Dynalets: A New Representation of Periodic Biological Signals and Spectral Data

Abstract

The biological information coming from electro-physiologic signal sensors like ECG or molecular signal devices like mass spectrometry has to be compressed for an efficient medical use by clinicians or to retain only the pertinent explanatory information about the mechanisms at the origin of the recorded signal for the researchers in life sciences. When the signal is periodic in time and/or space, classical compression processes like Fourier and wavelets transforms give good results concerning the compression rate, but bring in general no supplementary information about the interactions between elements of the living system producing the studied signal. Here, we define a new transform called dynalet based on Liénard differential equations susceptible to model the mechanism that is the source of the signal and we propose to apply this new technique to real signals like ECG, pulse activity and protein spectra in mass spectrometry.

Full text

Dynalets: a new representation of periodic biological signals and
spectral data
J. Demongeot1,2*, A. Hamie1, A. Glaria2, C. Taramasco2
Abstract — The biological information coming from electro-
physiologic signal sensors like ECG or molecular signal devices like
mass spectrometry has to be compressed for an efficient medical use by
clinicians or to retain only the pertinent explanatory information about
the mechanisms at the origin of the recorded signal for the researchers
in life sciences. When the signal is periodic in time and/or space,
classical compression processes like Fourier and wavelets transforms
give good results concerning the compression rate, but bring in general
no supplementary information about the interactions between elements
of the living system producing the studied signal. Here, we define a new
transform called dynalet based on Liénard differential equations
susceptible to model the mechanism that is the source of the signal and
we propose to apply this new technique to real signals like ECG, pulse
activity and protein spectra in mass spectrometry.
Keywords: Fourier transform; Wavelets; Dynalets; Signal
Processing; ECG; Pulse signal; protein mass spectrum
I. INTRODUCTION
The different ways to represent a biological signal aim to
both i) explain the mechanisms having produced it and ii)
facilitate its use in medical applications or in life sciences
research. The biological signals come from electro-
physiologic signal sensors like ECG, EEG,…, or from
molecular signal devices like mass spectrometry, bio-
arrays,…, and have to be compressed for an efficient
medical use by clinicians or to retain only the pertinent
explanatory information about the mechanisms at the origin
of the recorded signal for the researchers in life sciences.
When the signal is periodic in time and/or space, the
classical compression processes like Fourier and wavelets
transforms give good results concerning the compression
rate, but bring in general no supplementary information
about the interactions between elements of the living system
producing the studied signal. Here, we define a new
transform called Dynalet based on Liénard differential
equations, which are susceptible to model the mechanism
that is the source of the signal and we propose to apply this
new technique to real signals like ECG, pulse and mass
spectrometry. We will recall in Sections 2. and 3. the
classical Fourier and wavelets transform in a differential
equation view, then we present in Section 4. the prototype of
the Liénard equations, that is the van der Pol system. In
Section 5., we will define the Dynalet transform, and after
Manuscript received 15th November 2012.
1University J. Fourier Grenoble, AGIM CNRS FRE 3405, Faculty of
Medicine, 38700 La Tronche, France
*corresponding author : Jacques.Demongeot@agim.eu,
e-mail: stat_hamie@hotmail.com, antonio.glaria@uv.cl,
carla.taramasco@polytechnique.edu
in Section 6. an application to the ECG signal and, in
Sections 7. and 8., perspectives to the processing of
pulse and single cardiac cells activity and protein
spectrum coming from mass spectrometry.
II. FOURIER AND HALEY WAVELET TRANSFORMS
1) The Fourier transform comes from the aim by J. Fourier
to represent in a simple way the functions used in physics,
notably in the heat propagation (in 1807, cf. [1]). He used a
base of functions made of the solutions of the simple
pendulum differential equation (cf. a trajectory in Fig. 1):
dx/dt = y, dy/dt = -ω2x,
its general solution being:
x(t) = k cosωt, y(t) = - k ωsinωt.
By using the polar coordinates θ and ρ defined from the
variables x and z = –y/ω, we get the new differential system:
dθ/dt = ω, dρ/dt = 0,
with θ = Arctg(z/x) and ρ2 = x2+z2. The polar system is
conservative, its Hamiltonian function being defined by
H(θ,ρ) = ωρ. The solutions x(t) = k cosωt, z(t) = k sinωt
have 2 degrees of freedom, k and ω, respectively the
amplitude and the frequency of the signal, and they
constitute an orthogonal base, when we choose for ω the
multiples (called harmonics) of a fundamental frequency ω0.
After the seminal theoretical works by Y. Meyer [2,4], I.
Daubechies [3] and S. Mallat [5], J. Haley defined a simple
wavelet transform for representing signals in astrophysics
(in 1997, cf. [6]). He used a base of functions made of the
solutions of the damped pendulum differential equation (cf.
a trajectory in Fig. 1):
dx/dt = y, dy/dt = -(ω2 + τ2) x – τy,
its general solution being:
x(t) = ke-τtcosωt, y(t) = -ke-τt(ωsinωt + τcosωt)
By using the polar coordinates θ and ρ defined from the
variables x and z=-y/ω-x/τ, we get the differential system:
dθ/dt = ω, dρ/dt = - τρ,
The polar system is dissipative (or gradient), its potential
function being defined by P(θ,ρ) = -ωθ + τρ2/2. The
solutions x(t)=ke-τtcosωt, z(t)=ke-τtsinωt have 3 degrees of
freedom, k, ω and τ, the last parameter being the
exponential time constant responsible of the pendulum
damping.

