Abstract
Transfer entropy (TE) is a widely used measure of directed information flows in a number of domains including neuroscience. Many real-world time series for which we are interested in information flows come in the form of (near) instantaneous events occurring over time. Examples include the spiking of biological neurons, trades on stock markets and posts to social media, amongst myriad other systems involving events in continuous time throughout the natural and social sciences. However, there exist severe limitations to the current approach to TE estimation on such event-based data via discretising the time series into time bins: it is not consistent, has high bias, converges slowly and cannot simultaneously capture relationships that occur with very fine time precision as well as those that occur over long time intervals. Building on recent work which derived a theoretical framework for TE in continuous time, we present an estimation framework for TE on event-based data and develop a k-nearest-neighbours estimator within this framework. This estimator is provably consistent, has favourable bias properties and converges orders of magnitude more quickly than the current state-of-the-art in discrete-time estimation on synthetic examples. We demonstrate failures of the traditionally-used source-time-shift method for null surrogate generation. In order to overcome these failures, we develop a local permutation scheme for generating surrogate time series conforming to the appropriate null hypothesis in order to test for the statistical significance of the TE and, as such, test for the conditional independence between the history of one point process and the updates of another. Our approach is shown to be capable of correctly rejecting or accepting the null hypothesis of conditional independence even in the presence of strong pairwise time-directed correlations. This capacity to accurately test for conditional independence is further demonstrated on models of a spiking neural circuit inspired by the pyloric circuit of the crustacean stomatogastric ganglion, succeeding where previous related estimators have failed.
Highlights
In analysing time series data from complex dynamical systems, such as in neuroscience, it is often useful to have a notion of information flow
Transfer Entropy (TE) is an information-theoretic measure commonly used in neuroscience to measure the directed statistical dependence between a source and a target time series, possibly conditioned on other processes
Along with measuring information flows, it is used for the inference of directed functional and effective networks from time series data
Summary
In analysing time series data from complex dynamical systems, such as in neuroscience, it is often useful to have a notion of information flow. We intuitively describe the activities of brains in terms of such information flows: for instance, information from the visual world must flow to the visual cortex where it will be encoded [1]. Information coded in the motor cortex must flow to muscles where it will be enacted [2]. Transfer entropy (TE) [3, 4] has become a widely accepted measure of such flows. It is defined as the mutual information between the past of a source time-series process and the present state of a target process, conditioned on the past of the target. (in discrete time), the transfer entropy rate [5] is: T_ Y!X
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