Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model

Abstract

Time-frequency and time dependence of the output signal morphology of nonlinear oscillator neuron based on Van der Pol model using analytical and numerical methods were investigated. Threshold effect neuron, when it is exposed to external non-stationary signals that vary in shape, frequency and amplitude was considered.

Full text

I.J. Intelligent Systems and Applications, 2018, 4, 8-17
Published Online April 2018 in MECS (http://www.mecs-press.org/)
DOI: 10.5815/ijisa.2018.04.02
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
Time Dependence of the Output Signal
Morphology for Nonlinear Oscillator Neuron
Based on Van der Pol Model
Vasyl Lytvyn, Victoria Vysotska, Ivan Peleshchak, Ihor Rishnyak
Information Systems and Network Department, Lviv Polytechnic National University, Lviv, Ukraine
E-mail: Vasyl.V.Lytvyn@lpnu.ua, Victoria.A.Vysotska@lpnu.ua, peleshchakivan@gmail.com,
Ihor.V.Rishnyak@lpnu.ua
Roman Peleshchak
Ivan Franko Drohobych State Pedagogical University, Department of General physics, Drohobych, Ukraine
E-mail: rpeleshchak@ukr.net
Received: 05 July 2017; Accepted: 21 October 2017; Published: 08 April 2018
Abstract—Time-frequency and time dependence of the
output signal morphology of nonlinear oscillator neuron
based on Van der Pol model using analytical and
numerical methods were investigated. Threshold effect
neuron, when it is exposed to external non-stationary
signals that vary in shape, frequency and amplitude was
considered.
Index Terms—Nonlinear oscillator neuron, frequency
modulation, morphology of the information signal,
resonance effect, encoding and decoding of information.
I. INTRODUCTION
At present, intensive research on the application of
neural networks is being conducted to solve a wide range
of Data Mining tasks (identification of non-stationary
chaotic processes, clusterization, classification,
intellectual management, biosystems states diagnostics,
prediction, emulation and recognition of multispectral
input images). Considerable interest in the study of
modern neurodynamics lies in the processes of
information encryption, decryption and processing that is
transmitted by neurons. In the early stages of sensory
information processing Wavelet analysis is an effective
tool to study the information component of the neural
signals that are registered. Traditionally such research
demands to analyze the structure of point processes, i.e.
the time-frequency dynamics analysis of neural reviews
[1]-[6], in which information carriers are times of pulses
(spikes) generating, but not their form [7]. The
mechanisms that lead to the spikes generation are
partially known [8]. But how neurons and their ensembles
transmit information about around world, so far has
practically been unexplored.
In the wavelet-neurophase network, the wavelet-
neuron is structurally close to the standard formal neuron
with N inputs, but instead of the usual synaptic
weights
ik

, there are wavelet-synaptic weights
i
WS
ik

( 1,2,..., )iN
, in which the adjustable parameters are not
only the weights
ik

, but also the scaling and offset
wavelets. It should be noted that in all of the above-
mentioned artificial neural networks, neurons were
considered without their own dynamics (their own
frequency of oscillations of the neuron
00
i


) and
without taking into account the accumulation of
impulses
0k
N
in the neurons that triggered when
0k ck
NN
(
ck
N
is the threshold value of the k-th neuron
pulses). The main advantages of self-developing artificial
nonlinear neural networks is the ability to adapt to
dynamic conditions and the speed of functioning, which
is especially important when operating in real-time. In
order to provide the highest class of efficiency (speed of
operation) and the quality of the recognition of input
spectral images, it is proposed to develop the architecture
of self-developing artificial nonlinear neural networks
and an algorithm for their training based on evolutionary
simulation methods that can adapt to dynamic conditions
when working in real-time.
Accordingly, to above-mentioned there is necessity to
solve the following problems:
 It is necessary to suggest the method of
inhomogeneous nonlinear differential equations
with quadratic non linearity of the unknown
function at the first derivative.
 It is necessary to define that nonlinear oscillator
neuron can act as a frequency modulator, which
can modulate the input information of non-
stationary signal.
 It is necessary to determine that nonlinear
oscillator neuron with threshold effect
significantly alters the structure of the input
information non-stationary signal different in
shape, frequency and amplitude.

Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model 9
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
 It is necessary to determine the existence of
resonance effects in nonlinear oscillator neuron
when frequency of the external non-stationary
signal and dynamics of the natural frequency of
the neuron are equal.
 It is necessary to suggest information coding with
nonlinear oscillator neuron on the basis of
frequency modulation and decoding using an
inverse operator, which acts on the output signal
vector.
II. RELATED WORKS
In [2], [3], [6], the authors analyzed the case of signals
conversion by sensory neurons (threshold device), but its
own dynamic neuron was not taken into consideration.
Using classical models threshold systems such as
―integrate-and-fire‖ [2] and ―threshold crossing‖ [3], [6],
it has been shown that various characteristics of complex
dynamics at the entrance sensory neuron are stored in the
point process structure [2]-[6], [9]-[11], [17]-[33].
Computer neurons dynamics modeling when exposed
to constant external signal was conducted in Van der Pol
approaching [12]. The authors [13] investigated
frequency-temporal dynamics of sensory neuron
(threshold device) using the technology of double wavelet
analysis and taking into account the interaction of its own
dynamics and the dynamics that was caused by the
influence of external non-stationary signal. In this
sensory neuron was modeled as a threshold device that
converts the input signal to pulse sequence output. This
pulse sequence was described by the sequence of Dirac
delta functions, each of which corresponds to the pulse
(adhesions) generation moment. These model pulses have
the same shape and amplitude, that’s why information
about the external effect of dynamic signal appears only
in the time intervals between the moments of their
generation.
III. SETTING OBJECTIVES
The aim of this work is to study the analytical and
numerical method of time-frequency and time-
dependence of the output signal morphology for
nonlinear oscillator neurons based on Van der Pol model
taking to account threshold effect neurons when it is
exposed to different in shape, frequency and amplitude of
external non-stationary signals.
IV. MATHEMATICAL MODEL
Nonlinear oscillator sensory neuron type Van der Pol
(threshold device) with its own dynamics
0k

, which can
generate pulses in the absence of external non-stationary
signals
( ( ) 0)
k
Vt
, when the number of available sensory
neuron impulses
0k
N
reaches a threshold value
ck
N
0
()
k ck
NN
was considered. Thus, such neuron can be
considered as a threshold device that converts the input
non-stationary signal
()
k
Vt
into a sequence of pulses
output (Fig.1) due to the ―imposition‖ of the dynamics of
input non-stationary signal
()
k
Vt
on its own neuron
dynamic. Consequently, the signals conversion process
analysis by nonlinear oscillator sensory neuron is
complicated. The complex dynamics of transformation of
input non-stationary signal received by biological sensor
with its own dynamics shows an experimental record
signal (Fig.1b) [13]-[16]. This signal generated by a
biological neuron without external signal action (interval
0< t <110s with a low-δ-pulse sequence). At the time
interval 110s< t <200s shown on (Fig. 1b) the result of
external signal interaction with the biological sensor with
its own dynamics which leads the formation of high-order
δ-pulses sequence.
To illustrate, let us consider encoding information by
touch nonlinear oscillator k-neuron which describes the
nonlinear equation of the form
2 2 2
00
[ ( ; )] ( )
k k k k k ck k k k k
X X p N N X X V t

   
(1)
where
22
0
00 2
( ; ) tanh( )
k ck
k k ck k
k
NN
p N N p



setting the
amplitude of the k-th neuron;
0
k


;
0k
N
,
ck
N
,
2
k

are the number of pulses that come to
k-th neuron, threshold pulse k-th neuron and variance
respectively;
2
0k

is natural frequency k-th nonlinear oscillator
neuron;
()
k
Vt
is non-stationary input signal goes to k-th
neuron.
Nonlinear oscillator neuron has its own dynamics and
generates pulses in the absence of external signals at
0k ck
NN
, because under this condition
02
tanh( ) 0
k ck
k
NN


and accordingly
20
( ; ) 0
k k ck
p N N 
[14]. Mathematical model (1) can also be used to study
the collective behavior of ensembles of neurons
interconnected synaptic connections
jk

. For this purpose
the second and third terms are necessary to replace
kk
XX


where
1
;
N
k k jk j
j
X X X




1,2,...,N.


