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Communications in Nonlinear Science and Numerical Simulation xxx (2007) xxx–xxx www.elsevier.com/locate/cnsns

Synchronization dynamics in a ring of four mutually coupled biological systems H.G. Enjieu Kadji

a,b

, J.B. Chabi Orou

a,*

, P. Woafo

b

a

b

Institut de Mathe´matiques et de Sciences Physiques, B.P. 613, Porto-Novo, Benin Laboratory of Nonlinear Modelling and Simulation in Engineering and Biological Physics, Faculty of Science, University of Yaounde´ I, P.O. Box 812, Yaounde´, Cameroon Received 19 June 2006; received in revised form 31 October 2006; accepted 9 November 2006

Abstract This paper considers the synchronization dynamics in a ring of four mutually coupled biological systems described by coupled Van der Pol oscillators. The coupling parameter are non-identical between oscillators. The stability boundaries of the process are ﬁrst evaluated without the inﬂuence of the local injection using the eigenvalues properties and the fourthorder Runge–Kutta algorithm. The eﬀects of a locally injected trajectory on the stability boundaries of the synchronized states are performed using numerical simulations. In both cases, the stability boundaries and the main dynamical states are reported on the stability maps in the (K1, K2) plane. Ó 2006 Elsevier B.V. All rights reserved. PACS: 05.45.Xt; 05.65.+b Keywords: Synchronization dynamics; Biological systems; Local injection

1. Introduction For many years, coupled nonlinear oscillators have been a source of growing interest in diﬀerent research ﬁelds, ranging from physics, chemistry, engineering to biology and social sciences [1–5]. Between these nonlinear dynamical systems, two main classes are to be distinguished: the regular and chaotic one. But independent to their state which can be regular or chaotic, one of the most interesting features displayed by coupled oscillators is synchronization. This widespread phenomenon is observed everywhere in nature and is deﬁned as an adjustment of rhythms due to a weak interaction. It oﬀers the possibility of masking information bearing signal by chaotic signal from electronics [6–9] or optical sources [10–13] and also it possesses potential applications in communication engineering [14–20] as in biological and chemical structures [21,22]. Such situations *

Corresponding author. Tel.: +229 97471847; fax: +229 20222455. E-mail addresses: [email protected] (H.G. Enjieu Kadji), [email protected] (J.B. Chabi Orou), [email protected] (P. Woafo). 1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.11.004

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can well explain why in the last few years, the study of coupled oscillators with ring connexion has increased. For example, a ring of two neurons has been used to study the mixed inhibitory and excitatory circuits coordinating the motion of locust wings during ﬂight [23]. It has also been shown that in quadrupedal mammals, the four oscillators controlling the limb may be coupled in some form ring [24,25]. Recently, Chembo and Woafo [26,27] investigated the spatiotemporal dynamics of a ring of diﬀusively coupled single-well Duﬃng oscillators with a positive nonlinear stiﬀness. Our recent contribution in that topic has been the investigation of synchronized states in a ring of mutually coupled electrical oscillators [28]. The coupling parameters between oscillators were identical and the properties of variational equations have been called to tackle the dynamics of the model. The inﬂuence of an external excitation locally injected on the stability boundaries of the synchronized states has also been performed. Such models ﬁnd many potential applications. For instance in living system, the possible physiological signiﬁcance of this model is that of a narrow tube segment for the duodenum or intestines. The model for the duodenum should comprise a tube of oscillators having cells both around the periphery and along the axes of the tube rather than a one-dimensional chain, and thus a ring of oscillators could represent a narrow cross-sectional slice through the duodenum. In biotechnology, the use of immobilized enzymes with ring topology greatly minimized the cost of production by permitting repeated use of the enzymes and substantially increased the stability of the enzymes themselves [29]. The enzymes are then related to the support and does not migrate in the product, so that one can easily separate them from the product. The ring connexion of these four oscillators can be viewed as a ring of four protomers of the oligomer of an allosteric enzyme evolving from its unfavorable conformational state (T) to its active state (R) [30]. But sometimes due to some internal parameters of the system, the coupling parameters can behave nonidentical between the oscillators enrolled in the process. In that event, it might happen that the nearest neighbouring oscillators are coupled to a considered oscillator through a same or diﬀerent energy of activation due to a conformational spread. For instance, when the ring of identical protomers in considered, each protomer can exist in two distinct conformational states (active and inactive). A promotor may also bind a single molecule of an eﬀector ligand, present in the surrounding solution. In such a case, thermodynamics consistency then induces the existence of two diﬀerent dissociation constants of the ligand related to active and inactive protomers [31,32]. Therefore, this result to non-identical coupling coeﬃcient. Here, we aim to shed some light on the synchronization dynamics of such a system by trying to answer the following questions: What kind of oscillation patterns will it display? Will it display synchronous oscillations (Full synchronization)? Under which conditions they are stable? What will be the inﬂuence on the stability boundaries of a locally injected signal? The paper is organized as follows: The next section deals with the model and the problem statement. In Section 3, the stability analysis of the process is investigated using analytical and numerical methods. The inﬂuence of an external sinusoidal force locally injected on the stability boundaries of the synchronization process is considered in Section 4. A conclusion is given in the last section. 2. Model and problem statement The biological model consists of an enzyme–substrate system with ferroelectricity behavior in brain waves [33]. On the physical basis used by Frohlich [34,35], it comes that the evolution equation of the activated enzyme molecules can be described by the well-known classical Van der Pol equation €x eð1 x2 Þ_x þ x ¼ 0:

ð1Þ

Such reactions are not isolated but coupled in the biological systems where there is a cooperative interactions between neighbouring enzymes. Therefore depending on certain biochemical considerations among which the time delay aﬀorded by multistep processing and cooperative regulation in many regulatory phenomena, protein phosphorylation on short signal sequences in target molecule, the coupling circuit of enzymes can behave diﬀerently [36]. Here, we are taking a look at coupled enzymes with ring connexion. For sake of exempliﬁcation, we consider a ring of four mutually coupled biological systems (see Fig. 1) within which each element is Please cite this article in press as: Kadji HGE et al., Synchronization dynamics in a ring of four mutually ..., Commun Nonlinear Sci Numer Simul (2007), doi:10.1016/j.cnsns.2006.11.004

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K2

K1

VdP1

Vdp

Vdp

2

4

K1

3

Vdp3

K2

Fig. 1. Ring of four mutually coupled biological systems.

described by Eq. (1). We assume that the coupling between two biological systems is described by the diﬀusion of the solute concentration [37]. Thus, the coupling is written as the diﬀerence in the concentration and it comes that the dynamics of the ring is described by the following second-order non-dimensional nonlinear differential equations: €x1 eð1 x21 Þ_x1 þ x1 ¼ K 1 ðx2 x1 Þ þ K 2 ðx4 x1 Þ; €x2 eð1 x22 Þ_x2 þ x2 ¼ K 1 ðx1 x2 Þ þ K 2 ðx3 x2 Þ; €x3 eð1 x23 Þ_x3 þ x3 ¼ K 1 ðx4 x3 Þ þ K 2 ðx2 x3 Þ;

ð2Þ

€x4 eð1 x24 Þ_x4 þ x4 ¼ K 1 ðx3 x4 Þ þ K 2 ðx1 x4 Þ; where the overdot denotes time derivative, e the parameter of nonlinearity and xi (i = 1, 2, 3, 4) the concentration of the ith biological system. Within each system, one has positive and negative direction of solute ﬂow which correspond respectively to the inhibitory and excitatory coupling. Because of both positive and negative direction of solute ﬂow, it comes that the coupling parameters K1 and K2 can be positive or negative. The synchronization of the model described by Eq. (2) is of interest in neural systems. Indeed, the notion of synchronization is also linked with several central issues of neuroscience [38]. For example, synchronization seems to be a central mechanism for neuronal information processing within a brain area as well as for communication between diﬀerent brain area [38,39]. Experiments show during the acts of cognition that millions of ﬁring neurons lock-in-phase, forming a powerful signal for a short period of time, and then a new pattern of neurons synchronizes for the next act [40]. Additionally, synchronization plays a signiﬁcant role in several neurological diseases [41–43], in the modeling of interaction between diﬀerent physiological (sub)systems demonstrating oscillating behavior [44–46] and also in our hearts where the sinoatrial node of about one thousand pacemaker cells collectively generate a rhythm to control beating of the rest of the heart [47]. A particular characteristic of the Van der Pol oscillator (VdPo) is the high sensitivity of its phase to initial conditions. Consequently, if four VdPo are launched with diﬀerent initial conditions, their trajectory will ﬁnally circulate on the same limit cycle but with four diﬀerent phases. The purpose of the synchronization is then to phase lock those oscillators (phase synchronization). Please cite this article in press as: Kadji HGE et al., Synchronization dynamics in a ring of four mutually ..., Commun Nonlinear Sci Numer Simul (2007), doi:10.1016/j.cnsns.2006.11.004