Figure 1. Top left: a simple pendulum trajectory. Top middle: a damped pendulum trajectory. Top right: van der Pol limit-cycle (µ=10). Middle: relaxation
oscillation of van der Pol oscillator without external forcing (µ= 5). Bottom: representation of the harmonic contour lines H(x,y) = 2.024 for different values of µ.
III. THE VAN DER POL SYSTEM
For Dynalet transform, we propose to use a base of
functions made of the solutions of the relaxation pendulum
differential equation (cf. a trajectory in Fig. 1 Top), which is
a particular example of the most general Liénard differential
equation:
dx/dt = y, dy/dt = - R(x)x + Q(x)y,
which is specified in van der Pol case by choosing R(x) = ω2
et Q(x) = µ(1-x2/b2). Its general solution being not algebraic,
but approximated by a family of polynomials [6-13]. The
van der Pol system is dissipative (or gradient), its potential-
Hamiltonian system, with P and H functions (Fig. 2 Top
left), H being for example approximated at order 4, when ω
= b = 1, by [9,12]: H(x,y) = (x2+y2)/2 − µxy/2 + µyx3/8
−µxy3/8, which allows to obtain the equation of its limit-
cycle (cf. Fig. 1 Bottom and [9]): H(x,y) ≈ 2.024. The van
der Pol system has 3 degrees of freedom, b, ω and µ, the last
an-harmonic parameter being responsible of the pendulum
damping. These parameters receive different interpretations:
- µ appears as an anharmonic reaction term: when
µ=0, the equation is that of the simple pendulum,
i.e., a sine wave oscillator, whose amplitude
depends on initial conditions and relaxation
oscillations are observed even with small initial
conditions (Fig. 1 & 2 Middle), whose period T
near the bifurcation value µ=0 equals 2π/Imß; ß is
the eigenvalue of the Jacobian matrix J of the van
der Pol equation:
0 1
J = ,
-1 µ
whose characteristic polynomial is equal to: ß2 - µß
+ 1= 0, hence ß = (µ ± (µ2-4)1/2)/2 and T ≈ 2π +
πµ2/8.
- b looks as a term of control: when x > b, the
derivative of y is negative, acting as a moderator on
the velocity. The maximum of the oscillations
amplitude is about 2b whatever initial conditions
and values of the other parameters. More precisely,
the amplitude ax(µ) of x is estimated by
2b<ax(µ)<2.024b, for every µ > 0, and when µ is
small, ax(µ) is estimated by: a
x(µ) ≈
(2+µ2/6)b/(1+7µ2/96) [6,7]. The amplitude ay(µ) is
obtained for dy/dt=0, that is approximately for x=b,
then ay(µ) is the dominant root of the following
algebraic equation: H(b,ay(µ))=2.024.
- ω is a frequency parameter, when µ is small. When
µ>>1, the period T of the limit cycle is determined
mainly by the time during which the system stays
around the cubic function where both x and y are
O(1/µ) and the oscillations period T is roughly
estimated to be T≈2πµ/ω, and the system can be
rewritten as: dχ/dt=ζ, dζ/dt=-ω2χ+µ(1-χ2/µ2)ζ≈-
ω2χ+µζ, with change of variables: χ=µx/b, ζ=µy/b.
x
t
y
x
y
x
y
x
x(t)
x
x
t

