9 87
IEEE T RA N SA CT I ON S ON B IOM ED I CA L EN GI N EERI NG , V OL . 46, NO . 8, A UGU ST 1999
A mplitudes and L atencies of Single-Trial ERP' s
Estimated by a M aximum-Likelihood Method
Piotr Jaskowski * and Rolf Verleger
Abstract The common approach in research on event-related
electroencephalogram (EEG) potentials is to assume that the
trigger-related signal is always the same and can be extracted
from EEG background activity by simple averaging. To check
the validity of this approach and to provide more exact results,
latencies and amplitudes of components have to be estimated in
single trials. Pham et al. applied a maximum-likelihood approach
to solve the more general model which assumes that the signal
In analyzing ERP's, the on-going EEG is treated as background activity or noise. Unfortunately, ERP signals are low
in amplitude with respect to the on-going EEG. Therefore,
some special approach is needed to obtain the ERP signal . The
simplest approach assumes that identical signals are produced
after every stimulus and that e, is the realization of a common
zero-mean stationary Gaussian process (cf . [4])
hidden in EE G background activity has the same shape and am-
plitude but may vary in its latency from trial to trial. Extending
their method we present a solution in which amplitude variability
is also allowed. The utility of the solution to estimate the P3
component in single trials was investigated both by extensive
pseudoreal simulations and in an application to real data. The
r, (t) = s(t) + e, (t).
(2)
U nder such ci r cum stances an unb i ased esti m ate of s can b e
obtained by averaging the sweeps r~
simulations showed some advantage of the method over two
other methods (W oody's m ethod and peak-picking) commonly
(t) = (t) = g p (t)
used in event-related potentials research. Application to real data
provided a plausible description of single-trial sequential effects
(3)
d= l
This model , referred to as standard model , i s commonly used
on the amplitude of the P3 component.
LECTRI CAL brain activity can be measured by electrodes fixed to the scalp. The spontaneous electrical signal
measured this way is called electroencephalogram (EEG).
in ERP research and has allowed many fruitful applications [1].
However, it is approximately true at best, because ERP's may
show significant differences between averages of subgroups of
sweeps [5] and may show systematic variation across sweeps
with regard to amplitude and latency of their components
[6] [8]. So other models have been considered (e.g., [9] [13]).
Pham et al. [ 14], taking into consideration the possibility
of trial-to-trial variations of signal latency, suggested a model
External or internal stimuli evoke responses which add to
which can be expressed in the f orm
I ndex Terms Event-related EEG potentials, maximal likelihood estim ation, single-trial analysis.
I . I N T ROD U CT ION
E
the spontaneous EEG. Such responses are called event-related
potentials (ERP). So, the j th repetition of the stimulus evokes
r, (t) = s(t + ~, ) + e, (t)
(4)
a signal, rd(t), which is assumed to be a sum of spontaneous where T, is the latency of the j th sweep. To find TI they
E E G an d E R P
rs(t) = sd(t) + es(t)
(j = 1, . . . , N ; t 6 [0, Tj) (1)
employed a maximum-likelihood approach.
The signal latencies are of special interest in ERP research,
in particular in research on the P3 component, for two reasons.
One reason i s to measure P3 l atency in the same way as
where sJ(t) is the ERP signal and ey(t) is the EEG signal,
both recorded at time t . Event-related potential s are commonly
employed in cognitive neuroscience as a very useful tool for
studying the timing and structure of brain processes [1]. They
have al so clinical applications (e.g., [2] and [3]). The most
widely studied component of the ERP is the P3 (or P300)
component, evoked by any stimulus if task-relevant and if
sufficiently well delineated from other stimuli [ 1].
Manuscript received December 15, 1997; revised January 27, 1999. Asteri sk
i ndi cates cor respondi ng author .
*P. Jagkowski is with the Department of Neurology, Medical University of
L uebeck , 23538 L uebeck, Germany , and the D epartment of Psychophysi ology ,
Pedagogical University of Bydgoszcz, 85-867 Bydgoszcz, Poland (e-mail :
j askow sk i p @neuro.M U -L uenbeck .de).
