Wavelet Analysis of EPs

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The wavelet method applied to the analysis of category-related
components in visual evoked potentials of men

KARL-HEINZ G. DITTBERNER¹, LEONARD FUHRY²,
OTTO-JOACHIM GRÜSSER , ANDRZEJ W. S. T. PRZYBYSZEWSKI³,
and MARGITTA SEECK²

Department of Physiology, Freie Universität Berlin, Germany
and
Biophysics Research Institute,
Department of Comparative Physiology, Rijksuniversiteit,
Utrecht, The Netherlands

First release:  March 1, 1993
Revised release:  February 12, 1995
First Web release:  February 20, 1999

1	Address: Dipl.-Ing. K.-H. Dittberner, Institut für Physiologie, Freie Universität Berlin, Arnimallee 22, D-14195 Berlin-Dahlem, Germany. E-Mail: dit@mail.grumed.fu-berlin.de [Ed - 10.12.2004: all addresses are obsolete – new e-mail]
2	Present address: Department of Neurology, Ludwig Maximilian Universität, D-81377 München. E-Mail: u7x31an@SunMail.lrz-muenchen.de
3	Present address: Department of Cognitive & Neural Systems, Boston University, MA, USA. E-Mail: przy@bu.edu

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    In the last few years face-responsive components have been demonstrated by means of electroencephalographic recording techniques in the visual evoked potentials (or event-related potentials, EPs) of healthy subjects (Bötzel and Grüsser, 1987 , 1989 ; Jeffreys and Musslewhite, 1987 ; Bötzel, Grüsser, Häussler and Naumann, 1989 ; Jeffreys, 1989 ; Seeck, Heusser and Grüsser, 1989 ; Grüsser, Kirchhoff and Naumann, 1990 ; Jeffreys and Tukmachi, 1992 ; Jeffreys, Tukmachi and Rockley, 1992 ; Seeck and Grüsser, 1992 ; Jeffreys 1993 ). Recently three different cortical sources of the face-related components were identified from the responses of the magnetoencephalogram (Lu, Hämäläinen, Hari, Ilmoniemi, Lounasman, Sams and Vilkman, 1991 ), indicating that the EEG-potentials evoked by face stimuli are a complex composed of neuronal signals distributed to different parts of the brain. In the course of a further analysis of averaged EPs we searched for statistically reliable methods to analyse the category-related components of the EPs. In addition to the traditional peak-analysis, we preferred methods in which no subjective decisions were necessary, like those required when certain EP-peaks with their amplitudes and latencies have to be determined. Since inspection demonstrated that the EPs following a change in the patterned visual stimuli consist of several aperiodic oscillatory response components, it was interesting to apply in addition a suitable method for separating the different temporal frequency ranges appearing in the EPs within a few hundred milliseconds after the stimulus onset. The so-called wavelet analysis seems to be an appropriate tool for this purpose.

    Combes, Grossman and Tchamitchian (1989 ) summarized the application of the wavelet transformation, a method developed predominantly for the detection and analysis of transient periodic signals in the physical sciences. Recently, one of us applied the wavelet method to the analysis of action potential sequences in the flash-evoked responses of retinal ganglion cells (Przybyszewski, 1991 ). We tested this method for the analysis of averaged visual EPs, which we obtained in studies in which complex black-and-white visual patterns, such as photographs of faces, persons, tools, flowers, animals etc., constituted the set of stimuli. We will describe this method and illustrate its applications to selected data, which have also been analysed according to traditional methods and published in part (Seeck and Grüsser, 1992 ). A comparison of the present analysis with the published data indicates the advantage of the wavelet method. The method described in the present report would also be applicable to the analysis of other continuously recorded biological signals, such as magnetoencephalograms, changes in nerve cell membrane potentials etc., provided that these changes are related to stimuli with a sharp onset. Short descriptions of the wavelet method applied to the analysis of EPs have been presented at previous meetings (Przybyszewski and Grüsser, 1991 ; Seidler, Przybyszewski, Dittberner and Grüsser, 1991 ) and in Grüsser and Landis (1991 ). Bartnik, Blinowska and Durka (1992 ) demonstrated that by means of adequately selected wavelet functions individual EPs can be analysed; in this study the wavelet selected was used as a highly specialized temporal frequency filter. Recently Tiitinen, Sinkkonen, Reinikainen, Alho, Lavikainen and Näätänen (1993 ) used a complex Gabor function (modulated Gaussian) to compute wavelet transforms of auditory evoked potentials.