Figure 2: Top left: representation of the functions potential P and Hamiltonian H on the phase plane axis (xOy). Middle left: limit-cycle of the van der Pol equation
for different values of µ (from [16]. Bottom left: fit between relaxation oscillations, e.g., van der Pol signal (in blue) and single cardiac cell activity (in white, from
[14]). Middle right: isochronal landscape surrounding the van der Pol limit-cycle (µ=2, ω=b=1; period T ≃ 7.642).
IV. THE DYNALET TRANSFORM
The dynalet transform consists in identifying a Liénard
system based interactions mechanisms between its variables
(well expressed by its Jacobian matrix) analogue to those of
the experimentally studied system, whose limit cycle is the
nearest (in the sense of the Δ set distance, based on the
Hausdorff point to set distance) to the signal in the phase
plane (xOy), where y=dx/dt. For example, the Jacobian
interaction graph of the van der Pol system contains a
couple of positive and negative tangent circuits (called
regulon [11]). Practically, for performing the dynalet
transform it is necessary to choose: i) the parameter µ such
as the period of the van der Pol signal equals the mean
period (either for the van der Pol or for the signal
referential, chosen as the ECG signal in Fig. 3) a translation
of the origin of axes, then rotate these axes to match the first
van der Pol point (circled in red in Fig. 3) equal, by
convention, to the first ECG point (corresponding to the
constant value of the ECG signal between QRS complexes).
x
y
x

