Neural Comput & Applic (2013) 22:1201–1209 DOI 10.1007/s00521-012-0886-5

ORIGINAL ARTICLE

Estimation of brainstem auditory evoked potentials using a nonlinear adaptive filtering algorithm Nurettin Acır

Received: 19 August 2011 / Accepted: 4 February 2012 / Published online: 23 February 2012  Springer-Verlag London Limited 2012

Abstract In this study, we introduce a novel nonlinear system not only for tracking of both the latency and amplitude variations in brainstem auditory evoked potential (BAEP), but also for reduction of single-trial numbers in BAEP pattern extraction process. Trial-to-trial variations in auditory evoked potential (AEP) are very important in quantifying dynamical properties of the nervous system and in specifying the group-specific effects in clinical applications. Due to the nonlinear dynamics of the AEP, a nonlinear adaptive filtering technique is considered as a powerful tool for tracking such variations. Therefore, we have designed a wavelet network-based nonlinear adaptive filter (WaNe-NAF) satisfying asymptotic stability in the sense of Lyapunov. The simulation results are verified that the proposed WaNe-NAF can effectively track the trial-totrial variations. We have also compared the WaNe-NAF with the most widely used ensemble averaging technique using real measured human BAEP data. The WaNe-NAF shows promise for requiring less number of ensembles than conventional ensemble averaging method to attain adequate signal quality. As a result, the proposed filtering system is suggested as a powerful tool in AEP acquisition and processing systems. Keywords Auditory evoked potential  Nonlinear adaptive filtering  Wavelet network  Lyapunov stability

N. Acır (&) Electrical and Electronics Engineering Department, Nigde University, 51245 Nigde, Turkey e-mail: [email protected]

1 Introduction Auditory evoked potentials (AEPs) are voltage changes of gross electrical activity in the brain due to the synchronously sensory stimulation [1]. Brainstem auditory evoked potential (BAEP) is the short-latency version of AEP signals [2]. The measured AEPs are generally defined as the combination of electrical activity generated by multiple sources in brain and noises together with interference from nonneural sources such as eye blinks and other artifacts [3]. The amplitude of the AEP signal is often lower than the background noise such as ongoing electroencephalogram (EEG). Although the most common way to measure AEP is the ensemble averaging (EA) technique for determining stimulus-related AEP components, it neglects the trial-to-trial variations. Many studies in the literature showed that AEP signals are usually nonstationary and their characteristics vary from trial to trial [1, 4, 5]. So, EA results in a loss of information related to variations among single-trial responses. Many studies have been presented for tracking trial-totrial variations in AEPs [6–16]. Although the simplest technique used in tracking variations in AEPs is the linear filtering [6], it is not successful for all BAEP applications due to the overlap of related signal and noise spectrums [3, 17]. The Wiener filtering is also used as a posteriori filter for providing an optimal solution in the mean square error (MSE) sense [8, 18]. On the other hand, the AEP is assumed as a stationary process in Wiener-like techniques. So, the obtained results could not be optimal in the MSE sense due to the nonstationary nature of BAEPs [18]. Some other techniques are also used in single-trial AEPs such as wavelet transform–based denoising procedure [12], subspace regularization–based methods [13], independent component analysis [11, 15], statistical analysis [6, 14] and parametric methods [9, 16]. Although they have improved

