model in [11] as follows
˙
x
1
=
f
1
(
x
3
)
−
g
1
(
x
3
)
x
1
(1)
˙
x
2
=
f
2
(
x
2
)+
p
1
g
2
(
x
2
)
x
3
(2)
˙
x
3
=
f
3
(
x
2
,x
3
)
−
u
(3)
where
f
1
(
x
3
)=
e
80(
x
3

0
.
55)
1+
e
80(
x
3

0
.
55)
+0
.
25
p
1
−
e
80(
x
3

0
.
35)
1+
e
80(
x
3

0
.
35)
P
1+
10
0
.
001+
x
3
3
(4)
g
1
(
x
3
)=
1
1+
10
0
.
001+
x
3
3
(5)
f
2
(
x
2
)=
−
p
2
x
2
(6)
g
2
(
x
2
)=1
−
x
2
(7)
f
3
(
x
2
,x
3
)=
p
2
p
3
x
2
−
p
1
p
3
x
3
(1
−
x
2
)
−
p
4
x
3
x
3
+
p
5
(8)
x
1
,
x
2
and
x
3
are the state variables,
p
i
(
i
=1
,
2
,
···
,
5)
are constant parameters,
u
is the control variable.
TABLE II.1
DESCRIPTION OF VARIABLES AND PARAMETERS
Variables/Parameters
Description
x
1
synaptic strength
x
2
fraction of buffer that are occupied by
Ca
2+
x
3
intracellular calcium concentration
p
1
forward binding rates
p
2
backward binding rates
p
3
total concentration of the buffer
p
4
rate of calcium removal by
Ca
2+
pump
p
5
modi±er of calcium removal by
Ca
2+
pump
u
control signal,
u
=

αI
Ca
+
I
NMDA
The de±nitions according to the model in [10], [11]
are given in Table II.1, and their chemical properties and
interpretation are described as follows:
(i) In this synaptic plasticity model, synaptic strength
x
1
is controlled by the intracellular calcium concentration
x
3
;
(ii) Intracellular calcium dynamics are described by (2) and
(3), and
p
3
is the total concentration of the buffer,
x
2
represents the fraction of buffer that are occupied by
Ca
2+
,
p
1
and
p
2
are forward and backward binding
rates respectively;
(iii)
I
NMDA
and
I
Ca
are currents carried by
Ca
2+
ions
through the NmethylDaspartate (NMDA) Receptor
channels and by voltage dependent calcium channels
(VDCC); and
(iv)
p
4
represents rate of calcium removal by
Ca
2+
pump,
and
p
5
is the modi±er in the
Ca
2+
pump expression.
Seizure is a disturbance of the neuronal electrochemical
activity that a set of neurons suddenly produce a repeti
tive, synchronous discharge. It has been proved that non
synaptic neural plasticity (i.e. calcium dependent afterhy
perpolarization in neurons) can regulate the frequency of
the dominant rhythm in EEG,
while synaptic potentiation
may be responsible for irregular bursting prior to seizure
termination [11], [15]. The underlying hypotheses is that
synaptic potentiation and afterhyperpolarization (which reg
ulate patterns of neuronal bursting) play signi±cant roles
in alteration of neural rhythmic activity during seizures.
Therefore, synaptic strength is a critical factor in regulating
seizure. By decreasing the synaptic strength, we aim to
restore the neural rhythmic activity, and further modeling
must take into account the way in which the synaptic strength
acts on regulate neural rhythmic.
From the models, it is reasonable to think of modifying
calcium concentration by NMDA receptor channels current
I
NMDA
and voltage dependent calcium channels current
I
ca
, which can be chosen as control signals and become
negative and positive feedback signal respectively. However,
I
NMDA
and
I
ca
can be acted upon by more conventional
means: drugs, such as calcium channel blocker (CCBs)
and antiepileptic drugs etc., which are known to have an
in²uence on
I
Ca
and
I
NMDA
[2]. According to this, we do
not consider the underlying mechanism of the relationship
between drugs and these two kinds of currents, and suppose
a parameter
u
that we are able to vary as a control signal
u
=
−
αI
Ca
+
I
NMDA
. From a strict automatic control point
of view, we are now dealing with a nonlinear system with
three state variables
x
1
,
x
2
,
x
3
and one control variable
u
=
−
αI
Ca
+
I
NMDA
.
III. CONTROL DESIGN METHODS
Let
x
0
=[
x
10
x
20
x
30
]
T
∈
R
3
≥
0
denote the health value.
Consequently, the control objective is to force
x
to converge
to
x
0
. We introduce the external control agent
u
to reduce
the strength of synaptic plasticity for preventing the seizures.
Before proceeding further, we need to study the properties
of the system.
A. Nonnegativity
For the synaptic plasticity system, it is easy to prove that
the states of this system are nonnegative. From the chemical
property, we know the variables
x
1
(
t
)
>
0
,
x
2
(
t
)
>
0
,
x
3
(
t
)
>
0
,
∀
t
, which will also be proved mathematically
later. Therefore, we can assume that the initial values
x
i
(0)
>
0
,
i
=1
,
2
,
3
.
It is obvious that
e
80(
x
3

0
.
35)
1+
e
80(
x
3

0
.
35)
<
1
. And it is easy to
show that if
x
3
>
0
, we have that
f
1
(
x
3
)
>
0
(9)
g
1
(
x
3
)
>
0
(10)
WeC18.1
2013