Then, in the same way, we can fit the x and y axes origin
and scales, by keeping fixed the first point match van der
Pol / ECG and identifting the new origin with this point
(translation in green on Fig. 3) and minimizing the Δ set
distance between 2 sets of 100 points sampled respectively
from the ECG signal (blue points X
i extracted from the
mean ECG signal on the Fig. 3 Top) and from the van der
Pol signal (red points Yi extracted from the van der Pol
limit-cycle on the Fig. 3 Bottom). We obtain a polynomial
approximation of the fundamental pitch signal, and then the
harmonics.
The polynomial coefficients represent both the potential
and Hamiltonian parts of the van der Pol calculated in the
original ECG axes (using the inverse of the phase plane
transformation done by rotating / scaling / translating the
original ECG axes).
The potential and Hamiltonian parts used for this
transform can be calculated following [8,9]. For example,
for µ=1 and µ= 2, by corresponding polynomials are
respectively:
P1(x,y) = -3x2/4+y2/4+3x4/32+y4/96-x2y2/16,
H1(x,y) = (x2+y2)/2-3xy/2+3yx3/8-y3x/24-2
and
P2(x,y) = -3x2/4+y2/4+3x4/32+y4/96-x2y2/16
H2(x,y) = (x2+y2)/2-3xy/8+3yx3/8-y3x/24-1/2.
Using this potential-Hamiltonian decomposition, we can
calculate as a polynomial of order 2+j an approximate
solution S(kj,µ/2j)(t) of the van der Pol differential system
corresponding to the jth harmonics of the dynalet transform:
dx/dt = y and dy/dt = - x + µ(1-k2x2)y/2j
We will search for example for the approximate solution
x(t)=S(1,1)(t) as a polynomial of order 3 in the case µ=1:
x(t) = c0 + c1t + c2t2 + c3t3, y(t) = c1 + 2c2t + 3c3t2
The coefficients ci’s above are obtained by identification in
the following equation [3-5]:
(x2+y2)/2 −3µxy/2 +3µyx3/8 −µxy3/24 − 2.023 = 0.
Then, we get: c0
2/2 + c1
2/2 -3c0c1/2 + 3c0
3c1/8 - c0c1
3/24=2,
c2c3-9c3
2/2-9c0c2
3+9c0c2
3/4+27c0
2c3
2/8-3c0c2c3
2/4-c2
4/24 = 0
⇔ c2c3 - 27c2
3/2 + 9c3
2 -3c2c3
2/2 - c2
4/24 = 0, which implies
c0 =2, c1=0, c2 ≈ 0.46 and c
3≈0.04, i.e., approximately the
values given in [6].
Because of the symmetry of the limit cycle, all the
solutions{S(kj,µ/2j)}j∈IN are orthogonal and we can
decompose any continuous function f on this base, thanks to
the Weierstrass theorem.
V. APPLICATION TO THE ECG SIGNAL
To summarize the dynalet transform approach, the whole
approximation procedure done for the ECG signal (see Fig.
2) involves the following steps:
1) to perform a symmetrizing of x axis in the case of
derivation V1 in order to get a signal similar the ECG V5
(cf. Fig. 3 Middle) and a transformation rotating / scaling
the x and y axes of the ECG signal, so as to adjust them to
the maximum and minimum x and y of the vdP (van der
Pol) signal,
2) to perform a translation of the origin of axes of the ECG
signal (green vector on Fig. 3 Bottom) by adjusting the base
line to a selected phase of a vdP limit cycle of same period T
(called pitch period) as the ECG period: this phase is said
phase 0, surrounded by a red circle on Fig. 3), keeping
scaled and fitted to the vdP signal the x and y amplitudes of
the ECG signal,
3) to finish the approximation matching the ECG points set
to the vdP, by minimizing the difference set distance Δ
between the interiors of the ECG and vdP cycles (denoted
respectively ECG and VDP, with interiors ECGo and
VDPo) in the phase plane:
Δ(ECGo,VDPo)=Area[(ECGo\VDPo)∪(VDPo\ECGo)],
by using for example a Monte-Carlo method for estimating
the area of the sets interior to the linear approximation of the
ECG and vdP cycles, calculated from the point samples
{Xi}i=1,100 and {Yi}i=1,100. We can also calculate the
transform minimizing the cost function:
Σi=1,100dH(Xi,VDP), where dH(Xi,VDP)=infj=1,100{d(Xi,Yj)},
4) to use the first quadric approximation of the vdP limit
cycle to deduce the fundamental vdP component of the
ECG signal,
5) to reconstruct the approximated temporal profile of the
ECG signal, by using the approximate velocity obtained in
step 3 at discrete successive times i=1,…,100,
6) for the first harmonics of period T/2, to repeat all the
previous process (from step 1 to step 5) done for the
fundamental component, by using a van der Pol system of
period T/2 (see Fig. 4) for fitting the difference between the
original ECG signal and the fundamental component, and
then do the same for the successive harmonics of increasing
order 2, 3,… ,
7) to stop the approximation process when the x and y
amplitudes will be less than a fixed threshold.