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the results in some extents, they neglect the latency shifting (time delay) and amplitude variations after filtering processes. Therefore, adaptive filtering is needed and presented as a powerful technique for tracking AEP variations in latency and amplitudes [5, 17, 19, 20]. Adaptive filters have the ability to adjust their own parameter settings automatically and require little or no a priori knowledge of the statistical characteristics of the signal or noise. Adaptive filters used in tracking of AEP variations are generally gradient descent–based algorithms. However, the gradient descent–based searching may be trapped at local minima of cost function, and, if the signal has a large bounded input disturbance, global minimum point may not be found. Artificial neural networks (ANNs) for both linear and nonlinear filtering systems are potentially useful in determining AEP response variability [5]. In ANN-based filter designs, backpropagation-type gradient descent algorithms are widely used. However, those type of gradient descent algorithms are highly nonlinear for implementing and they suffer from local minima problem in some sense [21]. The radial basis function network (RBF) is also proposed as an alternative to backpropagation-type ANN algorithms due to its some linear and straightforward computational properties [22]. But center determination is a crucial extra and an expensive procedure in RBF [21]. However, the wavelet network (WN) carrying frequency-specific properties is considered as a powerful tool for function approximation, system identification, adaptive filter design and other nonlinear dynamic modeling problems [23–26]. The WN is indeed a combination of feed-forward ANN implementation and wavelet decomposition procedure. In WN structure, the parallel processing capability of ANN and a better function approximation property of wavelet transform are combined. Although the LMS algorithm [27] is one of the most widely used methods in adaptive filter design, the convergence rate of it might be rather slow when the input signal is colored as in AEP estimation. A recursive least square (RLS) algorithm [27] may be an alternative to LMS, which has a faster convergence rate but with additional complexity in computation. On the other hand, the gradient descent–based algorithms (e.g., LMS and RLS) may have a slow convergence in practice, and the searching for the global minimum of a cost function may be trapped at a local minimum during gradient descent. Unlike the gradient descent–based algorithms, the lyapunov stability theory (LST)-based adaptive algorithm adapted to WNs in this paper searches the global minimum along the energy surface in the state space with a unique solution through the adaptive adjustment of the filter weights as timescale goes to infinity. The error between desired reference input and filter output then asymptotically converges to zero. The detail about LST can easily be found in literature [26, 28–31].

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The main objective of this study is to estimate single-trial BAEP signals using a wavelet network–based nonlinear adaptive filter (WaNe-NAF) satisfying asymptotic stability in the sense of Lyapunov. In our previous study [26], we had designed a similar adaptive filter having infinite impulse response. In this study, we have designed a finite impulse response (FIR) adaptive filter for tracking of both the latency and amplitude variations in BAEP signals. The proposed WaNe-NAF can accelerate the convergence rate and estimate the single-trial BAEPs perfectly without or a very little latency shifting. It also provides an independency from the statistical properties of input signal unlike many other adaptive algorithms. We have also compared the WaNeNAF with the most widely used EA technique using a real measured human BAEP data set. The simulation results show that the proposed WaNe-NAF algorithm can be used in a possible fast BAEP acquisition system after some standard clinical experiments done by the AEP medical experts. The outline of paper is composed of four sections. After an introduction, the proposed WaNe-NAF algorithm is presented in Sect. 2. Simulation results and related discussions for both simulated and real measured BAEP signals are presented in Sect. 3. In Sect. 4, the paper is concluded.

2 WaNe-NAF algorithm In this section, we first design the WaNe-NAF with a standard least mean square (LMS) algorithm [27] for a convenient presentation, and then, accordingly, we present the LST-based WaNe-NAF algorithm. It must be mentioned here that basic theory of WN can be found in our previous study [26]. As seen from Fig. 1, the proposed filter consists of two inputs: (i) the primary input, d(k), is the sum of a signal s(k) and a noise n(k), (ii) the reference input, x(k), is composed of a signal related to but not necessarily having the same waveform with s(k) and a noise unrelated to n(k). Both noise signals in d(k) and x(k) are uncorrelated with both signals and each other. Under these assumptions, an AEP signal has N trials and the length of each trial vector is M data points, x 2 RM . Thus, a measured BAEP signal is denoted as follows: dðkÞ ¼ ½d1 ðkÞ d2 ðkÞ    dM ðkÞT ;

k ¼ 0; 1; . . .; N  1:

The output of WN having N hidden nodes (number of wavelones) is as follows: yðkÞ ¼

N 1 X

wi ðkÞui ðxÞ ¼ UwðkÞ

ð1Þ

i¼0

where x is reference input, and filter coefficient w(k) is as follows:

