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IFAC PapersOnLine 51-33 (2018) 12–14
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Available online at www.sciencedirect.com
2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Peer review under responsibility of International Federation of Automatic Control.
10.1016/j.ifacol.2018.12.077
©
2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
10.1016/j.ifacol.2018.12.077 2405-8963
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗
,
∗∗ Alexander L. Fradkov ∗
,
∗∗
,
∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Danila M. Semenov et al. / IFAC PapersOnLine 51-33 (2018) 12–14 13
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Adaptive synchronization of two coupled
non-identical Hindmarsh-Rose systems by
the Speed Gradient method
Danila M. Semenov ∗,∗∗ Alexander L. Fradkov ∗,∗∗,∗∗∗
∗Institute for Problems of Mechanical Engineering Russian Academy
of Sciences, 61 Bolshoy Ave V. O., Saint Petersburg, 199178, Russia
(e-mail: semenovdm90@gmail.com, fradkov@mail.ru)
∗∗ Department of Theoretical Cybernetics, Saint Petersburg State
University, 198504 Russia
∗∗∗ ITMO University, 49 Kronverskiy Ave, Saint Petersburg, 197101,
Russia
Abstract: The adaptive synchronization problem between two coupled non-identical
Hindmarsh-Rose systems was considered. It was shown that the usage of the developed con-
troller, which is based on the speed gradient method, ensures to achieve synchronized behavior
of the studied systems. The obtained results were mathematically proved and confirmed by
the simulations.
Keywords: Synchronization, Adaptive control, Speed Gradient method, Neural dynamics,
Hindmarsh-Rose system.
1. INTRODUCTION
The numerous studies of synchronization in dynamical
systems have created a wide interdisciplinary area which
includes a variety of scientific fields with their applica-
tions (Blekhman, 1988; Fradkov, 2007; Pikovsky et al.,
2003). In particular, such fields are biology and medicine.
There are many biological and medical systems which can
demonstrate synchronous regimes in their behavior. The
examples of such systems are the coordinated activity
of cardiac pacemaker cells, a population of fireflies that
flashes synchonously within its swarm and a population of
birds that gathers in a flock (Peskin, 1975; Buck and Buck,
1968). The most important example of this type of systems
are the neuronal populations and their dynamics in the
brain of a human or an animal. Indeed, it is well known
that the synchronization of a large number of neurons of
the central nervous system plays a key role in the forma-
tion of the brain waves (Pikovsky et al., 2003; Strogatz and
Stewart, 1993). Futhermore, it was ascertained that many
pathological states and diseases of the central and periph-
eral nervous systems, such as essential tremor, epilepsies
and Parkinson’s disease, relate directly to the anomalous
synchronization of the certain groups of neurons (Milton
and Jung, 2013; Rosenblum et al., 2000; Uhlhaas et al.,
2009). Today the methods, which are relied on suppression
of pathological synchronization in the nervous system, are
actively used in the therapy of these diseases. Obviously,
the development of such methods requires to apply qual-
itative mathematical tools. Our approach is based on the
applying of the tools of control theory.
The work was performed in IPME RAS and supported by Russian
Science Foundation (grant RSF 14-29-00142)
Nowadays, there are many scientific works dedicated to the
synchronization of mathematical models of neurons. The
majority of these works is devoted to a non-adaptive syn-
chronization (Plotnikov, 2015; Castanedo-Guerra et al.,
2016; Steur et al., 2009), which requires accurate knowl-
edge of the model parameters to design the controller.
Moreover, biological neurons have different physiological
characteristics, that in turn lead to non-identical parame-
ters in the models (Plotnikov et al., 2016). Therefore, the
methods of adaptive control should be used for the effec-
tive control of synchronization between biological neurons.
In our study, we apply the method of adaptive control
which is called the speed gradient method. This method is
based on the usage of Lyapunov functions and requires to
define the control goal as a objective function (Fradkov,
2007).
The rest of this paper is organized as follows. In Section 2
we describe the mathematical model. Section 3 gives brief
exposition of the speed gradient method in the differential
form. Section 4 deals with the adaptive synchronization
problem of two interconnected Hindmarsh-Rose systems.
The results of numerical simulation are given in Section 5.
2. A MATHEMATICAL MODEL
In our work, we consider a model which consists of two
interconnected non-identical Hindmarsh-Rose systems
˙x1=y1−ax3
1+bx2
1−z1+λ(x2−x1)+u,
˙y1=c−dx2
1−y1,
˙z1=ε[s(x1−r1)−z1];
˙x2=y2−ax3
2+bx2
2−z2+λ(x1−x2),
˙y2=c−dx2
2−y2,
˙z2=ε[s(x2−r2)−z2].
