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Somatosensory Evoked Potentials (SEPs) are brain electrical physiological signals elicited by the direct electrical stimulation of peripheral nerves. In other words, SEPis viewed as the nerve electric response produced by spinal cord sending or receiving sensory information in response to a stimulus (Turner et al., 2003). SEPhas been widely used during the clinical testing and monitoring of the spinal cord and the central nervous system with the surface electrical stimulation. It can be said that the SEPis the most popular technique for intraoperative spinal cord monitoring in the operating room over 30 years (Nash et al., 1977; El-Hawary et al., 2006). However, in practice, the SEPsignals recorded in the operating theaters are always contaminated by severe background noises (Krieger & Sclabassi 2001). The factors which cause noises may be electrical, physiological, anesthetic, surgical or abrupt event such as cough, body movement or adverse response to the stimulus of the patients. Generally, the recorded SEPsignal is of a very poor signal-to-noise ratio (SNR) nature of the typical values between -20 dBto 0 dB(McGillem et al., 1981).
Literature review of SEPextraction techniques showed that the Ensemble Averaging (EA) is the most commonly used practical technique for SEPextraction (MacLennan & Lovely 1995). Research studies reveal that the EA-SEPapproach is a kind of stimulus-locked signal averaging method, which is able to enhance the SNRin evoked potential recordings when a huge number of independent stimulus trails are used (such as hundreds or more than one thousands stimuli). This means that the EA-SEPextraction may lengthen the surgical time and hinder the surgical procedures (El-Hawary et al., 2006). Furthermore, EA-SEPapproach is lack of ability to provide the timely warning of the eminent danger of cord injury in spine surgeon monitoring. In conclusion, the major drawbacks of EA-SEPapproach are: First, the assumption that the captured SEPsignals are truly deterministic and invariant between ensembles is dubious. Actually, a number of studies showed that SEPsare nonstationary and time-varying across stimulus trails (Nishida et al., 1993; Woody, 1967). Second, the procedure is very time-consuming, requiring up to 2000 ensembles to identify the SEPsignal, which causes the discomfort to the subjects, and brings larger opportunity for the interference to degrade the SEPextraction. Moreover, careful evaluation of the working principle of the EA-SEPmethod reveals that the averaging process may merge the details carrying the information of certain neurological function in SEP. With the analysis above, we can conclude that EA-SEPmethod may fail to track trial-to-trial variations both in latency and amplitude. A more effective and reliable technique is expected to minimize the number of trials for SEPextraction, and the single trail SEPextraction is desired.
A lot of researches have been carried out for SEPextraction and various signal processing techniques have been investigated, including parametric modelling, nonlinear filtering, wavelet transform, adaptive filtering and independent component analysis (ICA) (Lange & Inbar 1996; Wei et al., 2002). It can be seen that a large number of records are still required to obtain a qualitative estimation in parametric model, and the study by Lange and Inbar suggested that it may not be able to provide adequate estimation of SEP(Lange & Inbar 1996).
In recent years, the present authors and some other researchers intensively investigated on the SEPextraction using adaptive filtering technique (AF-SEP) (Lin et al., 2004) (Lin et al., 2004; Hu et al., 2005; Lam et al., 2005). Research results showed that AF-SEPperforms better compared with the EA-SEPor other parameter estimation approaches in either stationary or non-stationary situation, and AF-SEPwas recommended as the most appropriate method to improve SNRof SEP(Lam et al., 2005). Specifically, AF-SEPunder investigation usually employed the conventional linear transversal adaptive filter. There are two different structures have been proposed: one is the adaptive noise canceller (ANC) SEPextraction method (ANC-SEP) (Hu et al., 2005; Ren et al., 2009), another is the multi-filter SEPextraction method (MAF-SEP) for low SNRSEPestimation, where the ANCis used to remove the correlated noise in a primary signal and the uncorrelated noise, while the SEPcomponents enhancement is carried out by the adaptive signal enhancer (ASE). Experimental results have shown that MAF-SEPmethod can greatly reduce the number of input trials for SEPextraction (Lam et al., 2005). Adaptive filter theory tells that the different adaptive algorithms provide different filtering performances (Haykin, 2001). The least mean squares (LMS) based adaptive noise canceller SEPmethod (LMS-ANC-SEP) was found to be a fast, simple, and reliable SEPextraction method for intraoperative spinal cord monitoring (Lam et al., 2005). The LMSalgorithm is famous for its simplicity at the price of having a relatively slow convergence rate and sensitive to the noise disturbance. To speed up the convergence, a Recursive Least Squares (RLS) based ANC-SEP(RLS-ANC-SEP) extraction algorithm was developed and studied in (Ren et al., 2009), where the Least Square cost function has been employed. RLSis a stable and accurate adaptive filtering algorithm (Haykin, 2001) since it updates the estimate using all the past available information, instead of the instantaneous measurement and error values in LMS. Intensive experimental results demonstrate that the RLS-ANC-SEPextraction outperforms the EA-SEPand the LMS-ANC-SEP. It also showed that the RLS-ANC-SEPis much less sensitive to noise disturbance over its counterpart algorithms, but at the expense of a heavier computational load.