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1203 d ( k ) = s( k ) + n( k )

(Reference Signal) x( k ) + Z

−1

(Primary Signal) D0

ψ(⋅)

w0

D1

ψ(⋅)

w1

DN −1

ψ(⋅)

wN −1

−b0 +

+

y (k )

∑

e(k)

_

−b1 Z −1 +

−bN −1

Fig. 1 Block diagram of WaNe-NAF system for BAEP estimation

wðkÞ ¼ ½w0 ðkÞ w1 ðkÞ . . . wN1 ðkÞT

ð2Þ

and 2

u0 ðx1 Þ 6 u0 ðx2 Þ 6 U¼6 .. 4 . u0 ðxM Þ

u1 ðx1 Þ u1 ðx2 Þ .. .

... ... .. .

u1 ðxM Þ

. . . uN1 ðxM Þ

3

uN1 ðx1 Þ uN1 ðx2 Þ 7 7 7 .. 5 .

ð3Þ

ð8Þ

where g(k) is the adaptation gain function as given in (9);   U jeðk  1Þj gðkÞ ¼ 1  j ð9Þ jaðkÞj kU k2

aðkÞ ¼ dðkÞ  Uwðk  1Þ

i ¼ 0; 1; . . .; N  1

ð4Þ

where wðÞ :
ð5Þ

Using the conventional LMS algorithm [27], the weight vector w is updated once at each trial by the steepest descent optimization algorithm as presented in (6). wðk þ 1Þ ¼ wðkÞ þ gUT eðkÞ

ð6Þ

where g is the step size that regulates the speed and stability of adaptation. For algorithmic implementations, we define the wavelet coefficients at discrete scales a and discrete times b by choosing the set of parameters as; ai ¼ 2i and bi ¼ 2i q; i ¼ 0; 1; . . .; N  1 q ¼ 0; 1; . . .; N  1:

wðkÞ ¼ wðk  1Þ þ gðkÞaðkÞ

and a(k) is the a priori estimation error as defined in (10);

and ui ðxÞ ¼ wðDi ðx  bi ÞÞ;

wðkÞ ¼ ½w0 ðkÞ w1 ðkÞ w2 ðkÞ    wN ðkÞT , such that the system output y(k) estimates the underlying BAEP signal s(k). Consequently, the hidden layer output matrix, U, acts as the input for the conventional linear adaptive filter. Thus, adaptive learning algorithms can be used to adjust the weight vector. As an alternative to LMS adaptive algorithm, we have developed here an LST-based adaptive algorithm for AEP estimation. Using the results and theorems in [22, 29], for a given reference input and primary input pairs fxk ; dk gN1 k¼0 for AEP estimation problem, the weight vector w(k) can be updated via an update rule given in (8), and the system output y(k) can be generated as yðkÞ ¼ UwðkÞ:

and

ð10Þ

and adaptation gain rate j should be in the range of 0  j\1: Then, the error e(k) converges to zero satisfying asymptotic stability in the sense of Lyapunov, and analytical proof is shortly given as follows [22]; eðkÞ ¼ dðkÞ  UwðkÞ ¼ dðkÞ  U½wðk  1Þ þ gT ðkÞaðkÞ ¼ dðkÞ  Uwðk  1Þ  UgT ðkÞaðkÞ ¼ aðkÞ  gT ðkÞaðkÞwðkÞ   U jeðk  1Þj ¼ aðkÞ  1  j aðkÞwðkÞ jaðkÞj kU k2   aðkÞ jeðk  1Þj ¼ aðkÞ  1  j aðkÞ ¼ j jeðk  1Þj jaðkÞj jaðkÞj ¼ jjeðk  1ÞjsgnðaðkÞÞ jeðkÞj ¼ jjeðk  1Þj jeð1Þj ¼ jjeð0Þj jeð2Þj ¼ jjeð1Þj ¼ j2 jeð0Þj

ð7Þ

Here, it must be noticed that the dilation and translation coefficients can directly be determined from (7) for each trial of implementation. The primary input at the ith iteration is the one trial of the recorded BAEP signal represented by the vector, d(k). Here, the first step is to create a U matrix with a given reference input in hidden layer, and then the next step is to perform the algorithm to adjust the weight vector,

jeð3Þj ¼ jjeð2Þj ¼ j2 jeð1Þj ¼ j3 jeð0Þj .. .

