# Higher-order Spike Synchrony

My recent visit to Sonja Grün’s lab made me thinking about statistical measures that would detect higher-order mutual interactions among neurons that would not be explained by lower-order interaction. If there are neural assemblies that work together, the detection of such mutually dependent neurons would be a key evidence. For synchrony (the simplest type of dependence/interaction between spiking neurons) the difference between pairwise and triple synchrony can be illustrated by the following spike trains:

Spike train pairs(A, B) and (B, C) are synchronous as they have synchronized action potential timings (red and blue, respectively). However, A, B, C are not synchronous all together in this example. Hence, there’s a need for capturing this higher-order synchrony structure.

Even when the time window of analysis is small enough, such that in a single trial basis, synchrony in pairs (A,B) and (B,C) implies synchrony in (A,C) because the jitter allowed for synchrony detection is in the order of window size, pairwise synchrony does NOT imply higher-order synchrony because in some trials (A,B) may be synchronous and some other trials (B,C) may be synchronous.

Detection of mutual interaction of multiple neurons is a difficult problem, and we know pairwise analysis tools can only check necessary conditions but not sufficient. (Simple example: given 3 random variables, X, Y, Z, pairwise independence does not imply mutual independence, since P(X,Y,Z) = P(X)P(Y)P(Z) may not hold.) Nevertheless most analysis are done with pairwise analysis these days. I was impressed by the work by Sonja’s group on CuBIC method to detect the presence of these higher order synchrony [1]. The main idea is that amplitude distribution of the population rate, they call ‘*complexity distribution*‘, has a heavier tail when the higher order synchrony is present. The idea does not require a combinatorial search, and can be directly applied without spike sorting. The disadvantages are that the assumptions may not hold in real data, and only the presence is detected and the actual interaction is not directly captured.

References

- Benjamin Staude, Stefan Rotter, Sonja Grün. CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains. Journal of Computational Neuroscience (28 October 2009) dx.doi.org/10.1007/s10827-009-0195-x

Very nice post! You always know how to present advanced topics in an understandable way.