Solution to equation (1) with the analytical numerical
method and consistent approximation method in the form
( ) ( ) ( )
( ) ( )sin ( )
n n n
k k k
X t a t t


(2)
where n = 1, 2, 3, …, N is an iteration number;
( ) ( )
0
( ) ( )
nn
k k k
t t t
  

(3)

10 Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
Fig.1. a – Schematic representation of process for the input signal
()
k
Vt
conversion by non-linear oscillator sensory neuron (threshold device). Pulses
generating times on the output of threshold device
()
k
Xt
correspond to the crossing moments of the threshold level; b – An experimental sample
recording of signal which is generated by biological neurons [13].
()
()
n
k
at
and
()
()
n
kt

– functions of time, which are
selected so that the ratio (2) satisfies the equation (1). In
addition we impose the condition that
()
()
n
k
at
is a slowly
variable function, i.e.
()
()
()
lim 1.
()
n
k k k
n
tkk
a p t
a p t

 
(4)
But since there are two functions
()
()
n
k
at
and
()
()
n
kt

,
and one equation, this condition ambiguous defines the
function. We demand that the condition was fulfilled
( ) ( ) ( )
0( )cos ( )
n n n
k k k k
X a t t


(5)
where
()
() ()
.
n
nk
k
dX t
Xdt

Substituting (2) in (1) and given the condition (5) we
obtain the system of equations
()
()
n
k
at
and
()
()
n
kt

:
() ( 1) ( 1) 2 2 ( 1) 2 2 ( 1) ( 1)
0
( ) ( 1)
( 1) 2 2 ( 1) 2 ( 1) ( 1)
00
() ( ) ( ) sin ( ) ( ; ) cos ( ) ( )cos ( )
( ) ( )sin
( ) sin ( ) ( ; ) sin ( )cos ( )
nn n n n n
kk k k k k k ck k k k
nn
n n n n
k k k
k k k k k k ck k k
da t a t a t p N N t V t t
dt
d t V t
a t p N N t t
dt
   

    
    

   

   


   

( 1)
()
()
n
k
t
at








(6)
The right side of the equation system (6) for the period
2π at
( ) 0
k
Vt
for the rule [15] was averaged:
2( ) ( )
0
1( ) .
2
nn
kk
d


   

(7)
In zero approximation expressions for
(0) ()
k
at
and
(0) ()
kt

are found from the system of equations:
(0) (0)
(0) (0)
( ) ( )
( ( )), ( ( ))
kk
kk
da t d t
A a t B a t
dt dt


(8)
where
(0) 2 2
(0) ( 0) (0) 0
()
( ( )) ( ) , ( ( )) .
82
kk
k k k k k
ap
A a t a t B a t


   


Integrating the equation (8), a zero approximation
expressions for
(0) ()
k
at
and
(0) ()
kt

were worked out:
2
(0) 2
( ) ,
1kk
k
kpt
p
at e




(9)
(0) 0
()
kk
tt


(10)

Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model 11
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
where
(0) ()
k
at
satisfies the criterion (4) of slowly variable
function. To find expressions
(1) ()
k
at
and
(1) ()
kt

in the
first approximation (n = 1) we should substitute the
expression (9), (10) into the system of equations (6). As a
result of integrating we obtain first approximation
expressions for
(1) ()
k
at
and
(1) ()
kt

. The process of
iteration stops when the conditions are fulfilled:
(n) (n 1)
(n)
( ) ( ) 1,
()
kk
k
a t a t
at


(n) (n 1)
(n)
( ) ( ) 1.
()
kk
k
tt
t




V. RESULTS
For modeling example, in this problem external non-
stationary signal
()
k
Vt
is chosen as a sum of N ordinary
non-stationary signals, each of which is centered at the
point
L
tt
and characterized by system parameters L
[16]
0
1
0
( ) ( ) .
N
k Lk Lk L k
LL
V t v t t



  

(11)
By selecting system parameters L, we can to construct
mathematical models of complex non-stationary signal
()
k
Vt
, spectral properties of which change over time. The
mathematical model of unsteady external signal
()
k
Vt
will reflect the dynamics of the real signal that
characterizes a physical (biological) process.
An example of a simple non-stationary signal
()
Lk L
v t t
is an expression:
2
2
()
1
( ) exp cos( ( ) ),
4
2
L
Lk L L L L
L
L
tt
v t t t t





