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

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 in which we are interested in information flows come in the form of (near) instantaneous events occurring over time, including the spiking of biological neurons, trades on stock markets and posts to social media. 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 discrete-time estimator on synthetic examples. We also develop a local permutation scheme for generating null surrogate time series 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 -- signifying the lack of a causal connection under certain weak assumptions. Our approach is capable of detecting conditional independence or otherwise even in the presence of strong pairwise time-directed correlations. The power of this approach is further demonstrated on the inference of the connectivity of biophysical models of a spiking neural circuit inspired by the pyloric circuit of the crustacean stomatogastric ganglion, succeeding where previous related estimators have failed. AUTHOR SUMMARYTransfer 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 also 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. The currently-used technique for estimating TE on neural spike trains first time-discretises the data and then applies a straightforward or "plug-in" information-theoretic estimation procedure. This approach has numerous drawbacks: it is very biased, it cannot capture relationships occurring on both fine and large timescales simultaneously, converges very slowly as more data is obtained, and indeed does not even converge to the correct value. We present a new estimator for TE which operates in continuous time, demonstrating via application to synthetic examples that it addresses these problems, and can reliably differentiate statistically significant flows from (conditionally) independent spike trains. Further, we also apply it to more biologically-realistic spike trains obtained from a biophysical model of the pyloric circuit of the crustacean stomatogastric ganglion; our correct inference of the underlying connection structure here provides an important validation for our approach where similar methods have previously failed