The Kendall correlation was first introduced by Maurice Kendall in 1938 [1]. As a rank correlation, it takes into account the specific ordering of the elements of the sets it is correlating. A Kendall correlation, τ, equal to 1 is interpreted as the elements in the two sets being ordered in the same way. τ = −1 is interpreted as the elements in the two sets being ordered exactly oppositely. And τ = 0 is interpreted as the ordering of the two sets having no relation to one another.

Despite being used in a number of other scientific fields [2–4], it is only recently that the Kendall correlation has started to become appreciated, and implemented, in neuroscience. In particular, due to the usual sparseness of spike trains (i.e. the large number of zeros), the Kendall correlation has been shown to be particularly appropriate for computing pairwise correlations between spike trains, especially as compared to Pearson’s correlation [5–7]. Recently, it was used to explore the place field structure of place cells in the hippocampus [7], and generally pairwise correlations can be useful for revealing aspects of the behavior of the recorded, or constructed (in the case of computational/theoretical studies), networks. We note that for the remainder of the paper, by spike train we mean specifically a vector of length n whose i^th element is a 1 if the corresponding neuron fired at least once during the i^th time bin of the recorded interval and 0 otherwise. This is a frequently used way to talk about spike trains and is appropriate if firing is particularly sparse or if the time bin size is sufficiently small.

A simple, non-optimized, way of computing the Kendall correlation of two row vectors, X and Y, is MATLAB’s function, corr(X, Y, ‘Type’, ‘Kendall’). On MATLAB’s website [8], they define the Kendall correlation as τ = 2 K n ( n - 1 ) (1) where K = ∑ i = 1 n - 1 ∑ j = i + 1 n ξ * ( X i , X j , Y i , Y j ), and ξ * ( X i , X j , Y i , Y j ) = { 1 ( X i - X j ) ( Y i - Y j ) > 0 0 ( X i - X j ) ( Y i - Y j ) = 0 - 1 ( X i - X j ) ( Y i - Y j ) < 0 (2)

However, as additionally stated, MATLAB’s function also has a normalization constant in the calculation of τ that adjusts for ties [8]. A Kendall correlation that takes this additional consideration into account is often referred to as τ_b in the literature [9]. Therefore, the true way in which MATLAB calculates the Kendall correlation of the row vectors X and Y is τ = K ( n 0 - n 1 ) ( n 0 - n 2 ) (3) where n₀ = n(n − 1)/2, n₁ = ∑_i t_i(t_i − 1)/2, and n₂ = ∑_j u_j(u_j − 1)/2. The sums of n₁ and n₂ are over all the distinct values X and Y take (respectively), and t_i is the number of elements in X equal to the i^th distinct value of X (u_j is the same, but for Y).

As can be seen from the definition of K, calculating τ requires summing over many of the pairs of values in X and Y (in fact, n(n − 1)/2 pairs, which means that the run time is O(n²)). For large spike trains, this results in a large computation time. For this reason, a faster, O(n ln n) method was developed [10], which makes use of the existence of a mapping between sorting and Kendall correlation. Additional work has been done using sorting and balanced tree structures in cutting edge ways to decrease the run time of other O(n ln n) methods [11]. While these methods—we will below consider specifically Knight’s method [10]—have great power because they are valid for arbitrary vectors, like the O(n²) method implemented by MATLAB, the generality is unnecessary for computing the Kendall correlation of spike trains. Below, we specifically take the inherent structure of spike trains (that is, that their elements take values only from {0, 1}) under consideration to derive a faster method of calculating Kendall correlations specific to spike trains. We show that our new method is O(n) and then examine how much faster our method is than Knight’s method under various conditions.

As mentioned above, the motivating idea for the following method is that, since spike trains take values only in {0, 1}, by taking this fact under consideration, we might be able to speed up the calculation of the Kendall correlation. In particular, we show that we can write an explicit formula for K (from Eq (1)) that can be evaluated very quickly—in fact, in O(n).

Considering Eq (2), we see that there are two principle cases we need to consider when calculating K: the case where X_i and X_j are in the same order as Y_i and Y_j (i.e. where ξ*(X_i, X_j, Y_i, Y_j) = 1), and the case where they are in the opposite order (i.e. where ξ*(X_i, X_j, Y_i, Y_j) = −1). The third case, ξ*(X_i, X_j, Y_i, Y_j) = 0, obviously doesn’t contribute to the value of K. We now consider these two cases separately.

The results of comparing our method to Knight’s and MATLAB’s methods, are shown in Fig 1. Unsurprisingly, both our method and Knight’s method show considerable advantage over the O(n²) method that is implemented by MATLAB [8] (Fig 1a). However, our method is definitively faster. Importantly, this holds true for a range of sparseness values (Fig 1b), although our method shows a slight slowing down for larger sparseness values, while Knight’s method does not. Our method is on average ≈ 35 times faster for a sparseness of 25% and ≈ 60 times faster for a sparseness of 1%. Because a sparseness of 25%, the maximum we tested, is unrealistic for any neural simulation or recording, our method is faster than Knight’s in a neurally plausible regime.

Finally, for all the correlations between spike trains we computed, we checked that the two Kendall correlation values were within 10⁻¹² of MATLAB’s Kendall correlation function (see Table 1). Therefore, we feel confident that our method is correct and equivalent (up to machine error) to MATLAB’s method.

We have presented a novel method to calculate Kendall correlations of large spike trains, and have demonstrated its advantage (in terms of computation time) to the standard for fast Kendall correlation computation [10]. We achieved this by specifically taking the structure of spike trains (the fact that they are made up of 1s and 0s) into consideration, and deriving explicit formulas for the components of the Kendall correlation (Eqs (9), (12) and (13)). These formulas are all linear in n, meaning our method is O(n), unlike Knight’s method which is O (n ln n). We have also, by way of computation, provided evidence that our method is correct and equivalent (up to machine error) to MATLAB’s standard method.

With a significantly faster method to compute the Kendall correlation between large spike trains, we hope that the Kendall correlation will become a more accessible tool for neuroscience. While we know there are faster ways to implement algorithms similar to Knight’s (as was explored in [11]) that may be faster than the method provided here, the simplicity of our method (a few linear equations) makes it much more appealing to neuroscientists who have limited technical knowledge and/or interest in computer science. We imagine it will be especially useful in computational/theoretical studies where large, sparse spike trains are frequently generated and whose pairwise correlations provide insight into the complex properties of the network. We hope that the fact that pairwise correlations over significantly longer time intervals (or equivalently, between spikes trains of longer lengths) can now be calculated quickly, more in-depth analysis of generated networks (in addition to analysis of observed/recorded networks) will be achieved.

Finally, we hope that our results make clear the usefulness of considering specifically the structure of spike trains when calculating certain quantities. We’re sure many other measures can be significantly sped up when taking this into consideration.