A series of event times is a type of point process. A good introduction to measuring correlations between point processes, as applied to neuronal spike trains, is given by Brillinger [1976]. One of the early, seminal works on point processes is that of Cox [1955].
The simplest measure of association between two temporal point processes (let's call them $A$ and $B$) is probably the association number, $n$. To calculate $n$ a window of half-width $h$ is defined around each time in series $A$. The individual association number, $c$, is then the number of events in series $B$ that fall within a given window, and $n$ is then defined as
$$
n(h) = \Sigma_{i=1}^{N} c_i
$$
This is generally calculated for a range of time lags, $u$, such that we get $n(u,h=const)$ [see, e.g., equations 9 and 10, and figure 3, of Brillinger [1976]].
If series $A$ and $B$ are uncorrelated then the association number will fluctuate, as a function of lag, due to sampling variations, but will have a stable mean. If we normalize by $2hT$, where $T$ is the length of the interval from which our samples were drawn, then we get an estimate of the cross-product density. Correlations at different time lags can then be seen by inspecting the cross-product density for departures from 1 (if the processes are independent the cross-product density should be 1, which is expected at large lags for most physical processes).
Assessing the significance of these departures can be addressed in a number of different ways, but many assume that at least one of the processes is Poisson [Brillinger, 1976; Mulargia, 1992]. If those assumptions are met then 95% confidence intervals on the cross-product density can be estimated by [Brillinger, 1976]
$$
1 \pm \frac{1.96}{2\sqrt{}2hTp_Ap_B}
$$
where $p_A$ and $p_B$ are the mean intensities of series $A$ and $B$, given by $p_A = N/T$, where $N$ is the number of events in $A$ (similar for $B$). Excursions outside the C.I. are therefore indicative of a significant association between the event sequences at certain lags.
If neither series is Poisson then a bootstrapping approach can be used to estimate confidence intervals [Morley and Freeman, 2007]. When taking this approach it's important to understand the system as resampling the series $A$ and $B$ may not work without applying, say, a moving block bootstrap to preserve correlations in the spike trains. The approach taken by Morley and Freeman was to instead resample from the individual association numbers.
... we see that n(u, h) is a summation of the N individual associations c$_i$ for given u, h. Using this set of individual associations, we can construct a new series, c*$_i$, by drawing with replacement a random selection of N individual associations. Summing these N randomly-sampled associations gives a bootstrap estimate of the association number for given u, h. Repeating this for every lag u, we construct a bootstrap estimate of the association number with lag n*(u, h). Performing this bootstrapping procedure K times allows us to model the sampling variation in n(u, h).
A further treatment of assessing confidence intervals using bootstrapping techniques is given by Niehof and Morley [2012], but the above should work for two series of neuronal spike trains (or similar simple system).
References:
- Brillinger, D. R. (1976), Measuring the association of point processes: A case history, Am. Math. Mon., 83(1), 16–22.
- Cox, D. R. (1955), Some statistical methods connected with series of events, J. R. Stat. Soc., Ser. B, 17(2), 129–164.
- Mulargia, F. (1992), Time association between series of geophysical events, Phys. Earth Planet. Inter., 72, 147–153.
- Morley, S. K., and M. P. Freeman (2007), On the association between northward turnings of the interplanetary magnetic field and substorm onsets, Geophys. Res. Lett., 34, L08104, doi:10.1029/2006GL028891.
- Niehof, J.T., and S.K. Morley (2012). “Determining the Significance of Associations between Two Series of Discrete Events : Bootstrap Methods”. United States. doi:10.2172/1035497
