1

Time-frequency analysis methods to quantify the time-

varying microstructure of sleep EEG spindles: Possibility

for dementia biomarkers?

P.Y. Ktonas (1), S. Golemati (1), P. Xanthopoulos (2, 5), V. Sakkalis (2, 6),

M. D. Ortigueira (3), H. Tsekou (1), M. Zervakis (2), T. Paparrigopoulos (1),

A. Bonakis (4), N. T. Economou (1, 7), P. Theodoropoulos (1),

S. G. Papageorgiou (4), D. Vassilopoulos (4) and C. R. Soldatos (1)

(1) Sleep Research Unit, Department of Psychiatry, University of Athens, Greece

(2) Department of Electronic and Computer Engineering, Technical University of

Crete, Chania, Greece

(3) UNINOVA and Department of Electrical Engineering, University Nova, Lisbon,

Portugal

(4) Neurological Clinic, Department of Neurology, University of Athens, Greece

(5) Department of Industrial and Systems Engineering, University of Florida,

Gainesville, U.S.A.

(6) Institute of Computer Science, Foundation for Research and Technology

(FORTH), Heraklion, Greece

(7) Sleep Center-Neurocenter (EOC) of Southern Switzerland, Lugano, Switzerland

Corresponding Author: Periklis Y. Ktonas; 31 Samara St., Psychico 15452, Athens,

Greece; Tel. 00 30 210 6712560; Fax 00 30 210 6725949; e-mail: pktonas@uh.edu

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Abstract

The time-varying microstructure of sleep EEG spindles may have clinical significance

in dementia studies and can be quantified with a number of techniques. In this paper,

real and simulated sleep spindles were regarded as AM/FM signals modeled by six

parameters that define the instantaneous envelope (IE) and instantaneous frequency

(IF) waveforms for a sleep spindle. These parameters were estimated using four

different methods, namely the Hilbert Transform (HT), Complex Demodulation (CD),

Matching Pursuit (MP) and Wavelet Transform (WT). The average error in estimating

these parameters was lowest for HT, higher but still less than 10% for CD and MP,

and highest (greater than 10%) for WT. The signal distortion induced by the use of a

given method was greatest in the case of HT and MP. These two techniques would

necessitate the removal of about 0.4 sec from the spindle data, which is an important

limitation for the case of spindles with duration less than 1s. Although the CD method

may lead to a higher error than HT and MP, it requires a removal of only about 0.23

sec of data. An application of this sleep spindle parameterization via the CD method

is proposed, in search of efficient EEG-based biomarkers in dementia. Preliminary

results indicate that the proposed parameterization may be promising, since it can

quantify specific differences in IE and IF characteristics between sleep spindles from

dementia subjects and those from aged controls.

Keywords – Sleep spindles, AM/FM signals, instantaneous envelope, instantaneous

frequency, biomarkers, dementia

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1. Introduction

The sleep spindle waveform is one of the hallmarks of human stage 2 sleep

electroencephalogram (EEG) and is also one of the few transient EEG events unique

to sleep. It is commonly known as a group of rhythmic EEG waves within the

frequency range of 11-15 Hz, characterized by progressively increasing, then

gradually decreasing amplitude, which gives the waveform its characteristic name. It

is usually of 0.5-2 sec duration and it may be present in low-voltage background EEG,

or superimposed on delta EEG activity, or temporally locked to a K complex

(Rechtschaffen and Kales, 1968). Sleep spindles are generated by cortico-thalamo-

cortical neuronal networks and are hypothesized to play an active role in inducing and

maintaining sleep (Steriade and Amzica, 1998). They are affected by brain pathology

(e.g., spindle incidence decreases in cerebral impairment), as well as by normal and

pathological aging (e.g., dementia) (Principe and Smith, 1982; De Gennaro and

Ferrara, 2003; Petit et al., 2004). With normal aging, sleep spindles are less numerous

and less well formed. In dementia, the sleep EEG patterns suggest accelerated aging.

Accordingly, sleep spindles may become even less numerous and their morphology

appears to deteriorate further (Petit et al., 2004).

Sleep can be useful for the study of neurological disorders because of its well-

defined and consistent structure. In particular, sleep EEG measures seem promising as

objective indicators in neurodegenerative disorders, including dementia, where sleep

changes appear to be an exaggeration of changes that come normally with aging.

Thus, in dementia there is more sleep fragmentation with an increased number and

duration of awakenings and much less numerous sleep spindles which are poorly

formed, of lower amplitude and shorter duration (Petit et al., 2004). Therefore, given

the sleep alterations in dementia, where the presence of sleep fragmentation may be

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related to changes in spindle incidence and morphology (Steriade and Amzica, 1998),

and given the possibility of sleep spindle involvement in cognition and learning

(Clemens et al., 2005; Fogel et al., 2007), the study of sleep spindle phenomenology

in dementia is of particular interest.