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    Subjects. Eighteen unpaid students (19–28 years of age) of both sexes and mainly from the faculty of medicine, whose right-handedness had been confirmed by the Edinburgh Inventory (Oldfield, 1971 ), volunteered for the experiments. They were not informed about the purpose of the study, but participated in the experiments because of their interest in learning EEG recording techniques.

    Stimulation. The subject sat in a moderately comfortable chair and rested his/her head against the back. The recording and stimulation equipment was in a separate room (fig. 1 in Bötzel and Grüsser, 1989 ). The subjects were instructed to relax and fixate the centre of the vertical white projection screen, 120 ¥ 180 cm, placed two meters away at eye level. The projection field for the black-and-white slides was 3.9 ¥ 5.9 degrees visual angle. The stimuli consisted of a sequence of slides belonging to four stimulus categories (faces, persons, flowers and tools) and were projected in a semirandom order onto the projection field for 2.5 to 4 seconds, whereby no slide was followed by one of the same category. As a rule, 160 different slides, equally distributed over the stimulus categories, formed a set of stimuli. A few of the items shown were incomplete. In such a case the subjects had to press a lever. This procedure guaranteed a fairly constant level of attention during the 2 ¥ 10-minute recording periods (cf. Seeck and Grüsser, 1992 ).

    The slides were projected alternately from one of two carrousel slide projectors precisely onto the same field. A horizontally moving shutter, driven by a pneumatic system and computer-controlled electromagnetic valves, alternately opened and closed the beams of the two projectors (fig. 1 in Bötzel and Grüsser, 1989 ). Due to the construction of the shutter, the moment one light beam was turned off, the other was turned on. Switching from one stimulus pattern to the next took less than 6 ms; average luminance to successive stimuli was kept constant at about 6 cd/m². Shutter movement and slide change in the two carrousel projectors were controlled by a digital computer. By means of a small light source, the signals of two photocells and two “coding” holes in the slide frame, each slide could be assorted by the computer into one of four stimulus categories. The set of respective stimuli applied in the experiments was the same for all subjects and is described in RESULTS.

    Electroencephalographic recordings. The electroencephalogram was recorded by means of Beckmann Ag/AgCl-electrodes attached to the scalp of the subject with Beckmann EEG electrode paste. The skin was prepared beforehand by rubbing with a cleaning paste. After a few minutes the electrode resistance was measured and was usually below 2 kW. In all recordings the reference electrodes consisted of “linked mastoids” or “linked ear lobes”, i. e. two electrodes connected by 2.2 kW resistors. Responses were recorded from the electrodes Cz, Fz, F3, F4, T5, T6 (Jaspers, 1958 ). The EEG signals were amplified by Grass P 511 amplifiers (passband 0.1–100 Hz); in addition a DC-electrooculogram (EOG) was recorded (0–100 Hz). The EEG and EOG signals were digitized (12 bit resolution for the amplitude) with a sampling rate of 200 Hz per channel, stored on magnetic disc and evaluated off-line by a HP-1000 computer. The programs allowed backward and forward averaging of the EEG signals from the moment of slide change. Sweeps containing eye movements or blinks were rejected before averaging. EPs were averaged separately according to the stimulus categories beginning 100 ms before the stimulus change and continuing over 900 ms. Only EEG-responses to complete items were analysed in the present study. EPs to incomplete items to which the subjects were supposed to respond by pressing a lever were excluded; the same was true when false positive motor responses were evoked. The digitized data were stored on digital tape and could be used later for further computation (individual averages, grand averages and standard deviations or standard errors, category-related difference signals, difference signals between electrodes, i. e. “bipolar recordings”, etc.).

    From wavelet transforms to “contour-plots”. The rather general concept of a wavelet (Grossmann and Morlet, 1984 ; Mallat, 1989 ; Grossmann, Kronland-Martinet and Morlet, 1990 ; Mallat and Zhong, 1992 ) comprises all complex square integrable functions f(x) of finite energy if their Fourier transform F(w) is differentiable and fulfils the admissibility condition, which reads in the Fourier space

(1)

    This condition means essentially that f(x) is of zero mean. It implies that F(0) = 0, F(±•) = 0, and

(2)

    More recently Koenderink and van Doorn (1990 ) recommended using the term “wavelet” for the n-th order derivative of the Gaussian function g(x) with variance s and the mean value m:

(3)

which reads in the normalized form with m = 0:

(4)

