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Dynamic coupling enhances network synchronization ⁎
11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 on Nonlinear Control Systems 11th IFAC Symposium Available online at www.sciencedirect.com Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-16 (2019) 610–615
Dynamic Dynamic Dynamic Dynamic
coupling enhances network coupling enhances network coupling enhances network synchronization coupling enhances network synchronization synchronization synchronization ∗ W. de Jonge ∗∗ J. Pena Ramirez ∗∗ ∗∗ H. Nijmeijer ∗
W. de Jonge ∗∗∗ J. Pena Ramirez ∗∗ H. Nijmeijer ∗∗ ∗∗ H. Nijmeijer ∗ W. de Jonge ∗ J. Pena Ramirez ∗∗ ∗∗ ∗ W. de University Jonge J.ofPena Ramirez H. Nijmeijer ∗ Technology, Department of Mechanical ∗ Eindhoven Eindhoven University of Technology, Department of Mechanical ∗ ∗ Engineering, Box 513, 5600 MB Eindhoven, Netherlands Eindhoven P.O. University of Technology, DepartmentThe of Mechanical ∗ Netherlands Engineering, Box 513, 5600 MB Eindhoven, Eindhoven P.O. University of Technology, DepartmentThe of Mechanical (e-mail: [email protected]) Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]) ∗∗Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Physics Division. Department of Electronics (e-mail: [email protected]) ∗∗ CICESE. Applied CICESE. Applied Physics Division. Department of Electronics and and ∗∗ (e-mail: [email protected]) ∗∗ Telecommunications, Carr. Ensenada-Tijuana 3918, Zona Playitas, CICESE. Applied Physics Division. Department of Electronics and ∗∗ Telecommunications, Carr. Ensenada-Tijuana 3918, Zona Playitas, CICESE. Applied Physics Division. Department of Electronics and C.P. 22860, Ensenada, B.C. Mexico Telecommunications, Carr. Ensenada-Tijuana 3918, Zona Playitas, C.P. 22860, Ensenada, B.C. Mexico Telecommunications, Carr. Ensenada-Tijuana 3918, Zona Playitas, C.P. 22860, Ensenada, B.C. Mexico C.P. 22860, Ensenada, B.C. Mexico Abstract: Abstract: In In the the study study of of network network synchronization synchronization it it is is common common to to consider consider that that the the nodes nodes in the network interact through static and diffusive couplings. However, this type of couplings Abstract: In the study of network synchronization it is common to consider that the nodes in the network interact through static and diffusive couplings. However, this type of couplings Abstract: In the study of network synchronization it is topologies, common tothe consider that the nodes has a limitation: for certain systems and certain network maximum number of in the network interact through static and diffusive couplings. However, this type of couplings has a limitation: for certain systems and certain network topologies, the this maximum number of in the network interact through static and diffusive couplings. However, type of couplings nodes that can be synchronized is relatively small. This paper presents a potential solution has a limitation: for certain systems and certain network topologies, the maximum number of nodes that can be synchronized is relatively small. This topologies, a potential solution paper presents has a limitation: for certain systems and certain network the maximum number of for relaxing the aforementioned In particular, is demonstrated that if the static nodes that can be synchronized limitation. is relatively small. This it paper presents a potential solution for relaxing the aforementioned limitation. In particular, it is demonstrated that if the static nodes that can be synchronized is relatively small. This paper presents a potential solution couplings in the network by interconnections then number of for relaxing limitation. In particular, it is demonstrated if the static couplings in the the aforementioned network are are replaced replaced by dynamic dynamic interconnections then the the that number of nodes nodes for relaxing the aforementioned limitation. In particular, it is demonstrated that if the static that can be synchronized is increased. As particular example, a star network of Hindmarsh-Rose couplings in the network are replaced by dynamic interconnections then the number of nodes that can beinsynchronized isare increased. As particular example, a star network of number Hindmarsh-Rose couplings the network replaced by dynamic interconnections then the of nodes neurons is considered. The stability of the synchronous solution in the network is investigated that can is beconsidered. synchronized is increased. particular example, a star of Hindmarsh-Rose neurons The stability ofAs the synchronous solution in network the network is investigated that can is be synchronized is increased. particular example, a star oflargest Hindmarsh-Rose by using the Master Stability Function approach in combination with the transverse neurons considered. The stability ofAs the synchronous solution in network the network is investigated by using the Master Stability Function approach in combination with the largest transverse neurons is considered. The stability of the synchronous solution in the network is in investigated Lyapunov exponents. Ultimately, the obtained results are experimentally validated aa network by using the Master Stability Function approach in combination with the largest transverse Lyapunov exponents. Ultimately, the obtained results are experimentally validated in network by using the Master Ultimately, Stabilityneurons. Function approach in are combination with validated the largest transverse of electronic Hindmarsh-Rose Lyapunov exponents. the obtained results experimentally in a network of electronic Hindmarsh-Rose neurons. Lyapunov exponents. Ultimately, the obtained results are experimentally validated in a network of electronic Hindmarsh-Rose neurons. © electronic 2019, IFAC (International Federation of Automaticcoupling, Control) Hosting by Elsevier Ltd. All rights reserved. of Hindmarsh-Rose neurons. Keywords: Network Synchronization, Keywords: Network Synchronization, dynamic dynamic coupling, Hindmarsh-Rose Hindmarsh-Rose neuronal neuronal model model Keywords: Network Synchronization, dynamic coupling, Hindmarsh-Rose neuronal model Keywords: Network Synchronization, dynamic coupling, Hindmarsh-Rose neuronal model 1. INTRODUCTION This paper presents a novel network synchronization 1. INTRODUCTION This paper presents a novel network synchronization scheme, in all interconnections between 1. INTRODUCTION This paper presents network synchronization scheme, in which which all aathe thenovel interconnections between the the 1. INTRODUCTION This paper presents novel network synchronization nodes are dynamic rather than static. It is demonstrated scheme, in which all the interconnections between the Synchronization in networks of dynamical systems finds scheme, areindynamic rather than static. It is demonstrated which all the interconnections between the Synchronization in networks of dynamical systems finds nodes that aa network with dynamic interconnections has are dynamic than static. It is demonstrated interesting applications ranging from biology to technothat network with rather dynamic interconnections has a a better better Synchronization in networks of dynamical systems finds nodes nodes are dynamic rather than static. It is demonstrated interesting applications ranging from biology to technoperformance than aa dynamic network static para network with interconnections has aIn better Synchronization in networks of dynamical systems finds logical applications. For example, the synchronization of that than network with with static couplings. couplings. parinteresting applications to technothat a network with