Several methods have been proposed in the literature for the quantification of sleep

spindle electrographic parameters. The most common relate to spectral analysis

techniques based on Fourier transformation and other related methods (e.g., wavelet

transform, matching pursuit) for the estimation of the dominant frequency content in

the spindles (e.g., Dijk, 1995; Werth et al., 1997; Huupponen et al., 2003; Huupponen

et al., 2006). Other approaches have related to the estimation of the period and

amplitude of the individual waves making up a spindle (e.g., Principe and Smith,

1982; Principe and Smith, 1986; Zeitlhofer et al., 1997). These techniques mimic the

way an electroencephalographer detects spindles visually in the sleep EEG and they

have contributed to methodology for the automated detection of sleep spindles. Work

has also been done on the quantification of the time-varying microstructure of sleep

spindles (Ktonas and Papp, 1980; Zygierewicz et al., 1999; Durka et al., 2005).

This paper contributes to the modeling and quantification of the time-varying sleep

spindle microstructure. Particular attention is given here to the modeling of time-

varying patterns (e.g., waxing and waning) in both the spindle envelope and the intra-

spindle frequency. Previous work utilizing matching pursuit methodology (e.g.,

Zygierewicz et al., 1999) has not considered intra-spindle frequency changes, which

have been observed either visually or by utilizing computer-aided procedures (e.g.,

Principe and Smith, 1986). A set of parameters are presented, related to the time-

varying sleep spindle morphology and to possible neurophysiological mechanisms of

sleep spindle generation. These parameters may serve as metrics for the eventual

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development of effective EEG biomarkers for early detection, differential diagnosis

and treatment follow-up in dementia.

In this work, real and simulated sleep spindles are regarded as both amplitude-

modulated and frequency-modulated (AM/FM) signals modeled by six parameters

that define the instantaneous envelope and instantaneous frequency waveforms

corresponding to a given sleep spindle. These parameters are estimated using four

different time-frequency methods, namely Hilbert Transform (HT), Complex

Demodulation (CD), Matching Pursuit (MP) and Wavelet Transform (WT), and a

related comparison of the methods is presented. Error figures are given for the

estimation of the six parameters using the above methods with simulated data, and

signal distortion issues related to these methods are discussed. An example is

presented with real sleep spindle data on the use of the six parameters as possible

discriminating metrics in dementia.

2. Methods

2A. Signal Model for a Sleep Spindle

Sleep spindles, being narrowband signals, can be regarded as AM/FM signals and

so expressed as (Ktonas and Papp, 1980):

x(t) = A(t)cos[g(t)]; 0)t(A ≥ (1)

where A(t) and g(t) are the instantaneous envelope (IE) and instantaneous phase,

respectively. IE is assumed here to have the form )tf2cos(kA)t(A aaa0 θ+π+= ,

with 0a Ak ≤ , as a simple approximation based on electrographic characteristics of

real sleep spindles. The instantaneous phase is assumed here to have the form

)tf2cos(ktf2)t(g bbb0 θ+π+π= , as a simple approximation based on previous work

(Hu and Ktonas, 2004). The instantaneous frequency (IF), in rads/sec, is the time

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derivative of g(t). Observe that, in the above model of g(t), f0 is a “carrier frequency”,

which models the “dominant” frequency of the sleep spindle at 12 Hz (for frontal

spindles) or at 14 Hz (for central spindles) (De Gennaro and Ferrara, 2003). Of

course, A(t) and g(t) could be modeled in a more complicated way (e.g., superposition

of several sinusoidal or other functions). However, the above models for A(t) and g(t)

have been chosen for this work due to their simplicity and are based on real sleep

spindle characteristics (Hu and Ktonas, 2004). Furthermore, it is expected that the

above models will be especially useful in quantifying the observed changes of sleep

spindle characteristics in aging and dementia (e.g., apparent loss of well-defined

quasi-sinusoidal (“spindle shape”) trend in the envelope of the sleep spindle, which

may justify the visual impression of “poorly formed” spindles in dementia). Figure 1a

shows an example of the above overall model (simulation), while Fig. 1b shows a real

sleep spindle from a healthy subject. Observe that the time-varying morphology of

the real spindle justifies the AM/FM nature of the model.

(Figure 1 here)

Given the known neurophysiological mechanisms for sleep spindle generation (e.g.,

Steriade and Amzica, 1998), one may hypothesize that A(t) and its defining

parameters (A0, ka, fa, θa) model primarily cortical neural processes, while g(t) and its

defining parameters (f0, kb, fb, θb) model primarily thalamic or thalamo-cortical neural

processes. For simplicity, we can assume that both phases (θa, θb) can be set to zero.

Accordingly, A0, ka and fa may reflect dynamics of spatio-temporal summation of

cortical post-synaptic potentials, resulting from thalamo-cortical neural activity and

contributing to the power in the spindle signal. Similarly, f0, kb and fb may reflect

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dynamics of reticular neural circuits in the thalamus, as well as dynamics of thalamo-

cortical neural circuits, which are involved in the generation of the spindle frequency

spectrum (11-15 Hz). Given the expected brain changes due to dementia, which may

involve loss of neurons and alterations in neural circuit connectivity (see, e.g., De

Gennaro and Ferrara, 2003; Petit et al., 2004; Rose et al., 2008), the above

parameterization may prove especially useful since it may provide suggestions for

plausible neuropathological mechanisms involved in sleep spindle alterations.

