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Nonlinear oscillation models for the spike sorting of single units recorded extracellularly Tetyana I. Aksenova Preclinical Neuroscience U318 INSERM Pav B, CHU Albert Michalon 38043 Grenoble Cedex 9 France Institute of Applied System Analysis Ukrainian Academy of Sciences Pr. Peremogy 37 Kiev 0356 Ukraine E-mail:
[email protected]
Olga K. Chibirova Preclinical Neuroscience U318 INSERM Pav B, CHU Albert Michalon 38043 Grenoble Cedex 9 France E-mail:
[email protected]
Abstract— The present study is devoted to the problem of automatic sorting of extracellularly recorded action potentials of neurons. The classification of spike waveform is considered as a pattern recognition problem of segments of signal that corresponds to the appearance of spikes. Nonlinear oscillating model with perturbation is used to describe the waveforms of spikes. It allows characterizing the signal distortions in both amplitude and phase. The spikes generated by one neuron assumed to be described by the same equation and should be recognized as one class. The problem of spike recognition is reduced to the separation of mixture of normal distributions in the transformed feature space. An unsupervised iteration-learning algorithm that estimates the number of classes and their centers is developed. It scans the learning set in order to evaluate spikes trajectories in phase space with maximal probability density in their neighborhood. To estimate the trajectories the integral operators with piece-wise polynomial kernels were used that provides computational efficiency. The new algorithm was tested on simulated and real data sets. I. INTRODUCTION The recording of extracellular neuronal activity provides unique information about the regions of brain. In humans it is used as a guide to the localizing optimal sites for deep brain stimulation (DBS) for treatment of motor disorders, in particular Parkinson’s Disease (PD). The quality of the information gained during the surgery depends on the ability to separate from background noise a few spike trains from the same electrode. Each spike train is assumed to represent a unique time series action potentials (spikes) of a single neuron. During the extracellular recordings the signals of
Alessandro E. P. Villa Laboratory of Neuroheuristics, Dept. of Physiology, University of Lausanne, Lausanne, Switzerland Laboratory of Neurobiophysics, Inserm U318, University JosephFourier, Grenoble, France E-mail: Alessandro.Villa @ujfgrenoble.fr
number of neurons are registered in the same time. As a result the mixture of spike of different neurons is observed. To solve the problem of spike sorting the basic hypothesis is that all spikes generated by one specific neuron are characterized by a similar shape and it is unique and conserved during a stationary recording. Widespread template-matching technique based on the comparison the spikes to the templates. The usual practice is to use a “supervisor”, i.e. an experienced human operator, who can provide a preliminary classification of the waveforms to select the templates. Both extracellular and intracellular noise may affect the shape of the action potential. The extracellular noise is usually taken into account as additive noise. The intracellular noise may produce intrinsic variations in the spike waveform, and is more difficult to account for. This article presents a method for spike sorting that makes it possible to take into account both types of noise. It is assumed that the spike waveforms are described by the nonlinear oscillating model with perturbation that characterized the signal distortions in both amplitude and phase. The spikes generated by one neuron are described by the same model and are recognized according to the model as members of the same class. The classification of spike waveform is considered as a pattern recognition problem. The use of local variables allows reducing the problem to the separation of mixture of normal distributions in the transformed feature space. Unsupervised learning algorithm that estimates the number of classes and their centers according to the distance between spikes trajectories in phase space is proposed. The estimation of trajectories in phase space required calculation of the high order derivatives in presence of noise. To solve the problem computationally effective algorithms were developed using the integral operators with piece-wise polynomial kernels. This provided
ISBN: 0-7803-8360-5 © 2004 IEEE
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computational efficiency for the real time application. The new method of spike sorting was tested on simulated and real data. II. MATHEMATICAL STATEMENT OF PROBLEM Presented approach based on the use of the inverse methods of nonlinear oscillation theory originally proposed † by Gudzenko [8] for modeling and later developed to solve various problems of signal processing [1,3], in particular, signal recognition [6,9]. New problem of spike sorting required developing new algorithms and methods the most important of which is unsupervised learning algorithm [2]. x (t ) = x(t)+ x(t) is observed at We suppose that a signal ~ discrete times t=0,1,.... Here x(t) is a sequence of independent identically distributed random variables with zero mean and finite variance (sx2!
