Extraction of event-related signals from multichannel ... - IEEE Xplore

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 47, NO. 5, MAY 2000

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Extraction of Event-Related Signals from Multichannel Bioelectrical Measurements Allan Kardec Barros*, Ricardo Vigário, Veikko Jousmäki, and Noboru Ohnishi

Abstract—Independent component analysis (ICA) is a powerful tool for separating signals from their mixtures. In this field, many algorithms were proposed, but they poorly use a priori information in order to find the desired signal. Here, we propose a fixed point algorithm which uses a priori information to find the signal of interest out of a number of sensors. We particularly applied the algorithm to cancel cardiac artifacts from a magnetoencephalogram. Index Terms—Event-related signals, evoked potentials, independent component analysis, multichannel measurements.

I. INTRODUCTION

T

HERE is a number of bioelectrical signals which are generated by a physiological response to some stimuli, either external or internal. They are locked in time to the stimuli and therefore are called event related signals. Among these, there are the evoked potentials which are responses occurring in the brain to for example visual or auditory stimuli. Neural activity can be measured by different methods such as the classical electroencephalogram (EEG) or more recently the magnetoencephalogram (MEG) or magnetoneurogram (MNG), which record multiple time-series related to different physiological activities. However, these measurements usually suffer from undesired interferences associated with blinks, eye-movements, muscle movement, cardiac noise and a number of other artifacts. Therefore, it is important to avoid that those interferences alter the result of the experiments. In the literature, ensemble averaging is frequently used, but it may alter the final result due to smearing of the averaged signal [13]. Other techniques have been proposed such as event-related periodic signal filtering [2], but this technique may not work in the case of nonperiodic signals. Recently, independent component analysis (ICA) appeared as a promising technique for separating independent sources in biomedical signal processing [3], [13], [15]. ICA is based on the following principle. Assuming that the original (or source) signals have been linearly mixed, and that these mixed signals are available, ICA finds in a blind manner a linear combination

Manuscript received June 1, 1999; revised December 1, 1999. Asterisk indicates corresponding author. *A. K. Barros is with RIKEN BMC, 2271-130 Anagahora, Shimoshidami, Moriyama-ku, Nagoya, Aichi 463-0003, Japan (e-mail: [email protected]). R. Vigário and V. Jousmäki are with the Laboratory of Computer and Information Science and Brain Research Unit, Helsinki University of Technology, FIN-02015 HUT, Finland. N. Ohnishi is with RIKEN BMC, 2271-130 Anagahora, Shimoshidami, Moriyama-ku, Nagoya, Aichi 463-0003, Japan. Publisher Item Identifier S 0018-9294(00)03275-4.

of the mixed signals which recovers the original source signals, possibly re-scaled and randomly arranged in the outputs. However, extracting all the independent signals from for example an MEG measurement, which may output hundreds of recordings, could take a long time, either in minutes or hours basis. Therefore, it is important to extract only the desired component. For carrying out this task, we propose here to use a deflation algorithm along with the a priori information available about the signal of interest. The deflation approach is important because we can extract only one independent component out of the measured signals, instead of extracting all at once. However, as we said above, the independent components come out in a random fashion, thus any of the signals could come out as the first one. For solving this problem we use the a priori information available through a Wiener filter to initialize the algorithm. This manuscript is distributed as follows. First, we review some important aspects of Wiener filtering in Section II. Then, we introduce the independent component analysis theory along with an algorithm for separating quasiperiodic sources in Section III. In Section IV, we show the methods used and in the following one, the experimental setup for testing the algorithm either by simulations or in real world applications. In Section VI, we show the results and in Section VII, we present the conclusion. II. IMPORTANT ASPECTS OF WIENER FILTERING Let us first define the problem in a framework that will be used throughout this manuscript. Consider mutually indepenarriving at dent signals electrodes (or another type of receiver). Each electrode gets a linear combination of the signals. For simplicity, we will drop the iteration index and only use when necessary, then we have . The mixed signals are thus given by (1) invertible matrix.1 where is an We assume that we can observe only the mixture . Moreover, without loss of generality, let (2) . In practice, this operation will speed up so that the convergence of the algorithm to be presented in the next section, besides turning it simpler. 1This model can be more complex with the number of sensors different of that of sources, and can also include noise.