    From such a set of wavelets we selected as the analysing wavelet the negative second-order derivative of the normalized Gaussian function of time h"(t), which is called the “Mexican hat” function (fig. 1a):

(5)

Figure 1a: Normalized Gaussian function h(x) (dotted) and its negative second derivative, the normalized “Mexican hat” function h"(x) with s = 1. [ Download as (22 kByte) ]

    For s = 1 this function was simplified to the basic wavelet y₁(t), i.e. the unshifted and undilated wavelet function

(6)

    This wavelet function has the best possible simultaneous concentration in the time domain and in the frequency domain, which means that the time-bandwidth product is as small as possible. This property makes this wavelet most suitable as a variable temporal bandpass filter in the intended analysis of EPs. As required in general for the wavelet analysis, eqs. (5) and (6) fulfil the admissibility condition eq. (1) and the so-called compatibility property eq. (2). For the wavelet application we generated the other members of the wavelet family by translating y_s(t) by t (t > 0) and dilating y_s(t) by s, obtaining the collection

(7)

    We computed the wavelet transform W(t, s), defined as a modified convolution integral of the wavelets in eq. (7), with the averaged evoked potential e(t), whereby W(t, s) depended on the width s of the wavelet family:

(8)

    Hereby is t the time of the evoked potential analysed and t the shift between e(t) and the wavelet y_t,s(t) applied in the process of computing the convolution integral. In computing eq. (8), W(t, s) was replaced by its discrete version with e[n] as the sampled sequence of EEG potentials with N points sampled at the instants k Dt. The exponent m in eq. (9) was set at 3 (see below):

(9)

    For a selected set of wavelet transforms s was varied systematically and for the computation the width of the wavelets was set at ±4 s (fig. 1a). The digital convolutions of eq. (9) were computed and displayed in a three- dimensional W-diagram (fig. 5), yielding a continuous measure of frequency-specific components of the EPs over time. Varying s of eq. (9) also corresponded, of course, to the application of a variable temporal frequency filter. By trial and error we found the value 3 to be the most adequate for m in eqs. (8) and (9). In this computational procedure the smaller s, the higher is the gain of the frequency filter and the smaller the bandwidth (fig. 1b). The wavelet transforms with the “Mexican hat” function lack an orthonormal basis. Since no inverse transformation was intended, this does not affect the present analysis. Recently Mallat and Zhang (1993 ) and Davis, Mallat and Zhang (1994 ) demonstrated that orthogonality of the wavelet function, which considerably restricts the families of possible wavelet functions, is not a necessary condition when applying wavelet analysis. They suggested using “redundant dictionaries” of wavelet functions and selecting the most “effective” wavelet for the respective task on the basis of its analytic success.

Figure 1b: Relation between the amplitude of the wavelet transform and the width of the wavelet for sinewave signals of different frequencies (2–50 Hz). [ Download as (22 kByte) ]

    To perform the wavelet analysis we used a small VAX cluster (running the virtual multi-tasking system VMS) configured with a server (VAX-3500), a microVAX-II with two 1 GByte disks, a colour work-station (VAX-3200, running X-Windows), a Macintosh computer (Mac-II), and a PostScript laser printer, all netted together by Ethernet. The pre-checking of optimal algorithms was done on the Mac-II under Mathematica™ (Wolfram, 1991 ) using the Georgia Tech package SignalProcessing (Evans, McClellan and McClure, 1990 ). The final programming for production on the VAX computers was implemented under IDL™, the Interactive Data Language (Stern, 1991 ), a 4th generation language designed especially for signal processing purposes. IDL supports directly the discrete convolution of signals with a kernel, and the Fourier transform. All graphic output was realized in the PostScript™ language (Adobe Systems, 1986 ). For binding and resizing the plots into documents we used the portable representation as Encapsulated PostScript (EPS-files).

    The W-diagrams were computed either separately for the grand averages of EPs recorded through selected electrodes or pairs of electrodes (“bipolar recordings”) and for the different categories or from the category-difference curves obtained by subtracting the averaged EPs related to pairs of stimulus classes (e.g. “faces–tools”).