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3. Stability analysis of the process 3.1. Analytical treatment The resulting dynamical state of the system described by Eq. (2) is interesting only when it is stable. This requires that all the perturbed trajectories return to the original limit cycle. Therefore, it is of interest to perform criteria that guarantee the asymptotic stability of the process because one can tolerate synchronization to failed but not to be unstable because it could damage the system. Indeed, such a situation will lead to full instability in the process of catalytic reaction within the cells and progressively to the whole biological system. The stability of the dynamical states can be studied through the linearization of Eq. (2) around the unperturbed limit cycle xs according to: € n1 eð1 x2s Þn_ 1 þ ½1 þ K 1 þ K 2 þ 2exs x_ s n1 ¼ K 1 n2 þ K 2 n4 ; € n2 eð1 x2s Þn_ 2 þ ½1 þ K 1 þ K 2 þ 2exs x_ s n2 ¼ K 1 n1 þ K 2 n3 ; € n3 eð1 x2 Þn_ 1 þ ½1 þ K 1 þ K 2 þ 2exs x_ s n3 ¼ K 1 n4 þ K 2 n2 ; € n4 eð1

s x2s Þn_ 1

ð3Þ

þ ½1 þ K 1 þ K 2 þ 2exs x_ s n4 ¼ K 1 n3 þ K 2 n1 ;

where ni stands for the perturbation term. For small value of e, the behavior of one VdPo can be described by a pure sinusoidal trajectory of the form: xs ¼ A cosðXt UÞ;

ð4Þ

where A and X are respectively the amplitude and the frequency of the unperturbed limit cycle in the ﬁrst approximation. The values of A and X are A = 2.00 and X = 0.999 for e = 0.10. Using the solution (4), the variational Eq. (3) can be rewritten as follows: eA2 € n1 þ B þ cos 2s n_ 1 þ ½C exA2 sin 2sn1 ¼ K 1 n2 þ K 2 n4 ; 2 2 eA € n2 þ B þ cos 2s n_ 2 þ ½C exA2 sin 2sn2 ¼ K 1 n1 þ K 2 n3 ; 2 ð5Þ 2 eA 2 _ € n3 þ B þ cos 2s n3 þ ½C exA sin 2sn3 ¼ K 1 n4 þ K 2 n2 ; 2 eA2 € n4 þ B þ cos 2s n_ 4 þ ½C exA2 sin 2sn4 ¼ K 1 n3 þ K 2 n4 ; 2 2 with B ¼ e A2 1 , C ¼ 1 þ K 1 þ K 2 . According to the type of coupling here, diagonal variables (so-called Fourier modes) can not be applied to uncouple Eq. (5) and therefore to perform the dynamics of the synchronization process around the parametric resonances as reported in Ref. [28] for the case of identical coupling parameters. Nevertheless, the behavior of such a process can be estimated far from these resonant states. Thus, to investigate the stability analysis, the variational Eq. (5) can be rewritten as follows: n_ ¼ Wn;

ð6Þ

where 0

0 B C B B B 0 B B K1 W¼B B 0 B B B 0 B @ 0 K2

1 B 0 0 0 0 0 0

0 K1 0 C 0 K2 0 0

0 0 1 B 0 0 0 0

0 0 0 K2 0 C 0 K1

0 0 0 0 1 B 0 0

0 K2 0 0 0 K1 0 C

1 0 0 C C C 0 C C 0 C C: 0 C C C 0 C C 1 A B

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The stability of the process depends on the nature of the eigenvalues N of Eq. (6). Saying that the process is stable means here that the synchronization is really achieved. Using the MATHEMATICA algorithm, the characteristic equation is given as follows: a0 N8 þ a1 N7 þ a2 N6 a3 R5 þ a4 R4 þ a5 R3 þ a6 R2 þ a7 R þ a8 ¼ 0