dynamic interconnections has aIn better logical applications. For ranging example,from the biology synchronization of performance ticular, the results in this paper illustrate and performance than a presented network with static couplings. In parinteresting applications ranging from biology to technothe brain activity in lowerand higher-order sensory areas logical applications. For example, the synchronization of ticular, the results presented in this paper illustrate and performance than a network with static couplings. In parthe brain activity in lowerand higher-order sensory areas suggest that the number of nodes that can be synchroticular, the results presented in this paper illustrate and logical applications. For example, the synchronization of and brain in cortico-limbic emotion circuits during emotional thatresults the number of nodes that canillustrate be synchrothe activity in lowerand higher-order sensory areas suggest ticular, the presented in this paper and and in cortico-limbic emotion circuits during emotional nized is larger compared to the number of nodes that suggest that the number of nodes that can be synchrothe brain activity in lowerand higher-order sensory areas situations inside the human brain (Nummenmaa et al., nized is that largerthe compared toofthe number ofcan nodes that are are and in cortico-limbic emotionbrain circuits during emotional suggest number nodes that be synchrosituations inside the human (Nummenmaa et al., synchronized the classical static coupling is is largerwhen compared to the number of nodesscheme that are and in synchronization cortico-limbic emotion circuits during emotional 2012), of the generators in power grids to nized synchronized when the classical static coupling scheme is situations inside the human brain (Nummenmaa et al., nized is largerwhen compared to the number of nodesscheme that are 2012), synchronization of the generators in power grids to synchronized used. the classical static coupling is situations insideoperation the human brain (Nummenmaa et al., keep the normal of the grid and prevent outages 2012), synchronization of the generators in power grids to used. synchronized when the classical static coupling scheme is keep the normal operation of the grid and prevent outages used. 2012), synchronization of the generators in power grids to (Wang etnormal al., 2016), and coordinated motion of vehicles, keep theet operation of the grid and prevent outages used. The dynamic couplings are described by second order (Wang al., 2016), and coordinated motion of vehicles, keep theas operation of safety the grid andhighway prevent outages The dynamic couplings are described by second order known platooning, where and capacity (Wang etnormal al., 2016), and coordinated of vehicles, linear differential equations are on the couplings areand described order known as where safety andmotion highway capacity The lineardynamic differential equations are based basedby onsecond the so-called so-called (Wang et platooning, al., by 2016), and coordinated motion of vehicles, dynamic couplings areand described by second order are increased synchronizing the and velocity, acceleration, known as platooning, where safety highway capacity The Huygens’ coupling (Pena Ramirez, 2013). These couplings linear differential equations and are based on the so-called are increased by synchronizing the and velocity, acceleration, known asand platooning, where safety highway capacity Huygens’ couplingequations (Pena Ramirez, 2013). These couplings linear differential and are based on the so-called braking, steering of the vehicles (Lefeber et al., 2017). are increased by synchronizing the velocity, acceleration, are designed such synchronization is achieved, coupling (Penawhen Ramirez, 2013). These braking, and steering of the vehicles (Lefeber acceleration, et al., 2017). Huygens’ are designed such that that when synchronization is couplings achieved, are increased by synchronizing the velocity, Huygens’ coupling (Pena Ramirez, 2013). These couplings braking, and steering ofoccur the vehicles (Lefeber etthere al., 2017). the coupling signals vanish asymptotically. are designed such that when synchronization is achieved, For synchronization to it is essential that exists the coupling signals vanish asymptotically. braking, and steering of the vehicles (Lefeber et al., 2017). are designed such that when synchronization is achieved, For synchronization to occur it is essential that there exists the coupling signals vanish asymptotically. an interconnection between the systems. For example, For synchronization to occur it is essential that there exists A star network of neurons the coupling signals vanish asymptotically. an interconnection between the systems.that For example, star network of Hindmarsh-Rose Hindmarsh-Rose neurons is is considered considered Forthe synchronization to occur it is essential exists in examples presented above, the power grids are A an interconnection between the systems. Forthere example, in the analysis and the local stability of the A star network of Hindmarsh-Rose neurons issynchronous considered in the examples presented above, the power grids are in the analysis and the local stability of the synchronous an interconnection between the systems. For example, A star network of Hindmarsh-Rose neurons considered interconnected viapresented power cables, inthe a platoon the cars in the examplesvia above,in power grids are in solution is investigated by using the well-known master the analysis and the local stability of the issynchronous interconnected power cables, a platoon the cars solution is investigated by using the well-known master in the examples presented above, power grids are in the analysis and the local stability of the synchronous communicate tovia each othercables, and with infrastructure interconnected power inthe a the platoon the cars stability function approach. Additionally, experimental solution is investigated by using the well-known master communicate tovia each othercables, and with the infrastructure stability function approach. Additionally, experimental interconnected power in a platoon the cars isfunction investigated by using the well-known master via wireless communication mechanisms, or infrastructure in the case of solution communicate to each othermechanisms, and with the results on a network of electronic Hindmarsh-Rose neurons stability approach. Additionally, experimental via wireless communication or in the case of results on function a networkapproach. of electronic Hindmarsh-Rose neurons communicate to each other and with the infrastructure stability Additionally, experimental neurons, thecommunication interaction is through chemical orthe electrical via wireless mechanisms, or in case of are provided. on a network of electronic Hindmarsh-Rose neurons neurons, thecommunication interaction is through chemical electrical via wireless mechanisms, or inor case of results are provided. results on a network of electronic Hindmarsh-Rose neurons synapses. neurons, the interaction is through chemical orthe electrical are provided. synapses. neurons, the interaction is through chemical or electrical are The paper provided. synapses. paper is is organized organized as as follows. follows. First, First, Section Section 2 2 presents presents A common type of interconnections in a network of dy- The synapses. the preliminaries and the corresponding problem statepaper is organized as follows. First, Section 2 presents A common type of interconnections in a network of dy- The the preliminaries and as thefollows. corresponding problem stateThe paper is organized First, Section 2 