    


(12)
which is a product of the envelope Hauss form to
oscillating function and is described by five parameters L
( , , , , )
Lk L L L L
Lt
   

(13)
where
Lk

denotes weight connections of inputs v1,…,vN
of the k-th neuron;
0k

is weight shift signal of
communication with the k-th neuron;
2
LL
f


is
external carrier frequency oscillation in hertz (Hz),
L
t
is
center localization signal by the time in seconds,
L

is
typical localization signal time interval in seconds,
L

is
the initial phase in radians.
Equation (11) with (12) describes all inputs, including
offset signal, coming with weights
Lk

the adder k-th
neuron (incoming operator
in
f
). Input operator
in
f
converts weighted weights
Lk

inputs and presents them
to the operator activation
a
f
(fig.2). For nonlinear
oscillator sensory neuron activation operator
a
f
looks
like, this input operator
22 2 2
00
2( ; ) .
a k k k k ck k
dd
f X p N N dt
dt


   

(14)
01v
1
v
N
v
in
f
0K

0
LK

0
L K N

a
f
out
f
 
k
Vt
 
k
Xt
Fig.2. The structure of artificial nonlinear oscillator neuron
The output signal of nonlinear oscillator neuron
()
k
Xt
(Fig.2) is transformed by the source operator
out
f
the
output signal of service activation. Output operator
out
f
is required to represent the state of the neuron in the
desired field values. In most studies, this operator is not
isolated, and under output signal the neuron to understand
the signal after activation operator
a
f
. However, during
the analysis and synthesis of artificial neural networks
(ANN), which have different activation functions of the
various regions and areas of value determination, is
necessary taking into account the output operator.
Consequently, nonlinear operator transformation of input
signals
()
k
Vt
vector in the vector output signal
()
k
Xt
can be written as
( ) ( ( ( ( ), )))
k out a in k Lk
X t f f f V t


. (15)
By selecting different combinations of
( , , , , ),
Lk L L L L
Lt
   

we can construct a theoretical
model that adequately describes the real physical
(biological) processes in the interaction of external non-
stationary signal
()
k
Vt
with its own dynamic physical
(biological) nonlinear neuron.
VI. DISCUSSION
Figs. 3-4 shows a first term (N = 1) non-stationary
external signal
()
k
Vt
((11), (12)) depending on the time
00
0 0 0 0 0
0
0
2
2
()
( ) exp cos( ( ) ),
4
2
L k L
L k L L L L
L
L
tt
v t t t t





    



(16)
applied to nonlinear oscillator sensory neuron for two
parameter values
0
L
:
0
L
= (3, 4

, 12, 3, 0);
0
L
= (1,
6

, 21, 1, 0) in accordance. In the first case the signal
00
()
L k L
v t t
(Fig.3) has an amplitude
03
Lk


, and the

12 Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
second –
01
Lk


respectively, in the first case, the signal
has a frequency
0
L
f
= 2Hz, centered at a point in time
0
L
t
=12s with time-lapse localization signal
0
L

=3s and
the initial phase
0
L

=0, and the second –
0
L
f
=3Hz;
0
L
t
=21s,
0
L

=3s;
0
L

=0.
Fig.3. Morphology of the external information signal
00
()
L k L
v t t
(16) with parameter values
0
L
= (3, 4

, 12, 3, 0);
0.4
k
p
and
0.1
k


.
Fig.4. Morphology of the external information signal
00
()
L k L
v t t
(16) with parameter values L0= (1,
6

, 21, 1, 0); pk=0.4 and
0.1
k


.
In numerical calculations, signals
()
k
Xt
the output of
the nonlinear oscillator neuron with input signals (Figs. 3-
4) parameters
0.4
k
p
and
0.1
k


match.
Figs. 5-6 present graphs of frequency modulation
()
() ()
n
nk
k
dt
dt



, which changes the instantaneous
frequency of the carrier oscillation information
()
k
Vt
(11), which consists of a simple non-stationary signal
00
()
L k L
v t t
(16), in accordance with a change in signal
caused by the interaction of an external signal carrying
information with its own dynamics
00.2
k


