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Fig. 3. Compound muscle action potential�(t) was decomposed via theintermediate waveformsy(t), �(t), and�(t) into a component� (t) due to theIAP spike and a component� (t) due to the afterpotential.�(t) approximatesthe temporal weighting functionW (t).

V. DISCUSSION

We have presented a simple model of the IAP as the sum of a spikeand a slow repolarization phase, both of which are expressed as smoothanalytic functions. A key feature of the model is that the parametersa

andb (which specify the relative sizes of the IAP spike and afterpo-tential) are proportional to the quadrupole and dipole moments of thesource distribution associated with the spike of the wave of excitation.These two parameters (along with the temporal weighting function)largely determine the shape of the MUAP at distant recording sites.They can also be reliably estimated from the MUAP waveform usingthe parameter estimation method presented in Section IV. The other de-tails of the IAP spike, including its width and asymmetry, have only aminor effect on MUAP shape. (The shape of the IAP spike does havean important effect on the shape of the leading edge of the MUAP, asdescribed in [8]).

The IAP model has two practical uses. First, it provides an effi-cient way to compute the MUAP in forward simulations using (6) and(7). Second, it provides a way to analyze MUAPs, including MUAPsrecorded noninvasively on the skin surface, to extract information aboutmotor-unit architecture (reflected in the temporal weighting function)and the muscle-fiber membrane (reflected in the IAP parameters) [7],[8].

REFERENCES

[1] G. V. Dimitrov and N. A. Dimitrova, “Influence of the afterpotentials onthe shape and magnitude of the extracellular potentials generated underactivation of excitable fibers,”Electromyogr. Clin. Neurophysiol., vol.19, pp. 249–267, 1979.

[2] G. V. Dimitrov, Z. C. Lateva, and N. A. Dimitrova, “Model of the slowcomponents of skeletal muscle potentials,”Med. Biol. Eng. Comput.,vol. 32, pp. 432–436, 1994.

[3] G. V. Dimitrov and N. A. Dimitrova, “Precise and fast calculation ofthe motor unit potentials detected by a point and rectangular plate elec-trode,”Med. Eng. Phys., vol. 20, pp. 374–380, 1998.

[4] D. Dumitru, J. C. King, and W. E. Rogers, “Motor unit action poten-tial components and physiologic duration,”Muscle Nerve, vol. 22, pp.733–741, 1999.

[5] Z. C. Lateva and K. C. McGill, “The physiological origin of the slowafterwave in muscle action potentials,”Electroencephalogr. Clin. Neu-rophysiol., vol. 109, pp. 462–469, 1998.

[6] H. P. Ludin, “Microelectrode study of normal human skeletal muscle,”Eur. Neurol., vol. 2, pp. 340–347, 1969.

[7] K. C. McGill and Z. C. Lateva, “Decomposition of the compoundmuscle action potential into source and weighting functions,” inProc.18th Annu. Int. Conf. IEEE Eng. Med. Biol. Soc, 1996, pp. 938–940.

[8] K. C. McGill, Z. C. Lateva, and S. Xiao, “A model of the muscle actionpotential for describing the leading edge, terminal wave, and slow after-wave,” IEEE Trans. Biomed. Eng., pp. 1357–1365, Dec. 2001.

[9] A. Persson, “The negative after-potential of frog skeletal muscle fibers,”Acta Physiol. Scand. (suppl. 205), vol. 58, 1963.

[10] G. Slavcheva, V. Kolev, and N. Radicheva, “Extracellular action po-tentials of skeletal muscle fiber affected by 4-aminopyridine: a modelstudy,”Biol. Cybern., vol. 58, pp. 235–241, 1996.

[11] B. K. van Veen, H. Wolters, W. Wallinga, W. L. Rutten, and H. B. Boom,“The bioelectric source in computing single muscle fiber action poten-tials,” Biophys. J., vol. 64, pp. 1492–1498, 1993.

