Figure 1:
Comparison of thresholding methods using percentages of clusterable and missing spikes when: a) the threshold factor is varied and b) the total number of spikes detected is varied.
1Keck Center for Integrative Neuroscience and Department of Physiology, 2Sloan Center for Theoretical Neurobiology, 3Department of Otolaryngology, University of California at San Francisco, CA 94143-0444 4Institute for Sensory Research ,L.C. Smith College of Engineering and Computer Science, Syracuse University,Syracuse, NY 5Air Force Research Laboratory/IFGC, 525 Brooks Rd.,Rome, NY 13441-4505 rebrik@phy.ucsf.edu, bdwright@phy.ucsf.edu, emondi@phy.ucsf.edu, ken@phy.ucsf.edu, http://millerlab.ucsf.edu/ *Corresponding author.
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We are exploring new methods of spike detection to improve spike-sorting in tetrode recordings. Based on our observation that the four channels of the tetrode carry highly correlated signals, we propose the use of a hyperellipsoidal thresholding surface in the 4-dimensional space of the signal values to detect spikes. This surface is determined by the cross-channel covariance matrix and provides a better approximation of the equiprobable surface of the noise amplitude distribution compared to the traditionally used hypercubical thresholding surface. This spike detection procedure greatly improves the separation of signal clusters from the noise cluster around the origin. We have extended these approaches to automatic spike-sorting in both amplitude and full waveform spaces.
Keywords: Tetrode; Spike-sorting; Multi-electrode recordings
Tetrodes allow recording from many nearby cells simultaneously and thus reveal short-range interactions of neurons. Though sorting of spikes in tetrode recordings is generally more reliable than that of traditional single-electrode recordings [1], the problem of assigning spikes to different neurons remains complicated and challenging. Typically, processing of neuronal recordings consists of two major stages: 1) detection of spikes and 2) spike-sorting. Much effort has been put into the problem of spike-sorting, while the first step has received less attention. Nevertheless, this step is very important for adequate assessment of cell interactions [2]: e.g., loss of a noticeable fraction of spikes from one neuron can lead to errors in calculating its degree of synchrony or reliability of synaptic connections with other neurons.
A standard spike detection procedure compares the signal value (or a function thereof) with a preset threshold. When the threshold is crossed, a spike is detected. The threshold value is usually based on the estimate of the signal variance. The number of detected spikes grows rapidly as the threshold is lowered. If we assume a spherical Gaussian distribution of the amplitudes of spikes, a change in the threshold value from 4s to 2s (where s is the standard deviation of the Gaussian distribution) will result in a 33-fold increase in the number of detected spikes. In practice, the change is not that dramatic (5 times and more). This increase in the spike number is due to the increase in the number of low-amplitude spikes. Not all of these low-amplitude events can be clustered, since some of them are just noise outliers detected as spikes.
Setting the threshold too high leads to missed spikes, while setting it too low leads to detection of many small-amplitude spikes that cannot be classified. So an optimal setting of the threshold is one that allows detection of all clusterable (useful) spikes while keeping the number of non-clusterable (garbage) spikes minimal. Different detection procedures can be compared by the ratio of useful spikes to the total number of spikes measured at the optimal (for the given procedure and the dataset) threshold setting.
Our observations [2] show that noise in tetrode recordings in the cat visual cortex and in the LGN is highly correlated across channels (typical cross-channel correlation coefficient lies in the range of 0.7-0.95). The observed cross-channel correlation is due to the common signal detected by all 4 electrodes within the tetrode.
There are several possible origins of this common source: a) cross-electrode stray capacitances, b) variations in the potential of the reference ("ground") electrode, and c) truly biological noise coming from spiking activity of distant neurons. If the possibility (a) were realized, the tetrode would be incapable of producing any "stereo effect", i.e. registering spikes from the same neuron with significant difference in amplitudes across channels. Direct measurements in saline solution show that the capacitances of the electrode tips to the solution are several times bigger than the cross-electrode capacitances, ruling out origin (a).
To test how much noise comes from the reference electrode, we measured cross-channel covariances of two tetrodes separated by a distance of a few millimeters. Cross-channel correlation coefficients within the same tetrode appeared to be relatively high: 0.8-0.92 (as expected), while the coefficients of correlation across channels of different tetrodes were significantly smaller: 0.47-0.51. In the case (b) (the "floating ground" problem) one would expect the correlation to be the same both within the individual tetrodes and across the tetrodes. Thus we are left with the case (c) - a biological origin of the common source, presumably due to activity of the surrounding neuronal population.
Traditionally, thresholding is performed on each channel independently: if the threshold is crossed on at least one channel, a spike is detected. This algorithm corresponds to a hypercubical (box-shaped) thresholding surface that does not fit the spike amplitude distribution. The threshold crossing criterion is given by the expression: Any Vi > f RMS, where Vi is the voltage at the channel i, RMS is the estimate of the signal variance, and f is an arbitrarily chosen threshold factor.
Due to the cross-channel correlation, the distribution of amplitudes of noise events detected as spikes (noise cluster) is elongated along the diagonal of the 4-dimensional hypercube formed by the four axes (after normalizing signals on all channels to have common range). Since the goal of the detection procedure is to exclude noise events while preserving "useful" spikes, it is natural to use a thresholding surface that fits the shape of the noise cluster.
