On a dynamical model of genetic networks
INNA SAMUILIK
1
, FELIX SADYRBAEV
1,2
1
Department of Natural Sciences and Mathematics
Daugavpils University
Parades street1
LATVIA
2
Institute of Mathematics and Computer science
University of Latvia
Rainis boulevard 29
LATVIA
Abstract:We consider the model of a four-dimensional gene regulatory network (GRN in short). This
model consists of ordinary differential equations of a special kind, where the nonlinearity is
represented by a sigmoidal function and the linear part is present also. The evolution of GRN is
described by the solution vector 𝑋(𝑡), depending on time. We describe the changes that the system
undergoes if the entries of the regulatory matrix are perturbed in some way. The sensitive
dependence of solutions on the initial data is revealed by the analysis using the Lyapunov exponents.
Key-Words:nonlinear dynamical system, attractor, chaotic regime, Lyapunov exponents, Artificial
Neural Networks
Received: June 17, 2022. Revised: August 9, 2022. Accepted: August 29, 2022. Available online: September 21, 2022.
1. Introduction
Nonlinear dynamics plays an important
role in modern natural science and studies
such objects and phenomena as dynamic chaos
and various types of self-organization of
matter. Problems of nonlinear dynamics have
exact analytical solutions only in very rare
cases, which is why they have to be studied
using a computer experiment. In the study of
various natural phenomena, the construction
of useful mathematical models with their
subsequent study using the exact and graphical
methods of modern mathematics is of decisive
importance. As mathematical models widely
systems of differential equations are used.
The variety of dynamics observed in
nonlinear systems can be reduced to simple
regimes associated with some repetition for a
wide variety of systems by characteristic types
of solutions. These characteristic solutions
have the important property of invariance.
Moreover, many other solutions to the system
are attractive to them. Knowledge of such
solutions - attractors allows getting an idea of
the overall picture of the nonlinear system
dynamics [6]. Changing system parameters can
significantly change attractor type. In this case,
the system has a bifurcation [1]. The
bifurcation is a change in the dynamics system,
accompanied by the disappearance of some and
the appearance of other regimes. Firstly, the
stable point goes into the periodic regime (limit
cycle) [1], [7], then to the chaotic regime
(strange attractor) [2], [8].
Consider the general form of writing the
𝑛-dimensional dynamical system, that is
expected to model a genetic regulatory
network,
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𝑥=
⋯ −𝑣𝑥,
…
𝑥=
⋯ .−𝑣𝑥,
(1) (1)
where 𝜇>0,𝜃 and 𝑣>0 are parameters,
and 𝑤 are elements of the 𝑛×𝑛 regulatory
matrix 𝑊 [8]. System (1) appears also in the
theory of telecommunication networks. [3]
The sigmoidal function 𝑓(𝑧)=
is
used in (1). The sigmoidal function is a
mathematical function having a characteristic
’S-shaped’ curve or sigmoid curve. They are
many: logistic function [1], [2], [7], [8], [11],
Gompertz function [9], Hill function, inverse
trigonometric functions. A set of coefficients
𝑤 form the so called regulatory matrix
𝑊 = 𝑤 … 𝑤
… … …
𝑤 … 𝑤 (2) (2)
The set 𝑄=𝑥
𝑅∶ 0 <𝑥<
,𝑖=
1,…,𝑛is invariant [5, Definition 2, Section
2.5, Ch.2] with respect to system (1). This
follows from the properties of the sigmoidal
function (the value range is the interval (0,1))
and can be established by inspection of the
vector field, defined by (1).
Systems of the form (1), but with different
sigmoidal functions, appear when studying
neuronal networks. An example is considered
in the next section. This example and
comparison with systems from the theory of
genetic networks, was one of motivations for
this work.
2. Materials and methods
Our consideration is numerical and
geometrical. All processes take place in a
bounded parallelepiped and our main intent is
to use the 3D projections of the attractor on
different subspaces, to construct the graphs of
solutions for understanding and managing the
system. Computations, plotting the solution,
and the image of the projections of attractors
are performed using Wolfram Mathematica.
Also, we use Lyapunov exponents (LE) and a
high Kaplan–Yorke dimension, guaranteeing
chaotic behavior for longtimes. First, we
illustrate the usage of Lyapunov exponents
considering the modification of the model of
Artificial Neuronal Network as given in [17].
