A jackknife approach to quantifying single-trial correlation between covariance-based metrics undefined on a single-trial basis

The quantification of covariance between neuronal activities (functional connectivity) requires the observation of correlated changes and therefore multiple observations. The strength of such neuronal correlations may itself undergo moment-by-moment fluctuations, which might e.g. lead to fluctuations in single-trial metrics such as reaction time (RT), or may co-fluctuate with the correlation betwe'en activity in other brain areas. Yet, quantifying the relation between moment-by-moment co-fluctuations in neuronal correlations is precluded by the fact that neuronal correlations are not defined per single observation. The proposed solution quantifies this relation by first calculating neuronal correlations for all leave-one-out subsamples (i.e. the jackknife replications of all observations) and then correlating these values. Because the correlation is calculated between jackknife replications, we address this approach as jackknife correlation (JC). First, we demonstrate the equivalence of JC to conventional correlation for simulated paired data that are defined per observation and therefore allow the calculation of conventional correlation. While the JC recovers the conventional correlation precisely, alternative approaches, like sorting-and-binning, result in detrimental effects of the analysis parameters. We then explore the case of relating two spectral correlation metrics, like coherence, that require multiple observation epochs, where the only viable alternative analysis approaches are based on some form of epoch subdivision, which results in reduced spectral resolution and poor spectral estimators. We show that JC outperforms these approaches, particularly for short epoch lengths, without sacrificing any spectral resolution. Finally, we note that the JC can be applied to relate fluctuations in any smooth metric that is not defined on single observations.