2B. Time-Frequency Analysis Methods: Estimation of IE and IF Waveforms

Previous work on the analysis and parameterization of sleep spindles has included

the use of Hilbert Transform (HT) (Ktonas and Papp, 1980), Complex Demodulation

(CD) (Ktonas and Papp, 1980; Hu and Ktonas, 2004), Wavelet Transform (WT)

(Huupponen et al., 2006), and Matching Pursuit (MP) (Zygierewicz et al., 1999;

Durka et al., 2005, Huupponen et al., 2006) techniques. The above four time-

frequency analysis methods are based on two distinct approaches which allow a direct

estimation of the instantaneous envelope and instantaneous phase (or frequency) of a

signal. Accordingly, in one approach, the envelope and phase are estimated by using

the analytic signal concept (HT and CD). In the other approach, they are estimated by

convolution with a complex basis function (WT and MP).

In this paper, we compare the estimation of six parameters from the IE and IF

waveforms, which can model the time-varying microstructure of sleep spindles based

on the model of Eq. (1), using HT, CD, WT and MP. Artificial signals, simulating

sleep spindles, as well as real spindles are utilized. As a result, practical

recommendations are provided for the use of HT, CD, WT and MP in analyzing sleep

spindles for clinical applications. Previously, a similar comparative study has

8

involved MP and WT, but for only one of the above described six parameters (f0) and

utilized artificial signals only (Huupponen et al., 2006). In the present paper, an

example is also presented with real data, of the use of the proposed model and the

corresponding six parameters towards the definition of promising EEG biomarkers in

dementia.

Hilbert Transform. The HT is a well-known transformation which is used to

determine the instantaneous characteristics (i.e., IE and IF) of a real signal x(t).

Essentially, it is used to compute a complex signal Y(t), called the analytic signal

corresponding to x(t), defined by:

( ) ( ) ( )tx̂itxtY ⋅−= (2)

where ( )tx̂ is the HT of x(t) (Thomas, 1969).

It is interesting to observe that the analytic signal Y(t) has a Fourier transform that

is zero for negative frequencies (one-sided spectrum). The IF function of a real signal

x(t) is defined as:

( )dt)t(d

21tIF φ

π= (3)

where φ (t) is the phase function of the analytic signal Y(t) corresponding to x(t).

The IE function of x(t) can be computed from the magnitude of the analytic signal

Y(t):

( ) ( ) ( )tx̂txtIE 22 += (4)

Complex Demodulation. The procedure for applying CD to a given real signal x(t),

which is assumed to be narrowband and with no spectral contribution at the origin

(this is a necessary condition for implementing CD), can be described as follows:

(1) The frequency spectrum (Fourier transform) of the signal is shifted to the left

(towards the origin) by the demodulating frequency (assumed to be f0 for the

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simulated sleep spindle signals in our study; however, for the real signals in our study,

the demodulating frequency was taken to be the “center of gravity” of the Fourier

transform of the sleep spindle signal in the frequency band of 9.5-16.5 Hz).

(2) The resulting complex signal is filtered with a lowpass filter (for our case, an

FIR filter of order 100, implemented using the FIR1 command of MATLAB—The

Mathworks Inc.) having a cut-off frequency of fct Hz. This cut-off frequency is chosen

such that, beyond that frequency, the frequency content of the complex signal around

the origin may be considered negligible (for our study, based on the analysis of

several real sleep spindles from healthy subjects, fct was taken to be 6 Hz).

(3) The frequency spectrum of the filter output is shifted to the right by the

demodulating frequency.

It is obvious that the objective of steps 1-3 above is to produce the analytic signal

corresponding to the real signal x(t) (see the HT description above). The IE and

instantaneous phase (as well as IF) functions of x(t) are subsequently calculated from

the magnitude and argument, respectively, of the analytic signal corresponding to x(t),

that results after the third step (see the HT description above).

Wavelet Transform. For a given real signal x(t), the Continuous Wavelet

Transform (CWT) W(α,τ) of x(t) is defined as the convolution between x(t) and

dilated and translated versions of a complex wavelet function ψ(t). Therefore,

dt)t(xt1),(W *∫∞

∞−$%

&'(

)ατ−

ψα

=τα (5)

where +ℜ∈α is the dilation or scale parameter, and ℜ∈τ is the translation

parameter. In the present study, the complex Morlet wavelet

!!!

"

#

$$$

%

&

=!!"

#$$%

&−

bc f

ttfj

bMorlet e

ft

221)(π

πψ was used, where fc=1 is the center frequency and fb=1

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is the bandwidth (variance). The scales examined were 23≤α≤60, with step equal to 1.

It is possible to relate the scales α to the frequencies f by approximating the center

frequency fc of the wavelet using the relation α=α /cs fff ; where fs is the sampling

frequency.

For a monochromatic signal, there is a scale αr(τ) at any given τ which corresponds

to a basis wavelet centered at τ whose frequency is equal to the local frequency of

x(t). The scale αr(τ) identifies the instantaneous frequency of the signal. As the CWT

is a linear operation, superimposed frequency components are manifested as different

scales αr(i)(τ) that locally maximize |W|. The curves formed by the points (αr

(i)(τ),τ) are

known as the “ridges” of the transform. The trajectory of each ridge can be used to

extract the IE and IF waveforms (Delprat et al., 1992).