A
B
Fig.1. Example of brain activity (A) and its derivative (B) with detected spikes
Let us assume that the spikes xi(t), 0!
dnx d n -1 x ˆ jÊ = f x ,..., Á ˜ + F (x ,..., t ), Ë dt n dt n -1 ¯
(1)
where n is the order of the equation, F( ) is a perturbation function and equation
Ê dnx d n -1 x ˆ j = f Á x ,..., n -1 ˜ Ë dt n dt ¯
(2)
describes a self-oscillating system with stable limit cycle
x 0j (t) = ( x10 (t),..., x n0 (t))¢ in phase space with codx d n-1 x , …, xn = n -1 for each j, 0£ j£ ordinates x1=x, x2 = dt dt p . T is the period of stable oscillations. The perturbation function F(x,…,t), bounded by a small value, is a random process with zero mean and small correlation time t*
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observation becomes generally irregular in time, i.e. t(qi+1)t(qi) ≠ 1. Let us consider a new feature space, with features x(t(qi)), 1!=!q1!
M(q): n(q),t(q) n(q) P(q)
The vector n(q ) = x(t(q )) - x 0j (q ) , x !Œ!Xj!!, has an
M(0) P0
Fig.2. New variables: phase q and normal deviation n(q). The thick line is the limit trajectory and the dash line is an arbitrary trajectory Eq.(1).
asymptotically normal distribution with mathematical †expectation close to zero for each class. In the new feature space we have obtained p normally distributed classes and the limit † cycle x j0 corresponds to the center of class X j. An example of two classes of spikes in phase space that were observed in real experiment is shown at Figure 3.
Let us denote g(q) = t(q)-q. The variables n(q) and g(q) characterize the deviation of an individual signal from the limit trajectory: n(q) normal deviation characterized the amplitude distortion, and g(q) characterize the phase shift. Restricting to the linier terms according the normal deviation the Eq. 1 can be rewriting in deviations [8] in the neighborhood of limit cycle. For the second order equation it holds dn/dq + Nj(q)n(q) = F(n)(q)
In [8] it was shown that vectors n i(q) are characterized by an asymptotically Gaussian distribution for any phase q in case of uncorrelated noise F(). It allows reducing our problem to well-known separation of a mixture of normal distributions. Let us refer to the space RT of features x(ti), ti!=!1,…,T as a standard feature space with dimension T. Let us consider the space Rn¥T , with dimension n¥ T, of features
x(t(q1))¢!|!x(t(q2))¢!|…|!x(t(qT) ) ¢ ) ¢ , x 0j (t) = x10 (t),..., x n0 (t) ¢
(
Our method requires the estimation of the signal trajectory in phase space. Higher-order derivatives of the signal should be calculated in presence of noise that is ill-poisd problem. In recent studies [4,5], we proposed to describe the signal trajectory in phase space using the convolution operators with piecewise polynomial kernel (Figure 4). k
k
Da x( t ) = Ú wa ( t - t )x( t )dt R
k
It was shown [4] that Da x estimates a smoothed derivative of the signal of the order k with parameter of regularization a . In [5] computationally efficient algorithm for calculations of derivatives is proposed.
wa1 (t )
wa2 (t )
-a
a
B. Feature space transformation
C. Estimation of the trajectory in the phase space.
4 a/ 2 3a /4
where F(n) and F(g) are the normal and orthogonal projections of perturbation function. For the high order equations system has the similar form according to vector n. Thus the system is determined in first approximation by the limit cycle and system (3). The equations (3) are linear according to the parameters. Therefore model determined by triplet (xj0(q),Nj(q),Qj(q)) of limit cycle and the parameters of systems in its neighbourhood. To solve the problem of pattern recognition of signal according to the model we should check whether at list one of the elements of triplet is different. Here we restricted to consider as different the model with different limit cycles.
Fig.3. Two classes of spike in phase space
a/
(3)
-3 a a/ 4 -a /2 -a /4
dg/dq = Qj(q) n(q) + F(g)(q),
-a/2
a/2
a
)
n-1
where ordinates
x1=x, x2 =
dx d x , …, xn = n -1 . It is dt dt
important to note that the partition of the interval of spike
†
1
2
Fig. 4. The kernel functions wa ( t ) and wa ( t ) are used to estimate the first and second derivative of the signal.
†
†
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D. Learning algorithm for spike recognition. An unsupervised learning algorithm for spike sorting is necessary in order to provide a fast and easy-to-use selection tool to a user during a real-time experiment or during human neurosurgery. The unsupervised learning algorithm scans a learning set formed by few tens of spike occurrences, usually corresponding to 30 seconds of recording time, and estimates the number of classes and their centers by measuring the distances between their trajectories in phase space [2]. The rationale is that in case of a Gaussian distribution the mean corresponds to the maximum of the probability density: the value x*!=!Ex provides the maximum of P(x-x*