An alternative to the only limit-cycle fit could be to use the
whole the isochron landscape (an isochron of phase φ being
the set of all initial conditions whose trajectory is
asymptotically synchronized with the oscillations starting on
the limit-cycle at the same phase φ, which constitutes a kind
of continuation of the limit-cycle inside the whole phase
space (see Fig. 2 Right for a display of the isochron
landscape). Such an approximation needs to observe the
empirical signal from different perturbed initial conditions
and not only on its asymptotic periodic behaviour, in order
to get an estimate of the empirical isochrons.
Figure 3: Top left: original ECG signal (V1 derivation). Top right: decomposition into two temporal profiles respectively of period T and T/2, whose
corresponding functions are orthogonal for the integral on [0,T[ vector product. Middle left: representation of different van der Pol limit-cycle, for different values
of µ (from µ=0,01 in red, to µ=4). Fourier. Middle right: comparison between Fourier and Haley wavelets decompositions of the ECG signal (V5 derivation),
showing a more rapid fit with wavelets (until the 4th harmonics) than Fourier (until the 17th harmonics). Bottom: from [16], representation of different waves from
van der Pol oscillator simulations (from [16]), from the symmetric type (left, for µ=0.4, ω=1, b=4) to the relaxation type (right, for µ=4, ω=1, b=4).
Harmonics
original
original
ECG
(V1 derivation)
Base line
Wavelets transform
Fourier transform

Figure 4: First row, from left to right: fundamental component extraction for the ECG signal, i) by identifying the phase 0 value on the original V1 derivation in
the phase plane xOy, where y=dx/dt, then on a sample of 100 points (in blue) extracted from the mean signal, and eventually on the van der Pol limit-cycle having
the same period as the pitch of mean ECG signal. Second row: ECG signal for different electro-physiologic derivations (from [17]). Third row: Match for getting
the fundamental component, with a transformation in the (xOy) phase plane consisting in rotating/scaling/translating the x and y axes to obtain the best fit for the
cost function based on the Hausdorff distance between sampled empirical mean points and the set made of the van der Pol limit-cycle. Fourth row: calculation of
the secondary signal by subtracting the fundamental component (in red) from the sampled original signal (in blue); the new data subtracted Zi are obtained by
subtracting (for each coordinate) from Xi the point Yi realizing the minimum of the Hausdorff distance between Xi and the fundamental component (left) to be
compared to a van der Pol limit cycle of period T/2 (right). Fifth row: calculation of the difference with its temporal profile during a period (left) and its
representation in the (xOy) phase plane (right).