(1)
5th IFAC Conference on Analysis and Control of Chaotic Systems
Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018
Copyright © 2018 IFAC 12
Here xi(t), yi(t)zi(t) are the state variables of the i-th sys-
tem; u(t) is the control variable; λis the coupling stength.
The Hindmarsh-Rose system is a dynamical model of a
biological neuron. In this model xi(t) describes the dynam-
ics of the membrane potentional, and yi(t), zi(t) illustrate
how to the sodium-potassium pump works. Since the rate
of changing zi(t) is determined by ε, such that 0 <ε1,
then yi(t) describes the dynamics of the slow potassium
current and zi(t) describes the dynamics of the fast sodium
current. The Hindmarsh-Rose model is a simplified version
of the Hodgkin-Huxley model. Nevertheless, this model is
able to demonstrate the most of the behavioral regimes of
a biological neuron such as spiking and bursting.
3. SPEED GRADIENT METHOD
In this section, we briefly describe the speed gradient
method. Consider the following dynamical system
˙x=F(x,θ,t),(2)
where x∈Rn— a state vector, θ∈Rm— an input vector
and F(x, θ, t) — a function, which is piecewise continuous
in tand continuously differentiable in x, θ. The control
goal can be defined by one of two following ways
Qt→0,while t→∞ (or Qt≤∆∀t≥t∗),(3)
where Qt≥0 is a goal function and ∆, t∗are constants.
In our study we consider the case when Qtis a local
functional, i.e. Qt=Q(x(t),t) and Qtis a smooth
scalar function. In case of a synchronization problem,
we can define the goal function as a quadratic form
Q(x)=(x1−x2)P(x1−x2), where P=PT>0 and
x={x1,x
2}is extended state space of the overall system.
In order to design a control algorithm, the derivative of the
goal function ˙
Qt=ω(x,θ,t) is calculated, i.e. the speed
(rate) of changing Qtalong trajectories of the system (2):
ω(x,θ,t)= ∂Q(x, t)
∂t +[∇xQ(x, t)]TF(x,θ,t).(4)
Then the gradient of ω(x,θ,t) is evaluated with respect to
the input vector θas
∇θω(x,θ,t)=∂ω
∂θ T
=∂F
∂θ T
∇xQ(x, t).(5)
Finally, the algorithm of changing θis determined by the
following differential equation
˙
θ=−Γ∇θω(x, θ, t),(6)
where Γ is a symmetric and positive definite matrix, e.g.,
Γ = diag{γ1,...,γ
m},γ
i>0.
The main idea of the algorithm (6) can be explained as
follows. In order to achieve the control goal (3), it is
necessary to change θin the direction of decreasing Qt.
However, it may be problematic since Qtdoes not depend
explicitly on θ. Instead of this, we may try to decrease
˙
Qtin order to achieve inequality ˙
Qt<0, which implies
decrease of Qt. Now we can write the algorithm (6) since
the function ˙
Qt=ω(x,θ,t) depends directly on θ.
The convergence of the speed gradient method depends
on a reasonable choice of θ. There are several conditions
which can help to make a correct choice. This conditions
can be found in (Fradkov and Pogromsky, 1996).
4. ADAPTIVE SYNCHRONIZATION OF THE
NON-IDENTICAL HINDMARSH-ROSE SYSTEMS
Let us do some transformations in (1) before formu-
lating of the adaptive synchronization problem. In the
beginning, we subtract the equations of the second
system from the equations of the first one, and set
ψ(t)=x2
1(t)+x1(t)x2(t)+x2
2(t), ϕ(t)=x1(t)+x2(t).
Then the following system holds:
˙
δx=−(aψ −bϕ +2λ)δx+δy−δz+u,
˙
δy=−dϕδx−δy,
˙
δz=ε(sδx−δz−sr),
(7)
where δx(t)=x1(t)−x2(t), δy(t)f=y1(t)−y2(t),
δz(t)=z1(t)−z2(t) and r=r1−r2. Next, we perform
the change of variables ex(t)=δx(t), ey(t)=δy(t),
ez(t)=δz(t)+sr, and get the system (7) in the new
coordinates:
˙ex=−(aψ −bϕ +2λ)ex+ey−ez+sr +u,
˙ey=−dϕex−ey,
˙ez=ε(sex−ez).
