Causality Testing, Directed Information and Spike Trains

Directed information has been used in information theory and in statistics as a functional that quantifies causal influences present in signals and empirical data. In this work, the causally conditional directed information (CCDI) rate is identified as a statistic for detecting causal relationships between discrete time series, in the presence of potential confounders. A hypothesis test is introduced for identifying the temporally causal influence of
$(x_{n})$
on
$(y_{\mathrm{Y}})$
, causally conditioned on a possibly confounding third time series
$(z_{n})$
. Under natural assumptions it is shown that the absence of temporally causal influence is equivalent to the CCDI rate being zero. The plug-in estimator for this functional is identified with the log-likelihood ratio test statistic for the desired test. This statistic is shown to be asymptotically normal under the alternative hypothesis and asymptotically
$\chi^{2}$
distributed under the null, facilitating the computation of p-values from empirical data. The resulting hypothesis test is employed in the analysis of spike train data recorded from neurons in the V4 and FEF brain regions of behaving animals during a visual attention task. The test results are seen to identify interesting and biologically relevant information.