Some research has shown than the conventional adaptive filters minimizing least squares (LS) or mean square error (MSE) are very sensitive to non-Gaussian or impulsive noise (Chan & Zou, 2004; Hazarika et al., 1997; Kong & Qiu, 1999). This is of increasing importance in biomedical signal processing field. Kong and Qiu (Kong & Qiu 1999) have done some preliminary research on a latency change detection and estimation algorithm under α-stable noise condition. They showed that the adaptive time delay estimation (TDE) algorithms based on the least mean square criterion failed to give an accurate estimation of the latency changes in the EPsignal, and they employed the direct least mean square (DLMS) adaptive TDEalgorithm derived based on the direct least mean p-norm criterion proposed by Etter and Stearn (Etter & Stearn, 1981). Theoretical analysis and simulation studies concluded that the DLMSalgorithm is robust to the noises in EPsignals with both Gaussian and non-Gaussian distributions.
SEPsignals recorded in the operating room have illustrated the impulsive characteristics under certain circumstance, such as some orthopedic manipulations using saw, drill, bone taps or bone bits. Based on our knowledge, there is no research carried out for the SEPextraction under the impulsive noise environment.
In this research, we will investigate the incorporation of robust M-estimator in the adaptive noise canceller structure for the SEPextraction. A recursive least M-estimate SEPextraction algorithm named as RLM-ANC-SEPhas been developed by minimizing a robust M-estimator cost function. The performance of the RLM-ANC-SEP, RLS-ANC-SEP, LMS-ANC-SEP, and EAmethods regarding to SEPextraction will be evaluated and compared quantitatively.
In this section, the framework and the working principle of the adaptive noise canceller (ANC) using the finite impulse response (FIR) filter for SEPextraction is introduced. The SEPextraction system setup and data generation is presented accordingly. The SEPextraction methods using least mean square algorithm (LMS-ANC-SEP) and recursive least square algorithm (RLS-ANC-SEP) are provided for completeness and comparison purpose. The SEPextraction using the recursive least M-estimate (RLM) algorithm is derived and discussed at last.
2.1. Adaptive noise canceller (ANC) for SEP extraction
In Figure 1 (a), a block diagram of ANCfor SEPextraction is illustrated, which mainly consists of a primary channel and a reference channel. The primary channel receives the source signal which refers to the raw SEPrecording and can be modelled as
where x(n) is the true SEPsignal and v(n) represents the background noise and interferences. In Figure 1, the reference channel represents a noise source denoted as r(n), and e(n) is the output of the ANCsystem, which is considered as the estimated version of the true SEPsignal which can be formulated as
where the output of the adaptive filter is denoted as y(n)=wT(n)r(n). r(n)=[r(n),…,r(n-M)]Tand w(n)=[w0(n),w1(n),…,wM(n)]Tare the output, input data vector and weight vector of the adaptive FIRfilter (AF), respectively. The derivation of the adaptive filtering algorithm is governed by a meaningful cost function. As the result, after the convergence of the AF, the difference between the filter output and the desired response will be minimized.
