Astrocyte-Inspired Controller Design

for Desynchronization of Two Coupled Limit-Cycle Oscillators

Ghazal Montaseri1, Mohammad Javad Yazdanpanah

1,2 and Mahmood Amiri

1

1School of Electrical and Computer Engineering, University of Tehran,

Tehran, Iran

e-mail: montaseri@ut.ac.ir, ma.amiri@ece.ut.ac.ir 2Control and Intelligent Processing Center of Excellence, School of ECE, University of Tehran

Tehran, Iran

e-mail: yazdan@ut.ac

Abstract—In this paper, a biologically inspired controller is

proposed for desynchronization of two coupled limit-cycle

oscillators. Recently, researchers have shown that “astrocyte”

has the potential to desynchronize the synchrony between two

coupled neurons. So, first, based on the astrocyte model, the

structure of the dynamic controller is suggested. Then,

controller parameters are tuned through an optimization

algorithm. The suggested controller has 3 important

properties: 1) the controller desynchronizes the oscillators

without any undesirable effects on the inherent behavior of

oscillators (e.g. stopping, annihilating or starting divergent

oscillations); 2) it requires little effort to maintain the desirable

desynchronized state; and 3) the controller is robust with

respect to parameters variations. Simulation results reveal the

efficiency of the proposed controller.

Keywords-astrocyte-inspired dynamic controller; coupled

limit-cycle oscillators; desynchronization; optimization algorithm

I. INTRODUCTION

Synchronization is an adjustment of rhythms of coupled oscillators due to their weak coupling. Synchronous flashing of fireflies, coordinated firing of cardiac pacemaker cells, synchronization in ensembles of electrical oscillators are examples of synchronized phenomena in biology, physics and engineering [1]. Sometimes, synchronization may be harmful. For example, although synchronization of different neurons plays an important role in biological information processing, diseases like Parkinson’s disease, essential tremor and Epilepsies are related to the pathologically enhanced synchronization of neurons [2].

There is a significant clinical need for new effective stimulation techniques in order to suppress the collective synchrony. Generally, the proposed method for suppression (control) of synchrony can be divided into two categories; non-feedback techniques [3,4] and feedback techniques [1,2,5-8]. Unlike these control strategies, in this study, we proposed a biologically inspired controller. The inspiration is based on the dynamic model of astrocytes.

Astrocytes are the most abundant type of glial cells. They control the content of extracellular fluid, electrolyte homeostasis and regulate neurotransmitter [9] and have a role in neural synchrony [10,11]. In [12], we show that

astrocyte has the potential to desynchronize the synchrony between two coupled neurons by changing the synchronization threshold. So, astrocyte can be a good candidate to propose a controller for desynchronization of coupled oscillators.

In this paper, we consider the problem of desynchronization of two coupled limit-cycle oscillators as the minimal network model. First, the structure of the dynamic controller is proposed based on the inspiration from the astrocyte model proposed in [13]. Then, the controller parameters are tuned by an optimization algorithm. The suggested astrocyte-inspired controller has three important properties: 1) when the controller output is applied to the coupled synchronized oscillators, it disturbs the synchrony state without having any undesirable effects on the oscillators. It means that the stimulation does not cause oscillators to stop oscillation or to start divergent oscillation. In fact, the stimulation breaks the synchrony just by shifting one of the oscillators’ phase about π radian; 2) the

controller can break the synchrony state and maintain the desired desynchronized behavior with a little effort. So, the stimulation has “demand-controlled” character; and 3) finally, the stimulation is robust with respect to oscillator and controller parameters. In other words, when there are no model mismatch/uncertainties and/or noise (nominal model), the suggested controller with optimal parameters can effectively desynchronize the coupled oscillators. When oscillator or stimulation parameters vary (e.g. due to noise or aging, respectively), although the stimulation may not be as effective as the nominal case, we show through mathematical analysis that oscillators’ states and stimulation still remain bounded.

The remainder of this paper is organized as follows: preliminaries and problem statement are described in Section II. Section III explains how the structure of the dynamic controller is constructed based on the biological inspiration. Analysis of the closed-loop system behavior is done in Section IV. An optimization algorithm for computing optimal values of controller parameters is proposed in Section V. Section VI illustrates the simulation results. Finally, the conclusion is presented in Section VII.

