**Abstract:** This
letter addresses ~~the ~~a system of nonlinear
robust adaptive control~~, ~~~~by~~ that
utilizes~~ing~~
linear matrix inequalities~~,~~ (LMIs) for asymptotic
synchronization of ~~the ~~two coupled chaotic
FitzHugh-Nagumo neurons ~~with~~under 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.

**Key****words:
**External Electrical Stimulation (EES); Chaos Synchronization; Robust Adaptive
Control; FitzHugh-Nagumo (FHN) Neurons

**1.**** I****ntroduction**

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 to~~as a means of understanding
~~functioning
of the ~~neural system functions and ~~to~~of
improving~~e~~
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, t~~The
FitzHugh-Nagumo (FHN) neuron model[JH1]
~~is
one of~~has been intensively studied and
extensively employed [JH2] ~~m~~~~odel of
neuron~~as a synchronization-investigative
tool ~~due~~owing to its ~~capability~~utility
~~to~~in
representing
neuronal
behavior ~~of neurons ~~under ~~the ~~sinusoidal
EES [4].

Various
FHN
neuron studies ~~for~~~~ ~~~~the ~~~~FHN
neurons, like~~concerning chaos and its
control, noise effects and filtering~~,~~ ~~and~~as well
as tracking and synchronization have been ~~investigated~~carried
out [5-11]. The ~~E~~effects
of the frequency
of the stimulation current on neural dynamics, ~~showing~~for example the
chaotic behavior of ~~the ~~FHN neurons ~~at~~under
certain frequencies, also have been investigated ~~in the literature~~[JH3] ~~ ~~[4].
Further,
~~T~~the dynamics of ~~the ~~identical
coupled FHN neurons under EES, ~~have been reviewed in the
previous works [12-14], explaining that~~by which
neuronal
synchronization, ~~of the neurons ~~~~can be
achieved ~~~~by ~~~~attaining~~given
a sufficiently large gap junction conductance, can be achieved, have
been reviewed [12-14]. Recently researchers have applied ~~different~~various[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
with~~achieve[JH5]
synchronization of both ~~the ~~coupled and ~~the ~~uncoupled
chaotic FHN neurons [11, 15-18]. These techniques, however, are based on ~~tota~~~~lly~~well
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 ~~part~~aspect
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 of~~certain neurons. The
stimulation current that ~~arriving~~arrives
at two different neurons has different phase shifts ~~due~~according
to the different
path lengths from the electrode to the ~~each ~~neurons.
~~Moreover,
t~~The amplitudes of the
stimulation current ~~also varies~~vary for each neuron as
well, due to the different medium losses. As
these medium losses and path lengths are ~~hard~~difficult
to measure, the amplitudes and the phase shifts of the stimulation current,
for both neurons, are uncertain. ~~Additionally~~Moreover,
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
~~the~~a
coupled
FHN neuron
model [JH7] ~~of the
coupled FHN neurons ~~~~with~~for an uncertain stimulation
current and ~~p~~~~resent~~provide
the necessary condition[JH8]
for neuronal
synchronization. ~~of the neurons. ~~Then,
in order to cope with the biological restrictions[JH9] , ~~W~~we
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. ~~W~~We
develop a linear
matrix inequalities (LMIs)-based sufficient condition
that guarantees asymptotic synchronization of ~~the ~~FHN
neurons under ~~the~~uncertain stimulation current
~~uncertain
~~amplitudes and phase shifts ~~of the stimulation
current ~~~~in addition to~~and ~~ ~~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 ~~is~~are
summarized below.

(1) ~~By~~To the
best of ~~the~~our knowledge,
~~of
authors, we are investigating~~this paper represents the first
~~time
~~synchronization of ~~the ~~FHN neurons ~~with~~under
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
providing~~report of ~~the~~a
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
developing~~report of ~~the~~a
linear
matrix inequality (LMI)-based FHN neuron synchronization strategy
~~for
synchronization of ~~~~the ~~~~FHN neurons ~~~~by~~with
which the controller parameters can be selected easily, without any tuning
effort, by utilizing available LMI~~-~~ routines.