Matching Pursuit with moving window. In MP the aim is to decompose a given

signal x(t) into a linear combination of N waveforms (atoms) chosen from a

predefined dictionary D = {dk(t): k = 1, 2,…, Q} so that:

∑=

+=N

1iNii )t(R)t(dw)t(x (6)

where RN(t) is the decomposition residual. In our case, we used the following cosine-

based atoms:

)tf2cos()t(d kkk φ+π= (7)

where 16:05.0:10fk = and ππ

=φ 2:10:0k

In this work, MP was performed within a moving 50-sample window, shifted every

5 samples (with N=1 in order to find the ( )iii tf2cos)t(d φ+π= that best matched the

signal within the window). The parameters obtained from the application of MP in

every window (frequencies fk and weights wi) were then interpolated and smoothed

with a cubic smoothing spline in order to produce the IF and IE waveforms,

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respectively. The value of 50 samples can be justified as follows: Given the sampling

frequency of 512 Hz used in this work, and the frequency range of 11-15 Hz for the

individual waves making up a sleep spindle, a 50-sample window may include from

one to almost two cycles of these waves, thus providing enough time resolution for

the quantification of “local” (within the sleep spindle) frequency changes via MP.

2C. Curve-fitting of IE and IF Waveforms Using MP, and Model Parameter

Estimation

Since the estimated IE and IF waveforms may not be purely sinusoidal, especially

for real sleep spindles, we performed MP-based curve fitting in order to estimate the

six parameters that define the spindle model (see Eq. 1).

For every waveform set (IE and IF) obtained with each of the four methods

described above, two atoms were chosen (one for IE and another for IF) having the

following form

)tf2cos()t(d fitfitfit φ+π= (8)

where 6:005.0:1.0ffit = and ππ

=φ 2:10:0fit

The six model parameters were then estimated from the MP-fitted sinusoids. More

specifically, fa and fb were taken directly from the sinusoid dictionary created within

the MP procedure. To estimate the remaining four parameters, the maximal and

minimal values of the IE and IF waveform-fitted sinusoids were detected. A0

corresponds to the average of the maximum and minimum value of the IE waveform-

fitted sinusoid, while f0 corresponds to the average of these two values from the IF

waveform-fitted sinusoid. Furthermore, ka corresponds to the amplitude of the IE

waveform-fitted sinusoid, while kb corresponds to the amplitude of the IF waveform-

fitted sinusoid divided by fb.

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If the waveform-fitted sinusoids are characterized by very low frequencies (fa, fb),

say, lower than 1 Hz, some of the model parameters may not be able to be estimated,

because an insufficient number of peaks and troughs may be present in the waveform-

fitted sinusoids. In such cases, A0 and f0 may be defined by averaging the IE and IF

waveforms, respectively. However, in these cases, no estimates of ka or kb are

possible.

3. Results

3A. Analysis of Simulated Sleep Spindles

An example of a simulated sleep spindle signal is shown in Figure 2a, where the

values of the six parameters used in the simulation are given below (the phase

parameter values, θα , θb, are assumed to be zero):

f0=13 Hz, A0=11.5 microvolts, ka=2.8 microvolts, fa=2.7 Hz, kb=0.27 rads, and

fb=4 Hz, where f0 is the central frequency (“carrier”) of the signal. These are typical

values for a real sleep spindle, based on preliminary work (Hu and Ktonas, 2004).

(Figure 2 here)

Figure 2a shows 500 points of the signal for a sampling frequency of 512 Hz.

Figure 2b shows the one-sided Fourier transform of the signal. In this graph, there is a

peak at 13 Hz, as expected. Two additional signal examples were produced, with

parameters within the range of typical values.

Table1 shows the average percentage errors in the estimation of the six model

parameters using HT, CD, WT and MP for the three simulated spindle signals,

without and with additive white, Gaussian noise. As we can see, HT had the lowest

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errors in the estimation of the parameters. The average errors for CD and MP were

lower than 10%. The WT method resulted in the highest errors (much higher than

10%). In all cases, the average percentage error increased as the signal-to-noise ratio

(SNR) became smaller.

(Table 1 here)

Figure 3 shows an example of IE and IF waveform estimation by utilizing the four

methods on a simulated signal (model in the figure). Observe the errors associated

with each method, in agreement with the error results in Table 1. As shown in Fig. 3,

there exists a distortion at the beginning and at the end in the estimated IE and IF

waveforms. This distortion results from the signal processing inherent in the four

methods and can be minimized by eliminating from the subsequent analysis

(estimation of the various parameters) a set of data points from the beginning and

from the end of the estimated IE and IF waveforms.

( Figure 3 here)

Since the Hilbert transform can be implemented as a filter, it will introduce

distortion due to its impulse response which has a slow convergence (Oppenheim and

Schafer, 1975). Simulation showed that eliminating 200 samples (100 from the

beginning and 100 from the end of the estimated IE and IF waveforms) could

minimize this distortion. This removal corresponds to removing about 0.4 sec from

the waveforms, for a sampling rate of 512 Hz. In the MP method, distortion may be

introduced at the beginning and at the end of the estimated IE and IF waveforms,

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since a smoothing spline is applied in order to interpolate these curves. This kind of

distortion is case-dependent and one cannot know a priori the signal amount that

should be removed. Empirical analysis showed that removing 100 samples from the

beginning and 100 samples from the end of the signal (a total of 200 samples, as in

the HT case) could effectively reduce this distortion.