presents namical systems are the so-called diffusive couplings, see A common type of interconnections in a network of dyment. After that, the proposed network synchronization the preliminaries and the corresponding problem statenamical systems are the so-called diffusive couplings, see ment. After that, the proposed network synchronization A common type of interconnections in a network of dythe preliminaries and the corresponding problem statee.g. (Steur et al., 2009; Pereira et al., 2014). However, for namical systems are the so-called diffusive couplings, see scheme with dynamic couplings is introduced in Section ment. After that, the proposed network synchronization e.g. (Steur et al., 2009; Pereira et al., 2014). However, for with that, dynamic couplings is introduced in Section namical systems are thePereira so-called diffusive couplings, ment. After the proposed network synchronization certain network topologies this et type of2014). coupling seemssee to scheme e.g. (Steur et al., 2009; al., However, for 33 and local stability of synchronous in with dynamic couplings introduced solution in Section certain network topologies this et type of2014). coupling seems for to scheme and the the local stability of the the is synchronous solution in e.g. to (Steur et al., 2009; Pereira al.,the However, scheme with dynamic couplings is introduced in Section fail induce synchronization when number of nodes certain network topologies thiswhen type of coupling seems to 3the network is presented in Section 4. Next, a numerical and the local stability of the synchronous solution in fail to induce synchronization the number of nodes theand network is presented in Section 4. Next, asolution numerical certain network topologies this type of coupling seems to 3 the local stability of the synchronous in in the network is large cf. (Huang et al., 2009). fail to induce synchronization when the number of nodes study is conducted in Section 5 and the corresponding the network is presented in Section 4. Next, a numerical in the is large cf. (Huang et the al., 2009). of nodes study is conducted in Section 5 and the corresponding fail to network induce synchronization when the network is presented in Section 4. Next, a numerical in the network is large cf. (Huang et al., number 2009). experimental validation is presented in the Section 6. Finally, study is conducted in Section 5 and corresponding in the network is large cf. (Huang et al., 2009). experimental validation is presented in the Section 6. Finally, study is conducted in conclusions Section 5 and corresponding a discussion and some areingiven in Section 7. experimental validation is presented Section 6. Finally, a discussion and some conclusions are given in Section 7. experimental validation is presented Section 6. Finally, Partially supported by CONACYT Mexico a discussion and some conclusions areingiven in Section 7. Partially supported by CONACYT Mexico a discussion and some conclusions are given in Section 7. Partially supported by CONACYT Mexico
supported by CONACYT Mexico Partially Partially©supported by(International CONACYT Mexico 2405-8963 2019, IFAC IFAC Federation of Automatic Control) Copyright © 2019 1085Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 1085Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 1085 10.1016/j.ifacol.2019.12.029 Copyright © 2019 IFAC 1085
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2. PRELIMINARIES AND PROBLEM STATEMENT
HR neuron. In other words, it is assumed that matrix C in (1) is 0 1 0 (8) C= 0 0 0 . 0 0 0 Finally, the synchronous solution of interest is given by (9) x1 (t) = x2 (t) = · · · = xN (t) = s(t), where s(t) corresponds to the solution of an isolated node.
This section briefly introduces basic concepts related to networks of dynamical systems with static couplings and presents the problem formulation. The notation used throughout this paper is as follows: bold symbols are used for matrices and/or vectors, whereas plain symbols denote scalar variables. 2.1 Preliminaries Consider a network of dynamical systems of the form (1) x˙ i = f (xi ) + ui , i n where x ∈ R is the state vector corresponding to node i, function f : Rn → Rn describes the isolated node dynamics, and the (control) input ui ∈ Rn is given by ui = −k
N
Gij Cxj ,
(2)
j=1
where Gij determines the strength of interaction between nodes i and j, the matrix C ∈ Rn×n describes which variables are used in the coupling, and k is the overall coupling strength. By defining 1 x x2 x= . ..
, F =
f (x1 ) f (x2 ) .. .
, and G = [Gij ] ,
(3)
xN f (xN ) the network (1)-(2) can be written in the compact form x˙ = F (x) − kG ⊗ Cx, (4) where G is the Laplacian matrix and the symbol ⊗ denotes the Kronecker product. Throughout this paper, it is considered that the isolated node dynamics are described by the Hindmarsh-Rose (HR) neuronal model, which is described by (Hindmarsh and Rose, 1984); xi yi + bx2i − ax3i − zi + I, i i 2 , x = yi , (5) f (x ) = c − dxi − yi , zi r [s(xi − x0 ) − zi ] .
where a, b, c, r ∈ R+ , x0 ∈ R− . The membrane potential is represented by xi , whereas yi and zi represent internal states. On the other hand, I is the bifurcation parameter and plays an essential role in determining the dynamic behavior of the neuron. Furthermore, the following assumptions are considered. A-1 The networks of HR neurons has a star network topology such that the Laplacian matrix G is given by N − 1 −1 −1 · · · −1 −1 1 0 · · · 0 −1 0 1 · · · 0 , G= (6) . .. .. . . .. .. . . . . −1 0 · · · 0 1 which has the following eigenvalues σ1 = 0, σk = 1, for k = 2, N − 1, and σN = N. (7) A-2 The neurons interact through the yi variable and the control input (2) only affects the xi -equation of the i
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2.2 Problem statement A network of HR neurons described by (4)-(5) with G and C as given in (6) and (8) respectively, has the following limitation: the synchronous solution (9) becomes unstable when the number of nodes N in the network is relatively large. For example, for the parameter values a = 1, b = 3, c = 1, d = 5, s = 4, r = 0.005, x0 = 0.005, I = 3.25 the maximum number of nodes that can be synchronized in the network is 4, cf. (Huang et al., 2009). This work elaborates on a potential solution to relax this limitation. Specifically, it is demonstrated that if the static coupling between the nodes, see Eq. (2) is replaced by a suitably designed second order dynamic coupling, then the number of nodes that can be synchronized in the network is increased. 3. PROPOSED NETWORK WITH DYNAMIC COUPLINGS This section introduces a novel network synchronization scheme in which all the interactions between the nodes is dynamic rather than static. The proposed network with dynamic interconnections has been inspired in our previous work (Pena Ramirez et al., 2018) and is described by x˙ i = f (xi ) + B 1 hi , i h˙ = Ahi − k
N
Gij B 2 xj ,
(10) i = 1, 2, . . . , N, (11)
j=1
where hi ∈ R2 is the state vector of the dynamic coupling, B 1 ∈ Rn×2 and B 2 ∈ R2×n are coupling vectors, which are assumed to only have one entry equal to one and all the other entries of these vectors are zero. The assumption on B 1 indicates that only one equation of the dynamical system will be directly affected by the coupling and likewise, the choice of B 2 comes from the assumption that only one state variable is available for measurement at node i. Finally, A ∈ R2×2 is a matrix containing the “control parameters” of the dynamic coupling and, as stated before, is based on the Huygens’ coupling. This matrix is given by −α 1 , (12) A= −γ1 −γ2 where α, γ1 , and γ2 are positive parameters to be chosen.