(8)
Now we are ready to formulate the adaptive synchroniza-
tion problem. Obviously, the stability of the system (8)
indicates the presence of synchronization between the sys-
tems (1). Then, the control goal can be defined as
ex(t)→0,e
y(t)→0,e
z(t)→0,while t→∞.(9)
In order to achieve this control goal, we propose to use a
controller in the following form:
u(t)=−(γ0−θ1ϕ)ex+θ2ϕey+θ3,(10)
where γ0is a controller gain; θ1(t), θ2(t) and θ3(t) are
tunable parameters. We will adjust these parameters by
the speed gradient method. To do this, we define the
objective function (11) which corresponds to the control
goal (9).
Q(e(t)) = 1
2e2
x(t)+e2
y(t)+ 1
εse2
z(t),(11)
where e(t) = col (ex(t),e
y(t),e
z(t)). We find its derivative
according to the system (8) and get the following
˙
Q=−(aψ −bϕ +2λ)e2
x+ (1 −dϕ)exey+
+e2
y+1
se2
z+srex+uex.(12)
Since ψ(t)≥0 and εQ (e(t)) ≤1
2e2
x(t)+e2
y(t)+e2
z(t)/s
hold ∀t≥0 then we can get the upper estimate of the
derivative (12):
˙
Q≤−εQ −(2λ−bϕ −1/2) e2
x+srex+
+uex+ (1 −dϕ)exey+e2
y/2+e2
z/2s. (13)
Now we substitute the control (10) in the last inequality
and write the result in the following form
˙
Q≤−εQ +e∗We+(θ1−b)ϕe2
x+
+(θ2−d)ϕexey+(θ3+sr)ez,(14)
where
W=−(γ0+2λ−1/2) 1/20
1/2−1/20
00−1/2s.
Let us find γ0and λfor which the quadratic form in
the inequality is negative definite. Using the Sylvester’s
criterion, we obtain the following condition
γ0+2λ−1≥0.(15)
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14 Danila M. Semenov et al. / IFAC PapersOnLine 51-33 (2018) 12–14
If the condition (15) is satisfied, the inequality takes the
form ˙
Q≤−εQ +ω(e, θ).(16)
where θ= col(θ1,θ
2,θ
3) and
ω(e, θ)=(θ1−b)ϕe2
x+(θ2−d)ϕexey+(θ3+sr)ez.
In order to adjust θ1,θ2and θ3, we find the gradient of
ω(e, θ) with respect to θ:
∇θω(e, θ)=exϕex
ϕey
1.(17)
Whence we get equations for the adjusting of θ1,θ2and θ3:
˙
θ=−γ∇θω(e, θ),(18)
where γis a gain of the speed gradient method. Finally,
using the well-known theorems of the speed gradient
method, we can establish an asymptotic stability of the
system (8). It, in turn, indicates the achievement of the
control goal (9). This fact can be formulated as a theorem.
Theorem 1. Suppose that λis the coupling strength of the
systems (1), and γ0,γare gains in the control algorithm.
Then, the controller u(t) ensures the achievement of the
control goal (9) ∀xi(0) (i=1,2) and ∀θ(0) of the systems
if the condition (15) is satisfied.
Thus, the theorem 1 is a sufficient condition for the
synchronization of the systems (1). In addition, we have
the asymptotic synchronization with respect to xi(t) and
yi(t)(i=1,2). Moreover, from the inequality (16), we can
conclude that the synchronization has an exponential rate.
5. SIMULATION
We perform a numerical simulation to confirm the correct-
ness of the theorem.
0 200 400 600
-2
0
2
4
0 200 400 600
-0.01
0
0.01
0 200 400 600
-10
-5
0
5
0 200 400 600
-0.01
0
0.01
0 200 400 600
-0.5
0
0.5
1
1.5
0 200 400 600
0.7995
0.8
0.8005
0.801
Fig. 1. Synchronization of two coupled Hindmarsh-Rose
systems. (a), (c) and (e) are dynamics of the systems;
(b), (c) and (b) are the synchronization errors.
For the simulation we choose the following values of the
paremeters: a= 1, b= 3, c= 1, d= 5, ε=3·10−3,
r1=−1, r2=−0.8, λ=0.4, γ0= 4. Since γ0and λ
satisfy condition (15), according to the theorem 1, the
controller (10) ensures the achievement of the goal (9),
hence the system are synchronized, as shown in Fig. 1.
6. CONCLUSION
The adaptive synchronization problem of two non-identical
interconnected Hindmarsh-Rose systems is considered in
the article. In our study we used the speed gradient
method to adjust the control parameters. It was mathe-
matically proven that such synchronization is achievable
through the controller which is given in our paper. More-
over, there is the asymptotic synchronization with respect
to xi(t) and yi(t)(i=1,2). The numerical experiments are
confirmed the adequacy of obtained theoretical results.
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