Abstract: This
letter addresses the a system of nonlinear
robust adaptive control, by that
utilizesing
linear matrix inequalities, (LMIs) for asymptotic
synchronization of the two coupled chaotic
FitzHugh-Nagumo neurons withunder unknown parameters, and the uncertain
stimulation
current amplitudes and phase shifts. of the stimulation
current. The results of the proposed control strategy,
are
demonstrated through numerical simulations, are presented herein.
Keywords:
External Electrical Stimulation (EES); Chaos Synchronization; Robust Adaptive
Control; FitzHugh-Nagumo (FHN) Neurons
1. Introduction
Synchronization
of chaotic neurons under external electric stimulation (EES) like(e.g.
deep brain stimulation), has
attracted increasing research attention over the past decade in
order toas a means of understanding
functioning
of the neural system functions and toof
improvinge
outcomes of the external therapies for cognitive
disorders [1-3],. has attracted increasing
attention of scientists and researchers in the last decade. Neuronal
synchronization, in enabling
coordination between different areas of the brain, plays an
important role in neural signal transmission.
of
the neural signals by developing coordination between different parts
of the brain. To investigate synchronization of neurons, tThe
FitzHugh-Nagumo (FHN) neuron model[JH1]
is
one ofhas been intensively studied and
extensively employed [JH2] model of
neuronas a synchronization-investigative
tool dueowing to its capabilityutility
toin
representing
neuronal
behavior of neurons under the sinusoidal
EES [4].
Various
FHN
neuron studies for the FHN
neurons, likeconcerning chaos and its
control, noise effects and filtering, andas well
as tracking and synchronization have been investigatedcarried
out [5-11]. The Eeffects
of the frequency
of the stimulation current on neural dynamics, showingfor example the
chaotic behavior of the FHN neurons atunder
certain frequencies, also have been investigated in the literature[JH3] [4].
Further,
Tthe dynamics of the identical
coupled FHN neurons under EES, have been reviewed in the
previous works [12-14], explaining thatby which
neuronal
synchronization, of the neurons can be
achieved by attaininggiven
a sufficiently large gap junction conductance, can be achieved, have
been reviewed [12-14]. Recently researchers have applied differentvarious[JH4]
feedback-linearization-,
uncertainty-observer-, fuzzy-logic- and neural-network-based nonlinear,
robust and adaptive control techniques based on feedback
linearization, uncertainty observers, fuzzy logic and neural networks in
order to cope
withachieve[JH5]
synchronization of both the coupled and the uncoupled
chaotic FHN neurons [11, 15-18]. These techniques, however, are based on totallywell
known FHN
neuron parameter values, of parameters of the FHN
neuron and so their application is limited
to deal
with the lumped uncertainty associated with the nonlinear partaspect
of the
neuronal dynamics.
In
invasive deep brain stimulation, an electrode is implanted in the skull of a
patient in order to stimulate a portion ofcertain neurons. The
stimulation current that arrivingarrives
at two different neurons has different phase shifts dueaccording
to the different
path lengths from the electrode to the each neurons.
Moreover,
tThe amplitudes of the
stimulation current also variesvary for each neuron as
well, due to the different medium losses. As
these medium losses and path lengths are harddifficult
to measure, the amplitudes and the phase shifts of the stimulation current,
for both neurons, are uncertain. AdditionallyMoreover,
the parameters
of the
FHN neurons, owing to the pertinent biological
restrictions[JH6] ,
are mostly unknown. due to the biological
restrictions. In this letter, first, we first present
thea
coupled
FHN neuron
model [JH7] of the
coupled FHN neurons withfor an uncertain stimulation
current and presentprovide
the necessary condition[JH8]
for neuronal
synchronization. of the neurons. Then,
in order to cope with the biological restrictions[JH9] , Wwe
address computationally efficient robust adaptive control for synchronization
of the
chaotic FHN neurons with all neural parameters unknown, in
order to cope with the biological limitations, by using
the knowledge of parametric bounds. WWe
develop a linear
matrix inequalities (LMIs)-based sufficient condition
that guarantees asymptotic synchronization of the FHN
neurons under theuncertain stimulation current
uncertain
amplitudes and phase shifts of the stimulation
current in addition toand unknown neural parameters. And
finally, the results of numerical
simulations of
coupled chaotic FHN neuron
synchronization for unknown parameters and an uncertain stimulation
current are provided as a
demonstration of
the effectiveness of the proposed methodology.[JH10] Our
main contributions isare
summarized below.