VI. PERSPECTIVES TOWARD THE SINGLE CARDIAC
CELL AND PULSE SIGNALS
Biological rhythms other than the ECG can be interpreted
and compressed using Liénard equations and the dynalet
transform, like the single cardiac cell activity, which
represent a good example of relaxation wave (cf. Fig. 2
Bottom left and [14]). Pulse activity models (cf. Fig. 5)
constitute also an application field for the dynalet transform.
Figure 5. Pulse ultrasound volume signal with temporal profile (top) and
representation in the (xOy) phase plane
Figure 6. An overview of van der Pol equation in a biological models
landscape. The interaction graph of these systems contains in general a
couple of positive and negative tangent circuits (called regulon [11]).
Since 50 years and the first models by Noble [18,19], 45
models have been proposed [20], many of them being based
on Hodking-Huxley like models, which are closely related
to the van der Pol equation (Fig. 6).
VII PERSPECTIVES TOWARD A NON-PERIODIC SIGNAL:
PROTEIC MASS SPECTRUM AND PROTEIC
“STETHOSCOPE”
Another perspective of application is the compression of
non-periodic signal. It is indeed fruitful to represent any
signal (periodic or not) as depending on the mechanism
which gave birth to it. If the mechanism of signal production
is of Liénard type, even if the signal is a single wave, it can
approximated by the dynalet transform, especially if the
wave is asymmetrical, type relaxation. By periodising it, we
can obtain the approximate polynomial representation of the
desired order, in the space of solutions of the Liénard
equation. A good example of this type of signal is formed by
the spectrum of a protein, observable by mass spectrometry
[21]. Identification of proteins by their spectrum allows to
build complex regulatory genetic networks, such as those at
work in controlling the immune system [22-26] .
Figure 7. Top and Middle: protein spectral data from mass spectrometry
showing cancer patient (top) and normal patient (middle) responses (from
[24]). Bottom: periodisation of the spectrum signal and transformation in a
audible sound.
100 L. Forest et al. / C. R. Biologies 330 (2007) 97–106
consider the new variable: u=(a −c/s)x(1−x)+(b −
cr/s)x (1−y).
Fig. 2 shows a trajectory of this system, which we
can algebraically approach by using its PH-decomposi-
tion [1],fora−c/s =0.1999, b−cr/s =0.2, d−
f/s =−4, and e−fr/s =0.1695.
3. Liénard systems as a paradigmatic model for
biological regulatory systems
3.1. Definition of a Liénard system
A Liénard system consists of two-dimensional or-
dinary differential equations (2D ODEs) defined by:
dx/dt=y,dy/dt=−g(x) +yf (x),wheregand fare
polynomials. We address here to the applications of the
Liénard equations and we refer to the previous note [1]
for the mathematical properties of the PH-decompo-
sition. Liénard equations are still actively studied by the
community of mathematicians, because they serve as a
reference model for the resolution of the XVIth Hilbert
problem [11–18].
3.2. Classical examples of Liénard systems in
biological modelling
Liénard equations have been used in physiology to
simulate both the heart and the respiratory system (van
der Pol equations original [19] and modified [20])and
the nerve impulse (FitzHugh–Nagumo equations [21,
22]). FitzHugh–Nagumo equations are just a 2D ap-
proximation of the Hodgkin–Huxley equations, funda-
mental in neurobiology [23–25].
FitzHugh–Nagumo equations can be approached by
the Wilson–Cowan system [26,27]. It has, in certain
parametric circumstances, the same behaviour as the
Hopfield equations. In addition, the kinetics of in vitro
self-assembly and disassembly of microtubules [28],
major elements of the cytoskeleton, can be expressed in
the form of a Liénard system. Finally, Liénard systems
are used for modelling oscillatory chemical reactions,
for example the Belousov–Zhabotinsky reaction (Noyes
equations [29,30]). These examples illustrate the univer-
sality of the Liénard systems (Fig. 3), which definitively
constitute the natural mathematical framework in which
many biological and chemical equations can be imbed-
ded. As presented in the previous note [1], one of their
main properties is their ability to model periodic behav-
iours with simple isochronal patterns (Fig. 4) suscepti-
ble to explain the entrainment of biological systems by
instantaneous periodic stimulations.
3.3. The example of the regulon
The regulon structure is frequently observed in biol-
ogy. It is made up of a loop of activation and inhibition
Fig. 3. An overview of Liénard systems in biological models.
Pulse Volume
x(t)
t
x
y=dx/dt

Key proteins in these networks are effectors of the genic
expression (activators or inhibitors) and may be subject to
pathologic conditions, leading to up- or down-expressions
and these regulatory interactions leading to abnormal
protein concentrations can be taken into account by a
Liénard type modelling. Of course, other alternative
techniques for estimating protein spectra already exist, like
kernel functional estimation tools [27-28], but there are
not based on the mechanism of the production of the
protein signal. The corresponding tool, a real protein
stethoscope, would give sense to numerous protein data,
which, although very heavy in terms of information (about
5 Go per patient in a modern hospital), are in general not
queried by clinicians (especially in emergency) in patient
centred data bases, and constitute therefore often true
cemeteries full of unused data.
VIII. CONCLUSION
Generalizing compression tools like Fourier or wavelets
transforms is possible, if we consider that non symmetrical
biological signals are often produced by mechanisms based
on interactions of regulon type (i.e., possessing at least one
couple of positive and negative tangent circuits inside their
Jacobian interaction graph). In this case, we can replace
the differential systems giving birth to biological signals
by a Liénard type equation, like the van der Pol system
classically used to model relaxation waves. The
corresponding new transform, called dynalet transform,
has been built in the same spirit as the wavelet transform
for turbulent systems like Burger equation [29-30] and as
the new tools of biological information design [31-33].
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