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TRENTOOL : an open source toolbox to estimate neural directed interactions with transfer entropy
(2011)
To investigate directed interactions in neural networks we often use Norbert Wiener's famous definition of observational causality. Wiener’s definition states that an improvement of the prediction of the future of a time series X from its own past by the incorporation of information from the past of a second time series Y is seen as an indication of a causal interaction from Y to X. Early implementations of Wiener's principle – such as Granger causality – modelled interacting systems by linear autoregressive processes and the interactions themselves were also assumed to be linear. However, in complex systems – such as the brain – nonlinear behaviour of its parts and nonlinear interactions between them have to be expected. In fact nonlinear power-to-power or phase-to-power interactions between frequencies are reported frequently. To cover all types of non-linear interactions in the brain, and thereby to fully chart the neural networks of interest, it is useful to implement Wiener's principle in a way that is free of a model of the interaction [1]. Indeed, it is possible to reformulate Wiener's principle based on information theoretic quantities to obtain the desired model-freeness. The resulting measure was originally formulated by Schreiber [2] and termed transfer entropy (TE). Shortly after its publication transfer entropy found applications to neurophysiological data. With the introduction of new, data efficient estimators (e.g. [3]) TE has experienced a rapid surge of interest (e.g. [4]). Applications of TE in neuroscience range from recordings in cultured neuronal populations to functional magnetic resonanace imaging (fMRI) signals. Despite widespread interest in TE, no publicly available toolbox exists that guides the user through the difficulties of this powerful technique. TRENTOOL (the TRansfer ENtropy TOOLbox) fills this gap for the neurosciences by bundling data efficient estimation algorithms with the necessary parameter estimation routines and nonparametric statistical testing procedures for comparison to surrogate data or between experimental conditions. TRENTOOL is an open source MATLAB toolbox based on the Fieldtrip data format. ...
Background: Transfer entropy (TE) is a measure for the detection of directed interactions. Transfer entropy is an information theoretic implementation of Wiener's principle of observational causality. It offers an approach to the detection of neuronal interactions that is free of an explicit model of the interactions. Hence, it offers the power to analyze linear and nonlinear interactions alike. This allows for example the comprehensive analysis of directed interactions in neural networks at various levels of description. Here we present the open-source MATLAB toolbox TRENTOOL that allows the user to handle the considerable complexity of this measure and to validate the obtained results using non-parametrical statistical testing. We demonstrate the use of the toolbox and the performance of the algorithm on simulated data with nonlinear (quadratic) coupling and on local field potentials (LFP) recorded from the retina and the optic tectum of the turtle (Pseudemys scripta elegans) where a neuronal one-way connection is likely present.
Results: In simulated data TE detected information flow in the simulated direction reliably with false positives not exceeding the rates expected under the null hypothesis. In the LFP data we found directed interactions from the retina to the tectum, despite the complicated signal transformations between these stages. No false positive interactions in the reverse directions were detected.
Conclusions: TRENTOOL is an implementation of transfer entropy and mutual information analysis that aims to support the user in the application of this information theoretic measure. TRENTOOL is implemented as a MATLAB toolbox and available under an open source license (GPL v3). For the use with neural data TRENTOOL seamlessly integrates with the popular FieldTrip toolbox.
Understanding causal relationships, or effective connectivity, between parts of the brain is of utmost importance because a large part of the brain’s activity is thought to be internally generated and, hence, quantifying stimulus response relationships alone does not fully describe brain dynamics. Past efforts to determine effective connectivity mostly relied on model based approaches such as Granger causality or dynamic causal modeling. Transfer entropy (TE) is an alternative measure of effective connectivity based on information theory. TE does not require a model of the interaction and is inherently non-linear. We investigated the applicability of TE as a metric in a test for effective connectivity to electrophysiological data based on simulations and magnetoencephalography (MEG) recordings in a simple motor task. In particular, we demonstrate that TE improved the detectability of effective connectivity for non-linear interactions, and for sensor level MEG signals where linear methods are hampered by signal-cross-talk due to volume conduction.
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.