R. Verleger is with the Department of Neurology, Medical University of
L uebeck , 23538 L uebeck , Germany .
Publ i sher Item Identi fi er S 00 18-9294(99)05775-4.
manual response time, i .e., as a score in every trial (review:
[ 15]). A second reason, here of particular concern, is that
latency j itter across sweeps can affect the signal shape and
amplitude as estimated from (3). Thus, a smaller amplitude of
the averaged ERP for an experimental condition or for some
clinically defined group might be an effect of larger latency
j itter rather than of real variation of single-trial amplitude [16].
Variation of P3 latency across trials has so far been addressed by two other methods. These are Woody' s [ 17] method
and peak-picking. The former calculates the cross correlation
or (more often) the cross covariance between a template which
resembles the shape of the P3 component of the averaged ERP
(generally a half-sine wave) and the low-pass filtered singletrial sweep (e.g., [16] and [18]). Then the time by which
the template should be shifted to obtain the maximal cross
correlation or cross covariance corresponds to the latency shift
0 0 18 92 94/ 9 9 $ 10 ,0 0 )
199 9 I E E E
I EEE T RA N SA CT I ON S O N BI OM ED ICA L EN GI NEERIN G , V O L . 46, N O . 8, A U GU ST 1999
988
of a given trial . The latter method, peak-picking, is simply
looking f or the largest positive peak in the P3 time interval in
a given low-pass fi ltered trial (e.g., [ 19]).
U sing " pseudoreal" simulations (cf . below ) Pham et al.
[ 14] showed that their method provided better estimates of
single-trial latencies ~~ for early components (the Nl P2
complex) of the occipitally recorded visual ERP than Woody ' s
method. So far, their method has not been applied to the P3
component. In our simulations designed to find l atency j itter
of the P3 complex we found that it is only slightly better
than Woody ' s and the peak-picking method [20] . The reason
f or the advantage of Pham et al.' s method with the early
visual ERP i s probably that the biphasic N l P2 complex much
more resembles random fl uctuations of the background EEG
than the broad P3 component. Thus, with increasing noise
the performance of Woodys and of the peak-picking method
d et er i or at ed m u c h f ast er t h an P h am et a l . ' s m et h o d .
M odel (4) seems to be reasonable for short-latency ERP
components which are assumed to be automatically evoked by
external stimuli and depend mainly on their physical aspects.
H owever, late ERP' s depend strongly on i nternal cognitive
processes and may fl uctuate in amplitude even if the physical
stimuli and subj ects' task remain identical . For example, even
i n the simple two-stimulus oddball task, the amplitude of
the P3 complex to target stimuli depends on the preceding
sequence of target and nontarget stimuli [21] [23] .
Theref ore, in this report we want to show the maximumlikelihood solution of the problem w ith a more general model
i n whi ch also amplitude j itter i s allowed, i .e., we assume that
the signal while having an invariant shape, can vary both in its
l atency and in its amplitude. This can be expressed as f ollows:
r, (t) = a, a(t + T, ) + e, (t).
A. Esti mati on of Latenci es and Amp li tudes of the Si ngle Tri als
As suggested by Pham et al. [14], iterative Fisher scoring
[26] was used to find a~ and r ~. After the pth iteration,
a"+ = a" + 6a; and T"+ = ~" + &~, where
1
6a, = Reg
S*(ur) exp(ter,)
f(~) Ri(I a) a~
r)exp(riced
j~S
~(u
j~)
~ [S(~)[2
f (~ )
an d
1
S
~,=R
e+f(~)R
,(cu)a,S
(a)exp(turv.)
I
a2(u2~S(a)) )2
f (~)
while S and f , estimates of S and f , can be calculated from
these equations'
Q R(w) exp(uuTi )
S(u))
(5)
W e w i l l al so sh o w h o w t h e t w o o t h er m eth o d s b eh av e i n th i s
situation in comparison to the method developed below .