~~N~~~~umerical
simulations ~~~~for~~~~ ~~~~synchronization
of the ~~~~coupled chaotic FHN neuron~~~~s~~~~ ~~~~with~~~~
unknown parameters and uncertain stimulation current are ~~~~also ~~~~provided
~~~~in order to~~~~ demonstrat~~~~e~~~~
effectiveness of the proposed methodology~~.

This letter is
organized as follows. Section 2 presents ~~the model of ~~the
two-coupled-FHN-neuron~~s~~ model ~~with~~for different stimulation current amplitudes and
phase shifts, ~~of the stimulation
current ~~and ~~present~~derives ~~a~~the 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 ~~provides~~describes 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 ~~taken~~given
as[JH13] dimensionless quantities [4, 10-11].

~~The~~Two
neurons’ ~~amplitudes of the ~~stimulation current amplitudes
~~for two neurons ~~under EES can differ
due to different medium losses. Similarly, ~~the~~an
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 ~~M~~medium
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 _{}. ~~Using~~For these conditions,
~~in
(1-2),~~[JH15] ~~ ~~~~we obtain~~~~ed that~~

_{}~~,~~ (3)

is required for synchronization of
the FHN neurons. ~~It~~This implies that _{}~~,~~ and _{}~~,~~ are the necessary ~~conditions
~~(but not sufficient) conditions
for
synchronization of the coupled FHN neurons~~.~~, ~~It~~which 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 a~~Address 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 neuron~~s~~ model becomes

_{} (4)

_{} (5)

**Assumption 1: **The parameters of the FHN
neurons are bounded ~~as~~such
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 ~~is~~was to develop ~~the~~a robust adaptive control law _{} for synchronization of
~~the
~~FHN neurons (4-5) under assumptions (1-2)[JH17] , ~~which~~to guarantee~~s~~ asymptotic convergence of the synchronization
errors _{}~~,~~ and _{}~~,~~ to zero.

**3. Robust a**

In
~~the
~~biological systems, ~~parameters of the ~~model parameters
~~are mostly~~generally[JH19] are
unknown,
~~due
to~~given the infeasibility of
experimental measurement. ~~The~~And parameter prediction ~~of
parameters ~~can be incorrect or deviate from the expected values[JH20] . Usually however, we have ~~an idea~~a sense
~~about~~of
the parametric ranges ~~which~~that are appropriate to, and therefore can
be helpful ~~for~~in 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 i~~Incorporating
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 using~~that is, _{}, and _{}, are written as

_{} (10)

where _{}~~,~~ and _{}. (11)

Before ~~going towards~~proceeding
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 ~~for~~to
handle~~ing~~
their variations.
~~in
these parameters. ~~The uncertain gap junction conductance,
_{}, is associated with the linear
part of the synchronization error dynamics. The ~~R~~robustness
of ~~the
~~control law _{} with respect to ~~the ~~parameter
_{} can be ensured straightforwardly,
~~which~~as
is ~~also
~~essential for ~~reduction of ~~reduction of the number of computations~~s~~.~~.~~
~~The
p~~Parameter _{} is associated with the
nonlinear ~~part~~component 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 order~~so 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) S*

*(ii) The convergence
of the
adaptive parameters *

*The controller parameter _{} is given by _{}.*

**Proof:** ~~Using~~Incorporating
(12) into
(10), the error dynamics become~~s~~

_{} (18)

Constructing the Lyapunov function (see for example [19-20])

_{}, (19)

with _{}, _{}, _{}, and _{}~~.~~, the ~~D~~derivative
of (19) is given by

_{}. (20)

~~Using~~Incorporating
(18) into
(20), we obtain~~ed~~

_{}. (21)

Using the adaptation laws (13-15) in (21), we get

_{}. (22)

_{}. (23)

_{}. (24)

For any _{},

_{}. (25)

Using (24) and
(25), we obtain~~ed~~

_{}. (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 obtain~~ed~~ the LMIs of (16-17). ~~Hence~~Thus,