The chaotic behavior is confirmed for the
specific values of parameters. Next, we
consider the four-dimensional genetic model
(8) composed of two independent 2D systems.
This new 4D system is shown to have an
attractor, which is stable under small
perturbations. The zero blocks in the
regulatory matrix (9) were then filled with
non-zero elements and new coupled 4D
system was considered with the regulatory
matrix (10). Chaos was not confirmed by the
analysis using the Lyapunov spectrum. Finally,
the system (8) was considered with the
regulatory matrix (11), where the entries were
randomly selected. Chaotic behavior was
observed.
3. Lyapunov exponents
The Lyapunov exponents are an
important tool for the characterization of an
attractor of a finite-dimensional nonlinear
dynamic system and their excessive sensitivity
to initial conditions.[12] The Lyapunov
exponent is an approach to detect chaos, and it
is ameasure of the speeds at which initially
nearby trajectories of the system diverge.[13]
Relationships between the Lyapunov
exponents and the propertiesand types of
attractors:
1. (𝐿𝐸,𝐿𝐸,𝐿𝐸,𝐿𝐸)=(−,−,−,−) -
stable fixed point;
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2. (𝐿𝐸,𝐿𝐸,𝐿𝐸,𝐿𝐸)=(0,−,−,−)-
periodic solutions (limit cycles);
3. (𝐿𝐸,𝐿𝐸,𝐿𝐸,𝐿𝐸)=(0,0,−,−)-
quasiperiodic solution (2 torus);
4. (𝐿𝐸,𝐿𝐸,𝐿𝐸,𝐿𝐸)=(+,0,−,−) -
strange attractor;
5. (𝐿𝐸,𝐿𝐸,𝐿𝐸,𝐿𝐸)=(+,+,0,−) -
hyperchaotic attractor.
Properties of Lyapunov exponents:
1. The number of Lyapunov exponents
is equal to the number of phase space
dimensions, or the order of the system of
differential equations. They are arranged in a
descending order.[14]
2. The largest Lyapunov exponent of a
stable system does not exceed zero.[13]
3. A chaotic system has at least one
positive Lyapunov exponent, and the more
positive the largest Lyapunov exponent, the
more unpredictable the system is.[13]
4. To have a dissipative dynamical
system, the values of all Lyapunov exponents
should sum to a negative number.[14]
5. A hyperchaotic system is defined as a
chaotic system with at least two positive
Lyapunov exponents. Combined with one null
exponent and one negative exponent, the
minimal dimension for a hyperchaotic system
is four.[15]
The formula for the Kaplan–Yorke
dimension is
𝐷=𝑗+∑
(3) (3)
4. Artificial Neural Networks
In 1943 American neurophysiologist and
cybernetician Warren SturgisMcCulloch and
American logician Walter Harry Pitts modeled
a neuronas a switch that receives input from
other neurons and, depending on the total
weighted input, is either activated or remains
inactive.[16]
Definition 1. A dynamical model
inspired by the connectivity and behavior of
neurons in the brain is an Artificial Neural
Network.
One example of which is
𝑥′=tanh∑𝑎𝑥−𝑏𝑥
, (4)
where N is the number of neurons, each of
which represents a dimension of the
system.[17] The hyperbolic tangent is a
sigmoidal function.
Consider the regulatory matrix
𝑊 = 0 −1 0 1
1 0 0 1
1 1 0 −1
0 −1 1 0(5)
and the system
⎩
⎪
⎨
⎪
⎧
𝑥′=tanh(𝑥−𝑥)−b𝑥,
𝑥′=tanh(𝑥+𝑥)−b𝑥,
𝑥′=tanh(𝑥+𝑥−𝑥)−b𝑥,
𝑥′=tanh(𝑥−𝑥)−b𝑥,(6)
at b = 0.03. The initial conditions are
𝑥(0)=1.2; 𝑥(0)=0.4; 𝑥(0)=
1.2; 𝑥(0)=−1. (7)
This system is studied numerically
(Wolfram Mathematica), providing description
of the phase space and images of 2D and 3D
projections. The method of Lyapunov
exponents was used to analyze the system and
to obtain evidences of sensitive dependence of
solutions on the initial data.
