Waveform drift

Spike sorting

Let \(X\) be a \(M\times N\) multi-channel timeseries array representing the signal acquired during a recording session. Here \(M\) is the number of channels and \(N\) is the number of timepoints in the recording. A spike sorting algorithm applied to this data will yield a sequence of firing events:

\[(t_1,k_1),\dots,(t_L,k_L)\]

where \(t_j\) is the timepoint of the \(j^{th}\) event and \(k_j\) is the corresponding neuron label

\[k_j \in {1,\dots,K}.\]

Ideally, assuming that each firing event produces a signal spike whose shape depends only on neuron label, we would have

\[X(m,t)=\sum_{j=1}^{L} W_{k_j}(m,t-t_j) + \eta(m,t)\]

where \(W_k\) is the waveform corresponding to the \(k^{th}\) neuron, and \(\eta\) is noise. We can estimate the average spike waveforms, or templates, from the output of spike sorting as follows:

\[W_k(m,t)=\sum_{j,k_j=k}{X(m,t_j+t)}\]

Drift

In practice the spike waveform for a given neuron will change slightly over time. This is known as drift. Our signal model becomes more complicated:

\[X(m,t)=\sum_{j=1}^{L} W_{k_j,t_j}(m,t-t_j) + \eta(m,t)\]

where the waveform \(W_{k,t}\) for the \(k^{th}\) neuron now has a time dependency.

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