195978-1-4577-1124-4/11/$26.00 c©2011 IEEE

II. PRELIMINARIES AND PROBLEM STATEMENT

Consider a limit-cycle oscillator [6] described as:

2,1,)(1

1 22 =+−−

= iy

xyx

y

x

y

x

i

iii

i

i

i

i

i

i

ω

ω (1)

Oscillators’ states ),( ii yx asymptotically converge to the

stable circular limit cycle with radius 1. They oscillate with the frequencies 21,2,1, ωωω ≠=ii . Suppose oscillators are

coupled linearly with the coupling strength 0>C as follows:

kiki

yy

xxC

y

xyx

y

x

y

x

ik

ik

i

iii

i

i

i

i

i

i

≠∈

−

−++−

−=

},2,1{,

)(1

1 22

ω

ω

(2)

In the polar coordinates ( iij

i jyxer i +=θ ), (2) can be

represented as:

kikirrC

rCrCrr

ikikii

ikkiii

≠∈−+=

−+−−=

},2,1{,),(sin/

)(cos)1( 2

θθωθ

θθ (3)

To clarify the results, considering the proposed definition in [14], we define the concept of synchronization.

Definition 1: The complete synchronization (CS) of two coupled oscillators (2) takes place when the following relations are satisfied:

0)()(lim),0)()(lim) 2121 =−=−∞→∞→

tytyiitxtxitt

(4)

The CS is expected for two identical systems (i.e. systems with the same ODEs and parameters). Since, due to the presence of an external noise or parameter mismatch, the condition of achieving CS may not be fulfilled, the following definition may be used instead.

Definition 2: The imperfect complete synchronization (ICS) of two coupled oscillators (2) takes place when the following relations are satisfied:

221121 )()(lim),)()(lim) εε ≤−≤−∞→∞→

tytyiitxtxitt

(5)

where 1ε and 2ε are small parameters such that

)()(sup 211 txtx −<<ε and )()(sup 212 tyty −<<ε .

Using (3), it can be shown that for coupling strength

greater than 2/21 ωω − , the ICS occurs, i.e.

))2/((sin)()(lim 211

21 Cttt

ωωθθ −=− −

∞→. The ICS concept

gets close to the CS concept for 2/21 ωω −>>C . We define

“desynchronization” as the case where πθθ =−∞→

)()(lim 21 ttt

.

To quantify the synchronization concept, the following complex order parameter is defined [15]:

=ℜ=

N

i

tjtj ieNet1

)()( )/1()(θΩ (6)

where )(tℜ is the synchronization index, )(tΩ is the mean

phase and N is the size of oscillators’ population (here

2=N ). It is clear that 1)(0 ≤ℜ≤ t . 1)( =ℜ t corresponds to

CS (coincidence of phases) and desynchronization is characterized by 0)( =ℜ t .

In this paper, the main purpose is to design a dynamic controller that can break the synchrony observed between

two oscillators due to the large coupling strength. In what follows, we illustrate how the controller structure is constructed based on biological inspirations.

III. ASTROCYTE-INSPIRED CONTROLLER DESIGN

In this section, first, the astrocyte function and, then, the proposed astrocyte-inspired controller are explained.

A. Astrocyte Dynamic Model

During the last decade, basic research in biology confirmed that glial cells are active players in neuronal activity and information processing [16,17]. The most abundant type of glial cells is star-shaped astrocytes. Although astrocytes do not exhibit electrical excitability, they are excitable with respect to intracellular calcium [17]. Increasing the intracellular calcium levels in astrocytes initiates the release of glutamate and ATP that are capable, by a feedback mechanism, of modulating synaptic strengths between nearby neurons [9].

At the cellular level, the main mechanisms underlying the tripartite synapse are as follows: neurotransmitters such as glutamate, released from presynaptic neuron during its activation, are bound to the glutamate receptors of the astrocytes adjacent to synaptic terminals. This triggers the production of the second messenger, inositol -trisphosphate (IP3) (1) and release of calcium (Ca

2+) into astrocyte

cytoplasm from endoplasmic reticulum (ER) (2). These calcium elevations propagate into nearby astrocytes as intercellular calcium waves with the passage of the second messengers through gap junctions [18]. As a consequence of the increased intracellular Ca

2+ concentration, the astrocyte

releases gliotransmitters, such as glutamate and ATP, into the extracellular space (3,4,5) and, thereby, it can regulate pre- and postsynaptic neurons [9,16,17]. The aforementioned processes are summarized in Fig. 1.