It is worthy to note that for the ANCapproach for SEPextraction, there are some important assumptions for achieving global convergence of the adaptive filter and the unbiased estimation of the desired signal. Firstly, the desired signal (x(n)) is corrupted by an additive interference (or noise) (v(n)) to form the primary signal s(n); Secondly, if the reference signal (r(n)) is a correlated version of the interference signal (v(n)), then a FIRfilter can be applied to transform r(n) to approximate v(n) and then suppress v(n) from s(n), which is illustrated in Figure 1(b); Thirdly, the reference signal (r(n)) must not contain a correlated component of x(n), otherwise, the SEPsignal component may also be cancelled at the output of the ANC.Therefore,it can be concluded that in the study of SEPextraction under ANCframework, the SEPrecording and the reference signal generation must be designed carefully to satisfy the above requirements. Some discussion of the SEPextraction system setup will be presented in the next section.
2.2. SEP Extraction system setup and data generation
2.2.1. SEP extraction system setup and signals
In our SEPextraction study, the SEPextraction system setup is illustrated in Figure 2. The SEPsignals were collected over Cz’ (2 cm posterior to Cz, 10-20 international system for EEGelectrode placement) versus the Fzof the 10-20 system. The stimulation for SEPrecording was applied on the posterior tibial nerve with the duration of 0.3 ms, the rate of 5.1 Hz and the constant current of 10 to 30 mA. The signals were amplified one hundred thousand times, bandpass filtered at 20-3000Hz. The SEPsignals were acquired and recorded to a computer with 12-bit resolution and the sampling rate of 5 kHz. We collected 500 trials for one subject and then the average of these trials is taken as a standard SEPtemplate in our study, which is shown in the first row of Figure 3 as xn.
2.2.2. The primary channel signal and reference channel signal generation
It is observed that continuous EEGis the major source of noise found in the primary SEPrecording channel and its variation version in the reference channel (Lam et al., 2005). In our study, EEGis recorded over Czand A1(auricular) versus Fz, respectively from the awaken subjects at a sitting position in a quiet environment. This is because the A1-Fzrecording signal has much less SEPcomponent, and A1-Fzrecording is suitable used for the generation of the reference channel signal r(n) (MacDonald et al., 2005). Meanwhile, the Cz-Fzrecording was used to superimpose onto the SEPtemplate to generate the EEG-contaminated SEPsignal, that is, continuous EEGsignals of different SNRlevels were added to each SEPtemplate to generate the EEG-contaminated SEPtrials (the primary channel signal s(n) at different SNRlevel) for algorithm performance evaluation purpose. An autoregressive moving average (ARMA) filter was employed to simulate the correlated WGNnoise for the reference channel. One example of the A1-Fz EEGsignal and Cz-FzEEGsignal recorded from scalp is plotted in Figure 3 as vnand rn, respectively. The SNRof EEG-contaminated SEPwas set to −10, −15 and −20 dB. The last row of Figure 3 (sn) shows the simulated primary channel SEPsignal at -15dB with EEGand WGN.
2.3. Adaptive noise canceller for SEP extraction using mean square estimation
The SEP extraction method under the adaptive noise canceller (ANC) framework derived from the Mean-Square-Error was firstly introduced and evaluated by some of the present authors in (Lam et al., 2005). From the adaptive filter theory (Haykin, 2001) and the configuration of the ANCshown in Figure 1, the SEPextraction problem can be solved as a linear ANCproblem since the FIRadaptive filter is used (named as ANC-SEP approach in this study). The commonly used error measure is the Mean-Square-Error (MSEdefined as JMSE=E[e2(n)], where E[.] represents the ensemble operator). The minimisation of the MSEresults in the Wiener normal equation under some statistically independent and signal wide-sense stationary (WSS) assumptions. The optimal solution of Wiener normal equation can be denoted as:
where Pmseis the cross-correlation vector between s(n) and r(n), and Rmseis the autocorrelation matrix ofr(n), which can be written as
As discussed in many literatures, the well-known least mean squares (LMS) algorithm is a stochastic gradient based adaptive algorithm to obtain the optimal solution of JMSE. The updating of the adaptive filter coefficient vector can be denoted as (Haykin, 2001)
where μlmsis the stepsize which is one of the most important factors that controls the initial convergence rate and steady state error of the LMS-ANCfor SEPextraction. Generally, a big stepsize yields rapid convergence but larger steady-state misadjustment error. A small stepsize yields slow convergence but a corresponding smaller steady-state misadjustment error. There exists a theoretical lower and upper bound of the choice of μlms(details can be referred to (Haykin, 2001). Usually, the choice of μlmsis suggested by the following condition in the LMSalgorithm (Haykin, 2001)
where Pinand Mis the input power and the order of the adaptive FIRfilter, respectively. In principle, the selection of the stepsize not only depends on the desired steady-state error level but also the statistical properties of the input signal of the adaptive filter. In other words, the convergence rate of the LMSalgorithm is greatly affected by the dynamic range of the eigenvalues of the autocorrelation matrix Rmse. Considering this essential limitation, it is not difficult to understand that the performance of the ANC-SEPapproach using LMSalgorithm may suffer from the conflict to the WSSassumption for s(n) and r(n) and the nonstationary property of the r(n).