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Our purpose is to find from the mixed vector one given component of the source signal , using some a priori information . included in a signal , correlated with , i.e., For carrying this out, we use all the components of the input vector in a linear combination as shown in Fig. 1. Thus, we , where is a weight vector to be estimated by have the algorithm, is the reference signal and the error is given by . The weights are updated by the minimization of the . There are a number mean-squared error (mse) given by of algorithms which use the mse as the cost-function. One of the simplest is the LMS, which estimates the weights in an on-line fashion, given by

Fig. 1. Diagram of a linear combiner for estimating the weights from the inputs.

(3) where is the step-size which controls the speed and the error in the convergence. However, we are interested here in estimating the weight vector in a way which is independent on parameters chosen by the user such as in (3). Thus, let us minimize the mse given by (4) and

where

. We have the gradient as (5)

Thus, from (5), we see that the optimum weight, which min. Because of (2), the optimum imizes the mse is weight (also called the Wiener weight) is then given by (6) A. Theorem 1 , where is the Frobenious norm, . If for a given and , then , where , and is a canonical base vector, i.e., and . Proof: From (6) we have that

Let and

(7) leads to . Normalizing From this theorem, we can see that in the presence of a reference, second order statistics are enough to separate signals. We shall discuss further implications of this theorem in Section III.

arating the signals in a blind manner2 . Within this framework, ICA algorithms find a linear combination of the elements of which gives the most independent components as the output. Usually this output is found using a matrix , so that the eleare mutually independent. ments of , different cost functions were used in the To find matrix literature, usually involving a nonlinearity which shapes the probability density function of the source signal. But higher order statistics, such as the kurtosis was widely employed as of a zero-mean random variable is well. The kurtosis defined as (9) However, we are here willing to extract or remove some given signal from a number of measured signals. Therefore, instead of matrix and from the output pick up finding an the desired signal (as in [14]), we are rather interested in finding only one component of . Therefore, this component is given , where is one of the rows of . In graphical by terms, this can be understood as the left side of the dotted line in Fig. 1. Most of the works in this field, called sequential blind extraction of sources were based on a cost function minimization or maximization which was a function of the kurtosis (see, for example [8], [6], and [9]). Here, we also use the fourth-order moment, either using the kurtosis for nonperiodic signals, or a new cost function for periodic or quasiperiodic ones. In the later, our objective is to obtain as the first output a signal , so that it fits the characteristics of a quasiperiodic signal. Thus, we propose to maximize the following function:3

(10) III. INDEPENDENT COMPONENT ANALYSIS The final objective of ICA is also separate signals, under the assumption that they are mutually independent, in other words, the joint probability density of the source signals are the product of the marginal densities of the individual sources (8) The difference to the method in the previous section is that most of the algorithms in the literature were proposed for sep-

where is the discrete time index (iteration number), is a constant and is an integer delay. Therefore, instead of using the variance as a normalizer to the fourth-order moment, we use the autocorrelation of the output signal at delay . The reasoning for this proposal is that for 2There are some discussions about the meaning of the term blind, but are out of the scope of this paper. 3We will drop the indexes from now on, and use them only when necessary to avoid confusion

BARROS et al.: EXTRACTION OF EVENT-RELATED SIGNALS FROM MULTICHANNEL BIOELECTRICAL MEASUREMENTS

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a signal periodic in , this autocorrelation should have a high value, while for other signals which are not periodic at , this value should be smaller. On the other hand, is used to weight this normalization of the fourth-order moment. IV. METHODS We maximize (10) subject to the constraint we shall find a which maximizes the Lagrangian

. Thus, (11)

Therefore, we should have

Fig. 2. Block diagram of the signal enhancement/elimination based on Wiener filtering.

(12) where

(17) Otherwise (18)

(13) The same can be carried out to (9), to obtain the algorithm of Hyvarinen and Oja [9]

by its norm and update . • Divide , otherwise, change the current • Test if weight to the Wiener one, added to a small random deviation. This step is important to guarantee that the solution is spatially near the Wiener one. ap• Repeat the last three above steps until proaches 1 (up to a small error).

