robust adaptive control , by
linear matrix inequalities , for asymptotic
synchronization of the two coupled chaotic
FitzHugh-Nagumo neurons with unknown parameters, and the uncertain
amplitudes and phase shifts of the stimulation
current. The results of the proposed strategy
demonstrated through numerical simulations.
Keywords: External Electrical Stimulation (EES); Chaos Synchronization; Robust Adaptive Control; FitzHugh-Nagumo (FHN) Neurons
of chaotic neurons under external electric stimulation (EES)
deep brain stimulation , in
order to understand
of the neural system and to
outcomes of the external therapies for cognitive
disorders [1-3] , has attracted increasing
attention of scientists and researchers in the last decade. Neuronal
synchronization plays an
important role in transmission
the neural signals by developing coordination between different parts
of the brain. To investigate synchronization of neurons, the
FitzHugh-Nagumo (FHN) model[JH1]
one of intensively studied [JH2] m odel of
neuron due to its capability
behavior of neurons under the sinusoidal
for the FHN
neurons, like chaos and its
control, noise effects and filtering , and tracking and synchronization have been investigated [5-11]. Effects
of the stimulation current on neural dynamics, showing
chaotic behavior of the FHN neurons at
certain frequencies, have been investigated in the literature[JH3] The dynamics of the identical
coupled FHN neurons under EES have been reviewed in the
previous works [12-14], explaining that
synchronization of the neurons can be
achieved by attaining
a sufficiently large gap junction conductance. Recently researchers have applied different[JH4]
robust and adaptive control techniques based on feedback
linearization, uncertainty observers, fuzzy logic and neural networks in
order to cope
synchronization of both the coupled and the uncoupled
chaotic FHN neurons [11, 15-18]. These techniques are based on tota lly
known values of parameters of the FHN
neuron and limited
with the lumped uncertainty associated with the nonlinear part
invasive deep brain stimulation, an electrode is implanted in the skull of a
patient in order to stimulate
a portion of neurons. The
stimulation current arriving
at two different neurons has different phase shifts due
path lengths from the electrode to the each neuron.
the amplitudes of the
stimulation current also varies for each neuron due to different medium losses. As
these medium losses and path lengths are hard
to measure, the amplitudes and the phase shifts of the stimulation current
for both neurons are uncertain. Additionally,
are mostly unknown due to the biological
restrictions. In this letter, we first present
model [JH7] of the
coupled FHN neurons with uncertain stimulation
current and p resent
the necessary condition[JH8]
for synchronization of the neurons. We
address computationally efficient robust adaptive control for synchronization
chaotic FHN neurons with all neural parameters unknown, in
order to cope with the biological limitations , by We
matrix inequalities (LMIs)based sufficient condition
that guarantees asymptotic synchronization of the FHN
neurons under the
amplitudes and phase shifts of the stimulation
current in addition to is
best of the knowledge
authors, we are investigating first
synchronization of the FHN neurons with
uncertain and different phase
the stimulation current for both neurons[JH11] .
Synchronization of the FHN neurons with uncertain and
different amplitudes of the
stimulation current is up to this date.
to best of our knowledge, first
global robust adaptive control law for synchronization of the FHN
neurons with all parameters unknown.
of our knowledge, first
synchronization of the FHN neurons by
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
twocoupledFHNneuron s with different amplitudes and
phase shifts of the stimulation
current and present a necessary condition for synchronization. Section
3 demonstrates the LMI-based nonlinear robust adaptive control for synchronization
uncertain coupled chaotic FHN neurons. Section 4 provides numerical simulations. Section 5 draws
2. Model Description
Consider two coupled chaotic FHN neurons [4-6] under EES with uncertain stimulation current given by
where and are states of the master FHN
neuron, and and are 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
the slave neurons are represented by and , respectively, and the phase shifts are represented by and , respectively. Time t
and angular frequency , are taken
as[JH13] dimensionless quantities [4, 10-11]. The amplitudes of the stimulation current for two neurons under EES can differ
due to different medium losses. Similarly, the stimulus signal arriving at two neurons from
the electrode can also have different phase shifts
due to difference 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] Medium
losses and path lengths cannot be determined
due to which reason the parameters ,, , and are unknown. It can be easily
that the neurons (1-2) are not synchronous
if , or , for any integer . When synchronization of the neurons occurs, , and ; and the synchronization errors become , and . Using these conditions
(1-2),[JH15] we obtain ed that
is required for synchronization of
the FHN neurons.