ð11Þ

jeðkÞj ¼ jk jeð0Þj For 0  j\1, when timescale goes infinity, k ! 1, eðkÞ ¼ 0. For avoiding the discontinuities, a slack variable, k, is added into the (9) [19];

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gðkÞ ¼

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  jeðk  1Þj 1  j k þ jaðkÞj kU k2 U

ð12Þ

In our experiments (in Sect. 3), (12) is used and performed for sufficiently small k value. In accordance with the proposed Lyapunov update formulas (presented in (8), (10), and (12)), the computational algorithm for BAEP estimation is presented as follows: Algorithm Parameters: • • •

n = filter order j = adaptation gain rate, 0\j\1 k\\1 and k 2 Rþ

Initialization: wð0Þ ¼ 0 Given Data: •

fxk ; dk gN1 k¼0

where xk is reference signal input, and dk is desired signal input (measured single-trial BAEP). Computation: •

wðkÞ ¼ wðk  1Þ þ gðkÞaðkÞ •

ni ðkÞ ¼ 1:508ni ðk  1Þ  0:1587ni ðk  2Þ  0:3109ni ðk  3Þ  0:0510ni ðk  4Þ þ vðkÞ ð13Þ In (13), m(k) represents the Gaussian distributed white noise process. In order to satisfy the assumption of uncorrelatedness, each trial of EEG is simulated from the steady-state response of an independently initialized autoregressive process [19]. Figure 2b shows a typical EEG trial generated using (13). The covariance of the driven noise m(k) was scaled to provide an SNR of -20 dB. A set of 90 trials of responses thus formed into a matrix for implementation. In order to perform the proposed WaNe-NAF filtering, an appropriate reference signal x(k) is required. It is obtained by using (14) in all simulation examples. xi ðkÞ ¼

L1 X 1 xj ðkÞ L  1 j¼0; j6¼i

ð14Þ

where L is the number of single trials used for obtaining reference input. Since each noise signal is uncorrelated for

eðkÞ ¼ dðkÞ  UwðkÞ U jeðk  1Þj gðkÞ ¼ ð1  j Þ 2 k þ jaðkÞj kU k

•

aðkÞ ¼ dðkÞ  Uwðk  1Þ

• •

s(k) in desired signal) with five typical peak components is shown in Fig. 2a. It was obtained by averaging 1,024 trials of recorded BAEP data from a human subject and mixed with an ongoing EEG signal (corresponding to n(k) in primary input desired signal) simulated by the following autoregressive model [19];

yðkÞ ¼ UwðkÞ

3 Simulation results and discussion In these simulation experiments, two different data sets are used, which are simulated and real measured BAEP signals. We have first created and used the simulated BAEP signals for an objective evaluation of the proposed algorithm in tracking of trial-to-trail variations. We have then used the real human BAEP signals for showing the extraction of BAEP patterns with the reduced number of estimated single trials. 3.1 Results for simulated BAEP signal In order to evaluate the signal-to-noise-ratio (SNR) improvements and convergence performance of the proposed WaNe-NAF, a simulated data set is created. A simulated deterministic BAEP signal (corresponding to

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Fig. 2 Simulated data. a The simulated BAEP signal. b The simulated one trial of ongoing EEG signal (-20 dB)

