A new method for the robust characterisation of pairwise statistical dependency between point processes

The robust detection of statistical dependencies between the components of a complex system is a key step in gaining a network-based understanding of the system. Because of their simplicity and low computation cost, pairwise statistics are commonly used in a variety of fields. Those approaches, however, typically suffer from one or more limitations such as lack of confidence intervals requiring reliance on surrogate data, sensitivity to binning, sparsity of the signals, or short duration of the records. In this paper we develop a method for assessing pairwise dependencies in point processes that overcomes these challenges. Given two point processes $X$ and $Y$ each emitting a given number of events $m$ and $n$ in a fixed period of time $T$, we derive exact analytical expressions for the expected value and standard deviation of the number of pairs events $X_i,Y_j$ separated by a delay of less than $\tau$ one should expect to observe if $X$ and $Y$ were i.i.d. uniform random variables. We prove that this statistic is normally distributed in the limit of large $T$, which enables the definition of a Z-score characterising the likelihood of the observed number of coincident events happening by chance. We numerically confirm the analytical results and show that the property of normality is robust in a wide range of experimental conditions. We then experimentally demonstrate the predictive power of the method using a noisy version of the common shock model. Our results show that our approach has excellent behaviour even in scenarios with low event density and/or when the recordings are short.