Pa,
.I
and
f (to) =
1
2
P R~(~) adS(to) exp( i+T) .eqno(7)
2
I I . L OG L IK EL IH OOD
Follow ing Pham et al. [ 14] we will consider the Fourier
With regard to ~~ and ai , model (5) i s not completely specified,
since one needs a reference f or defi ning latency of the signal
transform of the sweeps ri (t) rather than their time-domain and its amplitude. The reason is that v~ provides information
representation
R,. ((u) = a,.S(~) exp( i u)T, ) + E, (a))
(Sa)
where capital letters stand f or Fourier transf orms of time-
on how the sweeps are shifted with respect to one another only
and likewise ai on how the sweeps are scaled i n rel ation to
one another only. In order to make v.i and ai uniquely defined
one needs two additional constraints, such as P . r,. = 0
domain quantities and ~ = 0, 2' / T , 4x / T , . . .. Now, we
and (I /N) g ,. a, = 1. The former condition is equivalent
can treat E~(to) as independent complex, normally distributed
to defi ning the signal ' s latency as the average latency of
the single signal s, while the latter condition means that the
amplitude of the averaged and latency-corrected single signals
i s equal to one.
variables with mean and variance f (to), where f is the spectral
density of e~. The log likelihood for such a case was given by
Brillinger ([24] ; see also [ 14] and [25]):
I = p pi 'os(f(~))+(f(~)) '
I I I . S IM UL AT ION S T U D Y
4 J
.(R,(~) a,s(~) exp( ~~r,)~'j. (6)
We looked for the set of
(a, , T
unknown parameters 8
S(co), f (co)) which maximize log likelihood.
A . P seud o r ea l Si m u l a ti o n s
To investigate how effi cient the method is i n fi nding amplitudes and latencies of the P3 compl ex in single trial s we tested
' S* denotes the complex conj ugate of S.
9 89
JASKOWSKI AND VERLEGER: AMPLITUDES AND LATENCIES OF SINGLE-TRIAL ERP'S
it with simulated data. The pseudoreal simulations introduced
B. Data Recordi ngs
by Gasser and coworkers [27], [28] seem especially suitable
The data obtained by Verleger et al. [29] for 16 healthy
for this purpose. In this method, the signal morphology and controls were used. Recordings were performed during an
noise structure is preserved as closely as possible to reality auditory oddball task with 216 "nontarget" tones (86%) and
and, at the same time, the situation is f ully controlled. In 34 "target" tones (14%) presented in random order with an
the original pseudoreal simulations, N noise sweeps were
recorded before presentation of the stimuli. Each participant' s
average ERP was used as the signal and added to his/her
own noise sweeps. This way, N pseudoreal sweeps could be
obtained for each participant for which the signal latencies and
signal-to-noise ratios (SNR's) were a priori known and could
interval of 1.5 s between successive tones. The subj ects' task
be systematically varied.
for artifacts by a program which looked for blinks, zero lines,
out-of-scale values, and fast shifts larger than 100 /JV . EEG' s
were corrected by linear regression for blinks recorded by the
EOG electrodes placed above and below the right eye. Only
sweeps after target tones recorded from Pz were used in the
We had to modify this method slightly as we had no data
with suitably long recordings made before presentations of
st i m ul i .
~ The average ERP was calculated from real data
was to press a response key whenever the target was presented.
EEG signals were recorded by Ag/AgC1 electrodes affixed
to the scalp at Fz, Cz, Pz, and Oz and were sampled at 100 Hz
from 0.1 s bef ore stimulus onset to 1.0 s after stimulus onset.
So every sweep consisted of 110 samples. Data were screened
simulation study.
M ore d etai l s can be f o und i n [29 ] .
C . Sta ti sti c s
r , low -pass fi ltered w ith cutoff frequency of 3.5 H z, was
defi ned to be the ERP signal, s.