    The problem in evaluating the W-diagrams was to find a reliable method to compute the statistical significance of the peaks and troughs. To this end we compared the category-related W-diagrams or the category- difference- related W-diagrams with W-diagrams obtained after random shuffling of the experimental data. A computer program was written that guaranteed the formation of 30 different sets of randomly shuffled averaged EPs for the EPs recorded during each experimental paradigm and electrode, related in approximately equal frequencies to the subjects participating in one set of experiments and to the 4 stimulus categories applied. This procedure of random shuffling was performed for the individual averaged “monopolar” or “bipolar” EPs related to the stimulus categories. It resulted in EPs related to a “change in stimuli” irrespective of the stimulus categories. The random shuffling across categories and subjects should result in identical grand averages for the “change in stimulus”-EPs. Due to the random fluctuation of the experimental data, however, the 30 randomly shuffled grand averages differed from each other. Computing algebraic means and standard deviations of a set of 30 randomly shuffled EPs provided a reasonable estimate of the “noise” present in our data analysis (fig. 4a).

    When category-difference curves were evaluated, random shuffling was performed from the set of all 12 possible category-related difference curves. With the latter procedure, 0-values are expected theoretically for the random- shuffled means. Indeed the randomly shuffled curves were near 0 (fig. 4b). What was interesting in this analysis, however, was the standard deviation found for the 30 means obtained by random shuffling.

    For the randomly shuffled averaged EPs as well as for the randomly shuffled category- related difference curves we computed the wavelet transforms for 20 £ 8 s £ 300 ms. The averaged W-diagrams resulting from these convolutions and the selected statistical limits (L-values) were displayed. As a rule, the ±3 s.d.-limits were taken as the critical significance range for the evaluation of the W-diagrams obtained from the category-related EPs or the category-difference curves respectively. The computer program, however, also enabled us to apply other L-values (2.6 s.d. £ L £ 4.0 s.d.). In our standard procedure all regions (peaks and troughs) exceeding the ±3 s.d.-limit of the randomly shuffled W-diagrams were considered to deviate significantly from the randomly shuffled EPs (P < 0.003) and were taken as indicators that the average EPs contained category-responsive components. In the display of the W-diagrams all parts exceeding the ±3 s.d.-limit or other L-values were enhanced (fig. 5). The same procedure was applied for the category-related difference curves (fig. 6).

    Finally the points of intersection between the category-related W-diagrams and the selected limits of the randomly shuffled W-diagrams were displayed in a two-dimensional contour plot projected to the plane formed by the width of the wavelet and the time t related to the onset of the stimulus (time-frequency domain of the EPs). In this C-diagram small wavelet widths correspond to high frequency components of the EPs and large wavelet widths to low frequency components of the EPs. These contour plots (C-plots, fig. 7) represent the regions of significant differences between the wavelet transform of the category-related EPs and the average randomly shuffled EPs. The procedure for the W-diagrams obtained for the category- related difference curves and their contour plots was the same (fig. 8).

    In the following we will demonstrate the feasibility of this method in analysing category- related visual evoked potentials using examples selected from our studies of the last few years (e.g. Seeck et al., 1989 ; Grüsser and Landis, 1991 ; Seeck and Grüsser, 1992 ).

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Category-related EP-components

Figure 2: Grand averages of category-related EPs (bipolar recordings through electrodes F4–T6) from 18 subjects and 4 stimulus categories (a: faces, b: persons, c: flowers, d: tools). The algebraic mean (thick curve) and the ±3 s.e.-range are displayed. The similarity in the pairs of curves presented in (a) and (b), and in (c) and (d) respectively is obvious. [ Download as (37 kByte) ]

    Fig. 2 illustrates the grand averages of EPs (±3 s.e.) obtained in 18 subjects in a set of experiments in which 4 different stimulus categories (black-and-white photographs of faces, persons, flowers, and tools, cf. Seeck and Grüsser, 1992 ) were applied. The EPs stem from bipolar recordings through the electrode pair F4–T6. Fig. 3 shows the six category-related subtraction curves of the EPs {faces–persons, faces–flowers, faces–tools, persons–flowers, persons–tools, flowers–tools}. It is evident from figs. 2 and 3 that category-related stimulus differences appear predominantly between 150 and 350 ms after the change in stimulus pattern. Hereby the difference in the EPs {faces–persons} was significantly smaller than the EP-differences for {faces–flowers} and {faces–tools}. The EP-difference {flowers–tools} was very small, while {persons–flowers} and {persons–tools} contained marked differences around 200 ms after stimulus change. Similar results were obtained with other electrode pairs, but the responses recorded through the electrodes F4–T6 showed the strongest category-related differences and will be used in this demonstration of the wavelet analysis as a very efficient method for obtaining two-dimensional pictures of a multiscale (an inside) view of the EPs.