(Fig. 5)
and
02
k


(Fig. 6) nonlinear oscillator neuron.
s

Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model 13
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
Fig.5. The time dependence of the instantaneous frequency of the carrier signal information
00
()
L k L
v t t
(16) with parameter values
0
L
= (3, 4

, 12, 3, 0);
0.4
k
p
and
0.1
k


,
00.2
k


.
Fig.6. The time dependence of the instantaneous frequency of the carrier signal information
00
()
L k L
v t t
(16) with parameter values
0(1, 6 , 21,1, 0)L


;
0.4
k
p
and
0.1
k


,
02
k


Figs. 7-8 present graphs of morphology signal at the
output of the nonlinear oscillator neuron
()
k
Xt
defined
by the nature of the interaction of dynamics neuron with
frequencies
00.2
k


;
02
k


and dynamic, caused by
external influence
00
()
L k L
v t t
(16).
Fig.7. Schedule of morphology signal at the output of the nonlinear oscillator neuron
()
k
Xt
with parameter values
0
L
= (3, 4

, 12, 3, 0);
0.4
k
p
and
0.1
k


,
00.2
k


.

14 Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
Fig.8. Schedule of morphology signal at the output of the nonlinear oscillator neuron
()
k
Xt
with parameter values
0(1, 6 , 21,1,0)L


;
0.4
k
p
and
0.1
k


,
02
k


.
Figs. 9-10 present graphs of morphology signal at the
output of the nonlinear oscillator neuron
()
k
Xt
, defined
by the nature of the interaction of dynamics neuron with
frequencies
00.2
k


;
02
k


and dynamic, caused by
external influence
00
()
L k L
v t t
(16).
Fig.9. Schedule of morphology signal at the output of the nonlinear oscillator neuron
()
k
Xt
with parameter values
0
L
= (3, 4

, 12, 3, 0);
0.4
k
p
and
0.1
k


,
00.2
k


.
Fig.10. Schedule of morphology signal at the output of the nonlinear oscillator neuron
()
k
Xt
with parameter values
0(1, 6 , 21,1,0)L


;
0.4
k
p
and
0.1
k


,
02
k


.

Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model 15
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
The interaction of external signal
00
()
L k L
v t t
with
parameters
0
L
= (3, 4

, 12, 3, 0);
0(1, 6 , 21, 1, 0)L


with its own dynamics of nonlinear oscillator neuron
04
k


or
06
k


a sharp increase in the amplitude
of the output signal
()
k
Xt
(Fig. 11) compared to the
output signal in Fig.9. That is a resonance effect in the
external signal frequency which matches the natural
frequency of oscillation of the nonlinear neuron,
(
00Lk


). Graph of modulation frequency
()
()
n
kt

if a
resonance effect is shown in Fig.12.
Fig.11. Schedule of morphology signal at the output of the nonlinear oscillator neuron
()
k
Xt
if a resonance effect with parameter values
0
L
= (3, 4

, 12, 3, 0);
0.4
k
p
and
0.1
k


,
04
k


.
Fig.12. The time dependence of the instantaneous frequency of the carrier signal information
00
()
L k L
v t t
(16) if a resonance effect with parameter
values
0
L
= (3, 4

, 12, 3, 0);
0.4
k
p
and
0.1
k


,
04
k


.
To decode the structure of output signal
()
k
Xt
, it is
important to know the code of program, which describes
the inverse operator acting on a vector output, i.e.
111
( ( ( ))) ( , ).
in a out k k Lk
f f f X t V t

 
(17)
Thus, the received results allow hypothesizing that the
process of encoding information of nonlinear neurons can
be considered in terms of modulation frequency as we
know in radio physics that modulation frequency is one
way of information transmitting.
VII. CONCLUSIONS
The method of inhomogeneous nonlinear differential
equations with quadratic non linearity of the unknown
function at the first derivative was suggested.
It was defined that nonlinear oscillator neuron can act
as a frequency modulator, which can modulate the input
information of non-stationary signal.
It was determined that nonlinear oscillator neuron with
threshold effect significantly alters the structure of the
input information non-stationary signal different in shape,
frequency and amplitude.
s