~ Noise was obtained in two steps. In the first step, s(t )
was subtracted from the sweep rJ.(t)
n (t ) = r .(t )
s(t ).
~ In the next step, ni (t ) was Fourier-transformed, its phases
w er e r an d o m i z ed an d t h e i n v er se F o u r i er t r an sf o r m w as
calculated. The noi se sweeps simul ated in such a way
will be denoted by rI,(t ).
~ pseudoreal sweeps were obtained by adding a percentage,
P, of noise sweeps that we will refer to as noise amplification, to the ERP signal as defined above with given
amplitudes a, and latency shifts ri
r (t ) = P n (t ) + a r (t + r ').
Signal shifting was done in the frequency domain by multiply-
ing the Fourier transform of rJ. by exp( i~rJ-). Then on each
side of the sweep, a cosine-taper was added, going smoothly
down to zero. These long sweeps had 256 samples instead of
As a statistic to appreciate the goodness of the method,
the correlation was computed between real and estimated
amplitudes and latencies. Further, for every data set we calculated the global SNR according to the formula given by
Fein and Turetsky [30, p. 390]. It should be recognized that
this estimator of SNR has only limited value for comparative
purposes because with noise and signal power held constant its
value depends on the time range of the signal within the data
epoch. Moreover, because the signal amplitudes vary from trial
to trial, SNR' s vary from trial to trial, too, while the formula
provides only one value for the whole data set.
D . Starti ng Condi tions, N umber of Iterati ons, and Components
The starting values of rJ. were set to zero and of aJ to one.
L ike Pham et al. [ 14] f ound with their original method,
performance of the present method was improved by iterating.
They argued that two iterations were enough but we found that
five iterations gave better estimates of the true values, with no
improvement by further iterations. Further, above a certain
frequency, the signal had negligible power. Thus, analysis
could be restricted to a few frequency components. After some
attempts, we found the best solution with ten components,
which corresponds to a cutoff frequency of 3.9 Hz: for small
110 samples of the original sweeps.
For every ERP data set we made 66 simul ations i n which the
noi se c ontam i nati on , correl ati on coeffi ci ents w ere even better
latency range (three levels) and noise amplification (11 levels)
did not converge, leading to very small correlations.
with less than ten components, but for l arge noise the process
were orthogonally varied. The latency shift for each trial
was sampled from a zero-mean positively skewed distribution
I V . R Es ULT s A N D D I s c v s s IQN
which w as the convolution of normal and exponential distributions. The standard deviations (SD ' s) of the distributions
In Fig. 1 the correlation coefficients between real and esti-
corresponding to the three levels of the latency range were 20,
mated values are plotted against log P and SD of the latency
45, and 72 ms. N oi se amplifi cation was varied from 0.0 1 to
100 (logarithmically : 2 to + 2), with one (logarithmically : 0)
range. Every point represents the average calculated from
the 16 subj ects' ERP data sets. As can be seen, correlation
denoting the relationship found in the real data. The single-
coeffi ci ent s w ere cl ose to one f or sm al l noi se contam i nati on
trial amplitudes for a given simul ation were sampled from a
normal di stribution with mean equal to one and SD equal to
0.5 (with negative values set to zero).
and dropped down dramatically when log noise amplification approached zero. They were better for estimates of (b)
ampl itude than of (a) latency, especially in the realistic log-
9 90
I EEE T RA N SA CT ION S ON BIOM ED ICA L EN G IN EERIN G, V OL . 46, N O . 8, A U G U ST 1999
1.0
1.0
0 .8
0 .8
0 .6
0 6
0
0 .4
3
I
0 .4
I
0
O
0 2
0 .2
0 .0
0 .0
-2 .0
- 1.0
0 .0
1.0
2 .0
logy
logy
(a)
( a)
1.0
1.0
0 .8
0 .8
0 .6
0 .6
0
C
0
at
at
Q4
L
Q4
0
L
o
L
0 .2
0
0 .2
0 .0
lat. range = 45 ms
0 .0
logy
(b)
(b)
Fig. l .