In the case of CD, the distortion is due to the lowpass filtering involved in the

method. Tests with simulated data showed that 60 samples from the beginning and 60

samples from the end of the signal (a total of 120 samples) should be removed to

minimize the distortion. This removal corresponds to about 0.23 sec of data, for a

sampling rate of 512 Hz. In the WT case, since one is dealing with finite-length time

series, errors will occur at the beginning and the end of the wavelet power spectrum,

since the implementation of the wavelet transform assumes that the data is cyclic. One

solution is to pad the end of the time series with zeroes before performing the wavelet

transform and then remove these points afterwards. In this study, the time series was

padded with sufficient zeroes to bring the total number of points up to the next-higher

power of two, thus limiting the edge effects while still allowing for fast

implementation of the WT. Padding with zeroes introduces discontinuities at the

endpoints and, as one goes to larger scales, decreases the amplitude near the edges as

more zeroes enter the analysis. The cone of influence (COI) is the region of the

wavelet spectrum in which edge effects become important and is defined here as the

e-folding time for the autocorrelation of the wavelet power spectrum at each scale

(Torrence and Compo, 1998). This e-folding time is chosen so that the wavelet power

for a discontinuity at the edge drops by a factor of e−2 and ensures that the edge

effects are negligible beyond this point. As a result of the above, we had to remove

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about 120 samples (60 from the beginning and 60 from the end of the signal, as in the

case of CD) to minimize distortion.

It should be mentioned that since the distortion introduced by the bandpass

filtering of the real data (see below) was found to involve about 50 samples at the

beginning and the end of the bandpassed signal (zero-phase filtering), no data removal

due to bandpass filtering was implemented because the distorted data samples were

removed anyway for each of the four time-frequency methods as described above.

3B. Analysis of Real Sleep Spindles

Table 2 lists the parameter values obtained from two real sleep spindle examples

with the four methods. Reasonable parameter values are indicated in bold and were

assessed based on a heuristic visual examination of estimated IE and IF waveforms.

As indicated in the table, most parameter values obtained by HT, CD and MP were

deemed “reasonable”, while WT appeared to be less successful.

(Table 2 here)

Figure 4 shows two examples of analysis of real sleep spindles from two young

healthy adults. Parts (a) and (b) show the actual spindle waveforms, the obtained IE

waveforms and the corresponding MP-fitted curves. Parts (c) and (d) show the

obtained IF waveforms and the corresponding MP-fitted curves. The IE and IF

waveforms were obtained via CD, since this method had a relatively small error and

required less data removal than the HT and MP methods (see Section 3A). Observe

the relatively accurate estimation of the IE waveforms and the successful fit of the

quasi-sinusoidal trends in the IE and IF curves by the MP method.

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(Figure 4 here)

There may be cases of real sleep spindles where the estimated IE or IF waveforms

deviate considerably from a quasi-sinusoidal shape. In these cases, the MP-based

curve-fitting method may fail, since the sleep spindle waveform may not be able to be

represented by the assumed models in Section 2A. However, the possibility exists for

interactive MP-fitting, under the supervision of the user. Accordingly, in cases where

the particular shape of the IE or IF waveform is not quasi-sinusoidal, one may wish to

subtract a sinusoidal “first fit” and perform iteratively further sinusoidal fits to the

residuals. Therefore, the (first) MP-fitted sinusoid can be subtracted from the original

waveform (IE or IF) and MP-based curve-fitting can be performed on the residual.

Model parameters may then be calculated from the residual-MP-fitted sinusoids, as

described above. It should be pointed out that values for A0 and f0 based on this

procedure are meaningless. Figure 5 shows an example of such a residual curve-

fitting for an IE waveform. It should be mentioned that the MP-based curve-fitting

utilized in the application example described below (Part 3C) was performed on the

estimated IE and IF waveforms and not on related residuals.

(Figure 5 here)

3C. Application Example in Dementia

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Night sleep EEG signals from 3 healthy elderly subjects (female, age range 72-76

yrs) and 3 patients with dementia were obtained from the Sleep Research Unit of the

Department of Psychiatry at the University of Athens (Aiginiteion Hospital). Standard

polysomnographic procedures were applied for the sleep EEG recording of two nights

for each subject, utilizing a Brain Quick (Micromed) data acquisition system at 512

Hz sampling rate. Data from the second recording night were used in this study, in

order to eliminate the “first-night effect” problem (Agnew et al., 1966; Kryger et al.,

2005). The dementia patients were drug-naïve in-patients at the Neurological Clinic of

the University of Athens, diagnosed for dementia through standard clinical,

neuroimaging and neuropsychological procedures. The dementia cases had different

etiologies, including progressive supranuclear palsy (female, 59 yrs), posterior

cortical atrophy (female, 68 yrs) and fronto-temporal dementia (male, 71 yrs). The

study protocol was approved by the appropriate ethics board of the Aiginiteion

Hospital and informed consent was obtained from patients and controls.