Coherence-Weighted Wiener Filtering of SomatosensoryEvoked Potentials

J. S. Paul, A. R. Luft, D. F. Hanley, and N. V. Thakor*

Abstract—In this paper, we present a Wiener filtering (WF) approachfor extraction of somatosensory evoked potentials (SEPs) from the back-ground electroencephalogram (EEG), with sweep-to-sweep variations in itssignal power. To account for the EEG power variations, WF is modifiedby iteratively weighting the power spectrum using the coherence function.Coherence-weighted Wiener filtering (CWWF) is able to extract SEP wave-forms, which have a greater level of detail as compared with conventionaltime-domain averaging (TDA). Using CWWF, the components of the SEPshow significantly less variability. As such, CWWF should be useful as animportant diagnostic tool able to detect minimal changes in the SEP. In anexperimental study of cerebral hypoxia, CWWF is shown to be more re-sponsive to detection of injury than WF or TDA.

Index Terms—Coherence weighted Wiener filter, noise power, Wienerfilter.

I. INTRODUCTION

The use of optimal filters in the mean-squared sense, i.e., Wiener fil-ters (WFs) have been widely used for the extraction of somatosensoryevoked potentials (SEPs) [1]–[3]. Assuming that the measured wave-form f(t) contains two components: the signal or evoked potential,s(t) and the noise or ongoing EEG activity,n(t), the following as-sumptions are made to derive the filter: 1) The expected value of thenoise is zero, 2) noise in one epoch is uncorrelated with the noise in anyother and 3) the average power is the same for each epoch of noise. Forthe case of nonstationary evoked potentials (EPs), de Weerd and Kapp[4], [5] derived a time-varying filter based on the Wiener approach.They obtained time-varying power spectra by means of a bank of filters

Manuscript received January 18, 2001; revised July 5, 2001 This work wassupported by the National Institutes of Health (NIH) under Grant NS24282.Asterisk indicates corresponding author.

J. S. Paul is with the Department of Biomedical Engineering, Johns HopkinsSchool of Medicine, Baltimore, MD 21205 USA.

A. R. Luft and D. F. Hanley are with the Department of Neurology, Anesthe-siology, and Critical Care Medicine, Johns Hopkins School of Medicine, Balti-more, MD 21205 USA.

*N. V. Thakor is with the Department of Biomedical Engineering, Johns Hop-kins School of Medicine, 720 Rutland Ave., Baltimore, MD 21205 USA (e-mail:nthakor@bme.jhu.edu).

Publisher Item Identifier S 0018-9294(01)10242-9.

0018–9294/01$10.00 © 2001 IEEE

1484 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 12, DECEMBER 2001

with bandwidth proportional to their center frequency. The problem ofnonstationarities have also been addressed by extending the Wiener fil-tering in the framework of time-varying digital filter, by using discreteWavelet transform [6]. None of these articles, however, have addressedthe basic question about the sweep-to-sweep variations in the powerof background EEG. The most frequently used method in such casesis the technique of weighted averaging [7], [8], whereby the epochswith low noise activity are weighted more heavily than epochs withhigh noise activity. The performance of weighted averaging is largelyaffected by the random nature of the empirical weighting factor. In thispaper, we formulate the problem of weight determination based on thecoherence estimated from two consecutive epochs in the ensemble. Weshow that the use of coherence weighted averaging in the framework ofWiener filters [coherence-weighted Wiener filtering (CWWF)] tends toimprove the signal-to-noise ratio (SNR) of inhomogeneous ensembleof epochs such as that present during transient hypoxic condition. Ouroverall goal is to demonstrate through experimental and analytical workthat CWWF is helpful in extracting brain’s electrical response to injury.

II. A POSTERIORIWIENER FILTER FOR EP EXTRACTION

In the evoked potential field, there have been some attempts to buildan optimal filter in the least mean-squared sense, i.e., the WF, under theassumptions of additivity and independence of signal and noise [1]. Thefilter originally formulated by Walter [2] has the transfer function

H1(!) =Ss(!)

Ss(!) + Sn(!)(1)

whereSs(!) is the spectral density of the signal andSn(!) is the spec-tral density of the noise. For processing the average event related po-tential, the Wiener filter becomes [3]

H2(!) =Ss(!)

Ss(!) +1

NSn(!)