(a) Correlation betw een real and esti mated values of single-trial
latency and (b) amplitudes as a function of noise contamination (log 6) and
of latency range (as expressed by SD in ms). Interindividual SD' s are marked
0 .8
by bars ( I SD ), for better legibil ity for the tw o outermost values onl y .
0 .6
0
noise range around zero. Of less importance, correlations were
somewhat better when l atency w as allowed to vary more
(70 ms) than less (20 ms), probably because some slight error
of estimation, e.g., by 10 ms, w as large in relation to the true
shift when the true range of variation w as small .
lo
0
o
0 .2
0 .0
l at. range = 70 ms
A . Compari son wi th Other M ethods
logy
For this comparison we calculated latencies of the simulated
signals as estimated by the original method of Pham et al. as
well as by Woody' s and the peak-picking method. B oth for
Woody ' s method and for peak-picking, the simulated sweeps
were fi rst fi ltered by a zero-phase digital fi lter with cutoff
fr equency of 3.5 H z. The template was half -sinusoid with the
half -period equal to 300 ms. Thi s template width i s commonly
used to estimate the latency of the P3 complex . Then, Woody' s
method w as applied twice, once to fi nd the maximum of
c r o ss c or r el at i o n an d o n c e t o fi n d th e m ax i m u m
0 4
I
o f cr o ss
variance. We tried also a template which was part of the
averaged ERP, with the w idth of 300 ms centered around
the maximal positive peak between 250 and 600 m s in the
averaged signal which was previously corrected by l atencies
calculated by Woody' s method with the half -sine template.
We found, however, that Woody ' s method w orked better with
covariance and, moreover, that the iteration did not improve
the results. This is in agreement with previously published
(c)
Fig. 2.
Comparison of correlation coeffi cients as a function of noi se contam-
ination (log CI) for four methods of single-trial analysis. Each plot illustrates a
different latency range (expressed by SD of the range). Interindividual SD is
marked by bars (I SD), for better legibility for the two outermost values only.
results (e.g., [ 14], [30], and [31]). So we do not show results
for these two l atter vari ants of Woody ' s method.
The results are displayed in Fig. 2, where correlation coefficients are plotted against noise contamination for the three
ranges of latency variation. The present method as well as the
original Pham et al. method performed better than the other
methods for small latency range while f or moderate and large
latency ranges all methods w ere equally good.
B. Eli mi nation of Trials wi th Null-Signal
To improve reliability of the single-trial latency estimation, Pfefferbaum and Ford [ 18] proposed to eliminate trials
99 1
JASKOWSKI AND VERLEGER: AMPLITUDES AND LATENCIES OF SINGLE-TRIAL ERP'S
1.0
1.0
0 .8
0 .8
0 .6
0 .6
e
C
0 .4
0 .4
0 .2
0 .2
0
0
O
O
0 .0
0 .0
O.4 j
log(ti
1 .0
1.0
0 .8
0 .8
0 .6
0 .6
C
e
0 .4
0 .4
0 .2
0 .2
0
0
O
O
0 .0
0 .0
-0 .2
-0 .2
-0 .4 i
iogg
lOg(tl
1.0 .
1.0
0 .8
0 .8
0 .6
0 .6
0 .4
0 .4
L
0 .2
0 .2
0
0
O
O
0 .0
0 .0
-0.2 J
O.4I
-0 .2
log/
O .4 J
log/
(b)
(a)
Fig. 3. Comparison of correlation coefficients between real and estimated latencies as a function of noise contamination (IogS)) for Pham er al.'s method
(0 ) and the present method ($) applied to the set of recordings to which (a) eight or (b) 17 null-signals were inserted. Crosses indicate results obtained
by the present method after elimination of null-signals. Each plot illustrates a different latency range (expressed by SD of the range). Interindividual SD
is marked by bars (I SD), for better legibility for the two outermost values only.