To model the dynamics of the intracellular Ca2+

waves produced by astrocytes, Postnov et al. recently proposed a dynamic model for the astrocyte in [13]. This is a generalized and simplified mathematical model for a small neuron-astrocyte ensemble which considers the main pathways of neuron-astrocyte. This model is explained with the following set of equations [13]:

( )( )

( )( ) GmmmGmGmmGm

SmmmsmsmmSm

eeee

eecc

mec

dGGhcSG

dSShZSS

ccccccccccccf

ccfc

Srccfccc

/1)](tanh[1

/1)](tanh[1

)/().1/()1/(),(

),(

),(

344

242222

1

4

−−−+=

−−−+=

−++−+=

=

++−−=

τ

τ

τε

βτ (7-a)

(7-b)

(7-c) (7-d)

Figure 1. The main pathways for neuron-astrocyte interactions

196 2011 Third World Congress on Nature and Biologically Inspired Computing

where c is the calcium concentration in the astrocyte

cytoplasm, ec denotes the calcium concentration within the

endoplasmic reticulum, and parameters cε and cτ together

define the characteristic time for calcium oscillations. The calcium influx from the extracellular space is sensitive to the production of secondary messenger mS (IP3), which is

controlled by the factor β . The initial state of the calcium

oscillation is controlled by the parameter r. The calciumexchange between the cytoplasm and endoplasmic reticulum is defined by the nonlinear function ),( eccf . Increasing

calcium concentration in the cytoplasm leads to the release of astrocyte mediator mG . The interaction between astrocyte

and neurons is denoted by the parameter Z that shows the synaptic activity of neurons. Based on the results in [10-12], the astrocyte could have a

key role in stabilizing neural activity. In [12], we show that

astrocyte has the potential to desynchronize the synchrony

between two coupled neurons by increasing the threshold

value of synchronization. So, astrocyte can be a candidate to

propose a controller for desynchronizing coupled oscillators.

B. Controller Structure Based on Astrocyte Model

Based on the astrocyte model (7), the structure of the bio-inspired controller is proposed as:

( )( ) ykykZky

ykkxkx

654

321

1)](tanh[1 −−−+=

++−= (8)

in which ( )TyxX = are the controller states. The controller

generates the proper stimulation based on the observation Z

of the system states. So, )( 2211 yxyxaZ +++= is the

controller input with the controller input gain a and ik s,

6..,2,1=i are positive parameters. The construction of (8)

from (7) is explained by steps 1 through 3 as follows: Step 1: Biological studies (introduced in Subsection III. A) show that variation of calcium concentration plays an important role in the neurons-astrocytes interactions. Thus, we propose the controller, mainly, based on the dynamics of calcium concentration and we omit ODE (7-d) which models the release of glutamate. Note that, the controller is an inspiration from the astrocyte model not the copy of it. The initial structure of the controller up to this step is:

( )( ) SmmmsmsmmSm

eeee

eecc

mec

dSShZSS

ccccccccccccf

ccfc

Srccfccc

/1)](tanh[1

)/().1/()1/(),(

),(

),(

344

242222

1

4

−−−+=

−++−+=

=

++−−=

τ

τε

βτ (9-a)

(9-b)

(9-c)

Step 2: According to [13], cSm ττ > and cε in (9-b) is a

small parameter. So, cccSm τεττ >> . It means that ec has

fast dynamics in comparison with c and mS and rapidly

reaches its steady-state (occurring when 0/ =dtdce and, thus,

0),( =eccf ). Therefore, 0),( =eccf can be set in (9-a). In

this step, the resulting reduced controller model is:

( )( ) SmmmsmsmmSm

mc

dSShZSS

Srcc

/1)](tanh[1 −−−+=

++−=

τ

βτ (10)

Step 3: Finally, by time scaling Smt ττ /= and defining

cSmk ττ /1 = , rrk cSm ττ /2 = , cSmk τβτ /3 = , smSk =4 ,

smhk =5 , Smdk /16 = and renaming cx = and mSy = , the

structure of the inspired controller is simplified from (10) to (8) where τddxx /= and τddyy /= .