Note that the main difference between the results presented here and our previous work (Pena Ramirez et al., 2018) is that here the focus is in networks.
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Next, the following synchronization errors are defined (19) ei = xi − s, h1 , i = 1, . . . , N. Note that all the states hi , i = 1, 2, . . . , N , corresponding to the coupling systems, are considered as “errors”. This comes from the fact that, when synchronized, the coupling signals will (asymptotically) vanish. Then, using Eqs. (17),(18) and Eq. (16), yields the following synchronization error dynamics Fig. 1. The concept of Super-Node. The original nodes, given by x˙ i = f (xi ) are ‘extended’ by adding the hi dynamics. As a consequence, the dimension of each Super-Node is n + 2. Furthermore, note that the interaction between the Super-Nodes is via the hi variables. By defining x, F , and G, as given in (3), and h = [h1 , h2 , . . . , hN ]T , the above set of equations can be written in the following compact form x˙ = F (x) + I N ⊗ B 1 h, h˙ = I N ⊗ Ah − kG ⊗ B 2 x,
(13) (14)
where I N ∈ RN ×N .
This equation describes a network consisting of N nodes interacting via dynamic couplings. The asymptotic synchronous solution in the network (13) is given by lim x1 = x2 = · · · = xN = s(t),
t→∞
and
(15) 1
2
lim h = h = · · · = h
t→∞ 2
N
= O,
where O ∈ R is a vector, which entries are all zero and s(t) is the solution of an isolated node, i.e. s˙ = f (s), (16) Remark 1. Note that in the proposed network with dynamic couplings every node, which is originally described by x˙ i = f (xi ) is converted into a ‘Super-Node’ by adding the second order dynamic coupling, see Eq. (11). Therefore the dimension of each Super-Node is n + 2. The concept of the Super-Node is schematically depicted in Figure 1. Remark 2. The dynamic couplings (11) have been designed such that when synchronization is achieved, the coupling signals hi vanishes asymptotically. As a consequence, the synchronous solution corresponds to a solution of an isolated node, see Eq. (15).
∂f (s) i e + B 1 hi , ∂xi N i h˙ = Ahi − k Gij B 2 ej . e˙ i =
First, Eqs. (10)-(11) are linearized around the synchronous solution (15). This yields ∂f (s) i (x − s) + B 1 hi , ∂xi N i h˙ = Ahi − k Gij B 2 (xj − s). x˙ i = f (s) +
(21)
j=1
It is important to note that, since by assumption, all the nodes in the network are identical, it holds that all the Jacobians of function f , evaluated on the synchronous solution s(t) are identical, i.e. ∂f (s) ∂f (s) ∂f (s) = = ... = . (22) 1 2 ∂x ∂x ∂xN This fact, allows to write the error dynamics in the following compact form
e˙ I N ⊗ Df (s) I N ⊗ B 1 e = , (23) −kG ⊗ B 2 I N ⊗ A h h˙ where Df (s) is the Jacobian of function f , which describes the dynamics of an isolated node, evaluated on the synchronous solution s, see (16). Furthermore, e = [e1 , e2 , . . . , eN ]T and h = [h1 , h2 , . . . , hN ]T . As a next step, the following transformation is defined
:=
=
e h
V ⊗ I n O1 O2 V ⊗ I 2
¯ e ¯ , h
(24)
where V ∈ RN ×N is a matrix which columns are the eigenvectors of the Laplacian matrix G, I n ∈ Rn×n and I 2 ∈ R2×2 are identity matrices, and O 1 ∈ RnN ×2N and O 2 ∈ R2N ×nN are zero matrices. Since G is a symmetric matrix, the inverse of the proposed transformation (24) always exits and as a consequence the proposed transformation is valid. ¯ Eq. (23) takes the form In the new coordinates (¯ e, h),
4. LOCAL STABILITY ANALYSIS The stability of the synchronous solution (15) is investigated by using the Master Stability Function approach.
(20)
¯e˙ ˙ ¯ h
I N ⊗ Df (s) I ⊗ B 1 −kΓ ⊗ B 2 I N ⊗ A
¯ e ¯ , h
(25)
where I N ∈ RN ×N is the identity matrix and Γ ∈ RN ×N is a diagonal matrix containing the eigenvalues σi , i = 1, 2, . . . , N , of the connectivity matrix G. It should be noted that Eq. (25) is a block-diagonal matrix, composed by N blocks of the form
(17) (18)
¯e˙ i ¯˙ i h
i ¯ e Df (s) B 1 = ¯i , −kσi B 2 A h
˜ A
j=1
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Finally, the stability of the synchronous solution (15) is determined by computing the largest transverse Lyapunov exponent λ⊥max for each variational equation (26), as i
5
4
(27)
N
i i ˜ (t) − ln e ˜ (0) ln e , = lim t→∞ t ¯ i ]T . ˜i = [¯ ei , h where e λ⊥max i
Then, the following conditions for the local stability of the synchronous solution (15) are obtained • If λ⊥max < 0, for all i = 2, , . . . , N , the synchronous i solution is locally stable. • If λ⊥max > 0, for at least one i = 2, , . . . , N , the i synchronous solution is unstable. 5. NUMERICAL RESULTS This section presents some numerical results to illustrate the performance of the network with dynamic couplings. As a particular example, a star network of HindmarshRose neurons is considered. First, the case where the HR neurons interact through static couplings is considered. Consequently, the network dynamics are described by Eq. (4) with f (xi ), G, and C as given in (5), (6), and (8) respectively. The following parameter values are considered for the HR neurons: a = 1, b = 3, c = 1, d = 5, s = 4, x0 = −1.6, r = 0.005, I = 3.25. For these values, the limit behavior of the isolated HR neurons is chaotic bursting (Hindmarsh and Rose, 1984). According to (Huang et al., 2009), the variational equation for this network with static couplings is given by 2bxs − 3ax2s 1 − kσi −1 (28) e˙ i = −2dxs −1 0 ei , rs 0 −r where ei = xi (t) − s(t) and σi are the eigenvalues of the Laplacian matrix G.