(14) C. Algorithm Convergence A. On the First Extracted Component Now that we have obtained algorithms for the case of periodic and high-kurtosis signals as given by (13) and (14), respectively, we propose to initialize the weights by (15) which is the Wiener weight given in (6). However, in most of the cases Theorem 1 is not satisfied. In other words, there is a reference signal which is strongly correlated with the desired solution, but is also weakly correlated with some of the others. Therefore, we can assume the initialization as given by (15) to be spatially near the solution, depending on the reference input. Thus, we propose to search for the solution in the -dimensional hyper-plane so that (16) where

is the radius of a hyper-sphere centered at

It has be shown that (9) has a number of maximum and minimum which corresponds to the independent components and therefore (14) is globally consistent [9], [8]. Here, we show that the algorithm (13) is locally consistent. Let us study then its be. To that aim, we make havior around the point and define the function the change of coordinates , with given by (10). to the solution, Adding a small perturbation we have the following approximation: (19) where the derivative is given by

(20) and where we defined The substitution of (20) into (19) yields

.

.

B. Proposed Algorithm From the above reasoning, we propose the following fixedpoint algorithm. • Carry out principal component analysis (PCA) in vector in order to have , where . . Let the • Take the initial vector . iteration number Fc , where Fc is the • If the signal is periodic, find fundamental frequency of the signal to be extracted. • For a periodic signal, let

(21) Thus, we can see that is a minimum/maximum of that the following condition is satisfied:

given (22)

for a given source signal

.

D. Signal Enhancement/Elimination using the algorithm After obtaining the output described above, one may be interested either in keeping this signal for posterior manipulation or analysis, or in removing it from the sensors. In order to accomplish this last option, we can use simply the Wiener filter as proposed above. This can

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Fig. 3. Signals used in the simulation. (a) Electrocardiogram. (b) Respiratory artifact. (c) Electrodes artifact. (d) Random signal. (e) High kurtosis (transient) signal. (f)–(g) Periodic signals with different frequencies.

Fig. 4. One result of the simulation. (a). Desired signal. (b) Reference input. (c) Wiener filter output. (d) Algorithm output.

be carried out by estimating signal using the previous ICA method, and estimating its contribution to each element of as shown in Fig. 2. Thus, one can either have a vector of the , or the sensors with contribution of to each sensor as eliminated from it given by . From (6), we find that the elements of are estimated by (23) V. EXPERIMENTAL SETUP As a first test to the algorithm proposed in this paper, we have separated artificially generated mixed signals. Seven signals

were used. Three of them were taken from the MIT-BIH noise stress test database, which are standard for testing electrocardiographic (ECG) analyzers. They were composed of one ECG signal and two common artifacts frequently observed while measuring the ECG: respiratory and electrode artifacts. We added then a random, a high kurtosis signal4 (to simulate a transient disturbance) and two sine waves with different frequencies. They can be observed in Fig. 3. As the reference signal, we have used a train of pulses synchronized with the peaks of the signal we wanted to detect. With this, we wanted 4We will refer as “high” or “low” kurtosis signal

(s) so that (s)=E [s

]

s as one which has a kurtosis

1 or (s)=E [s ] 0, respectively.

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Fig. 5. Comparison between two algorithms for extraction of a periodic component. The results are shown in the order this component appeared. (a). Results for the periodic algorithm (PA) with a random initial weight. (b). Results for the kurtotic algorithm (KA) with a random initial weight. (c) Results for PA after initializing the weight to the Wiener solution.

