronic address: b.schelter@abdn.ac.uk.
authors.forename: Björn
authors.initials: B
authors.lastname: Schelter
mesh_terms: D000465:Algorithms; D000818:Animals; D058256:Brain Waves / Q000502:physiology; D003198:Computer Simulation; D004569:Electroencephalography; D006801:Humans; D051379:Mice; D008959:Models, Neurological; D015233:Models, Statistical; D009415:Nerve Net / Q000502:physiology*; D009474:Neurons / Q000502:physiology*
keywords: Analytical covariance matrix; Expectation-maximisation algorithm; Granger-causality; Incomplete data likelihood; Kalman filter; Observational noise; Statistics