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## Spike trains

A spike train is a sequence of N events (for example neuronal spikes, or heart beats) occurring at times . Variations in the events beyond their timing are ignored. Let us first note that this very common kind of data is fundamentally different from the case of unevenly sampled time series studied in the last section in that the sampling instances are not independent of the measured process. In fact, between these instances, the value of s(t) is undefined and the contain all the information there is.

Very often, the discrete sequence if inter-event intervals is treated as if it were an ordinary time series. We must keep in mind, however, that the index n is not proportional to time any more. It depends on the nature of the process if it is more reasonable to look for correlations in time or in event number. Since in the latter case we can use the standard machinery of regularly sampled time series, let us concentrate on the more difficult real time correlations.

> Figure: Heart rate fluctuations seen by plotting the time interval between consecutive heart beats (R waves) versus the beat number. Note that the spread of values is rather small due to the near-periodicity of the heart beat.

> Figure: Binned autocorrelation function of an RR interval time series. Upper panel: and are practically indistinguishable. Lower: Autocorrelation for a random scramble of the data. Note that most of the periodicity is given by the fact that the duration of each beat had rather little variation during this recording.

In particular the literature on heart rate variability (HRV) contains interesting material on the question of spectral estimation and linear modeling of spike trains, here usually inter-beat (RR) interval series, see e.g. Ref. [51]. For the heart beat interval sequence shown in Fig. 16, spectral analysis of versus n may reveal interesting structure, but even the mean periodicity of the heart beat would be lost and serious aliasing effects would have to be faced. A very convenient and powerful approach that uses the real time t rather than the event number n is to write a spike train as a sum of Dirac delta functions located at the spike instances:
With , the periodogram spectral estimator is then simply obtained by squaring the (continuous) Fourier transform of s(t):
Other spectral estimators can be derived by smoothing or by data windowing. It is possible to generate surrogate spike trains that realise the spectral estimator Eq.(28), although this is computationally very cumbersome. Again, we can take advantage of the relative computational ease of binned autocorrelations here. Introducing a normalisation constant , we can write . Then again, the binned autocorrelation function is defined by . Now we carry out both integrals and thus eliminate both delta functions. If we choose such that C(0)=1, we obtain:
Thus, all we have to do is to count all possible intervals in a bin. The upper panel in Fig. 17 shows the binned autocorrelation function with bin size sec up to a lag of 6 sec for the heart beat data shown in Fig. 16. Superimposed is the corresponding curve for a surrogate that has been generated with the deviation from the binned autocorrelations of the data as a cost function. The two curves are practically indistinguishable. In this particular case, most of the structure is given by the mean periodicity of the data. The lower trace of the same figure shows that even a random scramble shows very similar (but not identical) correlations. Information about the main periodicity is already contained in the distribution of inter-beat intervals which is preserved under permutation.

As with unevenly sampled data, the choice of binning and the maximal lag are somewhat delicate and not that much practical experience exists. It is certainly again recommendable to avoid empty bins. The possibility to limit the temporal range of is a powerful instrument to keep computation time within reasonable limits.

Next: Questions of interpretation Up: Various Examples Previous: Uneven sampling