ð7Þ

with a0 ¼ 1; a1 ¼ 4B; a2 ¼ 6B2 þ 4C; a3 ¼ B4 þ B3 þ 4B2 C þ 10BC; a4 ¼ ð2K 21 þ 2K 22 12B2 C 2C2 C B4 Þ; a5 ¼ ð4BK 21 5BC2 4B3 C þ 4BK 22 BCÞ; a6 ¼ ½ð2B2 þ 4CÞðK 21 þ K 22 Þ þ ð1 3B2 CÞC2 ; a7 ¼ ½4BCðK 21 þ K 22 Þ þ K 1 K 2 C2 ; a8 ¼ ½ðK 21 þ 2K 22 þ K 1 K 2 BÞC2 ðK 21 K 22 Þ2 : The corresponding eigenvalues are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 800K 2 399 þ 1 ; N1 ¼ 20 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 800K 2 399 1 ; N2 ¼ 20 pﬃﬃﬃﬃﬃﬃﬃﬃ 399j þ 1 N3 ¼ ; 20 pﬃﬃﬃﬃﬃﬃﬃﬃ 399j 1 N4 ¼ ; 20 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 800K 2 800K 1 399 þ 1 ; N5 ¼ 20 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 800K 2 800K 1 399 1 ; N6 ¼ 20 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 800K 1 399 þ 1 ; N7 ¼ 20 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 800K 1 399 1 N8 ¼ 20

ð8Þ

with j a complex number. The stability of the process is guaranteed if and only if each eigenvalue possesses a negative real part. Thus, the process stability depends exclusively on N2, N6 and N8 since Nl, (l = 1, 3, 4, 5, 7) have a real negative part for all K1 and K2. Consequently, the process is stable if K1 and K2 satisfy the following criteria: K 2 2 0:5; 0½ [ 0; þ1½; K 1 2 0:5 K 2 ; 0½ [ 0; þ1½;

ð9Þ ð10Þ

K 1 2 0:5; 0½ [ 0; þ1½:

ð11Þ

Otherwise, the synchronization is unstable and as t increases, ni(t) never goes to zero but has a bounded oscillatory behavior or goes to inﬁnity. Please cite this article in press as: Kadji HGE et al., Synchronization dynamics in a ring of four mutually ..., Commun Nonlinear Sci Numer Simul (2007), doi:10.1016/j.cnsns.2006.11.004

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3.2. Analytical and numerical results The criteria given through Eqs. (9)–(11) are used to identify analytically the stability boundaries of the synchronization process. Thus, Eq. (9) provides the validity domain where the coupling strength K2 must be chosen while Eq. (10) and (11) give the corresponding range authorized of K1 for which both K1 and K2 lead to the achievement of the process. Consequently in the case where the coupling parameter K2 leads to an excitatory process (K2 < 0), the critical stability boundaries of the process varies (see Eq. (10)). For instance, when K2 = 0.44, the ring displays full synchronization for K1 2 ] 0.07, 0[ [ ]0, +1[ while for K2 = 0.10, the same phenomenon is observed but for K1 2 ] 0.40, 0[ [ ]0, +1[. On the other hand, when the coupling coefﬁcient K2 leads to an inhibitory process (K2 > 0), the critical boundary of K1 deﬁning the synchronization domain does not vary (see Eq. (11)). In order to check the accuracy and to supply what is lacking in the analytical results, numerical simulations through the fourth-order Runge–Kutta algorithm are carried out. A time step Dt = 0.01 with the following initial conditions ðx1 ð0Þ; x_ 1 ð0ÞÞ ¼ ð1:0; 1:0Þ, ðx2 ð0Þ; x_ 2 ð0ÞÞ ¼ ð1:5; 1:5Þ, ðx3 ð0Þ; x_ 3 ð0ÞÞ ¼ ð2:0; 2:0Þ, ðx4 ð0Þ; x_ 4 ð0ÞÞ ¼ ð3:0; 3:0Þ are considered. Synchronization between two oscillators a and b is called with a condition that the distance of the phase trajectories d ab ¼ ja bj < 105 :

ð12Þ

Synchronization among all the oscillators involved in the process is realized if the total separation (TS) of all pair oscillators is smaller than a precision, namely X TS ¼ d ab < 105 : ð13Þ pairsðabÞ