The attractor as shown in Figure 1and
Figure 2 and oscillatory solutions in Figure
3and Figure 4.
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Figure 1.The projection of the attractor on 2D
subspace on(𝑥,𝑥).
Figure 2.The projection of the attractor on 3D
subspace (𝑥,𝑥,𝑥).
Figure 3. Solutions (𝑥,𝑥).
Figure 4. Solutions (𝑥,𝑥).
It was pointed out in [17] that the
minimal dissipative artificial neural network
that exhibits chaos has 𝑁 = 4 and is given
by (6) at 𝑏 = 0.043 and an attractor as
shown in Figure 5 and Figure 6.For specific
parameters this system solution has a chaotic
trajectory as shown in Figure 7 and Figure 8.
[17] Dynamics of Lyapunov exponents are
shown in Figure 9.
Figure 5.The projection of the attractor on 2D
subspace on (𝑥,𝑥).
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Figure 6.The projection of the attractor on 3D
subspace (𝑥,𝑥,𝑥).
Figure 7.Solutions (𝑥,𝑥).
Figure 8. Solutions (𝑥,𝑥).
Figure 9.
(𝐿𝐸,𝐿𝐸,𝐿𝐸,𝐿𝐸)=
(0.03,0.01,−0.09,−0.13)- strange (chaotic)
attractor, Kaplan–Yorke dimension is
𝐷=2.62.
In the next section, following this example, we
investigate our problem.
5. Results
Consider the four-dimensional system
⎩
⎪
⎪
⎨
⎪
⎪
⎧
𝑥′=
−𝑣𝑥,
𝑥′=
−𝑣𝑥,
𝑥′=
−𝑣𝑥,
𝑥′=
−𝑣𝑥.
(8)
Periodic solutions can exist in systems of
the form (8). Consider the system (3) with the
matrix
𝑊 = 0.5 2 0 0
−2 0.5 0 0
0 0 0.5 2
0 0 −2 0.5 (9)
and 𝜇=𝜇=𝜇=𝜇=10 ; 𝑣=𝑣=
𝑣=𝑣=1; 𝜃=1.25,𝜃=−0.75,𝜃=
1.25,𝜃=−0.75 .The initial values𝑥(0)=
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0.42,𝑥(0)=0.39,𝑥(0)=0.4,𝑥(0)=
0.395.
This system consists of two identical but
independent two-dimensional systems. The 4D
system has, by the appropriate choice of 𝜃-s,
the critical point of the type an unstable 4D
focus at the center of a unit cube. Both two 2D
systems have a stable limit cycle. The (𝑥,𝑥)
and (𝑥,𝑥) projections coincide with the 2D
limit cycles. The (𝑥,𝑥)projectionis depicted
in Figure 10. Any of these two
two-dimensional systems has a single critical
point of the type unstable focus. This is the
result of Andronov-Hopf bifurcation in 2D
systems. The unique critical point of the type
stable focus loses its stability, and instead, an
attractor in the form of a limit cycle emerges.
The limit cycles (Figure 10) with the same
periods exist. There are single critical points
inside both limit cycles, and both critical points
are of the type unstable focus.
Figure 10. The attractor of (8), projection on
(
𝑥
,
𝑥
)
.
Figure 11. The attractor of (8), projection
on
(
𝑥
,
𝑥
,
𝑥
)
.
Now fill in all zero elements of the regulatory
matrix (9) with values 0.1, so the regulatory
matrix becomes
𝑊 = 0.5 2 0.1 0.1
−2 0.5 0.1 0.1
0.1 0.1 0.5 2
0.1 0.1 −2 0.5 (10)
There is exactly one critical point at
(0.5472, 0.448, 0.5472, 0.448). The standard
linearization analysis provides the
characteristic numbers 𝜆,=−0.0099 ±
4.944𝑖; 𝜆,=0.4852 ± 4.944𝑖.Consider the
system (8), which is no longer uncoupled, with
the same parameters and initial conditions for a
solution. The dynamics of Lyapunov
exponents are shown in Figure 14.
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Figure 12. The projection of the attractor (10)
on 2D subspace
(
𝑥
,
𝑥
)
.