The sleep EEG record of each subject was visually scored for sleep stages

according to standard procedures (Rechtschaffen and Kales, 1968). Each record was

divided into three parts of equal length. Visually well-defined sleep spindles from the

largest sleep stage 2 of each part, from central scalp locations (C3 or C4), were

obtained for automatic analysis. Accordingly, the spindle waveforms were filtered

through a digital band-pass zero-phase filter with frequency cut-offs at 5 and 22 Hz,

in order for the spindle waveform components in the pass-band of 11-15 Hz to be

preserved with unity gain. After band-pass filtering, IE and IF waveforms were

obtained for each spindle, utilizing CD methodology. CD was chosen, since it

involved a relatively small error and required less data removal than HT or MP due to

distortion (see Section 3A). The IE and IF waveforms obtained via CD for each

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spindle were examined visually, and spindles exhibiting IE and IF waveforms with

quasi-sinusoidal trends, to agree with the sleep spindle model utilized in this paper,

were chosen for further analysis and parameter estimation. In particular, the IF

waveforms had to have most of their duration within the spindle frequency band (11-

15 Hz). As a result of the above analysis, 11 spindles from elderly controls and 11

spindles from dementia patients were chosen for parameter estimation. Since the

objective of this work was to show a promising application of the described

methodology, the above number of spindles analyzed, although small, was

nevertheless deemed sufficient for illustration purposes.

In general, sleep spindles from patients were less well-defined visually than sleep

spindles from the elderly controls. In addition, they had less overall amplitude.

However, any more detail as to modulation patterns in instantaneous envelope and

instantaneous frequency was not easily discernible visually. Figure 6 shows two sleep

(Figure 6 here)

spindle examples, each one from a different control subject, while Figure 7 shows

(Figure 7 here)

two sleep spindle examples, each one from a different dementia patient. Observe that

all spindles are about 1-sec long, as expected due to the aging process (compare with

the longer spindles from the young adults in Fig. 4). In addition, observe that the

spindles from the dementia patients exhibit faster oscillations in the sinusoids fitted to

19

the IF waveforms than the spindles from the controls. Incidentally, the fitted sinusoids

in all cases are the ones chosen as first in the MP curve-fitting procedure.

Table 3 shows parameter values for the examples in Figs. 6 and 7. EC1 and EC2

(Table 3 here)

are the two sleep spindles from the elderly control subjects (corresponding to the

spindle on the left and on the right, respectively, in Fig.6), while D1 and D2 are the

two sleep spindles from the dementia patients (corresponding to the spindle on the left

and on the right, respectively, in Fig. 7). Notice in Table 3 that both dementia spindles

had fb values much higher than those of the control spindles, in agreement with Figs.

6 and 7.

Table 4 shows average results across all spindles (11 from elderly control and 11

(Table 4 here)

from dementia patients) analyzed in this example. It is shown that, on the average,

there exist statistically significant differences (P<0.05) between elderly controls and

dementia patients in spindle parameters A0 and fb, in agreement with the results in

Table 3.

4. Discussion

This paper presents an AM/FM signal model for a sleep spindle where the

instantaneous envelope (IE) and instantaneous frequency (IF) waveforms are

20

assumed to be relatively simple mathematical expressions defined by six parameters

based on neurophysiological concepts which relate to sleep spindle generation. The

preliminary clinical application results justify such neurophysiology-based

assumptions for the model structure. Furthermore, this paper compares the application

of four different time-frequency methods for the estimation of the proposed six

parameters. Although two of the methods (MP and WT) have already been compared

in the literature for the estimation of one of the six parameters (f0), this is the first time

to the best of our knowledge that all four methods are compared for the estimation of

all the parameters. In addition, this is the first time to the best of our knowledge that

MP is used in a windowed manner to estimate IE and IF, and for WT to be used to

estimate IE.

The results of the clinical application example presented in this work indicate that

the proposed parameterization of the time-varying microstructure of sleep spindles,

based on the model in Section 2A, is promising. Indeed, as shown in Figs. 6 and 7 and

in Tables 3 and 4, the proposed sleep spindle model can quantify subtle differences in

this microstructure which may have clinical significance. Sleep spindles from

dementia patients, when compared to those from aged controls, were found to have

less average amplitude in the spindle envelope, indicating less spindle power, and

more variability in instantaneous frequency dynamics, indicating more intra-spindle

frequency “instability”. The above results point to the possibility of extracting

important information related to sleep spindle morphology which could lead to the

definition of useful dementia biomarkers. However, more work is needed with larger

data sets to assess the specificity of such sleep spindle characteristics to dementia. The

preliminary results presented in this paper imply differences in sleep EEG generation

mechanisms between dementia patients and healthy controls. Accordingly, the spindle

21

power difference implies a possible difference in cortical neural dynamics due to the

loss of cortical cells or cortical synapses in the dementia group. Furthermore, the

intra-spindle frequency difference implies a possible difference in thalamo-cortical

neural dynamics (e.g., compromised capability of thalamic “pacing” in the dementia

group). In fact, recent work based on diffusion tensor imaging has indicated the

possibility of changes in thalamo-cortical radiations in dementia (Rose et al., 2008),

which may contribute to the intra-spindle frequency instability. It is of interest to

speculate that the calculated changes in both sleep spindle model parameters may

relate to the visually obtained impression that sleep spindles in dementia exhibit a less

“regular” morphology, when compared to spindles from aged controls. Therefore, the

utilization of the proposed sleep spindle model parameters as potential EEG-based

biomarkers in dementia appears promising. However, as mentioned earlier, additional

studies involving larger data sets are needed to explore and justify this promise

further.