2.4. Adaptive noise canceller for SEP extraction using least square estimation
Motivated by the performance enhancement of the ANC-SEPmethod using LMSalgorithm compared with EA-SEP(Hu et al., 2005; Lam et al., 2005; Cui et al., 2008), some investigations of the ANC-SEPusing RLSalgorithm have been carried out and presented in (Ren et al., 2009). Instead of using MSEcost function, a conventional least square (LS) cost function is employed and the optimal solution of JLSis described as follows (Haykin 2001)
where, λis the forgetting factor with the value between 0 and 1, which controls the effective amount of data used in the averaging and hence the degree to which the RLSalgorithm can track the signal variation. The closer the value of λgoes to one, the lower will be the steady-state misadjustment error of the RLSalgorithm. Its tracking ability, however, will also be slower. RLS(n) is the autocorrelation matrix of the input vector at time index nand PLS(n) is the cross-correlation vector between the input vector and the reference signal at time index n. Generally, they can be estimated as
By applying the matrix inversion lemma to the optimal solution in (6), the famous recursive least square (RLS) algorithm can be derived, and it is summarized in Table 1 for the completeness (Interested readers can refer to (Haykin, 2001)). From Table 1, it is noted that the computational complexity of the RLSalgorithm is of order M2.
2.5. Adaptive noise canceller for SEP extraction using robust estimation
Carefully evaluating the properties of the recording SEPsignals in the operating room, it noted that these SEPsignals may have some nonstationary and impulsive like properties when the trial patients happen to the eye movement, cough and stimulus etc, which commonly exist. Under these kind of circumstances, the performance of the ANC-SEPmethods using LMSor RLSwill degrade or fail to extract SEPsignal due to the adverse effect of the noise. The new method is desired. Motivated by the research work done by Chan and Zou (Chan & Zou, 2004), a new error measure method based on the M-estimate has been introduced and the corresponding cost function instead of JMSEor JLSis used for providing the robustness in the algorithm, which is given as
where λis the positive forgetting factor and ρis an M-estimate function, which provides certain ability to suppress the adverse effect of impulsive noise on the cost function when the error signal becomes very large. In our study, the Huber M-estimate function and the related weighting function are used, which can be denoted as
where ξis the threshold parameter. The optimal solution w(n) for minimizing JR(n) can be obtained by differentiating (10) with respect to w(n) and setting the derivatives to zero. This yields the following M-estimate normal equation
where, RR(n) and PR(n) are called the M-estimate correlation matrix of r(n) and the M-estimate cross-correlation vector of r(n) and s(n), respectively. The adaptive algorithm for solving the normal equation (12) can be obtained in the same way as developing RLSalgorithm, and the resulting algorithm is called recursive least M-estimate algorithm (RLM) and it is summarized in Table 2. From Table 1 and Table 2, it can be seen that the computational complexity of RLMand RLSis similar except the cost to determine the weighting function q(e) in (15). It is also noted that when the signal is Gaussian distributed, RLSand RLMare identical. The contribution of the weight function q(e(n)) lies at the suppression of the adverse effects of the large estimation error due to the undesired impulsive interference on the adaptive filter weight vector w(n). The degree of this suppression is controlled by the parameter ξ,in our study, a recursive estimation approach is adopted which directly connects to the variance of the estimation error under the assumption of the interference is with contaminated Gaussian (CG) or alpha-stable distributions. The parameter ξhas been determined (shown in Table 2) when there is 95% confidence to detect and reject the impulses (Chan & Zou, 2004).