Figure 13. The projection of the attractor (10)
on 3D subspace
(
𝑥
,
𝑥
,
𝑥
)
.
Figure14.
(
𝐿𝐸
,
𝐿𝐸
,
𝐿𝐸
,
𝐿𝐸
)
=
(
0
,
−
0
.
33
,
−
0
.
55
,
−
0
.
72
)
- periodic solutions.
Consider the system (8) with the
regulatory matrix
𝑊 = 0.8 2 −0.8 0.5
−2 0.3 0.4 −0.7
−0.5 0.2 1.8 2
0.8 −0.7 −2 1.8(11)
andthe parameters 𝜇=𝜇=𝜇=𝜇=10;
𝑣=𝑣=𝑣=𝑣=1 and 𝜃, where
𝑖=1,2,3,4 are calculated as
⎩
⎪
⎨
⎪
⎧
𝜃=
,
𝜃=
,
𝜃=
,
𝜃=
. (12)
𝜃=1.25,𝜃=−1,𝜃=1.75,𝜃=
−0.05.The initial values𝑥(0)=0.4,𝑥(0)=
0.6,𝑥(0)=0.39,𝑥(0)=0.38.
The critical point is at (0.5,0.5,0.5,0.5).
The standard linearization analysis provides
the characteristic numbers 𝜆,=−0.44±
4.603𝑖; λ,= 4.33 ± 5.135 𝑖. The graphs of
chaotic solutions are shown in Figure 15. The
projection on three-dimensional subspaces is
depicted in Figure 16.The dynamics of
Lyapunov exponents are shown in Figure 17.
Figure 15.The graphs of 𝑥(𝑡),𝑖=1,2,3,4.
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Figure 16.The projection on 3D
subspace(𝑥,𝑥,𝑥).
Figure
17.(𝐿𝐸,𝐿𝐸,𝐿𝐸,𝐿𝐸)=
(0.2,0,−0.75,−0.92) – a chaotic solution.
6. Discussion
We have constructed the attractor for a 4D
dynamical system, arising in the theory of
genetic networks. This attractor is generated by
two limit cycles in the different 2D systems of
the same GRN kind (both limit cycles are
identical and have equal periods). A single
critical point is not attractive (the complex
characteristic values have positive real parts).
The 4D regulatory matrix at the beginning has
a block form and the 4D system is therefore
uncoupled. By filling the zero spaces in the
regulatory matrix with non-zero elements, the
system was made coupled. The attractor still
exists for sufficiently small perturbations. This
confirms the structural stability of a 4D system.
The GRN system with the regulatory matrix (6)
is shown to exhibit chaotic behavior. This is
confirmed by the analysis using the Lyapunov
spectrum. By using a similar procedure,
regulatory matrices of any dimension can be
constructed from lesser blocks. Combinations
of any attractors of lower dimensions are
possible. Filling zero spaces in the regulatory
matrix of block form often produces attractors
of a new structure.
7. Conclusions
In this note, we considered the creation of a
new 4D attractor from the two 2D periodic
attractors. Even for the 2D case, there are
several types of attractors in GRN systems,
namely, a single stable critical point, several
stable critical points, a limit cycle. All of them
in various combinations can be used to
construct attractors in GRN systems of higher
dimensions. They, in turn, can be used to
construct more complicated ones in higher
dimensions. Also, attractors generated by 3D
systems can be used to produce higher
dimensional ones. After mechanically
combining several low-dimensional matrices
into a single block matrix, this matrix can be
perturbed by filling zero zones with non-zero
entries. The resulting systems can reveal
different behaviors, including the generation of
a unique attractor. The chaotic behavior of
solutions seems to be generic in higher
dimensional GRN type systems. Their
investigation is possible by a combination of
qualitative and numerical methods. A
similarity of systems of differential equations
arising in models of GRN and neuronal
networks, is acknowledged.
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Acknowledgments
ESF Project No. 8.2.2.0/20/I/”Strengthening of
Professional Competence of Daugavpils
University Academic Personnel of Strategic
Specialization Branches 3rd Call”.
Conflicts of Interest
The authors declare no conflict of interest.