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Figure 3: Category-related difference curves (algebraic mean ±3 s.e.) as recorded from electrodes F4–T6 for the four stimulus categories (cf. fig. 2). While non-zero curves appeared within the first 400 ms whenever faces or persons were involved, the difference flowers–tools (f) exceeded the 0-values only marginally. [ Download as (53 kByte) ]

    Recordings as illustrated in figs. 2 and 3 led to the conclusion that the EPs did indeed contain category-related responses. In a preceding study (Seeck and Grüsser, 1992 ) these differences were evaluated by using traditional statistical methods such as a “running” Wilcoxon-test and the evaluation of the algebraic means of the amplitudes and the latencies of peaks and troughs. In particular for the evaluation of the peaks and troughs appearing in the individual averaged curves, a subjective decision as to what constitutes a “peak” or a “trough” was inevitable. We therefore computed W- and C-diagrams (cf. METHODS), a procedure by which no subjective selection had to be made. Before this procedure, the computation of randomly shuffled data as decribed in METHODS was necessary (fig. 4).

Figure 4: (a) Average curve for the EPs (F4–T6) shuffled across categories and subjects according to a random program. 30 randomly shuffled averaged EPs were computed and the algebraic mean ±3 s.d. of these 30 curves is presented. (b) Same procedure as in (a) but for category-related difference-curves (fig. 3) shuffled across all possible category combinations (cf. text). The average is about 0 as anticipated. The ±3 s.d.-limits of the 30 randomly shuffled curves used for later evaluations (computing the significant regions) are displayed. [ Download as (21 kByte) ]

Wavelet transforms, W-diagrams and C-plots

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Figure 5: W-diagrams W(t, s) computed for 20 £ 8 s £ 300 ms of the EPs recorded through electrodes F4–T6 (fig. 2, further explanation see text). The values of the W-diagrams exceeding the ±3 s.d.-limit of the W-diagram from the EPs after random shuffling across categories (fig. 4a) are shaded in darker red. For viewing details click on every figure. [ Download as (950 kByte) ]

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Figure 6: W-diagrams W(t, s) of category-related difference curves (fig. 3). Otherwise same conditions as in Figure 5. [ Download as (1391 kByte) ]

    Figs. 5a–d illustrate the W-diagrams obtained for the four categories in the responses recorded through the electrode pairs F4–T6, and figs. 6a–f the W-diagrams obtained for the category-related difference curves of fig. 3 computed for all 18 subjects and for electrodes F4–T6. In all W-diagrams the regions that exceeded the ±3 s.d.-range of the randomly shuffled W-diagrams – randomly shuffled across stimulus categories and subjects – are enhanced by darker colours (cf. METHODS). Figs. 7a–d and figs. 8a–f display the corresponding contour plots (C-plots). Six conclusions can be drawn immediately from these figures:

1. W-diagrams and their simplified representation in the C-plots reveal distinct and significant (P < 0.003) category-related components in the EPs.

2. In relation to the changing point in the stimulus pattern, the strongest category-related EP-components appeared about 80–130 and 160–230 ms after the stimulus change with faces and persons. The EPs evoked by flowers deviated only slightly around 170 and 230 ms from the randomly shuffled EPs, while with tools a significant peak appeared in the EPs around 300 ms.

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Figure 7: Contour plots of fig. 5. Dark grey areas correspond to ranges of the W-diagrams significantly below the set of randomly shuffled W-diagrams, those in light grey to ranges above. For further explanation see text. [ Download as (22 kByte) ]

3. The face- and person-related components in the EPs were strongest in wavelet transforms with 100 £ 8 s £ 300 ms, i.e. in a temporal frequency range between 12 and 35 Hz (see below).

4. For the temporal frequency domain above 35 Hz approximately, no indication was found that a distinct frequency – i.e. a 40 Hz-signal for example – contributed essentially to the category- related averaged EPs. This frequency band should have appeared quite easily, since the higher the frequency, the more the wavelet analysis applied enhances the frequency components of the responses (see above).

5. The W-diagrams and C-plots of the category-related difference curves indicated that EPs obtained with faces and persons were very similar to each other and revealed small differences around 90, 160, 400 and 460 ms (fig. 8a). Similarly, only minor differences in the EPs appeared when flowers and tools were the stimuli (fig. 8f) around 230, 310, 410 and 480 ms. All other category differences were very prominent and extended along four or five bands between 80 and 500 ms (figs. 8b–e).

6. When the width of the wavelets was changed to 100 £ 8 s £ 750 ms, no new significant bands appeared in the W-diagrams and the C-plots.