16 Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
The existence of resonance effects in nonlinear
oscillator neuron when frequency of the external non-
stationary signal and dynamics of the natural frequency
of the neuron are equal is determined.
Information coding with nonlinear oscillator neuron on
the basis of frequency modulation and decoding using an
inverse operator, which acts on the output signal vector
were suggested.
REFERENCES
[1] T. Sauer, ―Reconstruction of dynamical systems from
interspike intervals.‖ Phys. Rev. Lett., Vol. 72, pp. 3811-
3814, 1994.
[2] D.M. Racicot, A. Lonytin, ―Interspike interval attractors
from chaotically driven neuron models.‖ Physic D., Vol.
104, pp. 184-204, 1997.
[3] R. Castro, T. Sauer, ―Correlation dimension of attractors
through inters pike intervals.‖ Phys. Rev. E., Vol. 55, pp.
287-290, 1997.
[4] T. Sauer, ―Nonlinear Dinamics and Time Series.‖ Eds C.
Culter and D. Kaplan, Fields Institute Communications.
American Math. Society, Providence, RI, Vol. 11, pp. 63-
75, 1997.
[5] R. Hegger, H. Kantz, ―Embedding of sequences of time
intervals.‖ Europhys. Lett., Vol. 38, pp. 267-272, 1997.
[6] R. Castro, T. Sauer, ―Chaotic Stochastic Resonance:
Noise-Enhanced Reconstruction of Attractors.‖ Phys. Rev.
Lett., Vol. 79, pp. 1030-1033, 1997.
[7] A. Pavlov, A. Khramov, A. Koronovskyy, E. Sytnykova,
V. Makarov, A. Ovchynnykov, ―Veyvlet-analyz v
neyrodynamyke.‖ Uspekhy fyzycheskykh nauk, Vol.
182(9), pp. 905-939, 2012.
[8] H.C. Tuckwell, ―Introduction to Theoretical
Neurobiology.‖ Cambridge University. Press, Cambridge,
1998.
[9] N. Janson, A. Pavlov, A. Neiman, V. Anishchenko,
―Reconstruction of dynamical and geometrical properties
of chaotic attractors from threshold-crossing interspike
intervals.‖ Phys. Rev. E., Vol. 58, pp. R4-R7, 1998.
[10] A. Pavlov, O. Sosnovtseva, E. Mosekilde, V.
Anishchenko, ―Extracting dynamics from threshold-
crossing interspike intervals: Possibilities and
limitations.‖ Phys. Rev. E., Vol. 61, pp. 5033-5044, 2000.
[11] A. Pavlov, O. Sosnovtseva, E. Mosekilde, V.
Anishchenko, ―Chaotic dynamics from interspike
intervals.‖ Phys. Rev. E., Vol. 63, pp. 036205(5), 2001.
[12] S. John, Hooshang Hemami Bay, ―Modeling of a Neural
Generator with Coupled Nonlinear Oscillators.‖ IEE
Transactions biomedical engineering, V.BME, Vol. 34(4),
pp. 297-306, 1987.
[13] A. Pavlov, O. Pavlova, ―Prymenenye veyvlet-analyza v
yssledovanyyakh struktury tochechnykh protsessov.‖
Pysʹma v ZHTF, Vol. 32(21), pp. 11-17, 2006.
[14] V.Y. Suhakov, ―Osnovy synerhetyky.‖, Kyyiv, 2001.
[15] M. Boholyubov, Y. Mytropolskyy, ―Asymptotychni
metody v teoriyi neliniynykh rivnyan‖, M.: Nauka, 1992.
[16] S.V. Bozhokyn, ―Nepreryvnoe veyvlet-preobrazovanye y
tochno reshaemaya modelʹ nestatsyonarnykh syhnalov.‖
ZHTF, Vol. 82(7), pp. 8-13. 2012.
[17] T. Basyuk, ―The main reasons of attendance falling of
internet resource.‖, in Proc. of the X-th Int. Conf.
Computer Science and Information Technologies,
CSIT’2015, pp. 91-93, 2015.
[18] E. Burov, ―Complex ontology management using task
models.‖ International Journal of Knowledge-Based and
Intelligent Engineering Systems, Amsterdam: IOS Press,
Vol. 18(2), pp. 111-120, 2014.