The scheme of the closed-loop system composed of two oscillators and controller is shown in Fig. 2 based on which the closed-loop system is described as:

( )( ))(,

1)](tanh[1

1

1)(

1

1

1

1)(

1

1

2211654

321

221

21

2

222

22

2

2

2

2

2

2

112

12

1

121

21

1

1

1

1

1

1

yxyxaZykykZk

ykkxk

y

x

xyy

xxC

y

xyx

y

x

y

x

xyy

xxC

y

xyx

y

x

y

x

+++=−−−+

++−=

−−

−++−

−=

+−

−++−

−=

γω

ω

γω

ω

(11)

The controller input ( Z ) triggers the y -dynamics, then y

affects the x dynamics through the term yk3 . Since x has a

crucial role in (8) ( c in (7) respectively), we consider x as

the controller output. Finally, x controls (desynchronizes)

the two coupled oscillators through the terms x1γ and x2γ−

(called stimulation). Note that, among gliotransmitters that astrocyte releases, glutamate has excitatory but ATP has an inhibitory effect. This fact is modeled by considering a positive sign for excitation and negative sign for inhibition. In the rest of this paper, we divide controller parameters into internal (structural) parameters (i.e. 6,...,2,1, =iki ) and

external parameters ( 2,1, =iiγ and a ).

Next, we want to show that, despite the aforementioned simplification steps, the main and essential properties of the astrocyte model (7) are preserved in the structure of the proposed controller. To do that, the astrocyte and the

controller input ( Z ) are set to )1.0sin( tZ = and the value of

all parameters is selected to be 1. Fig. 3 shows the phase portrait of astrocyte states in ( mSc − ) space (left panel) and

the controller states in ( yx − ) space (right panel).

It is seen that, in both non-autonomous systems, the phase portraits ultimately converge to the approximately same limit cycle. So, it is expected that the controller, similar to an astrocyte, has a potential to desynchronize a synchrony behavior. Further simulations (Section VI) reveal the efficiency of the proposed inspired controller in the desynchronization of the synchronized oscillators.

IV. CLOSED-LOOP SYSTEM BEHAVIOR ANALYSIS

In this section, we prove that for every system and controller parameter: i) the proposed controller has the asymptotically

Figure 2. The general scheme of the closed-loop system

2011 Third World Congress on Nature and Biologically Inspired Computing 197

1 1.1 1.2 1.3 1.4 1.5 1.60

0.1

0.2

0.3

0.4

0.5

Sm

c

1 1.1 1.2 1.3 1.4 1.5 1.60

0.1

0.2

0.3

0.4

0.5

y

x

Figure 3. Phase portrait of astrocyte states in ( mSc − ) space (left panel)

and the controller states in ( yx − ) space (right panel)

asymptotically stable equilibrium point and ii) oscillators’ states are bounded.

A. Stability Analysis of Controller

The astrocyte-inspired controller (8) has the equilibrium point 1326 /)(),/( kykkxkMMy +=+= where

)](tanh[1 54 kZkM −+= . Note that 20 << M . By considering

the coordinate transformation xxx −=~ and yyy −=~ , the

equilibrium point is shifted to the origin and, in the new coordinates, (9) can be represented as the cascade system

(12) with the y~ driving subsystem and x~ driven subsystem:

)~(~)(~),~,~(~~~26131 yfykMyyxfykxkx =+−==+−= (12)

1f and 2f are both Lipschitz in their arguments.

Differentiating the Lyapunov function 21

~5.0 yV = leads to

26

261

~~)( ykykMV −<+−= . So, y~ -subsystem is globally

exponentially stable. In addition, since the unforced x~ -

subsystem (i.e., xkx ~~1−= ) is globally exponentially stable, it

is input-to-state stable with y~ as an output (see Lemma 4.6

of [19]). Finally, using Lemma 4.7 of [19], it is concluded that the origin is a globally asymptotically equilibrium point of the total system (12). Thus, for all values of ik , the

controller (8) is globally asymptotically stable.

Next, we want to calculate an upper bound of )(tx which

will be used in the next subsection. Let the initial condition of (8) be )0,0(),( 00 =yx . Differentiating the Lyapunov

function 25.02 xV = along the x -subsystem of (8) leads to:

][)1(

)(

1322

1

322

12

xkykkxxk

ykkxxkV

ϑϑ −++−−≤

++−= (13)

where 10 << ϑ . Since y -subsystem is globally

exponentially stable, we have 0,)( >∀≤ tyty (Note that y

is positive). So:

])[/1(,)1(

][)1(

3212

1

1322

12

ykkkxxk

xkykkxxkV

+≥∀−−≤

−++−−≤

ϑϑ

ϑϑ (14)

using Theorem 4.19 and Lemma 4.6 of [18], it is concluded that:

0,1

)(6

32

1

>∀+

+≤ tkM

Mkk

ktx

ϑ (15)

the closer ϑ is to 1, the less conservative upper bound is

obtained for )(tx . L is defined as the supremum of )(tx (i.e.