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Figure 8: Contour plots of fig. 6. For further explanation see text. [ Download as (36 kByte) ]

C-plots obtained with different statistical limits

    It was possible, of course, to apply other L-values characterizing the statistical limits instead of the ±3 s.d.-range of the randomly shuffled W-diagrams. Fig. 9 illustrates in one selected example {faces–flowers} how the C-plots change with different L-values. As in all statistical analyses, the investigator decides which limits of uncertainty he is willing to accept for data evaluation. The advantage of the C-plots obtained with different L-values varying between ±2.0 s.d. and ±5.0 s.d. is the availability of a graphical representation, easily indicating the consequences of the statistical certainty factor on data evaluation.

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Figure 9: Example of contour plots using different statistical limits L varying between ±2.0 s.d. (P < 0.05) and ±5.0 s.d. (P < 0.000001). [ Download as (36 kByte) ]

C-plots obtained with different recording sites

    Fig. 10 illustrates how the C-plots of bipolar EP recordings with different electrode combinations and the category difference {faces–tools} vary. It is evident that in the time-frequency domain, as determined by the time after stimulus change (abscissa in the C-plot) and the width of the wavelet applied (ordinate), varying regions were found to be significantly related to the category difference selected to illustrate the methods. Finally, of course, graphs such as those in fig. 10 can be constructed for all category differences and recording sites.

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Figure 10: Distribution of the category-related EP components across the scalp using the example {faces–tools}. The contour plots are obtained from different bipolar recordings of the EPs using the ±3.0 s.d.-limit (P < 0.003). Dark grey areas correspond to ranges of the W-diagrams significantly below the set of randomly shuffled W-diagrams, those in light grey to ranges above. For further explanation see text. [ Download as (44 kByte) ]

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    Applying the wavelet method to the analysis of category-related EPs necessitates some effort in programming. With suitable programs the computation of the W- and C-diagrams can be accomplished within a few minutes using a medium-sized work-station. Without any specific assumptions on the position of peaks and troughs, as necessary in the traditional analysis of the peaks in N1, N2, N3 and P1, P2, P3, it provides an analysis of category-related components in evoked potentials, event-related potentials or category-related EP-difference curves computed from the category-related EPs. W- and C-diagrams are easily readable graphic presentations. The statistical significance of the category-related components in the W-diagrams is obtained by relating the W-diagrams to those resulting after random shuffling across categories and subjects. The wavelet analysis is also applicable to non-averaged EPs (Bartnik et al., 1992 ), but noise in the recording will limit its use in human visual evoked potentials related to complex visual stimuli. It will be shown in a later report that the wavelet method is also applicable to the analysis of individual EPs from the epicortically recorded electroencephalogram of the monkey brain.

    Computing W- and C-diagrams is naturally not restricted to EPs or category-related difference curves derived from EPs. It can be applied to all time-variable continuously recorded biological signals evoked by external or internal stimuli. W- and C-diagrams allow a conjecture as to within which temporal frequency band the category-related stimulus deviations appear and at what point in time after the stimulus change. They provide graphic displays of the essential features of the responses.

    By varying s of eqs. (8) and (9) the wavelet method operates as a band-pass filter with variable bandwidths. To test the filter properties we computed W-diagrams for sinewave signals with frequencies relevant to the EP analysis. As demonstrated by Chui (1992 ) a linear relationship between the sinewave period T and the wavelet width (8 s) was valid for the maximum of the wavelet transform (in our case T = 8 s/1.155 [seconds]), which indicates that all relevant temporal frequencies of the EPs were covered by the W-diagrams and the C-plots when our analysis condition 20 £ 8 s £ 300 ms was applied (fig. 1b).

Acknowledgement: The work was supported by a grant from the Deutsche Forschungsgemeinschaft (Gr 161/35-1) and a “Postdoktoranden-Stipendium” of the DFG to M. S. (Se 520). During some of the research work the senior author (O.-J. G.) held the F. C. Donders-Professorship at the University of Utrecht. A. W. P. was supported in part by the Department of Comparative Physiology (Prof. Dr. W. van de Grind), Biophysics Research Institute of the University of Utrecht. We thank Mrs. J. Dames for expert help in the English translation of the manuscript, Mrs. I. Knierim for typing, Mr. P. Holzner for the photographic reproductions of the slides used as stimuli, Mr. J. Lerch and Dipl.-Ing. N. Nitert for mechanical and electronic assistance with the stimulation equipment.

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