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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.
Network or graph theory has become a popular tool to represent and analyze large-scale interaction patterns in the brain. To derive a functional network representation from experimentally recorded neural time series one has to identify the structure of the interactions between these time series. In neuroscience, this is often done by pairwise bivariate analysis because a fully multivariate treatment is typically not possible due to limited data and excessive computational cost. Furthermore, a true multivariate analysis would consist of the analysis of the combined effects, including information theoretic synergies and redundancies, of all possible subsets of network components. Since the number of these subsets is the power set of the network components, this leads to a combinatorial explosion (i.e. a problem that is computationally intractable). In contrast, a pairwise bivariate analysis of interactions is typically feasible but introduces the possibility of false detection of spurious interactions between network components, especially due to cascade and common drive effects. These spurious connections in a network representation may introduce a bias to subsequently computed graph theoretical measures (e.g. clustering coefficient or centrality) as these measures depend on the reliability of the graph representation from which they are computed. Strictly speaking, graph theoretical measures are meaningful only if the underlying graph structure can be guaranteed to consist of one type of connections only, i.e. connections in the graph are guaranteed to be non-spurious. ...