0,)( >∀< tLtx ) as:

++=

6

32

1

1

kM

Mkk

kL (16)

B. Boundedness Analysis of Oscillators’ States

In this subsection, we show that when the proposed controller is applied to both oscillators (see Fig. 2), regardless of the controller parameters’ values, the oscillators’ states remain bounded. This property is the first and most important control objective which ensures that divergent oscillations are not possible for oscillators.

Define 2,1,222 =+= iyxr iii . Differentiating the

Lyapunov function )(5.0 22

21

rrV += along the trajectories of

the closed-loop system (11) results in:

)()()1()1( 22211122

22

21

21

yxxyxxrrrrV +−++−+−≤ γγ (17)

On the circle 2,1,1,2 =>= iRRri , RyxR ii 22 ≤+≤− .

So, on the circle, we have:

xRRRRRV )(2)1()1( 21 γγ ++−+−≤ (18)

Using the inequalities RR < (since 1>R ) and Lx <

(according to (16)), (18) is simplified to:

)])(22[( 2122 γγ +++−+−≤ LRRRV (19)

By choosing

oRLR =++> )(22 21 γγ (20)

the second term of (19) gets negative; thus 2RV −≤ .

Theorem 1 summarizes the results of this section.

Theorem 1: Consider the closed-loop system (11). Suppose all the parameters are positive and the initial condition of the dynamic controller is )0,0(),( 00 =yx . Then,

i) The astrocyte-inspired controller is asymptotically stable.

ii) The set }2,1,,{ 2 =≤ℜ∈ℜ∈= iRryx oiiiΩ , where

)(22 21 γγ ++= LRo and L is defined as (16), is a

positively invariant set for the oscillators’ states. Property ii means that oscillators’ states ultimately enter the

circle oi Rr =2 and remain there for all future times. This

property shows controller robustness with respect to parameters’ variations. In other words, when system and controller parameters vary from their nominal values due to noise or aging, respectively, Theorem 1 guarantees that oscillators’ states and stimulation still remain bounded.

Although for all values of structural and external parameters, boundedness of states is guaranteed, desynchronization as the main control objective is not feasible unless a suitable set of controller parameters, especially iγ , is selected. In the next section, we will

propose an optimization algorithm for finding optimal values of 2,1, =iiγ .

198 2011 Third World Congress on Nature and Biologically Inspired Computing

V. COMPUTING OPTIMAL VALUES FOR CONTROLLER

PARAMETERS

In this section, we propose an optimization algorithm for computing optimal values of controller parameters. Here, optimality refers to the minimum synchronization index

)0( =ℜ and minimum stimulation energy, which means that,

among all feasible parameters’ values, those resulting in the minimum ℜ and minimum energy consumption are

selected. Among all controller parameters, iγ s are of great

importance. Structural controller parameters affect the intrinsic characteristic (such as dynamic velocity) of the controller. However, iγ s directly influence the oscillators’

behavior. They are controller output gain. So, the focus is on these parameters in the optimization algorithm.

We propose the following cost function:

dtxQxQyyxxQTtt }][][])()[({min

223

212

221

22110

, 21

γγγγ

+++++==

(21)

where iQ s, 3,2,1=i are weighting coefficients. Using (6),

221

2212211 )()(5.0)(5.0 yyxxjyxjyx +++=+++=ℜ . So,

minimization of ])()[( 221

221 yyxx +++ leads to the

minimum value of ℜ and, thus, desynchronization onset.

The last two terms in (21) penalize the energy of stimulation. Simulation results and further analysis (that are explained

later) showed that to have a permanent oscillation, the values of iγ should be restricted within a bound. For iγ s that are

beyond this bound, oscillation may stop. As was assumed before, the lower bound on iγ is zero; i.e. 0>iγ .