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i = j, it can readily be verified that (14) provides a reference signal with a high SNR. By the (14), the assumption of uncorrelatedness is also satisfied. To illustrate the tracking performance of the proposed algorithm, the generated 90 trials of BAEP signals were reformed with both latency and amplitude variations by a systematically shifting and scaling procedure and used in these simulations. Figure 3a shows the simulated BAEP signal (s(k)) with artificially introduced latency and amplitude variations. Since the latency variation is not abrupt in real single-trial recording series, the latency variations in the simulation data are made gradually. On the other hand, considering the amplitude variations in real recordings could be abrupt, the amplitudes are abruptly amplified at five selected regions. The overall signal mixed with ongoing EEG at -20 dB is shown in Fig. 3b. Although we have used this degenerated signal, one more modification is also possible. To illustrate tracking performance of the proposed WaNe-NAF algorithm, it is comparatively performed for both LMS- and LST-based algorithms. In the filtering application, j = 0.3 and k = 0.001 are chosen for WaNeNAF with LST and g = 0.01 is chosen for WaNe-NAF with LMS. Although the proposed algorithm’s performance increases with the filter order increment in the MSE sense, it is observed that the U matrix in algorithm becomes more seriously ill-conditioned along with the increase in the filter order greater than 30. Accordingly, the filter order is set to 10 after several trials. Quadratic biorthogonal B-spline wavelet function [32] is used as a nonlinear function in both algorithms due to its characteristic availability to BAEP signals. The algorithm is coded and performed by using Matlab software on a PC computer with the speed of 1.83 GHz. Figure 4 shows the filter outputs. The missed peaks in contaminated signal are successfully extracted from the contaminated signal (Fig. 3b) by using WaNe-NAF algorithms. As shown in Fig. 4, LST-based WaNe-NAF filter output has a much better consistency in peak components than the LMS-based WaNe-NAF filter output. It means that the ongoing EEG has all been eliminated by the proposed WaNe-NAF with LST. When we compare the filtered BAEP responses in Fig. 4a with the filtered responses in Fig. 4b, the LST-based algorithm performs much better than the LMS-based one. The tracking performance of the proposed WaNe-NAF algorithm with LST is evaluated by comparing the five BAEP components’ (e.g., Wave I–Wave V) latencies and amplitudes between the simulated BAEP signal and the filter outputs. Figure 5a shows the latency tracking performance, and Fig. 5b shows the amplitude tracking performance. The straight lines in Fig. 5 are simulated (original) latency and amplitude variations. The dotted

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Fig. 3 a The simulated BAEP signal with artificially introduced latency and amplitude variations. b Raw BAEP data obtained mixing the simulated BAEP with the simulated EEG data at SNR of -20 dB

Fig. 4 3D view of estimated BAEP signals, respectively, by a WaNe-NAF with LMS and b WaNe-NAF with LST

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Fig. 5 Comparison of tracking performance of the proposed WaNe-NAF with LST algorithm: a tracking performance on latency variation and b tracking performance on amplitude variation. Solid lines represent original simulated BAEP signals’ latencies and amplitudes, and dashed lines represent the estimated BAEP signals’ latencies and amplitudes

lines represent estimated latency and amplitude variations by the proposed LST-based WaNe-NAF algorithm. As shown in Fig. 5, the solid lines and dashed lines are coincided, which means that the WaNe-NAF with LST perfectly tracks the latency and amplitude variations. The performance of the proposed WaNe-NAF algorithm is evaluated for different noise levels. Different SNRs for the simulated BAEP recordings are created to compare the filter performance for different noise levels. The SNR is varied from -35 to 0 dB with the increments of 5 dB. The number of trials is 256. The conventional LMS [27], RLS [27] and the proposed LST-based algorithm are comparatively performed in WaNe-NAF to process the above-used simulated recordings. The mean MSE across trials is calculated for each algorithm at different noise levels (Fig. 6). As a result, the proposed

