Network inference algorithms are valuable tools for the study of large-scale neuroimaging datasets. Multivariate transfer entropy is well suited for this task, being a model-free measure that captures nonlinear and lagged dependencies between time series to infer a minimal directed network model. Greedy algorithms have been proposed to efficiently deal with high-dimensional datasets while avoiding redundant inferences and capturing synergistic effects. However, multiple statistical comparisons may inflate the false positive rate and are computationally demanding, which limited the size of previous validation studies. The algorithm we present—as implemented in the IDTxl open-source software—addresses these challenges by employing hierarchical statistical tests to control the family-wise error rate and to allow for efficient parallelization. The method was validated on synthetic datasets involving random networks of increasing size (up to 100 nodes), for both linear and nonlinear dynamics. The performance increased with the length of the time series, reaching consistently high precision, recall, and specificity (>98% on average) for 10,000 time samples. Varying the statistical significance threshold showed a more favorable precision-recall trade-off for longer time series. Both the network size and the sample size are one order of magnitude larger than previously demonstrated, showing feasibility for typical EEG and magnetoencephalography experiments.

We present IDTxl (the Information Dynamics Toolkit xl), a new open source Python toolbox for effective network inference from multivariate time series using information theory, available from GitHub (https://github.com/pwollstadt/IDTxl).Information theory (Cover & Thomas, 2006;MacKay, 2003;Shannon, 1948) is the mathematical theory of information and its transmission over communication channels. Information theory provides quantitative measures of the information content of a single random variable (entropy) and of the information shared between two variables (mutual information). The defined measures build on probability theory and solely depend on the probability distributions of the variables involved. As a consequence, the dependence between two variables can be quantified as the information shared between them, without the need to explicitly model a specific type of dependence. Hence, mutual information is a model-free measure of dependence, which makes it a popular choice for the analysis of systems other than communication channels.

Information theory allows us to investigate information processing in neural systems in terms of information transfer, storage and modification. Especially the measure of information transfer, transfer entropy, has seen a dramatic surge of interest in neuroscience. Estimating transfer entropy from two processes requires the observation of multiple realizations of these processes to estimate associated probability density functions. To obtain these necessary observations, available estimators typically assume stationarity of processes to allow pooling of observations over time. This assumption however, is a major obstacle to the application of these estimators in neuroscience as observed processes are often non-stationary. As a solution, Gomez-Herrero and colleagues theoretically showed that the stationarity assumption may be avoided by estimating transfer entropy from an ensemble of realizations. Such an ensemble of realizations is often readily available in neuroscience experiments in the form of experimental trials. Thus, in this work we combine the ensemble method with a recently proposed transfer entropy estimator to make transfer entropy estimation applicable to non-stationary time series. We present an efficient implementation of the approach that is suitable for the increased computational demand of the ensemble method's practical application. In particular, we use a massively parallel implementation for a graphics processing unit to handle the computationally most heavy aspects of the ensemble method for transfer entropy estimation. We test the performance and robustness of our implementation on data from numerical simulations of stochastic processes. We also demonstrate the applicability of the ensemble method to magnetoencephalographic data. While we mainly evaluate the proposed method for neuroscience data, we expect it to be applicable in a variety of fields that are concerned with the analysis of information transfer in complex biological, social, and artificial systems.

Information processing performed by any system can be conceptually decomposed into the transfer, storage and modification of information-an idea dating all the way back to the work of Alan Turing. However, formal information theoretic definitions until very recently were only available for information transfer and storage, not for modification. This has changed with the extension of Shannon information theory via the decomposition of the mutual information between inputs to and the output of a process into unique, shared and synergistic contributions from the inputs, called a partial information decomposition (PID). The synergistic contribution in particular has been identified as the basis for a definition of information modification. We here review the requirements for a functional definition of information modification in neuroscience, and apply a recently proposed measure of information modification to investigate the developmental trajectory of information modification in a culture of neurons vitro, using partial information decomposition. We found that modification rose with maturation, but ultimately collapsed when redundant information among neurons took over. This indicates that this particular developing neural system initially developed intricate processing capabilities, but ultimately displayed information processing that was highly similar across neurons, possibly due to a lack of external inputs. We close by pointing out the enormous promise PID and the analysis of information modification hold for the understanding of neural systems.

The disruption of coupling between brain areas has been suggested as the mechanism underlying loss of consciousness in anesthesia. This hypothesis has been tested previously by measuring the information transfer between brain areas, and by taking reduced information transfer as a proxy for decoupling. Yet, information transfer is a function of the amount of information available in the information source—such that transfer decreases even for unchanged coupling when less source information is available. Therefore, we reconsidered past interpretations of reduced information transfer as a sign of decoupling, and asked whether impaired local information processing leads to a loss of information transfer. An important prediction of this alternative hypothesis is that changes in locally available information (signal entropy) should be at least as pronounced as changes in information transfer. We tested this prediction by recording local field potentials in two ferrets after administration of isoflurane in concentrations of 0.0%, 0.5%, and 1.0%. We found strong decreases in the source entropy under isoflurane in area V1 and the prefrontal cortex (PFC)—as predicted by our alternative hypothesis. The decrease in source entropy was stronger in PFC compared to V1. Information transfer between V1 and PFC was reduced bidirectionally, but with a stronger decrease from PFC to V1. This links the stronger decrease in information transfer to the stronger decrease in source entropy—suggesting reduced source entropy reduces information transfer. This conclusion fits the observation that the synaptic targets of isoflurane are located in local cortical circuits rather than on the synapses formed by interareal axonal projections. Thus, changes in information transfer under isoflurane seem to be a consequence of changes in local processing more than of decoupling between brain areas. We suggest that source entropy changes must be considered whenever interpreting changes in information transfer as decoupling.