al.' s maximum-likel ihood method, because the present method
either eight or 17 null-signal sweeps to the set of 34 simulated
sweeps and estimated P3 latencies by the original Pham et
al. method and the present method. The results are presented
in Fig. 3. As could be expected, the methods performed much
worse for 17 than for eight null sweeps. (Compare these results
also with those presented in Fig. 2.) Such a weak performance
results obviously from decreased SRN with increasing number
of null-signal trials. Then we eliminated those latencies of
those single-trials whose relative amplitudes were estimated
by the present method to be smaller than 0.1 and computed
the correlation coeffi cients from the remaining nonnull signal
trials (crosses in Fig. 3). The improvement of performance is
provides estimates of the single-trial amplitudes. We added
very clear.
containing very low (if any) signals. In order to do this,
they matched the template in Woody's method not only
in the window where P3 (signal range) is expected but
also in a later window where no P3 is expected (noise
range). They rej ected those trials for which the fit in the
signal range was not better than in the noise range. Smulders
et al. [32] extended this idea to peak-picking by rej ect-
ing signals when the amplitude of the maximal positive
wave in the signal range was not larger than in the noise
r an g e .
This may also be done with the present extension of Pham et
992
I EEE T RA N SACT ION S ON B IO M ED ICA L EN GIN EERI NG, V OL . 46, N o . 8, A U GU ST 1999
3 .0
2 .5
2 .0
c t.
E
1.5
1.0
0 .5
0 .0
T r ia l n u m b e r
Fig. 4. Relative amplitude ny estimated by the present method in real data, for the fi rst 100 trials of an oddball task. Vertical arrows indicate targets.
Val ues are the means over 16 subj ects.
C. Appli cati on to Real Data Target
Eff ect and Sequenti al Eff ects
A s a dem onstr ati on of how the m ethod w or k s w i th r eal d ata,
we considered the variation of single-trial amplitudes by the
so-called target eff ect and by sequential effects.
I n the oddball task , one stimulus (target) is presented
randomly in some trials only (20% of tri als in the present
data), while the other (nontarget) is presented in the rest of
trials. Target stimuli evoke a large P3 complex, nontarget
stimuli a small one. However, as was found by averaging
stimuli separately for different preceding sequences, nontarget
stimuli evoke a considerable P3 if directly preceded by a target
stimulus (alternation effect [21] plus a possible precedingtarget effect [22]) and, on the other hand, when preceded by
an unusually long sequence of nontargets [33] . To show these
effects in single trials, we applied the method described in
this paper to all trials, i.e., both target and nontarget trials.
Because the sequence of targets and nontargets was the same
for every subj ect we could average the obtained single-trials
amplitude over subj ects. The results are presented in Fig. 4
f or the first 100 stimuli . The target eff ect i s clearly visible:
Every target stimulus, marked by a vertical arrow , resulted in
a transient increase of signal amplitude. But also the sequential
effects on nontargets can be distinguished: First, nontarget P3' s
are relatively large when immediately preceded by targets.
a sum of some overlapping components with their relative
positions varying from trial to trial . This seems to become
particularly important f or long-latency ERP' s where higher
cognitive processes like attention, memory and learning are
involved. The problem of reasonable decomposition is more
or less successf ully addressed in some other studies (e.g.,
[ 10] , [34], and [35] ; see also [36] where an attempt was
undertaken to separate two overlapping components of P300
by experimental manipulation rather than by mathematical
analysis).
M oreover, both in the traditional approach and in our
method, zero-mean noise is assumed. This assumption is
hardly true. A lthough the ongoing EEG was f ound to be
normally distributed i n resting state, it was also shown that
the distribution may become non-Gaussian in other states,
e.g., during performance of a mental arithmetic task [37] .
Therefore, noi se may noticeably vary during the experiment as
it is hard to believe that subj ects are in the same state all the
experiment long. I n our simulation, we tried to prepare sweeps
which were as similar to real ones as possible. In particular,
w e d i d not m odel the noi se as Gau ssi an . N ev er thel ess, the
method gives reasonable estimates of true values of both
l atency and amplitude. A l so f or real data (Section IV -C)
the results obtained by the method are plausible from a
psychophysi ological point of view .