Using both numerical and analytical results enable us to derive a stability map showing the main states of the system and their corresponding stability boundaries in the plane (K1, K2). Our results are reported in Fig. 2 and the coming results are observed. Three main domains D1, D2 and D3 are found. In the ﬁrst domain (D1), the synchronization process is unstable numerically but analytically stable. Within that domain, the amplitude of oscillations are bounded as shown for instance in Fig. 3 for K1 = 0.06 and K2 = 0.06. It should also be

D2

D1

D3

0.5 0.4 0.3 0.2

K2

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

K1 Fig. 2. A stability boundaries of synchronization process without the local injection. D1: Area where the synchronization process is analytically achieved but numerically unstable. D2: Area where the synchronization process is both analytically and numerically achieved. D3: Area where the synchronization process is both analytically and numerically non-achieved.

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7

1 0.9 0.8 0.7

TS(t)

0.6 0.5 0.4 0.3 0.2 0.1 0 100

120

140

160

180

200

Time(s) Fig. 3. Time history of TS(t) displaying bounded oscillations in the process of synchronization for K1 = 0.06 and K2 = 0.06.

underlined that for some particular values of the coupling coeﬃcients in this ﬁrst region, the phenomenon of clusters synchronization (oscillators cleave into two subgroups or attractors, called clusters such that all oscil-

x2- x4

x1- x4 x2- x3

x1 - x2 x1- x3

x3-x4

4

2

0

-2

-4

0

10

20

30

40

50

60

Time(s) Fig. 4. Time history of all pair of oscillators displaying clusters synchronization for K1 = 0.39 and K2 = 0.1.

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lators within one cluster move in perfect synchrony but the motion of two clusters is not synchronized at all) not forecasted by the analytical treatment is also obtained as we show in Fig. 4 for K1 = 0.48 and K2 = 0.06. The two clusters obtained are (x1 = x3) and (x2 = x4). For clustering , the inequality (13) is not fulﬁlled since the distance of phase trajectories of few oscillators are smaller than the precision. The second region (D2) corresponds to both analytical and numerical area where full synchronization is found in the ring. Therefore all four oscillators display the same dynamics as plotted in Fig. 5 for K1 = 0.20 and K2 = 0.42. In such a case,

8 7 6 5

TS(t)

4 3 2 1 0 -1 0

20

40

60

80

100

120

140

Time(s) Fig. 5. Time history of TS(t) displaying full synchronization in the ring for K1 = 0.20 and K2 = 0.42.

120

100

TS(t)

80

60

40

20

0 0

2

4

6

8

10

12

14

16

18

20

Time(s) Fig. 6. Time history of TS(t) displaying instability in the process of synchronization for K1 = 0.51 and K2 = 0.30.

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x1 = x2 = x3 = x4 and the emitted signal is more powerful. The last domain (D3) stands for the intersection between both analytical and numerical instability areas. The corresponding oscillations are not bounded as in (D1) but are very large and tend to inﬁnity as t increases. An illustration of such a behavior is represented in Fig. 6 for K1 = 0.50 and K2 = 0.30. Good agreement is found between analytical and numerical results. 4. Inﬂuence of a locally injected signal on the stability boundaries As we have mentioned before, diﬀerent dynamical states are observed in several identical coupled oscillators such as clustering or complete synchronization. However, sometimes, due to environmental constraints or because of its potential application, the system can be coupled to an external independent oscillator or excitation. Indeed, real systems always contains impurity or irregularity in the form of natural ﬂuctuation associated with dissipation as well as in the form of random external environment. In self-sustained systems, there are many circumstances causing irregularity. For example, the cardiovascular systems of humans display undamped self-sustained vibrations but due to the inﬂuence of various sources of irregularity, the cardiac vibration of humans and animal are not perfectly periodic [48]. This is commonly achieved through the local injection technique consisting of a unidirectional coupling between the external command oscillator and a ﬁxed representative of the nonlinear coupled system [49]. An interest of local injection in biological systems is the impulse transmission in neurons and the synchronization of neural network [50]. When the local injection is taken into account in our model, Eq. (2) becomes: €x1 eð1 x21 Þ_x1 þ x1 ¼ K 1 ðx2 x1 Þ þ K 2 ðx4 x1 Þ gðx1 xc Þ; €x2 eð1 x22 Þ_x2 þ x2 ¼ K 1 ðx1 x2 Þ þ K 2 ðx3 x2 Þ; €x3 eð1 x23 Þ_x3 þ x3 ¼ K 1 ðx4 x3 Þ þ K 2 ðx2 x3 Þ;

