L-ratio (l_ratio)

Calculation

This description assumes a tetrode is used as an example.

L-ratio uses 4 principal components (PCs) for each tetrode channel (the first being energy, the square root of the sum of squares of each sample in the waveform, followed by the first 3 PCs of the energy normalised waveform). This yields spikes which are each represented as a point in 16 dimensional space.

Define, for each cluster \(C\), \(D_{i,C}^2\), the squared Mahalanobis distance from the centre of cluster \(C\) for every spike \(i\) in the dataset (similarly to the calculation for isolation distance above). Assume that spikes in the cluster distribute normally in each dimension, so that \(D^2\) for spikes in a cluster will distribute as \(\chi^2\) with 16 degrees of freedom. This yields \(\textrm{CDF}_{\chi^2_{\mathrm{df}}}\), the cumulative distribution function of the \(\chi^2\) distribution. Define for each cluster \(C\), the value \(L(C)\), representing the amount of contamination of the cluster \(C\):

\[L(C) = \sum_{i \notin \mathrm{C}} 1 - \mathrm{CDF}_{\chi^2_{\mathrm{df}}}(D^2_{i, C})\]

\(L\) is then the sum of probabilities that each spike which is not a cluster member of \(C\) should be included in the cluster. Therefore the inverse of this cumulative distribution yields the probability of cluster membership for each spike \(i\). \(L\) is then normalised by the number of spikes \(N_s\) in \(C\) to allow larger clusters to tolerate more contamination. This yields L-ratio, which can be expressed as:

\[L_{\mathrm{ratio}}(C) = \frac{L(C)}{N_s}\]

Expectation and use

Since this metric identifies unit separation, a high value indicates a highly contaminated unit (type I error) ([Schmitzer-Torbert] et al.). [Jackson] et al. suggests that this measure is also correlated with type II errors (although more strongly with type I errors).

Example code

References

Cluster quality metrics computed from principal components.

spikeinterface.qualitymetrics.pca_metrics.mahalanobis_metrics(all_pcs, all_labels, this_unit_id)

Calculates isolation distance and L-ratio (metrics computed from Mahalanobis distance)

Parameters
all_pcs2d array

The PCs for all spikes, organized as [num_spikes, PCs].

all_labels1d array

The cluster labels for all spikes. Must have length of number of spikes.

this_unit_idint

The ID for the unit to calculate these metrics for.

Returns
isolation_distancefloat

Isolation distance of this unit.

l_ratiofloat

L-ratio for this unit.

A well separated unit should have a low L-ratio (Schmitzer-Torbert et al.). Since this metric identifies unit separation, a high value indicates a highly contaminated unit (type I error) (Schmitzer-Torbert et al.). Jackson et al. suggests that this measure is also correlated with type II errors (although more strongly with type I errors) (Jackson et al.).

Literature

Introduced by Schmitzer-Torbert et al.. Early discussion and comparison with isolation distance by Jackson et al..

Citations

Schmitzer-Torbert

Schmitzer-Torbert, Neil, and A. David Redish. “Neuronal Activity in the Rodent Dorsal Striatum in Sequential Navigation: Separation of Spatial and Reward Responses on the Multiple T Task.” Journal of neurophysiology 91.5 (2004): 2259–2272. Web.

Jackson

Jadin Jackson, Neil Schmitzer-Torbert, K.D. Harris, and A.D. Redish. “Quantitative Measures of Cluster Quality for Use in Extracellular Recordings.” Neuroscience 131.1 (2005): 1–11. Web.