(2)

whereN is the number of responses used to obtain the average. Theseformulations are noncausal in that their inverse transforms are nonzerofor negative time. This estimate of the WF is a zero phase-shift digitalfilter, ranging from zero to one, which passes those frequency regionswhere the signal has a larger spectrum than the background noise notphase-locked to the stimulus. There are a number of subtleties involvedin the application of WFs to the EP data. A major source of difficulty isthe necessity of estimating the spectral densities of the EP and the noise(ongoing EEG). Frequently, these estimates are obtained by averagingthe magnitude square of the spectra of the individual responses. Let themeasured waveform for theith stimulus be

xi(t) = s(t) + ni(t) (3)

and let the ensemble average�x(t) be

�x(t) =1

N

N

i=1

xi(t)

=s(t) +1

N

N

i=1

ni(t): (4)

For the case of uncorrelated noise and signal, it is readily shown that

S�x(!) =Ss(!) +1

NSn(!) (5)

Sx(!) =Ss(!) + Sn(!) (6)

whereS�x(!), Ss(!),Sn(!)n, andSx(!) are the spectral densities ofthe ensemble average�x(t), the signals(t), the noisen(t)n, and the

random processx(t), respectively. Solving these equations forSs(!)andSn(!) gives

Ss(!) =N

N � 1S�x(!)�

Sx(!)

N � 1(7)

Sn(!) =N

N � 1[Sx(!)� S�x(!)]: (8)

The WF function is then given by

H(!) =Ss(!)

Ss(!) +1

NSn(!)

: (9)

The power spectral densities estimated as the squared magnitude ofthe Fourier Transform tend to have unnecessarily large variances. Asan alternative, we may use the Fourier transform of the autocorrelationfunction as an estimate of the respective power spectrum.

III. M ODIFIED AVERAGING USING COHERENCEWEIGHTING

The accuracy of the filtered output is increased if the filter is ableto account for those frequency regions with a larger amount of back-ground noise. To achieve this, the power spectrum is calculated itera-tively with the inclusion of each additional recording into the ensemble.With this procedure, the effect of outliers or other artefacts entering intothe ensemble is reduced. The coherence function xy of two stationarytime sequencesx(k) andy(k) is defined as [9]

xy(!) =Sxy(!)

[Sxx(!)Syy(!)](10)

whereSxx(!) andSyy(!) are the autopower spectra of the signalsx(k) andy(k) andSxy(!) is the cross spectrum betweenx(k) andy(k). The coherences represent the degree of correlation between thedifferent frequency components of the two sequences. In the processof averaging, it is important to give more weighting to the frequenciesthat are highly correlated than the rest. This is accomplished by multi-plying the power spectrum of each vector in the ensemble with coher-ence spectrum estimated between the new time sequence and the recentaverage. Additionally, the noise spectrum is weighted in a complemen-tary fashion to reduce the influence of noise for those frequencies withlesser degree of correlation. Thus the ensemble averaging equations forthe ith ensemble becomes

S�x(!; i) =i� 1

iS�x(!; i� 1) +

1

i (!; i)Sx(!; i) (11)

and

S�n(!; i) =i� 1

iS�n(!; i 1) +

1

i(1� (!; i))Sx(!; i) (12)

where (!; i) is the spectral coherence computed for the recentmemberSx(!; i) and the previous averageS�x(!; i � 1). With theiterative update, the variance of noise should ideally reduce with thenumber of iterations. The averaging decreases the noise in a mannerdirectly proportional to the standard deviation of noise in theith

step and inversely proportional to the number of replications. In theabsence of any outliers in the data, the variances of the noise estimatesare related by

�2i � �2i�1 = mean!

(1� (!; i))Sx(!; i) �

�2i�1i

(13)

where�2i = 1=2� S�n(!; i)d! is the noise variance in theith

step. Since the noise variance has to reduce with each iteration, theoutliers in the ensemble can be eliminated by enforcing the condition

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 12, DECEMBER 2001 1485

Fig. 1. Sample somatosensory SEPs processed with one of three methods (TDA, WF, and CWWF) are shown for time steps of 5 min. CWWF enhances lowamplitude peaks as compared with conventional Wiener filtering or TDA.

TABLE I

�2i�1=i > mean

!(1� (!; i))Sx(!; i) . With the presence of

excess amount of noise or outliers, the average noise power tends toincrease for the next iteration. The coherence spectrum is therefore

weighted by a forgetting factor� before inclusion in (11) and (12).At each iteration, the filter is constructed using

H(!; i) =Ss(!; i)

Ss(!; i) +S (!;i)

N

: (14)

The filter function is obtained as the IDFT ofH(!; i). The powerspectrumSx(!; i+ 1) is then computed using the filtered estimate.