It should be pointed out that the results presented in this methodological paper are

preliminary and are based on only three patients. Although the patients exhibited

different etiologies for their dementia, all three cases involved cortical dementias,

which imparts a certain uniformity in the patient data. The sample of sleep spindles

analyzed was preselected so that (1) there was no “first-night effect” and (2) the

respective instantaneous envelope and instantaneous frequency waveforms were

quasi-sinusoidal to agree with the sleep spindle model utilized in this paper. More

work is needed to explore the need for any modifications in the sleep spindle model

(see below) which can allow the analysis of other type of spindles as well.

Furthermore, the sample of sleep spindles analyzed was small, since the objective was

to show preliminary results. Accordingly, only three sleep stage 2 periods, distributed

22

throughout the night, were analyzed for each subject. Future studies should include

spindles from all such periods during the night sleep recording.

Most of the time-frequency analysis methods discussed in this work appear to

estimate the proposed sleep spindle model parameters with an acceptable degree of

accuracy. It was shown in simulated data that HT, CD and MP exhibit average errors

of less than 10%. Although HT had the lowest error, its use for parameter estimation

may not be appropriate in clinical sleep spindle studies, since it requires a removal of

200 samples from the spindle data, which amounts to about 0.4 sec of data for a

sampling frequency of 512 Hz. This may be problematic in the analysis of short sleep

spindles (e.g., less than 1-sec long) which may be observed in dementia. The

heuristics-based MP method was found to produce low error rates as well. It is of

interest that an MP-based method for determining the dominant frequency in

simulated sleep spindle data provided the best overall performance when compared to

other methods, including the discrete Fourier transform and WT (Huupponen et al.,

2006). However, the MP method described in the present work appears to require the

same removal of data samples as the HT method. Even if the WT method leads to

reasonable values for some parameters in real spindles and is robust in noisy signals,

it has the poorest performance. WT suffers from intrinsic edge artifacts (Torrence and

Compo, 1998), because the Morlet wavelet is not completely localized in time, which

affects the parameter estimation. A more smooth choice of scales around an expected

f0 could improve the results. However, such fine tuning would lead to lack of

generality in the sense that a predefined frequency is targeted. Although the CD

method may lead to a higher error rate than HT and MP, it could be utilized in the

analysis of short spindles, since it requires a removal of only 120 samples (about 0.23

sec for a sampling rate of 512 Hz). Therefore, for this sampling rate spindles of at

23

least 0.8 sec in length could be analyzed with the CD method, since at least 0.5 sec of

data are needed to investigate the presence of at least 2 Hz “oscillations” in the IE and

IF waveforms with the methodology described in this work.

We believe that no model may be capable to faithfully capture the variability of

physiological signals. The intent in this paper was to present a relatively simple

deterministic approach in the modeling of sleep spindles. This was the reason that IE

and IF were chosen to be sinusoidal. Indeed, real sleep spindles do exhibit “rich

morphology changes” which may not be able to be faithfully modeled by simple

deterministic models, nor by stochastic models either. We could have proposed a

more elaborate model, with more parameters, that could capture more variability in

the signal. However, in that case, the parameter estimation would have become more

involved. Furthermore, a high-dimensional parameter space would have doubtful

practical significance. We believe that a given model shows its usefulness in its

applications. The preliminary results presented in this paper concerning model

parameter changes in dementia contribute to the justification of the model.

Although the results reported here appear promising, more work is needed in

developing appropriate curve-fitting methodology for parameter estimation, especially

in case the IE and IF waveforms happen to exhibit very low frequency content or to

deviate from a quasi-sinusoidal shape. Although an interactive curve-fitting procedure

could be utilized, as described in Section 3B, the proposed sleep spindle model may

not hold in such cases. Therefore, additional investigation should take place with real

data to examine if the proposed model is sufficient to capture important details in the

time-varying microstructure of sleep spindles, or if appropriate modifications will be

needed. Of interest here could be the use of empirical mode decomposition (EMD)

techniques (Rato et al., 2008). In fact, preliminary analysis done with this method led

24

to several AM/FM components for a given sleep spindle, with only one of them being

of interest as far as typical sleep spindle characteristics in IE and IF are concerned.

This will be the subject of further research.

This paper addresses important methodological issues related to the definition and

use of a simple mathematical model for a sleep spindle. Furthermore, the preliminary

clinical results presented point to the promise of the model as a tool in dementia

studies. It should be mentioned that since the parameters of the proposed model may

relate to plausible neurophysiological mechanisms (see Section 2A), aside from

clinical applications this model could lead to further exploration, possibly involving

animal studies, of the underlying neurophysiological mechanisms for sleep spindle

generation in healthy and pathological conditions.

Acknowledgement

This work was supported in part by the EU NoE BIOPATTERN project (project no.