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WaNe-NAF with LST is superior to the other two algorithms at all SNR levels. To compare the convergence dynamic of the proposed WaNe-NAF algorithm, we have performed the WaNe-NAF with LST for different j values ranging from 0 to 1 with the increment of 0.1. In Fig. 7, we just give the convergence performance of the algorithm for three different j values. Filter order is set to 10 for the comparison. It has been observed from (11) that the adaptation gain rate parameter j should be in range, 0 \ j \ 1. Otherwise, the filter tends to diverge. Although small positive j close to zero may lead to a fast convergence rate, no significant change is observed on the performance of the proposed algorithm for j values smaller than 0.3. So, we set j value to 0.3 in our experiments. It is also clear from the Fig. 7 that a few numbers of trials are enough to reach an acceptable MSE level. So, this

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1207

LMS RLS LST

MSE (dB)

20 10 0 -10 -20 -35

-30

-25

-20

-15

-10

-5

0

SNR (dB)

Fig. 6 Performances of the WaNe-NAF with three different algorithms for different noise levels. The MSE values are calculated trial by trial and then were averaged to produce the plot

0

MSE (dB)

-4

κ=0.8 -8

κ=0.5 -12

κ=0.3 -16 -20

0

20

40

60

80

Number of Iterational Trials

Fig. 7 Convergence performance of the WaNe-NAF algorithm

result infers that the proposed algorithm can be used for obtaining AEP patterns with reduced number of single trials. So, the overall recording time will also be drastically reduced. It will be a great advantage in AEP acquisition for clinical applications. 3.2 Results for human BAEP signal

data points (or equivalently 10 ms) were stored for later processing. The first set of 200 single trials recorded from first subject was drawn from a female normal hearing person who is 24 years old. Figure 8a shows the ensemble averaged of measured 200 single-trial signals, and Fig. 8b shows the averaged of first 64 single-trial estimates with LST-based WaNe-NAF. Although the single-trial numbers of estimated BAEP (64 single trial) are less than the singletrial numbers of ensemble averaged BAEP (200 single trial), the estimated BAEP waveform (Fig. 8b) containing five peak components is very clear, and the peak latency and amplitude can be readily measured. This result is quiet consistent with ensemble averaged waveform. Thus, the number of single trials is drastically reduced for obtaining a BAEP waveform. It indeed means that the measuring time will be reduced if the proposed filtering approach is used in BAEP data acquisition systems. The second set of human data was recorded from second subject who is a male person age 54 years at six different stimulation levels. Each set of 200 trials were used. Here, the validation of the proposed WaNe-NAF is evaluated by the comparison between WaNe-NAF results and conventional ensemble averaged BAEP signals at six different stimulation levels. Figure 9a and b show the original raw BAEP signals obtained by using a conventional ensemble averaging method with 200 trials and estimated BAEP signals with 64 trials, respectively. As seen from the

(a) 0,5

V

III

0,4

Amplitude

30

IV

I II

0,3

VI

0,2 0,1 0,0

0

2

4

6

8

10

8

10

Latency (ms)

(b)

0,5

V

III

0,4

Amplitude

In this subsection, the proposed LST-based WaNe-NAF algorithm is performed to estimate real measured human BAEP signals. The BAEP trials to be used were recorded from two different subjects. BAEP recordings were originally measured by using SMART-EP interface that is a dual processor evoked potential acquisition and processing system [33]. The two adult normal hearing subjects were elicited with rarefaction clicks (100 ls) monaurally at 20-Hz repetition levels. The recordings were obtained differentially from (?) vertex and (-) ipsilateral mastoid with contralateral mastoid serving as ground. Measured signals were amplified 100,000 times. The BAEP signal was sampled at 40 kHz, and two sets of 200 trials were drawn from two subjects, separately. All records were obtained in a sound-treated chamber with subjects lying on a bed comfortably. For each response trial, the first 400

IV

I II

0,3

VI

0,2 0,1 0,0

0

2

4

6

Latency (ms)

Fig. 8 BAEP waveforms obtained by a ensemble average of measured 200 single trials and b ensemble average of estimated 64 single trials