As a next step, the largest transverse Lyapunov exponents λ⊥max , for i = 2, . . . , N are computed from the variational i equation (28) using the Wolf algorithm (Wolf et al., 1985). The obtained results are depicted in Figure 2. In the gray area it holds that λ⊥max < 0 and consequently, in i this region the synchronous solution (9) is locally stable, whereas in the white area λ⊥max > 0 and therefore, in i this region of the (k, N )-plane the synchronous solution (9) is unstable, i.e. synchronization is not guaranteed. Ultimately, from this figure it is clear that synchronization can be induced in the network for a narrow interval of coupling strength (0.35 < k < 0.55) and the maximum number of nodes than can be synchronized is N = 4. Next, the performance of the proposed network synchronization scheme with dynamic couplings is investigated. In this case, the network is described by Eq. (13) with f (xi ) as given in (5), Laplacian matrix G as given in (6). The considered parameter values for the HR neurons are as described above for the static network example and the parameter values of the entries of matrix A corresponding to the dynamic coupling, see (11), are chosen as α1 = 5, γ1 = 3, and γ2 = 3k. (29)
613
3 0
0.2
0.4
k
0.6
0.8
1
Fig. 2. Largest transverse Lyapunov exponent λ⊥max , for i i = 2, . . . , N computed as a function of the coupling strength k and the number of nodes N for a network with static couplings. Gray area: λ⊥max < 0. White i area: λ⊥max > 0. Clearly, the maximum number of i nodes that can be synchronized with static couplings is N = 4. Furthermore, following assumption [A − 2], see Section 2 and matrix C given in (8), it is assumed that the dynamic coupling is applied to the xi -equation of each HR neuron and that only the yi variable is available for measurement. Consequently, the coupling vectors B 1 and B 2 in (13)-(14) are chosen as follows 0 1 0 0 0 . (30) B1 = 0 0 , B2 = 0 1 0 0 0 For this network, the corresponding variational equation, is as given in (26) with 1 2bxs − 3ax2s 1 −1 0 −2dxs −1 0 0 0 ˜ = (31) A rs 0 −r 0 0 , 0 0 0 −α 1 0 −kσi 0 −γ1 −γ2
where xs is the x variable of an isolated node, see (16) and σi are the eigenvalues of the connectivity matrix G and are as given in (7).
Again, the largest transverse Lyapunov exponents λ⊥max , i for i = 2, ...N (27) are computed—as a function of the couplings strength k and the number of nodes N in the network—from the variational equation (26) by using the Wolf algorithm. The obtained results are shown in Figure 3. In the gray region it holds that λ⊥max < 0, and therefore the syni chronous solution (15) is locally stable, whereas in the white regions λ⊥max > 0 and consequently the syni chronous solution (15) is unstable. Fom this figure it can be seen that the maximum number of nodes that can be synchronized is N = 13. Remark 3. From a quick comparison between the results presented in Figure 2 for a network with static couplings and the results depicted in Figure 3 for the proposed network with dynamic couplings it becomes evident that the synchronizability in the network is enhanced when using dynamic couplings, in the sense that a larger number of nodes can be synchronized. Furthermore, it is also important to note that a smaller coupling strength may be required for inducing synchronization when dynamic
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20
N
15 10 5 0
0.2
0.4
k
0.6
0.8
1
Fig. 4. Experimental set-up at CICESE.
, Fig. 3. Largest transverse Lyapunov exponent λ⊥max i for i = 2, . . . , N computed as a function of the coupling strength k and the number of nodes N for the proposed network with dynamic couplings. Gray area: λ⊥max < 0. White area: λ⊥max > 0. In this case, the i i maximum number of nodes that can be synchronized is N = 13. couplings are used. For the example at hand, the minimum value of the coupling strength k for which synchronization is induced in a network with static couplings is k > 0.35, as shown in Figure 2, whereas for the case of a network with dynamic couplings, the minimum coupling strength is k > 0.04, see Figure 3. This feature may be attractive from an energetic viewpoint in the sense that a smaller coupling strength implies less consumption of energy to synchronize the network. Also, we have observed that the coupling strength k has a strong influence in the convergence speed to the synchronization manifold: the convergence is slow when k is small and is faster for larger k.
dynamically coupled neurons is presented. From this figure it is clear to see that the neurons achieve synchronization. Furthermore, the experimental results are in good agreement with the numerical results shown in Figure 5a. The main difference is that in the experiment, perfect synchronization is not achieved due to the unavoidable small differences between the electronic neurons. However, the synchronization errors e = y1 −yi , i = 2, . . . , N remain small, as shown in Figure 5c.
6. EXPERIMENTAL RESULTS
(a) Simulation.
0 -2 -4 -6 -8 7000
7200
7400
7600
7800
8000
0
This section presents the experimental set-up that is used and the corresponding experimental results.
-2
6.1 Experimental set-up
-4
The experimental set-up is depicted in Figure 4 and consists of three parts: 1) ten electronic HR neurons, which design and construction is explained in our previous work (Velasco Equihua and Pena Ramirez, 2018), 2) a dSpace data acquisition card with a sampling rate of 10KHz, which measures the signals from the electronic neurons and sends back the corresponding dynamic coupling signals, 3) a computer, in which the dynamic couplings are implemented in the software MATLAB.
-6 -8 0
0.2
0.4
0.6
0.8
1
(b) Experiment.