In the polar coordinates, the phase dynamics of oscillators are described by:

)]sin())[cos(/()(sin)/(

)]sin())[cos(/()(sin)/(

2222122122

1111121211

θθγθθωθ

θθγθθωθ

−−−−=

−+−+=

rxrrC

rxrrC (22)

First, we focus on 1θ . The maximum value of 1θ is:

11121max1 /2/ rxrrC γωθ ++= (23)

which is obtained at )4/,4/( 21 πθπθ =−= . On the other

hand, for )4/,4/3( 21 πθπθ == , 1θ takes its minimum as:

11121min1 /2/ rxrrC γωθ −−= (24)

Since 0max1 >θ , if min1θ is also positive, the right hand side

of 1θ will be positive for all values of 1θ and 2θ which

means that 1θ is an increasing variable. In other words, from

the dynamical system point of view, 01 =θ does not have a

solution or 1θ does not have an equilibrium point (stationary

state). Consequently, having 01 >θ for all values of 1θ and

2θ ensures permanent oscillations of the first oscillator.

In the more conservative case, the upper bound of )(tx

(i.e. L ) results in a minimal value for min1θ with respect to

x . So, if 1γ satisfies (25), 1θ is always increasing.

)2/()( max2111 xrCr −< ωγ (25)

To have a constraint independent of 2,1, =iri , it is assumed

that 121 ≅≅ rr in (25). This is due to the observation that the

controller does not alter the amplitude of oscillation significantly. Finally, the constraint on 1γ is:

)2/()( 11 LC−< ωγ (26)

Similarly, 2γ should satisfy:

LC 2/)( 22 −< ωγ (27)

Constraints (26) and (27) determine the conservative bound on iγ . These constraints guarantee that the controller does

not disturb oscillators’ natural behavior and just force them to oscillate in a desynchronized manner.

After calculating the constraints on iγ s, the optimization

problem for computing the optimal values of iγ s (i.e., *iγ s)

is proposed as follows:

)11(

2,1),2/()(0:

}][][])()[({min 02

232

122

212

211, 21

systemloopclosedThe

iLCtosubject

dtxQxQyyxxQ

ii

Ttt

−

=−<<

+++++==

ωγ

γγγγ

(28)

In (28), the oscillators’ and controller’s states are predicted based on the closed-loop system model (11). We suppose that controller structural parameters are fixed and are selected suitably to satisfy the desired performances such as dynamic velocity of the oscillators, actuators’ limitations and considerations. Note that, although in the suggested optimization problem, optimization is done subject to

2,1, =iiγ , controller structural parameters ( 6,...,2,1, =iki )

may also be considered as optimization variables. Once the

optimal values *iγ are computed off-line by solving (28),

they can be used in the proposed controller (8). In the next section, we will illustrate the results of

applying the astrocyte-inspired controller (8) with the

optimal values of iγ resulting from (28).

VI. SIMULATION RESULTS

Consider the closed-loop system (11) with *1γ and *

2γ

that are computed by the optimization problem (28). The parameters values are given in Table I. The optimal values of

1γ and 2γ are 0.420.14, **21

== γγ . Simulation results are

shown in Fig. 4. The top panel shows the coupling strength. For sec1000 << t , the two oscillators are uncoupled (i.e.

0=C ) and they are coupled for sec100≥t . Suppose the

stimulation is applied to the oscillators at sec500=t when

the oscillators are completely synchronized. This means that,

2,1,0 == iiγ for sec500<t and *ii γγ = for sec500≥t .

The second panel shows the oscillators’ states 1x and 2x .

As the second panel indicates, for sec1000 << t , the two

uncoupled oscillators oscillate independently with their natural frequencies. By increasing the coupling strength, for

sec100≥t , the coupled oscillators get synchronized and

oscillate with the mean frequency )(5.0 21 ωωω += . After

2011 Third World Congress on Nature and Biologically Inspired Computing 199

applying the stimulation at sec500=t , it is observed that the

proposed controller is able to break the synchronized behavior. This observation agrees with the synchronization index plotted in the third panel. Finally, the controller output ( x ) is depicted in the bottom panel. The important point

about the proposed controller is that the controller generates the appropriate stimulation with little effort (compare the amplitude of x before and after the stimulation). So, the

proposed controller has a demand-controlled character, i.e. as soon as the desired desynchronized state is obtained, the maintenance of this state requires the minimal amount of controller force. The reason is that, after achieving desynchronization, the controller input ( Z ) gets close to zero and this leads to a decrees in the controller output ( x ).

