Our separation algorithm calculates models for all possible neuron combinations and time shifts. But instead of using a probabilistic description to find the most appropriate model as in [6] we estimated the best fitting model by calculating cross-correlations of models and overlapping spikes.

Figure 4 Separation results for different noise amplitudes and different numbers of signal channels. Each bar shows the standard error of the mean.

The models are computed with help of the mean spikes from each neuron. The mean spikes are shifted over a distinct period against each other and summed up to obtain a linear superposition (see Fig. 2).

The model with the highest correlation will be assigned to the observed overlapping spike. The active neurons (units) and the spike times forming the overlapping spike will then be defined as the units and the spike times which are included in the model which fits best.

The calculation of the number of required models leads to a combinatorial problem that can be described as the number of ways to choose *k* overlapping units out of a set of *n* units. The number of possible combinations *c* can be calculated with the binomial coefficient.

$$c=\sum _{k=2}^{m}\left({}_{k}^{n}\right)$$(1)In Eq. 1 *n* is the number of units included in the signal and *k* is the number of units that can be included in one overlap thus cannot be smaller than two. If really all theoretically possible combinations ought to be considered it must be *m* =*n* with m the maximum number of units that can be contained in an overlap. All combinations of units from two up to m have to be considered, *m* can be chosen smaller than *n*, hence 2 < = *k* < = *m* < = *n*.

In the second step it must be considered that the waveforms forming an overlapping spike can occur within different intervals. Thus the degree of overlap has to be considered (see Fig. 3 (C)).

Since the real spike times are usually not known and spike alignment introduces additional temporal shifts, several time shifts have to be introduced into the model calculation. The number of considered shifts *s* is set to the number of samples within a spike waveform. The number of models *M* is then

$$M=\sum _{k=2}^{m}\left({}_{k}^{n}\right){s}^{k}.$$(2)To find the model which resembles the overlapping spike as good as possible, the cross-correlation is applied to each model and each overlapping spike.

Seeing that the spike alignment, which is applied during a common spike sorting, modifies the position of the waveforms within the extracted window, we incorporate the coefficients from shifts of ±*s*/2 between model and overlapping spike for the selection of the best fitting model (see Fig. 3 (D and E)).

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