3. Simulation study and discussion
As discussed above, we have introduced the SEPextraction approaches under the ANC framework by using different adaptive filtering algorithms. Specifically, employing LMS, RLSand RLMalgorithms to update the weighting vector of the adaptive FIRfilter in ANC results in the LMS-ANC-SEPmethod, RLS-ANC-SEPmethod and RLM-ANC-SEPmethod, respectively. In this section, the performance of these adaptive filtering methods for SEPextraction under Gaussian and impulsive noise environment has been evaluated and compared by intensive simulation experiments.
3.1. Experiment 1: SEP extraction under Gaussian noise
In this section, we aim to visually illustrate the SEPextraction performance of the algorithms discussed above under Gaussian noise condition. The detailed performance comparison between EA-SEP, LMS-ANC-SEP, RLS-ANC-SEPand RLM-ANC-SEPmethods under EEGand WGNcontamination can be referred to the work presented in papers (Lam et al., 2005) and (Ren et al., 2009). Here, we only illustrate one set of the SEPextraction results for reader’s favorite review. Figure 4 shows the SEPextraction results from 50 SEPtrails by different algorithms. In this experiment, the SEPtemplate (xn), simulated primary signals (sn, vn) and reference signal (rn) are the same as those shown in Figure 3 at SNR=-15dB. The order of the adaptive filter Mis set to be 10, the step size μof the LMS-ANC-SEPis chosen as 2x10-4, the forgetting factor of the RLS-ANC-SEPand RLM-ANC-SEPalgorithms is set to be 0.99. The parameters for RLM-ANC-SEPin Table 2 are set as λσ=0.9 and Nw=7. From Figure 4, it is clear to see that the signals extracted from 50 trials by EA-SEPand LMS-ANC-SEPare difficult to detect the positive and negative peaks required for quantitative analysis and diagnosis of the SEPsignal. More precisely, the positive peak around 35ms and the negative peak around 40ms, which are two most commonly-used criteria for the online monitoring during the spinal surgery, are still buried in the heavy background noise, so that their latencies and amplitudes cannot be measured accurately. On the other hand, we can see that the performance of RLM-ANC-SEPis almost the same as that of RLS-ANC-SEP, which outperforms than other two algorithms. It is apparent that two peaks around 35ms and 40 ms can be easily observed and their latencies and amplitudes can be precisely measured in the results using RLS-ANC-SEPand RLM-ANC-SEPmethods. All these findings in practice can be well explained in theory. That is, the RLS/RLM-based algorithms have a fast convergence rate than LMS-based algorithm. Furthermore, the RLM-ANC-SEPalgorithm is comparable to RLS-ANC-SEPalgorithm under EEGand WGNenvironment. We next test and compare their performances when few SEPtrials are contaminated with impulsive noises.
3.2. Experiment 2: SEP extraction under impulsive noise
This simulation is set up to compare the SEPextraction performance of the EA-SEP, LMS-ANC-SEP, RLS-ANC-SEPand RLM-ANC-SEPunder EEGand individual impulse contaminated noise environment. Generally, the impulsive noise can be generated by a contaminated Gaussian (CG) model proposed in (Haweel & Clarkson, 1992). The impulses are generated individually with arrival probability Par=2×10-3 and the variance is chosen as 200. In our study, only for performance illustration purpose, the positions of the impulses are assumed to occur at 19ms, 28ms, 35ms, 44ms, and 78ms, respectively (which is not necessary to fix the position of the impulses, but here it is for us to gain the better performance visualization for different algorithms). The SEPtemplate (xn), one sample primary interference (vn) with impulses, one sample of the reference signal (rn) and the resultant primary signal (sn) at -15dB are shown in Figure 5. The difference between Figure 3 and Figure 5 only lies at several impulses added in the primary interference signal (vn). In this case, the primary signal is composed of a SEPtemplate, an A1-Fz EEGcomponent, and a contaminated Gaussian noise.