[19] V. Lytvyn, V. Vysotska, L. Chyrun, L. Chyrun, ―Distance
Learning Method for Modern Youth Promotion and
Involvement in Independent Scientific Researches.‖ in
Proc. of the First International Conference on Data
Stream Mining & Processing, DSMP, pp. 269-274, 2016.
[20] J. Chen, D. Dosyn, V. Lytvyn, A. Sachenko, ―Smart Data
Integration by Goal Driven Ontology Learning.‖
Advances in Big Data. Advances in Intelligent Systems
and Computing, Springer, pp. 283-292, 2016.
[21] V. Lytvyn, V. Vysotska, O. Veres, I. Rishnyak, H.
Rishnyak, ―Classification Methods of Text Documents
Using Ontology Based Approach.‖ Advances in
Intelligent Systems and Computing, Vol. 512, Springer, pp.
229-240, 2016.
[22] V. Teslyuk, V. Beregovskyi, P. Denysyuk, T. Teslyuk, A.
Lozynskyi, ―Development and Implementation of the
Technical Accident Prevention Subsystem for the Smart
Home System.‖ International Journal of Intelligent
Systems and Applications (IJISA), Vol. 10, No. 1, pp. 1-8,
2018.
[23] Zh. Hu, Ye.V. Bodyanskiy, O.K. Tyshchenko, and V.O.
Samitova,"Fuzzy Clustering Data Given in the Ordinal
Scale", International Journal of Intelligent Systems and
Applications (IJISA), Vol.9, No.1, pp.67-74, 2017.
[24] Zh. Hu, Ye.V. Bodyanskiy, O.K. Tyshchenko, V.O.
Samitova,"Possibilistic Fuzzy Clustering for Categorical
Data Arrays Based on Frequency Prototypes and
Dissimilarity Measures", International Journal of
Intelligent Systems and Applications (IJISA), Vol.9, No.5,
pp.55-61, 2017.
[25] Zh. Hu, Ye.V. Bodyanskiy, O.K. Tyshchenko, V.M.
Tkachov, ―Fuzzy Clustering Data Arrays with Omitted
Observations‖, International Journal of Intelligent
Systems and Applications (IJISA), Vol.9, No.6, pp.24-32,
2017.
[26] Zh. Hu, Ye.V. Bodyanskiy, O.K. Tyshchenko, and V.O.
Samitova,"Fuzzy Clustering Data Given on the Ordinal
Scale Based on Membership and Likelihood Functions
Sharing", International Journal of Intelligent Systems and
Applications (IJISA), Vol.9, No.2, pp.1-9, 2017.
[27] J. Su, V. Vysotska, A. Sachenko, V. Lytvyn, Y. Burov,
―Information resources processing using linguistic
analysis of textual content.‖ in Proc. of the 9th IEEE
International Conference Intelligent Data Acquisition and
Advanced Computing Systems: Technology and
Applications (IDAACS), Bucharest, Romania, pp. 573-
578, 2017.
[28] V. Lytvyn, V. Vysotska, Y. Burov, O. Veres, I. Rishnyak,
―The Contextual Search Method Based on Domain
Thesaurus.‖ Advances in Intelligent Systems and
Computing, Vol. 689, Springer, pp. 310-319, 2017.
[29] V. Pasichnyk,, T.Shestakevych, ―The model of data
analysis of the psychophysiological survey results.‖
Advances in Intelligent Systems and Computing, Vol. 512,
pp. 271-281, 2017.
[30] P. Zhezhnych, O.Markiv, ―Linguistic Comparison Quality
Evaluation of Web-Site Content with Tourism
Documentation Objects‖ Advances in Intelligent Systems
and Computing, Vol. 689, pp. 656-667, 2018.
[31] O. Bazylyk, P. Taradaha, O. Nadobko, L. Chyrun, T.
Shestakevych, ―The results of software complex OPTAN
use for modeling and optimization of standard engineering
processes of printed circuit boards manufacturing‖ in
Proceedings of the 11th International Conference
TCSET'2012, pp. 107 – 108, 2012.