VII. CONCLUSION

In this research, we focus on the impact of the astrocyte inspired controller on a minimal network of two coupled limit-cycle oscillators. Based on the mathematical analysis and simulation results, the proposed controller has the potential to desynchronize the coupled synchronized oscillators. What is done in this work may be considered as the first step in designing an efficient stimulation -based on the biological inspiration- for the therapy of neurological diseases with pathological synchronization. Further works should be done to improve the astrocyte based controller to be applicable for globally coupled population of neurons.

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TABLE I. THE PARAMETERS VALUE OF THE CLOSED-LOOP SYSTEM

(11) AND THE OPTIMIZATION PROBLEM (28)

1ω 1.0 C 02.0 1k 1 3k 5.1 5k 2

2ω a 5.1 2k 05.0 4k 1 6k 2

T 400 1Q 1 2Q 10 3Q 100 L 08.0

0 100 200 300 400 500 600 700 800 9000

0.01

0.02

C

0 100 200 300 400 500 600 700 800 900

-1

0

1

2

x1 ,

x2

0 100 200 300 400 500 600 700 800 900

0.2

0.4

0.6

0.8

ℜ

0 100 200 300 400 500 600 700 800 9000

0.2

0.4

0.6

Time(sec)

x

Stimulation on Coupling on

Figure 4. The effect of varying coupling strength C (top panel) on the

two coupled oscillators (second panel) with the controller output (bottom

panel) which is applied for sec500≥t .

[8] M. Luo, Y. Wu, and J. Peng, “Washout filter aided mean field feedback desynchronization in an ensemble of globally coupled Neural Oscillators,” Biological Cybernetics, vol. 101, 2009, pp. 241-246, doi: 10.1007/s00422-009-0334-5.

[9] M. Amiri, F. Bahrami, and M. Janahmadi, “Functional modeling of astrocytes in epilepsy: a feedback system perspective,” Neural Computing & Applications, 2010, doi:10.1007/s00521-010-0479-0.

[10] T. Fellin T, O. Pascual, S. Gobbo, T. Pozzan, PG. Haydon, and G. Carmignoto, “Neuronal synchrony mediated by astrocytic glutamate through activation of extrasynaptic NMDA receptors,” Neuron, vol 43, 2004, pp.729-743, doi:10.1016/j.neuron.2004.08.011.

[11] G. Seiferta, G. Carmignotob, and C. Steinhäusera, “Astrocyte dysfunction in epilepsy,” Brain Research Reviews, vol. 63, 2010, pp. 212-221, doi:10.1016/j.brainresrev.2009.10.004.

[12] M. Amiri, G. Montaseri, and F. Bahrami, “On the role of astrocytes in synchronization of two coupled neurons: a mathematical perspective,” unpublished.

[13] D. E. Postnov, R. N. Koreshkov,N. A. Brazhe, A. R. Brazhe, and O. V. Sosnovtseva, “Dynamical patterns of calcium signaling in a functional model of neuron-astrocyte networks,” Journal of Biological Physics, vol. 35, 2009, pp. 425-445, doi:10.1007/s10867-009-9156-x.

[14] A. Stefa ski, Determining thresholds of complete synchronization, and application. World Scientific Series on Nonlinear Science, Series A -vol. 67 2009.

[15] P. F. Pinsky and J. Rinzel, “Synchrony measures for biological neural networks,” Biological Cybernetics, vol. 73, 1995, pp. 129-137, doi: 10.1007/BF00204051.

[16] T. Fellin, O. Pascual, and P. G. Haydon, “Astrocytes coordinate synaptic networks: balanced excitation and inhibition,” Journal of Physiology, vol. 21, 2006, pp. 208-215, doi: 10. 1152/physiol. 00161.2005.

[17] M. M. Halassa, T. Fellin, and P. G. Haydon, “Tripartite synapses: roles for astrocytic purines in the control of synaptic physiology and behavior,” Neuropharmacology, vol. 57, 2009, pp. 343-346, doi:10.1016/j.neuropharm.2009.06.031.

[18] J. Hung and M. A. Colicos, “Astrocytic Ca2+ waves guide CNS growth cones to remote regions of neuronal activity,” PLoS ONE. 3(11):e3692, 2008, doi:10.1371/journal.pone.0003692.

[19] H. K. Khalil, Nonlinear Systems. New Jersey, Prentic Hall, 2002.

200 2011 Third World Congress on Nature and Biologically Inspired Computing