For this simulation, all parameter settings are the same as those used in Experiment 1. The SEPextraction results from 50 SEPtrails under impulsive noise by different algorithms are shown in Figure 6. If no impulsive noise occurs, the extraction results of four different methods should be approximately identical to their counterparts in Figure 4.As a result, Figure 5 can be regarded as a standard to evaluate the robustness of these methods when impulsive noises are added. From Figure 6, it is clear to see that the adverse impact of the impulses on the SEPextraction for EA-SEP, LMS-ANC-SEPand RLS-ANC-SEPalgorithms compared with their counterpart algorithms under WGNshown in Figure 4. More specifically, for the EA-SEPmethod, since the amplitudes of the impulsive noise are rather large compared to that of WGN, they cannot be averaged out completely using finite number of trials. As for the LMS-ANC-SEPand RLS-ANC-SEPmethods, which employ an LScriterion for the updating of the filter coefficients in ANC, their performances are degraded severely because the coefficient estimates in ANCare unstable and may be greatly deviated from the reasonable values when impulsive noise occurs. The performance degradation can be more easily observed in the result of RLS-ANC-SEPin Figure 6, where the adverse impacts of impulsive noises around 35ms and 44ms are distinct and its difference with RLS-ANC-SEPof Figure 4 is obvious. Unlike those methods based on averaging or LScriterion, RLM-ANC-SEPemploys an M-estimation function in ANCso that the impulsive noise can be detected and suppressed effectively. As the result, its harmful impact on SEPextraction is reduced considerably. The simulation results illustrate the advantage of RLM-ANC-SEP, and we can see that RLM-ANC-SEPshows its robustness to the impulsive interferences and its performance is close to that under WGNcondition. In Figure 6, we can hardly find the traces of impulsive noise in the RLM-ANC-SEPresult and peaks were clearly seen and measurable. In a simple word, impulsive noise which degrades the outputs of EA-SEP, LMS-ANC-SEPand other LS-based SEPextraction methods will do little harm to RLM-ANC-SEP.
As mentioned before, impulsive noise often occurs during spinal surgery in operating theatres and it will greatly decrease the quality of SEPrecording. Current SEP recording technique works in this way when some SEPtrials are contaminated with impulsive noise, they will be discarded. However, these trials with impulsive noise also contain useful SEPinformation, and the rejection of these trials will increase the time to record a useful SEPsignal, and make the recording and monitoring discontinuous, which is undesirable. Therefore, making use of SEPtrials contaminated with impulsive noise is necessary and robust SEPextraction method, such as the proposed RLM-ANC-SEPmethod, is advantageous. Our preliminary study and experimental results show that the RLM-ANC-SEPmethod has an excellent performance in impulsive noise environment, it may be taken as a good solution to achieve reliable and continuous SEPrecording for monitoring under Gaussian and impulsive noise environment.
Aiming at developing the efficient SEPrecording system, we have introduced the SEPextraction methods under the ANC framework using adaptive FIRfilter. A new SEPextraction method called RLM-ANC-SEPwas developed to obtain the the fast and robust performance under Gaussian and Contaminated Gaussian noise environment. RLM-ANC-SEPminimizes the modified Huber M-estimator based cost function instead of the conventional mean square error and least squares error based cost functions, which provides the robust ability when impulses occurring in the primary channel, and maintains the fast convergence as the RLS-ANC-SEPalgorithm. Simulation study proved that either RLM-ANC-SEPor RLS-ANC-SEPhas better and more robust convergence performance than LMS-ANC-SEP. The performances of RLM-ANC-SEPand RLS-ANC-SEPshowed equivalent under WGNcondition, but RLM-ANC-SEPpresented its robustness to the impulsive interferences. Clinical application and validation study could be our future work on this proposed SEPsignal extraction approach.
This work was partially supported by Shenzhen Science and Technology Program (No. 08CXY-01), Hong Kong ITF Tier 3 (ITS/149/08), and Research Grants Council of the Hong Kong SAR (GRF HKU7130/06E).
Yuexian Zou, Yong Hu and Zhiguo Zhang (July 5th 2011). Fast Extraction of Somatosensory Evoked Potential Based on Robust Adaptive Filtering, Adaptive Filtering Applications, Lino Garcia, IntechOpen, DOI: 10.5772/19293. Available from:
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