Time Dependence of the Output Signal Morphology for Nonlinear Oscillator Neuron Based on Van der Pol Model 17
Copyright © 2018 MECS I.J. Intelligent Systems and Applications, 2018, 4, 8-17
[32] A. Bondariev, M. Kiselychnyk, O. Nadobko, L. Nedostup,
L. Chyrun, T. Shestakevych, ―The software complex
development for modeling and optimizing of processes of
radio-engineering equipment quality providing at the
stage of manufacture‖ in Proceedings of the 11th
International Conference TCSET'2012, pp. 159, 2012.
[33] V. Riznyk, ―Multi-modular Optimum Coding Systems
Based on Remarkable Geometric Properties of Space‖
Advances in Intelligent Systems and Computing, Vol. 512,
pp. 129-148, 2017.
Authors’ Profiles
Vasyl Lytvyn was born on January 17,
1976. In 1997 he received the MS from Ivan
Franko National University of Lviv, major:
applied mathematics. He received the PhD
in Engineering Science and Doctor of
Science degrees from Lviv Polytechnic
National University, in 2003 and 2012,
respectively. He works at Lviv Polytechnic
National University as a head of information systems and
networks department.
The area of expertise includes construction of subject domain
ontologies, methods of knowledge extraction, ontological
engineering, and methods of computer-assisted ontology filling,
ontology languages, testing of software systems, design process
and software management, system analysis.
V.Lytvyn is an author of over 150 scientific papers.
Peleshchak Roman. In 1980 he received the
MS from Ivan Franko National University of
Lviv, major: radiophysics and electronics.
He received the Doctor of Physical and
Mathematical Sciences degrees in 2001. He
received the Professor rank in 2002. He
works at Ivan Franko Drohobych State
Pedagogical University as a head of the
Department of General physics. The area of expertise includes
construction of theoretical physics, physics of intense
nanoheterosystems with different dimensions, nonlinear neural
networks, encoding methods that using nonlinear neural
networks, pattern recognition, quantum dots in photodynamic
therapy. R. Peleshchak is an author of over 320 scientific papers.
Peleshchak Ivan was born on July 21, 1995.
He studies at Lviv Polytechnic National
University as a undergraduate student of
information systems and networks.
Victoria Vysotska, PhD, Deputy Head of
Information systems and networks
Department, Associate Professor of
Information Systems and Networks
Department, Institute of Computer Science
and Information Technology at Lviv
Polytechnic National University, Lviv,
Ukraine. In 2014 defended candidate thesis (PhD) ―Methods
and tools of information resources processing in the electronic
content commerce systems‖. Research interests: content,
information systems and networks, ecommerce, business-
process, information resources, commercial content, content
analysis, content monitoring, content search, electronic content
commerce systems, content management system, content
lifecycle, Internet newspaper, software systems, models,
algorithms, analysis, methods and strategies of systems design.
Victoria has over 18 years of teaching in Lviv Polytechnic
National University. She has published more than 270 scientific
papers in various national and international journals and
conferences, 4 monograph, 5 textbooks. Lector is mathematical
linguistics, discrete mathematics and numerical methods in
informatics, information resources processing. Information
about citations is available in http://orcid.org/0000-0001-6417-
3689, http://victana.lviv.ua/index.php/naukovi-statti or
https://scholar.google.com.ua/citations?hl=uk&user=-
MCARowAAAAJ&view_op=list_works. Detailed information
about Victoria Vysotska can be found on:
https://ua.linkedin.com/pub/victoria-vysotska/29/1b7/261.
Rishnyak Ihor, Senior lecturer of
Information Systems Institute Department in
Computer Science and Information
Technology at Lviv Polytechnic National
University
Research interests: Project management,
Project management, Intelligent Decision
Support Systems
How to cite this paper: Vasyl Lytvyn, Victoria Vysotska, Ivan
Peleshchak, Ihor Rishnyak, Roman Peleshchak, "Time
Dependence of the Output Signal Morphology for Nonlinear
Oscillator Neuron Based on Van der Pol Model", International
Journal of Intelligent Systems and Applications(IJISA), Vol.10,
No.4, pp.8-17, 2018. DOI: 10.5815/ijisa.2018.04.02