ð14Þ

€x4 eð1 x24 Þ_x4 þ x4 ¼ K 1 ðx3 x4 Þ þ K 2 ðx1 x4 Þ; where xc represents the dynamics of the external oscillator and also plays the role of the command signal and g the local injection strength. Throughout our study we take xc as the periodic solution of a Van der Pol equation. Then we have €xc eð1 x2c Þ_xc þ xc ¼ 0:

ð15Þ

In this case, the stability of the process is investigated through the following linearized equations: € n1 eð1 x2s Þn_ 1 þ ½1 þ K 1 þ K 2 þ 2exs x_ s n1 ¼ K 1 n2 þ K 2 n4 gn1 ; €2 eð1 x2 Þn_ 2 þ ½1 þ K 1 þ K 2 þ 2exs x_ s n2 ¼ K 1 n1 þ K 2 n3 ; n s € n3 eð1 x2 Þn_ 1 þ ½1 þ K 1 þ K 2 þ 2exs x_ s n3 ¼ K 1 n4 þ K 2 n2 ;

ð16Þ

s

€ n4 eð1 x2s Þn_ 1 þ ½1 þ K 1 þ K 2 þ 2exs x_ s n4 ¼ K 1 n3 þ K 2 n1 with the deviation ni = xi xc. Here, the derivation of eigenvalues can not allow one to estimate analytically the stability boundaries of the process because of the strong nonlinearity between the coupling parameters K1, K2 and the local injection strength g. Nevertheless, these boundaries are determined numerically through the resolution of Eq. (14) using the fourth-order Runge–Kutta algorithm. The synchronization criterion is still the one deﬁned by inequality (13). From our investigations, it comes that the most interesting eﬀect of a locally injected signal is to increase drastically the range of K1 where the synchronization process is achieved for a ﬁxed value of K2. Therefore, one has some modiﬁcations of the previous stability boundaries obtained without the local injection. Our results are reported on the stability map in the (K1, K2) plane (see Fig. 7) for g = 0.8. Indeed, when K2 = 0.44 for instance, the synchronization is eﬀective for K1 2 ]0.10, 0.004[ [ ]0.009, 0.435[ [ ]0.449, +1[ while without the local injection but with the same value of K2, full synchronization is obtained for K1 2 ]0.06, 0[ [ ]0, 0.438[ [ ]0.444, +1[. The inﬂuence of the local injection is more eﬀective for higher values of K2. Thus without the locally injected signal and when K2 = 0.18 for instance, synchronization is achieved for K1 2 ]0.492, 0.182[ [ ]0.176, 0.002[ [] 0.004, +1[. On the other hand, when the local injection is taken into account with the same value of K2, all the oscillators display the same motion Please cite this article in press as: Kadji HGE et al., Synchronization dynamics in a ring of four mutually ..., Commun Nonlinear Sci Numer Simul (2007), doi:10.1016/j.cnsns.2006.11.004

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D1

D2

0.5 0.4 0.3 0.2

K2

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0 K1

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 7. Eﬀects of the local injection on the stability boundaries of the synchronization process: g = 0.8. D1: Area where the synchronization process is analytically achieved but numerically unstable. D2: Area where the synchronization process is both analytically and numerically achieved.

x2- x4

x1- x4 x2- x3

x1 -x 2 x1-x3

x3- x4

4

2

0

-2

-4 0

40

80

120

160

Time(s) Fig. 8. Time history of all pair of oscillators displaying full synchronization for g = 0.80 with the parameters of Fig. 4.