0.1 0.05
6.2 Experiments 0
The experimental set-up is adjusted such that it mimics the network dynamics (13),(14) with connectivity matrix (6). The parameter values are the same as those used for generating Figure 3 and the coupling strength is fixed to k = 1. For this value of k it is expected from the results presented in Figure 3 that the network achieves the synchronous solution (15). The obtained experimental results are shown in Figure 5b, where the time series corresponding to the y-state of ten
-0.05 -0.1 -0.15 0
0.2
0.4
0.6
0.8
1
(c) Synchronization errors.
Fig. 5. Synchronization in a star network of 10 neurons.
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7. DISCUSSION AND CONCLUSIONS
REFERENCES
A network synchronization scheme, in which the nodes interact through a second order dynamic coupling has been presented. The obtained results suggest that dynamic couplings allow increasing the number of nodes in networks for which a static coupling induces synchronization for a limited number of nodes.
Hindmarsh, J. and Rose, R. (1984). A model for neuronal bursting using three coupled differential equations. Proceedings of the Royal Society of London, 221(1222), 87– 102. Huang, L., Chen, Q., Lai, Y.C., and Pecora, L.M. (2009). Generic behavior of master-stability functions in coupled nonlinear dynamical systems. Phys. Rev. E, 80, 036204. Lefeber, E., Ploeg, J., and Nijmeijer, H. (2017). A spatial approach to control of platooning vehicles: Separating path-following from tracking. IFAC-PapersOnLine, 50(1), 15000 – 15005. 20th IFAC World Congress. Nummenmaa, L., Glerean, E., Viinikainen, M., J¨a¨askel¨ainen, I.P., Hari, R., and Sams, M. (2012). Emotions promote social interaction by synchronizing brain activity across individuals. Proceedings of the National Academy of Sciences, 109(24), 9599–9604. Pena Ramirez, J. (2013). Huygens’ synchronization of dynamical systems: beyond pendulum clocks. Ph.D. thesis, Department of Mechanical Engineering. Pena Ramirez, J., Arellano-Delgado, A., and Nijmeijer, H. (2018). Enhancing master-slave synchronization: The effect of using a dynamic coupling. Phys. Rev. E, 98, 012208. Pereira, T., Eldering, J., Rasmussen, M., and Veneziani, A. (2014). Towards a theory for diffusive coupling functions allowing persistent synchronization. Nonlinearity, 27(3), 501–525. Steur, E., Tyukin, I., and Nijmeijer, H. (2009). Semipassivity and synchronization of diffusively coupled neuronal oscillators. Physica D: Nonlinear Phenomena, 238(21), 2119 – 2128. Velasco Equihua, G. and Pena Ramirez, J. (2018). Synchronization of Hindmarsh-Rose neurons via Huygenslike coupling. IFAC-PapersOnLine, 51(33), 186 – 191. 5th IFAC Conference on Analysis and Control of Chaotic Systems CHAOS 2018. Wang, B., Suzuki, H., and Aihara, K. (2016). Enhancing synchronization stability in a multi-area power grid. Scientific Reports, 6, 26596(1–11). Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A. (1985). Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3), 285 – 317.
For the particular case of a star network of HindmarshRose neurons, the number of nodes that can be synchronized is increased in almost three times when dynamic couplings are used instead of the classical couplings. Moreover, for other topologies like ring and all-to-all topologies (these results are not presented in this document) we have observed a similar increment in the number of synchronized nodes. However, it should be noted that, although the number of nodes is considerably increased using dynamic couplings, there still exists an upper limit in the number of nodes. Consequently, it would be interesting to investigate how to further modify the dynamic coupling such that synchronization is induced for an arbitrarily large number of nodes. On the other hand, the stability of the synchronous solution (15) in network (13)-(14) has been investigated by using a weak approach, namely the Master Stability Function. Hence, further studies are required in order to derive stronger stability results perhaps using Lyapunov stability theory. As a first step on this direction, one may consider the use of the Lyapunov theory for perturbed systems in order to tune the parameters of the dynamic coupling, see Eq. (12). Furthermore, a formal analysis is still necessary in order to demonstrate that the number of nodes in a network of dynamical systems can be increased when the static interconnections are replaced by dynamic couplings. This is in fact the topic of our ongoing research. It should also be emphasized that in the present work we have considered a second order dynamic coupling. As explained in the introductory section, this choice obeys to the fact that such coupling mimics the coupling used by Huygens to synchronize his pendulum clocks. However, it would be interesting to explore the influence of the order in the dynamic coupling on the onset of synchronization, i.e. to conduct an study for the general case where the coupling has order n. Finally, it should be noted that the obtained experimental results suggest that the proposed network synchronization scheme has certain degree of robustness. ACKNOWLEDGEMENTS The second author acknowledges the support from the mexican Council for Science and Technology CONACYT under project C´ atedra CONACYT No 888: “Sincronizaci´ on de sistemas din´ amicos interconectados: aplicaciones en f´ısica, biolog´ıa e ingenier´ıa.”