References
[1]. S. Atslega, F. Sadyrbaev, I. Samuilik. On Modelling
Of Complex Networks. Engineering for Rural
Development(ISSN1691-5976), 2021, pp. 10091014.
http://tf.llu.lv/conference/proceedings2021/Papers/
TF223.pdf
[2]. A.Das, A.B.Roy, Pritha Das. Chaos in a three
dimensional neural network. Applied Mathematical
Modelling, 24(2000), 511-522.
[3]. Y.Koizumi et al. Adaptive Virtual Network
Topology Control Based on Attractor Selection. J. of
Lightwave Technology, (ISSN :0733-8724), Vol.28
(06/2010), Issue 11, pp. 1720-1731
DOI:10.1109/JLT.2010.2048412.
[4]. C.Grebogi, Le-Zhi Wang et al. A geometrical
approach to control and controllability of nonlinear
dynamical networks, Nature Communications, Vol. 7,
Article number:11323 (2016), DOI:
10.1038/ncomms11323
[5]. L. Perko. Differential Equations and Dynamical
Systems. Springer, 2001.
[6]. E. Ott. Chaos in Dynamical Systems (2nd ed.).
Cambridge: Cambridge University Press, 2002.
doi:10.1017/CBO9780511803260
[7]. F. Sadyrbaev, I. Samuilik. On the hierarchy of
attractors in dynamical models of complex networks. 19
Intern.Confer.Numer.Analys. and Appl.Mathematics,
Rhodes, Greece, 20-26 September 2021, To appear in
AIP Conference Proceedings. https:
//aip.scitation.org/journal/apc
[8]. F. Sadyrbaev, I. Samuilik, V. Sengileyev. On
Modelling of Genetic Regulatory NetWorks. WSEAS
Transactions on Electronics,2021, Vol. 12, No. 1, 73.-
80.lpp. ISSN 1109-9445. e-ISSN 2415-1513.
doi:10.37394/232017.2021.12.10
[9]. I. Samuilik, F. Sadyrbaev. Modelling Three
Dimensional Gene Regulatory Networks. WSEAS
Transactions on Systems and Control. 2021, Vol. 12, No.
1, 73.-80.lpp. ISSN 1109-9445. e-ISSN 2415-1513.
doi:10.37394/232017.2021.12.10
[10]. J. C. Sprott. Elegant Chaos Algebraically Simple
Chaotic Flows. World Scientific Publishing Company,
2010, 302 pages. https://doi.org/10.1142/7183
[11]. K. Nantomah. On Some Properties of the Sigmoid
Function. Asia Mathematika, AsiaMathematika, 2019.
hal-02635089.
[12]. H. Gritli, N. Khraief, S. Belghith. Further
Investigation of the Period-Three Route to Chaos in the
Passive Compass-Gait Biped Model. In A. Azar, S.
Vaidyanathan (Ed.),Handbook of Research on Advanced
Intelligent Control Engineering and Automation (pp.
279-300),2015. IGI Global.doi:/10.4018/978-1-4666-
7248-2.ch010
[13]. K. Nosrati, Ch. Volos. Bifurcation Analysis and
Chaotic Behaviors of Fractional-Order Singular
Biological Systems. Nonlinear Dynamical Systems with
Self-Excited and Hidden Attractors, Springer, 2018.
Pages 3-44.
[14]. W. S. Sayed, A. G. Radwan, H. A. H. Fahmy. Chaos
and Bifurcation in Controllable Jerk-Based Self-Excited
Attractors. Nonlinear Dynamical Systems with Self-
Excited and Hidden Attractors, Springer, 2018. Pages
45-70
[15]. S. Vaidyanathan, V. Pham, Ch. Volos, A. Sambas.
A Novel 4-D Hyperchaotic Rikitake Dynamo System
with Hidden Attractor, its Properties, Synchronization
and Circuit Design. Nonlinear Dynamical Systems with
Self-Excited and Hidden Attractors, Springer, 2018.
Pages 345-364
[16]. A. Krogh. What are artificial neural networks?. Nat
Biotechnol. 2008 Feb;26(2):195-7. doi: 10.1038/nbt1386
[17]. J. C. Sprott. Elegant Chaos Algebraically Simple
Chaotic Flows. World Scientific Publishing Company,
2010, 302 pages. https://doi.org/10.1142/7183
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