Estimating Transfer Entropy in Continuous Time Between Neural Spike Trains or Other Event-Based Data

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.