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Dynamic Dynamic Dynamic Dynamic
coupling enhances network coupling enhances network coupling enhances network synchronization coupling enhances network synchronization synchronization synchronization ∗ W. de Jonge ∗∗ J. Pena Ramirez ∗∗ ∗∗ H. Nijmeijer ∗
W. de Jonge ∗∗∗ J. Pena Ramirez ∗∗ H. Nijmeijer ∗∗ ∗∗ H. Nijmeijer ∗ W. de Jonge ∗ J. Pena Ramirez ∗∗ ∗∗ ∗ W. de University Jonge J.ofPena Ramirez H. Nijmeijer ∗ Technology, Department of Mechanical ∗ Eindhoven Eindhoven University of Technology, Department of Mechanical ∗ ∗ Engineering, Box 513, 5600 MB Eindhoven, Netherlands Eindhoven P.O. University of Technology, DepartmentThe of Mechanical ∗ Netherlands Engineering, Box 513, 5600 MB Eindhoven, Eindhoven P.O. University of Technology, DepartmentThe of Mechanical (e-mail: [email protected]) Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]) ∗∗Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Physics Division. Department of Electronics (e-mail: [email protected]) ∗∗ CICESE. Applied CICESE. Applied Physics Division. Department of Electronics and and ∗∗ (e-mail: [email protected]) ∗∗ Telecommunications, Carr. Ensenada-Tijuana 3918, Zona Playitas, CICESE. Applied Physics Division. Department of Electronics and ∗∗ Telecommunications, Carr. Ensenada-Tijuana 3918, Zona Playitas, CICESE. Applied Physics Division. Department of Electronics and C.P. 22860, Ensenada, B.C. Mexico Telecommunications, Carr. Ensenada-Tijuana 3918, Zona Playitas, C.P. 22860, Ensenada, B.C. Mexico Telecommunications, Carr. Ensenada-Tijuana 3918, Zona Playitas, C.P. 22860, Ensenada, B.C. Mexico C.P. 22860, Ensenada, B.C. Mexico Abstract: Abstract: In In the the study study of of network network synchronization synchronization it it is is common common to to consider consider that that the the nodes nodes in the network interact through static and diffusive couplings. However, this type of couplings Abstract: In the study of network synchronization it is common to consider that the nodes in the network interact through static and diffusive couplings. However, this type of couplings Abstract: In the study of network synchronization it is topologies, common tothe consider that the nodes has a limitation: for certain systems and certain network maximum number of in the network interact through static and diffusive couplings. However, this type of couplings has a limitation: for certain systems and certain network topologies, the this maximum number of in the network interact through static and diffusive couplings. However, type of couplings nodes that can be synchronized is relatively small. This paper presents a potential solution has a limitation: for certain systems and certain network topologies, the maximum number of nodes that can be synchronized is relatively small. This topologies, a potential solution paper presents has a limitation: for certain systems and certain network the maximum number of for relaxing the aforementioned In particular, is demonstrated that if the static nodes that can be synchronized limitation. is relatively small. This it paper presents a potential solution for relaxing the aforementioned limitation. In particular, it is demonstrated that if the static nodes that can be synchronized is relatively small. This paper presents a potential solution couplings in the network by interconnections then number of for relaxing limitation. In particular, it is demonstrated if the static couplings in the the aforementioned network are are replaced replaced by dynamic dynamic interconnections then the the that number of nodes nodes for relaxing the aforementioned limitation. In particular, it is demonstrated that if the static that can be synchronized is increased. As particular example, a star network of Hindmarsh-Rose couplings in the network are replaced by dynamic interconnections then the number of nodes that can beinsynchronized isare increased. As particular example, a star network of number Hindmarsh-Rose couplings the network replaced by dynamic interconnections then the of nodes neurons is considered. The stability of the synchronous solution in the network is investigated that can is beconsidered. synchronized is increased. particular example, a star of Hindmarsh-Rose neurons The stability ofAs the synchronous solution in network the network is investigated that can is be synchronized is increased. particular example, a star oflargest Hindmarsh-Rose by using the Master Stability Function approach in combination with the transverse neurons considered. The stability ofAs the synchronous solution in network the network is investigated by using the Master Stability Function approach in combination with the largest transverse neurons is considered. The stability of the synchronous solution in the network is in investigated Lyapunov exponents. Ultimately, the obtained results are experimentally validated aa network by using the Master Stability Function approach in combination with the largest transverse Lyapunov exponents. Ultimately, the obtained results are experimentally validated in network by using the Master Ultimately, Stabilityneurons. Function approach in are combination with validated the largest transverse of electronic Hindmarsh-Rose Lyapunov exponents. the obtained results experimentally in a network of electronic Hindmarsh-Rose neurons. Lyapunov exponents. Ultimately, the obtained results are experimentally validated in a network of electronic Hindmarsh-Rose neurons. © electronic 2019, IFAC (International Federation of Automaticcoupling, Control) Hosting by Elsevier Ltd. All rights reserved. of Hindmarsh-Rose neurons. Keywords: Network Synchronization, Keywords: Network Synchronization, dynamic dynamic coupling, Hindmarsh-Rose Hindmarsh-Rose neuronal neuronal model model Keywords: Network Synchronization, dynamic coupling, Hindmarsh-Rose neuronal model Keywords: Network Synchronization, dynamic coupling, Hindmarsh-Rose neuronal model 1. INTRODUCTION This paper presents a novel network synchronization 1. INTRODUCTION This paper presents a novel network synchronization scheme, in all interconnections between 1. INTRODUCTION This paper presents network synchronization scheme, in which which all aathe thenovel interconnections between the the 1. INTRODUCTION This paper presents novel network synchronization nodes are dynamic rather than static. It is demonstrated scheme, in which all the interconnections between the Synchronization in networks of dynamical systems finds scheme, areindynamic rather than static. It is demonstrated which all the interconnections between the Synchronization in networks of dynamical systems finds nodes that aa network with dynamic interconnections has are dynamic than static. It is demonstrated interesting applications ranging from biology to technothat network with rather dynamic interconnections has a a better better Synchronization in networks of dynamical systems finds nodes nodes are dynamic rather than static. It is demonstrated interesting applications ranging from biology to technoperformance than aa dynamic network static para network with interconnections has aIn better Synchronization in networks of dynamical systems finds logical applications. For example, the synchronization of that than network with with static couplings. couplings. parinteresting applications to technothat a network with dynamic interconnections has aIn better logical applications. For ranging example,from the biology synchronization of performance ticular, the results in this paper illustrate and performance than a presented network with static couplings. In parinteresting applications ranging from biology to technothe brain activity in lowerand higher-order sensory areas logical applications. For example, the synchronization of ticular, the results presented in this paper illustrate and performance than a network with static couplings. In parthe brain activity in lowerand higher-order sensory areas suggest that the number of nodes that can be synchroticular, the results presented in this paper illustrate and logical applications. For example, the synchronization of and brain in cortico-limbic emotion circuits during emotional thatresults the number of nodes that canillustrate be synchrothe activity in