Measuring information-transfer delays

PLoS One. 2013;8(2):e55809. doi: 10.1371/journal.pone.0055809. Epub 2013 Feb 28.


In complex networks such as gene networks, traffic systems or brain circuits it is important to understand how long it takes for the different parts of the network to effectively influence one another. In the brain, for example, axonal delays between brain areas can amount to several tens of milliseconds, adding an intrinsic component to any timing-based processing of information. Inferring neural interaction delays is thus needed to interpret the information transfer revealed by any analysis of directed interactions across brain structures. However, a robust estimation of interaction delays from neural activity faces several challenges if modeling assumptions on interaction mechanisms are wrong or cannot be made. Here, we propose a robust estimator for neuronal interaction delays rooted in an information-theoretic framework, which allows a model-free exploration of interactions. In particular, we extend transfer entropy to account for delayed source-target interactions, while crucially retaining the conditioning on the embedded target state at the immediately previous time step. We prove that this particular extension is indeed guaranteed to identify interaction delays between two coupled systems and is the only relevant option in keeping with Wiener's principle of causality. We demonstrate the performance of our approach in detecting interaction delays on finite data by numerical simulations of stochastic and deterministic processes, as well as on local field potential recordings. We also show the ability of the extended transfer entropy to detect the presence of multiple delays, as well as feedback loops. While evaluated on neuroscience data, we expect the estimator to be useful in other fields dealing with network dynamics.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Algorithms
  • Animals
  • Brain / physiology
  • Computer Simulation
  • Electroretinography
  • Humans
  • Information Theory*
  • Models, Neurological*
  • Neural Conduction / physiology*
  • Neurons / physiology*
  • Retina / physiology
  • Synaptic Transmission / physiology*
  • Turtles

Grant support

MW, RV, and VP received financial support from LOEWE Grant “Neuronale Koordination Forschungsschwerpunkt Frankfurt(NeFF).” VP received financial support from the Max Planck Society. MW thanks the Max Planck Institute for Mathematics in the Sciences for funding a visit which contributed to this work. JL thanks the LOEWE Grant “Neuronale Koordination Forschungsschwerpunkt Frankfurt(NeFF)” for funding a visit which contributed to this work. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.