(1) ByTo the
best of theour knowledge,
of
authors, we are investigatingthis paper represents the first
time
synchronization of the FHN neurons withunder
uncertain and different stimulation current phase
shifts.
of
the stimulation current for both neurons[JH11] .
Synchronization of the FHN neurons with uncertain and
different stimulation current amplitudes of the
stimulation current is also remains
rare. up to this date.
(2) According
to best of our knowledge,This is the first-ever
time
providingreport of thea
global robust adaptive control law for synchronization of the FHN
neurons with all parameters unknown.
(3) By best
of our knowledge,This is the first-ever
time
developingreport of thea
linear
matrix inequality (LMI)-based FHN neuron synchronization strategy
for
synchronization of the FHN neurons bywith
which the controller parameters can be selected easily, without any tuning
effort, by utilizing available LMI- routines.
Numerical
simulations for synchronization
of the coupled chaotic FHN neurons with
unknown parameters and uncertain stimulation current are also provided
in order to demonstrate
effectiveness of the proposed methodology.
This letter is
organized as follows. Section 2 presents the model of the
two-coupled-FHN-neurons model withfor different stimulation current amplitudes and
phase shifts, of the stimulation
current and presentderives athe necessary condition for synchronization. Section
3 demonstrates the LMI-based nonlinear robust adaptive control for synchronization
of the
uncertain coupled chaotic FHN neurons. Section 4 providesdescribes numerical simulations and presents their results. Section 5 draws
conclusions.
2. Model Description
Consider
two coupled chaotic FHN neurons [4-6] under EES with an uncertain stimulation current, given by
(1)
(2)
where 
and 
are the states of the master FHN
neuron, and 
and 
are the states[JH12] of the slave FHN neuron. The gap junction
conductance between the master neuron and the slave neuron is represented by 
. The amplitudes of the external stimulation current for the
master and the slave neurons are represented by 
and 
, respectively, and the phase shifts are represented by 
and 
, respectively. Time t
and angular frequency 
, are takengiven
as[JH13] dimensionless quantities [4, 10-11].
TheTwo
neurons’ amplitudes of the stimulation current amplitudes
for two neurons under EES can differ
due to different medium losses. Similarly, thean
electrode’s stimulus signal arriving at two neurons from
the electrode can also have different phase shifts,
due to differences in the path lengths. To consider
these facts, the amplitudes and the phase shifts of the stimulation current for
the coupled FHN neurons (1-2) are different.[JH14] The Mmedium
losses and path lengths cannot be precisely determined,
exactly,
due to which reason the parameters ,, , and are unknown. It can be easily
be verified
that the neurons (1-2) are not synchronous
if 
, or 
, for any integer 
. When synchronization of the neurons occurs, 
, and 
; and the synchronization errors
correspondingly become 
, and 
. UsingFor these conditions,
in
(1-2),[JH15] we obtained that
, (3)
is required for synchronization of
the FHN neurons. ItThis implies that 
, and 
, are the necessary conditions
(but not sufficient) conditions
for
synchronization of the coupled FHN neurons., Itwhich shows that the neurons
(1-2) are very sensitive to the amplitudes and the phase shifts of the
stimulation current. Even a small difference in these amplitudes or phase
shifts can either desynchronize the synchronous neurons or prevent
synchronization of the non-synchronous neurons. To aAddress the problem of the synchronization of the neurons
(1-2) under these conditions, we use single control input 
, and the
overall model of the coupled FHN neurons model becomes
(4)
(5)
Assumption 1: The parameters of the FHN
neurons are bounded assuch
that[JH16]
, (6)
, (7)
, (8)
, (9)
where the subscripts min and max represent the minimum and maximum values of the parameters, respectively.