IV. EXPERIMENTAL PROTOCOL

Three adult male Wistar rats (250–350 g) were used. All procedureswere conducted according to NIH guidelines and were approved bythe institutional animal care committee. Anesthesia was induced withhalothane (3% in 50% O2/50% N2O) and maintained by constant-

1486 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 12, DECEMBER 2001

(a) (b)

Fig. 2. The figure compares variabilities for (a) amplitudes and (b) latencies of six distinct peaks in the somatosensory evoked potential (SEP) extracted by eitherTDA or CWWF applied to 17 sequential windows (100 sweeps each). CWWF is significantly more robust with regard to estimated latency and amplitude of SEPand is therefore considered superior to conventional TDA (p < 0:0001 for amplitude variation andp < 0:005 for latency variation).

volume ventilation (1 ml/100-g body weight) with 2%–2.3% Halothanein 30% O2/70% N2O. Stimulation electrodes (silver needles) wereplaced into the left forpaw, subcutaneously over the median nerve. Thescalp was longitudinally incised and reflected laterally to expose theskull. Four Burrholes were drilled into the bone at 3 mm lateral (left andright) and 1 mm anterior/3 mm posterior to bregma carefully leavingthe underlying dura mater intact. A fifth burrhole was placed in themidline, 7–9 mm posterior to bregma, to accommodate the ground elec-trode. Stainless steel epidural EEG electrodes were placed into the bur-rholes and connected to a cylindrical plug (Plastics One Inc, Roanoke,VA) via thin silver wires.The electrodes were connected to two pream-plifiers (gain�10) and amplifiers (Grass Telefactor Inc., MA, gain�20000, high-pass: 1-Hz, low-pass: 100 Hz, 60-Hz notch filter) usinga left versus right bipolar montage. All signals were digitized (Micro1401, CED Ltd., London, U.K.) and recorded onto computer hard disk.Electrical stimulation was applied using a constant voltage stimulator(Grass, MA) (5–10 V) at a frequency of 1 Hz. After recording baselineEEG for 30 min, oxygen concentration was reduced to 10% for 30 minto induce hypoxia.

V. EXPERIMENTAL RESULTS

All digitized SEP data were averaged from 100 trials with identicalstimulus parameters. Fig. 1 shows typical SEP waveforms obtainedby averaging 100 sweeps by 1) by time-domain averaging (TDA), 2)by usinga posterioriWF and, 3) using the CWWF. The latencies ofthe SEP turning points obtained using the three methods are listed inTable I. It is seen that peaks occurring after 25 ms are clearly enhancedby WF. In the TDA, the peaks following 25 ms are masked by thelow-frequency EEG artifacts. The peaks appearing in the 25–60 mswindow are the long-latency peaks due to the thalamo-cortical activity.It is observed that the CWWF is able to resolve these long-latencypeaks into components that represent different levels of cortical pro-cessing.

It is observed that the peaks N3 and P4 are clearly more visible inthe EP estimated using CWWF. To rule out the possibility that thesepeaks are due to noise, we computed the SEPs using windowed en-sembles placed at different time points in the record. Each ensembleconsisted of 100 sweeps. The variability in the latencies and amplitudes

obtained using the TDA and the CWWF are depicted in Fig. 2(a) and(b), respectively. Amplitudes of early components show significantlydiminished variability using CWWF as compared with TDA (F-test,P < 0:0001). With regard to latency, time-averaging gave stable re-sults for early components of SEP [Fig. 2(b)]. However, CWWF gen-erated more reproducible results for late-component latencies (F-test,P < 0:005).

Fig. 3(a)–3(d) demonstrates the changes in SEP waveforms observedduring transient hypoxia. Each figure represents SEPs averaged over100 epochs (i.e., 100 s). Since the ensembles are chosen from overlap-ping time windows, the four figures represent four overlapping timeframes extending for a period of 5 min following the onset of hypoxia.The variations in the peak amplitudes are brought about by the changesin the excitability of excitatory and inhibitory classes of cortical neu-rons resulting from the hypoxic insult. The general profile of amplitudevariation over a larger time period of 1600 s (approximately 26 min.)is shown in Fig. 4.