To build a hyperellipsoidal thresholding surface for this case, we first estimate the cross-channel covariance matrix, C, from random chunks of the data. For any given time the channel voltages can be represented as a 4-vector, V, and the threshold crossing criterion is given by the expression: V TC -1V > f2, where f is an arbitrarily chosen threshold factor. Note that different channel sensitivity is accounted for in the inverse of the covariance matrix.
To compare the two detection procedures outlined above we used
the following method. We
first marked and clustered spikes at a low value of the threshold, thus
obtaining the
number of all possibly clusterable spikes, Nc0.
Then we repeated the
same procedure for several gradually increasing values of the threshold
factor. For each
threshold value we measured the number of clusterable spikes, Nc
, the
number of spikes missing from the clusterable population, Nm
º Nc0
- Nc , and the total
number of detected spikes, Nt
.
Figure 1:
Comparison of thresholding methods using percentages of clusterable and
missing spikes
when: a) the threshold factor is varied and b) the total number of
spikes detected is
varied.
The results of the comparison are shown in Fig.1a.
An ideal detection procedure
should have the percentage of clusterable spikes, pc
º
Nc / Nt = 1,
and the percentage of missing clusterable
spikes pm º
Nm / Nt = 0.
In a real spike detector these values depend on the threshold factor f.
For very
low values of f both pc
» 0
and pm
» 0,
meaning that while no spikes are missing, most of the
detected spikes are just noise events. With increasing f,
both pc
and pm start to grow, and a
procedure that allows a bigger gap between pc
and pm has better
performance. It is clear from the graph that the
hyperellipsoidal thresholding surface outperforms the hypercubical one.
This becomes more
obvious when the values of pc
and pm are plotted as a
function of the total number of detected spikes (Fig.1b): for any given
number of detected
spikes, the hyperellipsoidal thresholding surface gives a bigger value
of pc
and smaller value of pm. For
example, to avoid missing more than 5% of
the clusterable spikes, one must analyze 50-100% more spikes in the
hypercubical case.
This method of thresholding using the shape of the noise surface is also of practical importance in automatic methods for spike-sorting. There are several problems in automatic spike-sorting that relate to thresholding methods. First, in the space of the four spike amplitudes, clusters that invade the threshold boundary will be clipped, making inference of clusters difficult and less robust. Second, in the traditional thresholding method, when one lowers the threshold low enough to avoid significant clipping, one quickly becomes overwhelmed by the sheer number of spikes. Handling this amount of data is computationally infeasible and/or inefficient and the large number of garbage events can "break'' some clustering algorithms. Finally, an important issue in automatic methods is how to deal with outliers and noise events. One usually introduces "garbage'' clusters to deal with this. In the case of hyperellipsoidal thresholding, the garbage cluster for low amplitude noise is well determined by the cross-channel covariance matrix. Based on our previous discussion, the hyperellipsoidal thresholding method clearly helps alleviate all of these problems.
We have implemented several automatic clustering methods, using our proposed thresholding technique. These include a mixture of Gaussians model to infer both the covariance structure of the clusters and the most probable number of cells in the recording, the latter determined using BIC as a complexity penalty (see e.g. [3]). We have also used this thresholding technique in the full 4-channel spike waveform space with an automatic clustering procedure using mixtures of probabilistic principal component analyzers [4-5]. We found marked differences in these automatic methods when coupled with the different thresholding techniques. Dramatic differences were seen in the proportion of spikes associated with a particular cluster and in the structure of the probability models describing clusters that were clipped by thresholding. These differences can potentially be important in, e.g., spike correlation analyses.
Tetrode channels are highly correlated by neurobiological sources. We have proposed a new spike-detection procedure that fits the empirical noise distribution in the 4-dimensional space. Using this approach allows a significant decrease in the number of noise events that are inevitably detected along with actual spikes. Because of their better properties in characterizing the noise distribution and reducing the amount of cluster clipping, our new thresholding method also leads to improved performance in automatic clustering.
Work supported by grant R01-NS33787 from the NINDS (KM), by
the Searles Scholars'
Program (KM), by a grant from the Alfred P. Sloan Foundation (KM) and
by the AFRL (AE).
[1] C. Gray, P. Maldonado, M. Wilson and B. McNaughton, Tetrodes markedly improve the reliability and yield of multiple single-unit isolation from multi-unit recordings in cat striate cortex, J. Neur. Meth., 63(1-2) (1995) 43-54.
[2] S. Rebrik, S. Tzonev and K.D. Miller, Analysis of Tetrode Recordings in Cat Visual System, in Proceedings of CNS97 (Computation and Neural Systems Meeting, Big Sky Montana, July 1997), J.M Bower, ed. (Plenum Press, 1998).
[3] B.D. Ripley, Pattern recognition and neural networks (Cambridge University Press, Cambridge, 1996).
[4] S. Roweis, EM Algorithms for PCA and SPCA, in Advances in Neural Information Processing Systems, v.10, p.626, 1998.
[5] M.E. Tipping and C.M. Bishop, Mixtures of Probabilistic
Principal Component
Analyzers, Aston University Neural Computing Research Group Technical
Report NCRG/97/003,
June 1997.