508803).

25

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29

(a)

(b)

Figure 1. (a) Simulated sleep spindle produced with the model of Eq 1,

and (b) Real sleep spindle after bandpass filtering (5-22Hz).

30

(a)

(b)

Figure 2. (a) Example of an AM/FM simulated sleep spindle signal, and (b)

its one-sided Fourier transform.

31

(a)

(b)

(c)

(d)

Figure 3a. Examples of IE waveform estimation using (a) HT, (b) CD, (c) WT, (d) MP.

32

(a)

(b)

(c)

(d)

Figure 3b. Examples of IF waveform estimation using (a) HT, (b) CD, (c) WT, (d) MP.

33

(a)

(b)

(c)

(d)

Figure 4. Two examples of IE and IF waveforms and related MP-fitted curves in sleep spindles

from two young control subjects. (a), (b) Sleep spindle and IE waveforms, and MP-fitted curves.

(c), (d) IF waveforms and MP-fitted curves. Solid lines correspond to CD-estimated IE and IF

waveforms and dotted lines correspond to MP-fitted curves.

34

(a)

(b)

Figure 5. Example of residual curve fitting on an IE waveform. (a) MP-based curve-

fitting of original waveform, (b) MP-based curve-fitting on residual waveform. Solid

lines correspond to waveforms on which MP-fitting is performed, and dotted lines

correspond to fitted curves

35

(a)

(b)

(c)

(d)

Figure 6. Two examples of IE and IF waveforms and related MP-fitted curves in sleep spindles

from two healthy elderly subjects. (a), (b) Sleep spindle and IE waveforms, and MP-fitted curves.

(c), (d) IF waveforms and MP-fitted curves. Solid lines correspond to CD-estimated IE and IF

waveforms and dotted lines correspond to MP-fitted curves.

36

(a)

(b)

(c)

(d)

Figure 7. Two examples of IE and IF waveforms and related MP-fitted curves in sleep spindles

from two dementia patients. (a), (b) Sleep spindle and IE waveforms, and MP-fitted curves. (c), (d)

IF waveforms and MP-fitted curves. Solid lines correspond to CD-estimated IE and IF waveforms

and dotted lines correspond to MP-fitted curves.

37

TABLE 1. AVERAGE PERCENTAGE OF PARAMETER

ESTIMATION ERROR OVER NOISE-FREE AND NOISY

SIMULATED SLEEP SPINDLE EXAMPLES

HT MP CD WT

Example 1

Noise-free 1.30 2.25 8.85 13.90

SNR: 10dB 4.28 5.37 9.10 14.10

SNR: 20dB 1.92 3.46 8.96 13.96

Example 2

Noise-free 0.78 4.82 5.56 21.98

SNR: 10dB 2.19 8.27 7.34 23.33

SNR: 20dB 2.01 7.59 6.60 22.63

Example 3

Noise-free 1.66 4.57 8.72 19.85

SNR: 10dB 5.12 5.49 9.02 21.59

SNR: 20dB 4.64 5.07 8.89 20.61

38

TABLE 2. ESTIMATED PARAMETER VALUES FOR REAL

SPINDLES. REASONABLE PARAMETER VALUES ARE IN BOLD

Parameters HT MP CD WT

Real spindle 1

A0 (µV) 9.60 9.78 9.49 10.43

fa (Hz) 2.38 2.39 2.32 2.39

ka (µV) 5.28 5.03 5.18 2.10

f0 (Hz) 12.10 12.31 12.35 11.95

fb (Hz) 2.61 2.49 2.12 2.46

kb (rads) 0.78 0.45 0.76 0.51

Real spindle 2

A0 (µV) 9.07 9.15 9.57 10.31

fa (Hz) 2.39 2.51 2.18 2.39

ka (µV) 5.36 5.01 4.91 4.92

f0 (Hz) 12.73 12.55 12.75 13.57

fb (Hz) 5.19 4.76 2.35 2.74

kb (rads) 0.53 0.19 0.70 0.31

39

TABLE 3. ESTIMATED PARAMETER VALUES IN ELDERLY CONTROLS (EC)

AND DEMENTIA PATIENTS (D) OF FIGS. 6 AND 7

EC1 EC2 D1 D2

A0 (µV) 8.1 14.7 7.4 5.9

fa (Hz) 0.66 1.9 1.4 0.9

ka (µV) 4.2 6.3 3.5 3.7

f0 (Hz) 11.2 12.6 13.9 13.7

fb (Hz) 1.3 2.0 4.2 4.9

kb (rads) 0.69 0.86 0.29 0.41

40

TABLE 4. SPINDLE PARAMETERS OF ELDERLY CONTROLS

VS DEMENTIA PATIENTS

Elderly

Controls

Dementia

Patients

P-value

(Wilcoxon test)

A0 (µV) 13.14±3.73 7.62±2.48 7.20E-04

fa (Hz) 1.09±0.46 1.35±1.00 0.6223

ka (µV) 7.21±2.52 5.82±3.44 0.1024

f0 (Hz) 11.91±1.07 12.63±1.20 0.1396

fb (Hz) 1.71±1.51 3.23±1.40 0.0151

kb (rads) 1.26±1.03 0.83±0.84 0.341