Assumption 2: The parameters (,, , and ) of the stimulation current are unknown constants.
The purpose of the present study iswas to develop thea robust adaptive control law for synchronization of
the
FHN neurons (4-5) under assumptions (1-2)[JH17] , whichto guarantees asymptotic convergence of the synchronization
errors 
, and 
, to zero.
3. Robust aAdaptive cC[JH18] ontrol
In
the
biological systems, parameters of the model parameters
are mostlygenerally[JH19] are
unknown,
due
togiven the infeasibility of
experimental measurement. TheAnd parameter prediction of
parameters can be incorrect or deviate from the expected values[JH20] . Usually however, we have an ideaa sense
aboutof
the parametric ranges whichthat are appropriate to, and therefore can
be helpful forin solving,
the
biological problems. Consequently, the parameters of the neural
model are not exactly known but we still have useful
information about the parametric
bounds.[JH21] By iIncorporating
this knowledge, makes possible the development of robust
adaptive control for synchronization of the FHN
neurons with uncertain parameters and stimulation currents.
can
be developed which is the main objective of this section. To
develop this control law, the dynamics of the synchronization errors for the coupled
FHN neurons (4-5), by usingthat is, , and , are written as
(10)
where 
, and 
. (11)
Before going towardsproceeding
to the design strategy, we must identify the parameters for
which adaptation laws are required. We are using
single control input , due to which fact, adaptation laws for the parameters
and 
cannot be developed,
so the control
strategy must be sufficiently robust forto
handleing
their variations.
in
these parameters. The uncertain gap junction conductance,
, is associated with the linear
part of the synchronization error dynamics. The Rrobustness
of the
control law with respect to the parameter
can be ensured straightforwardly,
whichas
is also
essential for reduction of reduction of the number of computationss..
The
pParameter 
is associated with the
nonlinear partcomponent of the
synchronization error dynamics, so we can use adaptation of for simplicity
of the sake of controller design
procedure
simplicity. Additionally, we use two adaptation
laws for the parameters 
and 
we use two adaptation
laws associated with the uncertain
time varying stimulation signals in orderso as to reduce both the
number of computations and the complexity of the controller
design procedure, rather than using four adaptation laws for the parameters
,, , and . The proposed controller is then given by
, (12)
where
, 
and 
are estimates of the parameters
, 
and , respectively. The
adaptation laws for these parameters are given by
, 
, 
, (13)
, 
, (14)
, 
. (15)
Note that the
control law (12) and the adaptation laws (13-15), in contrast to the conventional techniques [11,
15-18], do not require measurements
of the
neural states and . in contrast to the
conventional techniques [11, 15-18]. Now we provide the LMI-based
sufficient condition for asymptotic synchronization of the FHN neurons.
Theorem
1: Consider the FHN neurons (4-5)
with the synchronization error dynamics (10-11) satisfying the assumptions (1-2).
Suppose that the LMIs
,, 
, (16)
, (17)
are verified. Then, the nonlinear control
law (12) along with the adaptation laws (13-15) ensures:
(i) Ssynchronization of the coupled
FHN neurons with asymptotic convergence of the synchronization
errors 
and 
to zero;
(ii) The convergence
of the
adaptive parameters , and to 
, and , respectively, where is a suitable constant.
The controller parameter 
is given by 
.