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 u~~Using
_{}, in (13-15), ~~we obtain~~~~ed~~~~ ~~_{}, _{}, and _{}~~,~~ are satisfied in the steady state. ~~It~~This
further implies that

_{}, _{}, and _{}~~,~~ (32)

are satisfied in
the steady state, where _{}, _{} and _{} are the constant steady state
[JH22] values. Now, ~~putting~~inputting
the steady state conditions from (30-32) into (18), we obtain~~ed~~

_{}, (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 ~~exact~~precise
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 ~~mentioning~~noting
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 ~~required~~necessary
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 ~~choosing~~incorporating[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 ~~as~~the 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 using~~with 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 ~~ ~~are
converging,[JH24] to _{} and _{} _{}, and _{}, respectively. Fig~~.~~ure
3 plots the adaptive parameter _{}, which is converges~~ing~~
to a constant value by ~~applying~~application of the
proposed controller. ~~Hence t~~The
FHN neurons thus are
synchronized by ~~utilizing~~means of the robust adaptive
control methodology.[JH25]

**5. Conclusions**

This letter addresses the synchronization of ~~the ~~two
coupled chaotic FHN
neurons ~~with~~for unknown parameters~~,~~ and
uncertain stimulation current amplitudes and phase shifts. ~~in the stimulation
current. ~~By incorporating ~~the ~~knowledge
of parametric bounds, ~~the~~an LMI-based nonlinear robust adaptive control law ~~has
been~~was formulated ~~which~~that guarantees asymptotic convergence of the synchronization
errors to zero. Additionally, our strategy guarantees ~~exact~~precise adaptation of ~~the
parameters related to~~~~ the ~~external
stimulation current parameters. The proposed scheme ~~is~~was applied ~~for~~to the synchronization of ~~the ~~coupled
FHN neurons, ~~and~~the
simulation results for which ~~are~~were ~~demonstrated~~presented
herein.

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[10] D. Q. Wei, X. S.
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[22] M. Rehan, A.
Ahmed,

Fig. 1. ~~Shows the s~~Synchronization
error plots for the coupled chaotic uncertain FHN neurons under EES. The
controller is applied at _{}. Both synchronization errors ~~are ~~converge~~ing~~
to zero by ~~applying~~application of the robust adaptive
controller (a) synchronization error _{}, (b) synchronization error _{}

Fig. 2. ~~Shows c~~Convergence
of the adaptive parameters _{} and _{} to the stimulation
parameters _{} and _{}, respectively, by application~~ying~~ of
the robust
adaptive control

Fig. 3. ~~Shows c~~Convergence
of the adaptive parameter _{} to a constant value by
application~~ying~~ of
the robust adaptive control

[JH2]Delete this if
you prefer.

[JH3]implicit

[JH4]*OR (alternative
meaning): alternative

[JH5]*OR (alternative
meaning): manage

[JH6]OR (different
emphasis): limitations

[JH7] OR: “neural
model”

[JH8]*OR: condition__s__

[JH9]… OR (different emphasis): limitations

[JH10]… moved up to
here from below

[JH11](?) implicit

[JH12]In both
instances of “the states,” change to “states” (i.e. delete “the”) if there are
also other states for each neuron.

[JH14]??—couldn’t
understand

[JH15]Implicit /
already established

[JH16]OR (alternative
meaning): by

[JH17](?) Remove these
added parentheses here and *passim *if
necessary.

[JH18]… just for
consistency with your established protocol

[JH21]Doesn’t really
add very much to what has just been said, and in fact is mostly redundant
(repetitive)

[JH22]For the
adjectival form here and *passim*,
hyphenate (steady-state) if typically done so in your target journal.

[JH24](this comma is
necessary—not a typo)

[JH26]Neither included
in page count nor checked