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for K1 2 ]0.55, 0.185[ [ ]0.172, 0.004[ [ ]0.01, +1[. From these results, it comes that for some ranges of K2, the achievement of the synchronization process is now possible for K1 < 0.5, a situation forbidden for g = 0. Such an inﬂuence is more eﬀective in the case where K1 induces an excitatory feedback coupling (K1 < 0) while K2 leads to an inhibitory connection (K2 > 0) more than in the reverse situation (K1 > 0, K2 < 0). But when both K1 and K2 generate an inhibitory feedback coupling (K1 > 0, K2 > 0), the local injection is not really of interest. One can notice that the local injection also enables the transition from clusters states to full synchronization as shown in Fig. 8. As the local injection strength increases, the above eﬀects are still reported and become more enhanced. For example when g = 10 and K2 = 0.4, the ring is fully synchronized for K1 2 ]0.64, 0.404[ [ ]0.391, 0.004[ [ ]0.010, +1[. 5. Conclusion In this paper, we have studied the synchronization dynamics in a ring of mutually coupled biological systems without and with a locally injected signal. Without the local injection, eigenvalues properties has allowed us to derive the stability boundaries of the process. Numerical simulations have validated and complemented analytical results. When the local injection has been considered, numerical simulations permitted us to perform domains of achievement of the process. It has also been observed that a locally injected trajectory can be used to control regions of full synchronization for some speciﬁc nature of coupling, and also to achieve the transitions from clusters states to full synchronization. Acknowledgements A part of this work was done during the Victor Rothschild Fellowship visit (April–May 2006) of H.G. Enjieu Kadji at the African Institute for Mathematical Sciences (AIMS). He would like to express special thanks to AIMS. References [1] de Pinto Sandro E, Lopez Sergio R, Viana Ricardo L. Collective behavior in a chain of Van der Pol oscillators with power-law coupling. Physica A 2002;303:339–56. [2] Woafo P, Yamapi R, Chabi Orou JB. Dynamics of a nonlinear electromechanical system with multiple functions in series. Commun Nonlinear Sci Numer Simulat 2005;10:229–51. [3] Rand RH, Holmes P. Bifurcation of periodic motions in two coupled Van der Pol oscillators. Int J Non-Linear Mech 1980;15:387–99. [4] Murray JD. Mathematical biology. 3rd ed. New York: Springer; 2001. [5] Kozlowski J, Parlitz U, Lauterborn W. Bifurcation analysis of two coupled periodically driven Duﬃng oscillators. Phys Rev E 1995;51:1861–7. [6] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4; Pecora LM, Carroll TL. Driving systems with chaotic signals. Phyica A 1992;44:2374–83. [7] Oppenheim AV, Wornell GW, Isabelle SH, Cuomo K. Proceedings of the international conference on acoustics, speech and signal processing, vol. 4. New York: IEEE; 1922. p. 117. [8] Kocarev LJ, Halle KS, Eckert KS, Chua LO. Experimental demonstration of secure communications via chaotic synchronization. Int J Bifurcat Chaos 1992;2:709–13. [9] Cuomo K, Oppenheim AV. Circuit implementation of synchronized chaos with applications to communications. Phys Rev Lett 1993;71:65–8. [10] Winful HG, Rahman L. Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers. Phys Rev Lett 1990;65:1575–8. [11] Roy R, Thornburg KS. Experimental synchronization of chaotic lasers. Phys Rev Lett 1994;72:2009–12. [12] Sivaprakasam S, Shore KA. Critical signal strength for eﬀective decoding in diode laser chaotic optical communications. Phys Rev E 2000;61:5997–9. [13] Murakami A, Ohtsubo J. Chaos synchronization based on a continuous chaos control method in semiconductor lasers with optical feedback. Phys Rev E 2001;63. [14] Pecora LM, Carroll TL, Johnson GA, Mar DJ. Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos 1997;7:520–43. [15] Nakata S, Miyata T, Ojima N, Yoshikawa K. Self-synchronization in coupled salt-water oscillators. Physica D 1998;115:313–20. [16] Miyakawa K, Yamada K. Synchronization and clustering in globally coupled salt-water oscillators. Physica D 2001;151:217–27. [17] Strogatz SH, Stewart I. Coupled oscillators and biological synchronization. Sci Am 1993;269(6):102–9. [18] Reimann P, Ven den Broeck C, Kawai R. Nonequilibrium noise in coupled phase oscillators. Phys Rev E 1999;60:6402–6.