Proof: UsingIncorporating
(12) into
(10), the error dynamics becomes
(18)
Constructing the Lyapunov function (see for example [19-20])
, (19)
with , , , and ., the Dderivative
of (19) is given by
. (20)
UsingIncorporating
(18) into
(20), we obtained
. (21)
Using the adaptation laws (13-15) in (21), we get
. (22)
. (23)
. (24)
For any ,
. (25)
Using (24) and
(25), we obtained
. (26)
For asymptotic
convergence of the synchronization errors, 
. Hence
, (27)
where 
, (28)
and 
. (29)
By applying the
Schur complement [20-22] to the inequality (29) and further using
, we obtained the LMIs of (16-17). HenceThus,
asymptotic convergence of the synchronization errors to zero is ensured,
which completes the proof of the statement
(i) in the Theorem 1. In the steady state,
the synchronization errors and the states of neurons satisfy
, 
, (30)
and 
. (31)
By uUsing
, in (13-15), we obtained 
, 
, and 
, are satisfied in the steady state. ItThis
further implies that
, 
, and 
, (32)
are satisfied in
the steady state, where , 
and 
are the constant steady state
[JH22] values. Now, puttinginputting
the steady state conditions from (30-32) into (18), we obtained
, (33)
which can only
be true if 
, and 
, because the stimulus frequency 
cannot be infinity. Hence
the steady state values of the adaptive parameters 
and 
are equal to 
and 
, respectively. Thus, our adaptation laws guarantee exactprecise
estimation of the unknown parameters related to the stimulation current. This
completes the proof of the statement
(ii) in the Theorem 1. It is worth mentioningnoting
that convergence of the adaptive parameters and to the stimulation
current parameters and is ensured by
incorporating the steady state knowledge into the adaptation laws and the
synchronization error dynamics, but is not guaranteed by the
Lyapunov method.
It is often requirednecessary
to minimize the control efforts by minimizing the controller
gain [20]. In the present scenario, the control efforts can be minimized by
minimizing the controller parameter . For this purpose, we can transform the LMIs (16-17),
by choosingincorporating[JH23] 
, into the optimization problem
,
subject to
, 
. (34)
4. Simulation Results
For validation
of the proposed methodology, we choose the model parameters for
the chaotic FHN neurons asthe model
parameters 
, 
, 
, 
, 
, 
, 
, 
, and 
, with initial conditions 
, 
, 
, 
, 
, 
, and 
. By solving the Theorem 1, the controller
parameters 
, 
, 
, 
, and 
, are obtained for the parametric
ranges 
and 
. Fig.ure 1 shows the synchronization
error plots obtained by usingwith the proposed control law.
The controller is applied at 
. It is clear that, using the controller, both
synchronization errors are converging to zero. by using the controller. The
plots for the adaptive parameters and are shown in Fig. 2.
Both parameters 
, and 
, respectively. Fig.ure
3 plots the adaptive parameter , which is convergesing
to a constant value by applyingapplication of the
proposed controller. Hence tThe
FHN neurons thus are
synchronized by utilizingmeans of the robust adaptive
control methodology.[JH25]
5. Conclusions
This letter addresses the synchronization of the two
coupled chaotic FHN
neurons withfor unknown parameters, and
uncertain stimulation current amplitudes and phase shifts. in the stimulation
current. By incorporating the knowledge
of parametric bounds, thean LMI-based nonlinear robust adaptive control law has
beenwas formulated whichthat guarantees asymptotic convergence of the synchronization
errors to zero. Additionally, our strategy guarantees exactprecise adaptation of the
parameters related to the external
stimulation current parameters. The proposed scheme iswas applied forto the synchronization of the coupled
FHN neurons, andthe
simulation results for which arewere demonstratedpresented
herein.
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Fig. 1. Shows the sSynchronization
error plots for the coupled chaotic uncertain FHN neurons under EES. The
controller is applied at . Both synchronization errors are convergeing
to zero by applyingapplication of the robust adaptive
controller (a) synchronization error , (b) synchronization error 
Fig. 2. Shows cConvergence
of the adaptive parameters and to the stimulation
parameters and , respectively, by applicationying of
the robust
adaptive control
Fig. 3. Shows cConvergence
of the adaptive parameter to a constant value by
applicationying of
the robust adaptive control