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Fig. 9 BAEP signal estimates obtained from; a EA, b WaNe-NAF with LST. The signals are measured for six different stimulation levels such as S1 (80 dB nHL), S2 (70 dB nHL), S3 (60 dB nHL), S4 (50 dB nHL), S5 (40 dB nHL) and S6 (30 dB nHL)

figures, WaNe-NAF can effectively be used to enhance the BAEP signal with drastically reduced number of trials. This infers that one of the most important advantages of the proposed WaNe-NAF is the significant reduction in the trials and so more less recording time. This is a very important result for clinical applications. Although the linear and Wiener-like filter techniques have been used in tracking variations [6, 8, 18], they are not effective methods for improving AEP estimation due to the overlap of related signal and noise spectrums [3, 17]. Quiroga and Garcia [12] proposed a wavelet transformbased denoising procedure for event-related potentials. But the used signals are about one-second length with lowerfrequency synthetic signals. The patterns are clearly visible in long-latency event-related potentials even if it is raw, whereas the short-latency BAEP signals are quite noisy according to it. Some other techniques are also used in single-trial AEPs such as independent component analysis

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[11, 15], statistical analysis [6, 14] and parametric methods [9, 16]. Although they have improved the results in some extents, they all neglect the latency shifting (time delay) and amplitude variations after filtering processes. Qiu et al. [5] proposed an RBF-based AEP tracking system. Although they have improved the single-trial estimation, their method could not track the latency and amplitude variation at all [5]. ANN-based filter techniques [5, 19] are also used in AEP estimations, but they are generally based on gradient descent algorithms. However, the gradient descent–based searching may be trapped at local minima of cost function, and, if the signal has a large bounded input disturbance, global minimum point may not be found. Unlike the gradient descent–based algorithms, the proposed LST-based WaNe-NAF searches the global minimum along the energy surface in the state space with a unique solution through the adaptive adjustment of the filter weights as timescale goes to infinity. The error between desired reference input and filter output then asymptotically converges to zero. It must also be noticed here that we have also aimed the reduction in single trials for BAEP pattern extraction. On the other hand, all studies mentioned above did not aim such reduction process for extraction of BAEPs. It is an important advantage of the proposed system that differs from other studies mentioned above. Comparison of our WaNe-NAF tracking system with other tracking systems given in the literature is difficult due to the varieties in the system types, architectures and data sources (e.g., data types (short latency, middle latency and long latency), displaying montages, degrees of artifact presence, recording type, status of subject, etc.). Therefore, a direct comparison is difficult due to these reasons. Nevertheless, when we evaluate the performance of our system, the proposed WaNe-NAF with LST not only shows excellent performance to track trial-to-trial variability in BAEP signal, but also it contributes to obtain a perfect BAEP patterns by averaging single-trial estimates. The simulation results show that the proposed WaNe-NAF algorithm can be used in a possible fast BAEP acquisition system after some standard clinical experiments done by the AEP medical experts. It may be a future work as a realtime implementation in a neuro-sensory laboratory.

4 Conclusion In this paper, we have designed a nonlinear WaNe-NAF filtering system providing the stability in the sense of Lyapunov for tracking the latency and amplitude variations. The proposed system not only tracks both the latency and amplitude variations in BAEPs, but also reduces the number of single trials in BAEP pattern extraction process. Simulation results show that it can be used as a signal

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enhancer for a better illustration of some meaningful peak components having clinical values. It must also be noticed here that the proposed WaNe-NAF algorithm can be used in the extraction of BAEP by averaging the drastically reduced number of estimated single trials. Consequently, we suggest that it can be suited as an effective module (real time) for a faster data acquisition system. Because the reduction in single-trial numbers results in a less timeconsuming real-time measuring of BAEPs. Acknowledgments This study has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) with the ¨ zproject number of TUBITAK-105E084. We also appreciate Dr. O ¨ zdamar from University of Miami and his group in Neurosencan O sory Engineering Lab. for obtaining a part of human BAEP data.

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