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[19] Boccaletti S, Farini A, Arecchi FT. Adaptive synchronization of chaos for secure communication. Phys Rev E 1997;55:4979–81. [20] Denis Tsygankov, Spontaneous synchronization of Josephson junctions and ﬁber lasers. PhD thesis, Georgia Institute of Technology, School of Physics, 2005. [21] Mirollo RE, Strogatz SH. Synchronization of the pulse-coupled biological oscillators. SIAM J Appl Math 1990;50:1645–62. [22] Kuramoto Y. Chemical oscillations, waves and turbulence. Berlin: Springer; 1984. [23] Robertson RM, Pearson KG. Neural circuits in the ﬂight system of the locust. J Neurophysiol 1985;53:110–28. [24] Collins JJ, Richmond SA. Hard-wired central pattern generators for quadrupedal locomotion. Biol Cybern 1994;71:371385. [25] McClellan AD. Organization of spinal locomotor networks: contributions from model systems. Comments Theor Biol 1996;4:6391. [26] Chembo Kouomou Y, Woafo P. Generalized correlated states in a ring of coupled nonlinear oscillators with a local injection. Phys Rev E 2002;66:066201. [27] Chembo Kouomou Y, Woafo P. Transitions from spatiotemporal chaos to cluster and complete synchronization states in a shiftinvariant set of coupled nonlinear oscillators. Phys Rev E 2003;67:046205. [28] Woafo P, Enjieu Kadji HG. Synchronized states in a ring of mutually coupled self-sustained electrical oscillators. Phys Rev E 2004;69:046206. [29] Kamoun P, Lavoine AH, de Verneuil H. Biochimie et biolobie moleculaire, Flammarion, Paris 2003. [30] Audigie´ Cl, Zonszin F. Biochimie me´tabolique. Paris: Dion; 1993. [31] Duke TAJ, Le Nove`re N, Bray D. Conformational spread in a ring of proteins: a stochastic approach to allostery. J Mol Biol 2001;308:541–53. [32] Bray D, Duke T. Conformational spread: the propagation of allosteric states in large multiprotein complexes. Annu Rev Biophys Biomol Struct 2004;33:53–73. [33] Kaiser F. Coherent oscillations in biological systems, I, Bifurcation phenomena and phase transitions in an enzyme–substrate reaction with ferroelectric behavior. Z Naturforsch A 1978;294:304–33. [34] Frohlich H. Long range coherence and energy storage in a biological systems. Int J Quantum Chem 1968;641:649–52. [35] Frohlich H. In: Marois, editor, Quantum mechanical concepts in biology, in theoretical physics and biology, 1969. p. 13–22. [36] Wagner A. Circuit topology and the evolution of robustness in two-gene circadian oscillators. PNAS 2005;102:11775–80. [37] Rand RH, Holmes PJ. Bifurcation of periodic motions in two weakly coupled Van der Pol oscillators. Int J Non-Linear Mech 1980;5:387–99. [38] Singer W, Gray C. Visual feature integration and the temporal correlation hypothesis. Annu Rev Neurosci 1995;18:555–86. [39] Gray C, Singer W. Stimulus-speciﬁc neuronal oscillations in the cat visual cortex. Soc Neurosci 1987;404:3. [40] Schechter B. Now the brain gets rhythm. Science 1996;274:339–40. [41] Freund HJ. Motor unit and muscle activity in voluntary motor control. Physiol Rev 1983;63:387–436. [42] Elbe R, Koller W. Tremor. Baltimore: John Hopkins University; 1990. [43] Belykh I, de Lange E, Hasler M. Synchronization of bursting neurons: what matters in the network topology. Phys Rev Lett 2005;94:188101. [44] Glass L. Synchronization and rhythmic processess in physiology. Nature 2001;410:277–84. [45] Sturis J, Knudsen C, O’Meara NM, Thomsen JS, Mosekilde E, VanCauter E, Polonsky KS. Phase-locking regions in a forced model of slow insulin and glucose oscillations. Chaos 1995;5:193–9. [46] Neiman A, Pei X, Russel D, Wojtenek W, Wilkens L, Moss F, et al. Synchronization of the noisy electrosensitive cells in the paddleﬁsh. Phys Rev Lett 1999;82:660–3. [47] Michaels DC, Matyas EP, Jalife J. Mechanisms of sinoatrial pacemaker synchronization: a new hypothesis. Circulat Res 1987;61:704–14. [48] Vadivasova TE, Strelkova GI, Anishchenko VS. Phase-frequency synchronization in a chain of periodic oscillators in the presence of noise and harmonic forcings. Phys Rev E 2001;63:036225. [49] Hu G, Xiao J, Gao J, Li X, Yao Y, Bambi H. Analytical study of spatiotemporal chaos control by applying local injections. Phys Rev E 2000;62:R3043. [50] Doiron B, Chacron MJ, Maler L, Longtin A, Bastian J. Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli. Nature 2001;421:539–43.

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