Second, in the rather long nontarget sequences between the
fi rst and second and between the second and third target, the
amplitude of nontarget P3' s i s gradually i ncreasing. Thus, the
present method is able to confi rm the regul arities inferred from
averaged data.
D . Li mi tati ons of the Extended Pham et al. 's M ethod
The traditional approach of averaging evoked potential s
as a method for improvi ng the SNR i s based on several
strong assumptions. In particular, it i s assumed that in every
sweep the signal has a constant shape and is perfectly timel o c k ed t o an ex t er n al
st i m u l u s. I n o ur
m eth o d t h ese t w o
assumptions have been softened: 1) latency j itter i s allowed
and 2) although the general shape of the signal is assumed to
be constant, variations of its ampl itude w ere allowed. One
cannot , how ev er , excl ude that no nl i near v ar i ati ons such as
expansion and/or compression in the segments of evoked
potentials take place, or that the entire evoked potential i s
V . C ON CL U SION S
We proposed a generalization of the method originally
developed by Pham et al. [ 14] . B oth methods all ow for
variations of signal l atency. H owever, whereas Pham et al.' s
original method i s based on the assumption that both waveform
and amplitude of the signal are identical after every stimulus,
the present method allows also variations of signal ampl itude.
The extensive pseudoreal simul ations performed in the paper
showed some advantage of the method over two other methods
commonly used i n ERP research. It enables to estimate not
only latency variation of the P3 complex but also trial-totrial variations of its amplitude and, what is also important, if
the amplitude really varies, getting close to zero, the method
can rej ect these sweeps and thereby outperform the original
m eth o d .
I n summary , although the method is mathematically
more demanding, it seems to be a reasonable alternative
t o o th er m eth o d s .
993
JA SK OW SK I A N D V ERL EGER: A M PL IT U DES A N D L AT ENC IES OF SIN G L E-T RIA L ERP' S
[25] R. B. Davies, "Optimal inference in the frequency domain," in Hand-
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University of Poznan, Poland, where he received
the M.Sc. degree. He received the Ph.D. degree
in natural sci ences from the M edical A cademy of
Poznan, Pol and, in 1982.
He w orked in the M edical A cademy of Poznail
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until October 1998 and is currently a Professor of
Psychophysiology at the Pedagogical University of
Bydgoszcz, Poland. In 1992, he started a collaboration with R. Verleger which has been continued until now. In 1995, he
obtained a fellowship from the A. von Humboldt Foundation. His main
interests are in psychophysics and psychophysiology. His research interests
include perception of temporal order, response force, and event-related EEG
potentials.
Rolf Ver leger was born in Ravensburg, Germany,
in 195 1. H e received the diploma in psychol ogy
from the University of Konstanz, Germany, in 1976
and the Ph.D. degree in social sciences from the
University of Tuebingen, Germany, in 1986.
He w orked i n M annhei m and Tuebingen, Ger-
expectancies on reaction times and event-related potentials: Evidence
many, before j oining the Department of Neurology
f or automatic and controlled expectancies," J. Exp. Psychol.: LM C, vol .
18, pp. 810- 822, 1992.
[24] D . R . Bril li nger , " Some aspects of the analysis of evoked response ex-
in the M edi cal University of L uebeck , L uebeck,
periments," in Stati sti cs and Related Topi cs, H. Csorgo, D. A. Dawson, J.
N . K . Rao, and A . J. M d. E . Saleh, Eds.
N orth-H oll and, 1981, pp. 155 168.
A msterdam, the Netherlands:
Germany, in 1988, where he received the title of
Professor in 1998. H i s mai n interests are in cogni -
tive neurophysiology and neuropsychology. He has
been invol ved i n research on event-related EEG potenti als, both basic and
methodological aspects as well as clinical applications.