The amplitude measurements obtained by using CWWF indicate thatthe onset of hypoxia leads to a rapid decline of late SEP components[Fig. 4(b)], whereas early peaks remain unchanged [Fig. 4(a)]. The ini-tial reduction in the peaks referred to, as “decline” is probably due tothe reduced excitability of the cortical neurons with sustained feed-back from the inhibitory pathways in the initial transient period. Overa period of time, the excitability of the cortical neurons is graduallyincreased which explains the subsequent increase in the amplitudes ofthe late cortical peaks as shown in Fig. 4(b). This is in full accordancewith theoreticalexpectations: Late components which represent higherorder cortical processing of the stimulus, are more likely affected bymild hypoxia than early SEP components. Early components reflect thestimulus travelling through the afferent pathways; this transmission isless vulnerable to mild hypoxia.

VI. DISCUSSION

As compared with TDA, WF is shown to give more consistent esti-mates of amplitudes and latencies for SEPs extracted from successivesweeps. However, WF is not a promising solution for the cases wherenoise power varies during sweeps. Such variations can cause a reduc-tion in SNR of low amplitude peaks that occur at the trailing edge of

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 12, DECEMBER 2001 1487

Fig. 3. The changes in sample somatosensory SEPs processed with one of three methods (TDA, WF, and CWWF), during hypoxic condition, for time steps of 5min.

(a) (b)

Fig. 4. Temporal changes in the amplitudes during hypoxia, (a) for the less sensitive (the first three peaks) and (b) for the more sensitive (the last three peaks)peaks.

1488 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 12, DECEMBER 2001

the SEP. The information about such peaks could be an effective toolfor analyzing SEPs during injury. The major advantage of the CWWFis that it enhances the SNR of peaks shown by the conventional WF.

Coherence weighting fails to yield reliable estimates of the peaksin situations where the noise power changes abruptly from one sweepto the other. Such variations can be monitored during each step of theiteration. If the increase in noise power is within tolerable limits, themethod could still enhance the peaks by using an exponential weightingfactor in conjunction with the Coherence weights. Under stable ex-perimental conditions, it is a rare phenomenon for the noise powerto change abruptly during each sweep. With the availability of sev-eral sweeps for the analysis, it would be meaningful to eliminate thesweep where the changes cannot be accommodated into the frameworkof the iterative computations involved in the proposed CWWF filter.With slow variations in noise power, the CWWF serves as a better al-ternative than TDA or WF to enhance the finer details of the SEP.

REFERENCES

[1] J. P. C. De Weerd, “Facts and fancies abouta posteriori Wiener fil-tering,” IEEE Trans. Biomed. Eng., vol. BME-28, pp. 252–257, Jan.1981.

[2] D. O. Walter, “A posteriori Wiener filtering of averaged evokedresponses,”Adv. EEG Anal., Electroencephalogr. Clin. Neurophysiol.,pp. 61–70, 1969.

[3] D. J. Doyle, “Some comments on the use of Wiener filtering for the esti-mation of evoked potentials,”Electroencephalogr. Clin. Neurophysiol.,vol. 38, pp. 533–534, 1975.

[4] J. P. C. De Weerd and J. I. Kapp, “Spectro-temporal representations andtime-varying spectra of evoked potentials,”Biol. Cybern., vol. 41, pp.101–117, 1981.

[5] , “A posterioritime-varying filtering of averaged evoked potentials.II. Mathematical and computational aspects,”Biol. Cybern., vol. 41, pp.223–234, 1981.

[6] O. Bertrand, J. Bohorquez, and J. Pernier, “Time-frequency digitalfiltering based on an invertible wavelet transform: An application toevoked potentials,”IEEE Trans. Biomed. Eng., vol. 41, pp. 77–88, Jan.1994.

[7] M. Hoke, B. Ross, R. E. Wickesberg, and B. Lutkenhoner, “Weightedaveraging — Theory and application to electric response audiometry,”Electroenceph. Clin. Neurophysiol., vol. 57, pp. 484–489, 1984.

[8] B. Lutkenhoner, M. Hoke, and C. Pantev, “Possibilities and limitationsof weighted averaging,”Biol. Cybern., vol. 52, pp. 409–416, 1985.

[9] N. V. Thakor, X. Kong, and D. F. Hanley, “Nonlinear changes in brain’sresponse in the event of injury as detected by adaptive coherence esti-mation of evoked potentials,”IEEE Trans. Biomed. Eng., vol. 42, pp.42–51, Jan. 1995.