# A Comparison of Euclidean metrics in spike train space

Statistical analysis and inferences on spike trains are one of the central topics in neural coding. However, a fundamental obstacle is that

the space of all spike trains is not an Euclidean space. Over the past few years, two Euclidean-like metrics were independently developed to measure distance in the spike train space. One of these

has been further used in defining summary statistics (i.e. mean and variance)

• We state summary statistic definitions using the GVP metric and propose an efficient algorithm to compute them.

• We compare the two metrics and corresponding summary statistics on basis of theory, properties, and applications.

• We apply both inference frameworks in a neural coding problem for a recording in geniculate ganglion stimulated by different chemical tastes, demonstrating that consistent statistical

inferences can be conducted

We have systematically compared two Euclidean metrics, Elastic and GVP, on spike train space. We have shown the key differences and similarities between the Elastic and GVP metrics arising from the definition and mathematical consequences. Among these results the most relevant

are:

1. The mean spike train under the GVP metric is moreintuitive - the position of spikes in the mean is the average position of matched spikes in the set. Thus, the GVP metric is easy to understand and straightforward to use,

2. The distance and mean under the Elastic metric depends on the size of the time interval of the recording, whereas the GVP metric does not. On the other hand, the GVP distance has stronger dependence on the number of spikes due to the penalty term,

3. The Elastic metric has a more profound mathematical background and is easy to extend to current progress in computational information geometry.

4. Due to Euclidean properties both metrics allow to define basic summary statistics such as mean and variance and Euclidean Embedding. These statistics can be further used in statistical inference problems.

5. The classification score evaluation and computa-

tional costs comparison shows that the summary

statistics of Euclidean metrics not only allow to re-

duce computational cost in simple

classification problems, but also can significantly

improve the classification scores.