to mimic an external trigger which is usually available in the evoked potential experiments, for example. The radius of . interest around the initial Wiener guess was In a second simulation experiment, we compared the performance of the periodic and nonperiodic algorithms summarized in the updating rules (17) and (18). We will call them, for simplicity of the discussion, periodic algorithm (PA) and kurtotic algorithm (KA), respectively. The goal was to see how these algorithms isolate a desired signal when the initial weight is set to a random vector. To do so, we mixed linearly five signals: one sine wave, one artificial signal with very high kurtosis, the electrode artifact of the previous experiment, one artificial signal with low kurtosis and one white Gaussian signal. The length of the mixtures was of 10 000 points, and both algorithms were run 100 times, with random initial weights and mixing matrix. Finally, we tested PA for the extraction of cardiac artifacts from MEG data. The data comprised around two minutes of 122-channel MEG recording with sampling frequency of 297 Hz. Fig. 6 show a subset of eight channels. The highlighted signal, corresponding to a sensor on the lower left side of the sensor array, was used to identify the -waves, just in the way it is carried out for calculating heart rate variability.5 The resulting train of pulses synchronized with the -waves was used as reference. VI. RESULTS From the first experimental data, shown in Fig. 3, we have observed that the PA always found the desired sinusoidal component, namely the one in frame (f). Fig. 4 presents the results attained in that experiment. The train of impulses, used as reference, is presented together with the desired original signal, 5In particular, we high-pass filtered the signal at 0.9 Hz and used a threshold to detect the peak at each period.

Fig. 6. An illustrative subset of the 122-channel MEG data (used). The highlighted signal was used for the identification of the cardiac peaks.

the Wiener filter output and the independent component. Limitations in the verification of the conditions of Theorem 1 lead to a poor estimate by the Wiener filter. The sinusoidal shape is nevertheless present. The last frame show how PA accurately recovers the original signal. In the second artificial data set, the order of appearance of the desired sinusoidal signal was recorded, for both algorithms, and subsequently plotted in Fig. 5. Note that PA, using the periodic a priori information, consistently identifies the sinusoidal component among the first two to extract [see Fig. 5(a)]. On the other hand, KA privileges the kurtosis of the signals involved, rating the sinusoidal typically in third or fourth place [see Fig. 5(b)]. However, when initializing the weight to the

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generated signals, and experimental data. In both applications we have been able to select the independent component to be extracted, via an efficient initialization process, based on the Wiener weights associated to a reference signals. In the particular example of the extraction of cardiac artifacts from MEG data, we achieved a very accurate identification of the contaminating component, without the need of any additional electrical measurements. In event-related studies it is often the case that the external trigger may be available to the neuroscientist, supplying the both the information on the periodicity of the desired outputs, and a good reference signal for the initialization of the ICA algorithm. This suggests possible applications of the proposed periodic algorithm on evoked potential/field studies. Fig. 7. Independent component found on the MEG data, utilizing the periodic ICA algorithm. On top, three views of the sensor plane show the corresponding field pattern. An insert in the bottom magnifies a 10 s section of the component.

Wiener solution, the algorithm completely succeeded, as it can be seen in Fig. 5(c). We have included the very high kurtotic signal also to show that the proposed algorithm does not always extract the desired component at first, due to the mixing matrix condition. That is, the possibility of the desired signal to come out as the first one increases with its energy at the sensors. The measurement of event-related activity in the brain may be corrupted by cardiac artifacts typically exceeding the amplitude of the evoked response itself [11]. In order to solve this problem, Jousmaki and Hari [11] suggested to use electrodes measuring directly the ECG, and subsequent removal of this disturbance from the data. Using minimal a priori information on the MEG signals, Vigário et al. [15] suggested an efficient method to remove the cardiac artifacts, which was based on PA. The algorithm proposed in the present publication, utilizing the additional periodic information, achieves an even better result, as can be seen in Fig. 7. The independent component extracted using PA is shown in the middle of the figure, with an amplification insert containing only 10 s. On top, three views of the magnetic field patterns generated by such components are plotted. These correspond to the the estimate of the mixing vector corresponding to such a component. Equivalently, the magnetic field contours produced by a neural source with time activation represented by the independent component. The field patters, with a somewhat tilted characteristic agrees with the earlier works by Jousmaki and Hari [11] and Vigário et al. [15]. VII. CONCLUSION We have introduced a new algorithm, suitable for the extraction of independent components from their linear mixtures in the case of periodic or quasiperiodic signals. Experimental evidence of its functioning was shown, using both artificially

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