It implies that , and , are the necessary conditions
(but not sufficient) for
synchronization of the coupled FHN neurons . It 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 address synchronization of the neurons
(1-2) under these conditions, we use single control input and the
overall model of the coupled FHN neuron s becomes
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
The purpose of the present study
is to develop the robust adaptive control law for synchronization of
FHN neurons (4-5) under assumptions 1-2[JH17] , which guarantee s asymptotic convergence of the synchronization
errors , and , to zero.
biological systems, parameters of the model are mostly
to infeasibility of
experimental measurement. The prediction of
parameters can be incorrect or deviate from the expected values[JH20] . Usually, we have an idea
the parametric ranges which can
be helpful for solving
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 incorporating
this knowledge , robust
adaptive control for synchronization of the FHN
neurons with uncertain parameters and stimulation current
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 , and , are written as
, . (11)
going towards the design strategy, we must identify the parameters for
which adaptation laws are required. We are using
single control input , due to which adaptation laws for the parameters
and cannot be developed,
strategy must be robust for
these parameters. The uncertain gap junction conductance
is associated with the linear
part of the synchronization error dynamics. Robustness
control law with respect to the parameter
can be ensured straightforwardly
essential for reduction of computation s .
parameter is associated with the
nonlinear part of the
synchronization error dynamics, so we can use adaptation of for simplicity
of the controller design
procedure. Additionally, we use two adaptation
laws for the parameters and associated with the uncertain
time varying stimulation signals in order to reduce both computations and complexity of the controller
design procedure, rather than using four adaptation laws for the parameters
,, , and . The proposed controller is then given by
, and are estimates of
, 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) do not require measurements
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)
are verified. Then the nonlinear control law (12) along with the adaptation laws (13-15) ensures:
Synchronization of the coupled
FHN neurons with asymptotic convergence of the synchronization
errors and to zero;
adaptive parameters , and to , and , respectively, where is a suitable constant.
The controller parameter is given by .
(10), the error dynamics become s
Constructing the Lyapunov function (see for example [19-20])
with , , , and
of (19) is given by
(20), we obtain ed
Using the adaptation laws (13-15) in (21), we get
For any ,
Using (24) and
(25), we obtain
For asymptotic convergence of the synchronization errors, . Hence
where , (28)
and . (29)
By applying the
Schur complement [20-22] to the inequality (29) and
, we obtain ed the LMIs of (16-17). Hence,
asymptotic convergence of the synchronization errors to zero is ensured
which completes 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)
, in (13-15), we obtain ed , are satisfied in the steady state. It
further implies that
, , and
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
estimation of the unknown parameters related to the stimulation current. This
completes proof of the statement
(ii) in the Theorem 1. It is worth mentioning
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 not guaranteed by the
It is often
to minimize the control efforts by minimizing the controller
gain . 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[JH23] , into the optimization problem
, . (34)
4. Simulation Results
of the proposed methodology, we choose
the model parameters for
the chaotic FHN neurons as , , , , , , , , and , with initial conditions , , , , , , and . By solving the Theorem 1, the controller
parameters , , , , and , are obtained for the parametric
ranges and . Fig . 1 shows the synchronization
error plots obtained by using the proposed control law.
The controller is applied at . It is clear that 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 are
converging[JH24] to , and , respectively. Fig .
3 plots the adaptive parameter , converg ing
to a constant value by applying the
proposed controller. Hence the
FHN neurons are
synchronized by utilizing the robust adaptive
This letter addresses synchronization of
coupled chaotic FHN
neurons with unknown parameters , and
uncertain amplitudes and phase shifts in the stimulation
current. By incorporating the knowledge
of parametric bounds, the LMI-based nonlinear robust adaptive control law has
been formulated which guarantees asymptotic convergence of the synchronization
errors to zero. Additionally, our strategy guarantees exact adaptation of the
parameters related to the external
stimulation current. The proposed scheme is applied for synchronization of the coupled
FHN neurons and
simulation results are demonstrated.
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Shows the synchronization
error plots for the coupled chaotic uncertain FHN neurons under EES. The
controller is applied at . Both synchronization errors are converg ing
to zero by applying the robust adaptive
controller (a) synchronization error , (b) synchronization error
of the adaptive parameters and to the stimulation
parameters and , respectively, by appl ying
of the adaptive parameter to a constant value by
the robust adaptive control
[JH2]Delete this if you prefer.
[JH4]*OR (alternative meaning): alternative
[JH5]*OR (alternative meaning): manage
[JH6]OR (different emphasis): limitations
[JH7] OR: “neural model”
[JH9]… OR (different emphasis): limitations
[JH10]… moved up to here from below
[JH12]In both instances of “the states,” change to “states” (i.e. delete “the”) if there are also other states for each neuron.
“assumed to be”
[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
(1) “most model parameters”
[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