lowerand higher-order sensory areas suggest ticular, the presented in this paper and and in cortico-limbic emotion circuits during emotional nized is larger compared to the number of nodes that suggest that the number of nodes that can be synchrothe brain activity in lowerand higher-order sensory areas situations inside the human brain (Nummenmaa et al., nized is that largerthe compared toofthe number ofcan nodes that are are and in cortico-limbic emotionbrain circuits during emotional suggest number nodes that be synchrosituations inside the human (Nummenmaa et al., synchronized the classical static coupling is is largerwhen compared to the number of nodesscheme that are and in synchronization cortico-limbic emotion circuits during emotional 2012), of the generators in power grids to nized synchronized when the classical static coupling scheme is situations inside the human brain (Nummenmaa et al., nized is largerwhen compared to the number of nodesscheme that are 2012), synchronization of the generators in power grids to synchronized used. the classical static coupling is situations insideoperation the human brain (Nummenmaa et al., keep the normal of the grid and prevent outages 2012), synchronization of the generators in power grids to used. synchronized when the classical static coupling scheme is keep the normal operation of the grid and prevent outages used. 2012), synchronization of the generators in power grids to (Wang etnormal al., 2016), and coordinated motion of vehicles, keep theet operation of the grid and prevent outages used. The dynamic couplings are described by second order (Wang al., 2016), and coordinated motion of vehicles, keep theas operation of safety the grid andhighway prevent outages The dynamic couplings are described by second order known platooning, where and capacity (Wang etnormal al., 2016), and coordinated of vehicles, linear differential equations are on the couplings areand described order known as where safety andmotion highway capacity The lineardynamic differential equations are based basedby onsecond the so-called so-called (Wang et platooning, al., by 2016), and coordinated motion of vehicles, dynamic couplings areand described by second order are increased synchronizing the and velocity, acceleration, known as platooning, where safety highway capacity The Huygens’ coupling (Pena Ramirez, 2013). These couplings linear differential equations and are based on the so-called are increased by synchronizing the and velocity, acceleration, known asand platooning, where safety highway capacity Huygens’ couplingequations (Pena Ramirez, 2013). These couplings linear differential and are based on the so-called braking, steering of the vehicles (Lefeber et al., 2017). are increased by synchronizing the velocity, acceleration, are designed such synchronization is achieved, coupling (Penawhen Ramirez, 2013). These braking, and steering of the vehicles (Lefeber acceleration, et al., 2017). Huygens’ are designed such that that when synchronization is couplings achieved, are increased by synchronizing the velocity, Huygens’ coupling (Pena Ramirez, 2013). These couplings braking, and steering ofoccur the vehicles (Lefeber etthere al., 2017). the coupling signals vanish asymptotically. are designed such that when synchronization is achieved, For synchronization to it is essential that exists the coupling signals vanish asymptotically. braking, and steering of the vehicles (Lefeber et al., 2017). are designed such that when synchronization is achieved, For synchronization to occur it is essential that there exists the coupling signals vanish asymptotically. an interconnection between the systems. For example, For synchronization to occur it is essential that there exists A star network of neurons the coupling signals vanish asymptotically. an interconnection between the systems.that For example, star network of Hindmarsh-Rose Hindmarsh-Rose neurons is is considered considered Forthe synchronization to occur it is essential exists in examples presented above, the power grids are A an interconnection between the systems. Forthere example, in the analysis and the local stability of the A star network of Hindmarsh-Rose neurons issynchronous considered in the examples presented above, the power grids are in the analysis and the local stability of the synchronous an interconnection between the systems. For example, A star network of Hindmarsh-Rose neurons considered interconnected viapresented power cables, inthe a platoon the cars in the examplesvia above,in power grids are in solution is investigated by using the well-known master the analysis and the local stability of the issynchronous interconnected power cables, a platoon the cars solution is investigated by using the well-known master in the examples presented above, power grids are in the analysis and the local stability of the synchronous communicate tovia each othercables, and with infrastructure interconnected power inthe a the platoon the cars stability function approach. Additionally, experimental solution is investigated by using the well-known master communicate tovia each othercables, and with the infrastructure stability function approach. Additionally, experimental interconnected power in a platoon the cars isfunction investigated by using the well-known master via wireless communication mechanisms, or infrastructure in the case of solution communicate to each othermechanisms, and with the results on a network of electronic Hindmarsh-Rose neurons stability approach. Additionally, experimental via wireless communication or in the case of results on function a networkapproach. of electronic Hindmarsh-Rose neurons communicate to each other and with the infrastructure stability Additionally, experimental neurons, thecommunication interaction is through chemical orthe electrical via wireless mechanisms, or in case of are provided. on a network of electronic Hindmarsh-Rose neurons neurons, thecommunication interaction is through chemical electrical via wireless mechanisms, or inor case of results are provided. results on a network of electronic Hindmarsh-Rose neurons synapses. neurons, the interaction is through chemical orthe electrical are provided. synapses. neurons, the interaction is through chemical or electrical are The paper provided. synapses. paper is is organized organized as as follows. follows. First, First, Section Section 2 2 presents presents A common type of interconnections in a network of dy- The synapses. the preliminaries and the corresponding problem statepaper is organized as follows. First, Section 2 presents A common type of interconnections in a network of dy- The the preliminaries and as thefollows. corresponding problem stateThe paper is organized First, Section 2 presents namical systems are the so-called diffusive couplings, see A common type of interconnections in a network of dyment. After that, the proposed network synchronization the preliminaries and the corresponding problem statenamical systems are the so-called diffusive couplings, see ment. After that, the proposed network synchronization A common type of interconnections in a network of dythe preliminaries and the corresponding problem statee.g. (Steur et al., 2009; Pereira et al., 2014). However, for namical systems are the so-called diffusive couplings, see scheme with dynamic couplings is introduced in Section ment. After that, the proposed network synchronization e.g. (Steur et al., 2009; Pereira et al., 2014). However, for with that, dynamic couplings is introduced in Section namical systems are thePereira so-called diffusive couplings, ment. After the proposed network synchronization certain network topologies this et type of2014). coupling seemssee to scheme e.g. (Steur et al., 2009; al., However, for 33 and local stability of synchronous in with dynamic couplings introduced solution in Section certain network topologies this et type of2014). coupling seems for to scheme and the the local stability of the the is synchronous solution in e.g. to (Steur et al., 2009; Pereira al.,the However, scheme with dynamic couplings is introduced in Section fail induce synchronization when number of nodes certain network topologies thiswhen type of coupling seems to 3the network is presented in Section 4. Next, a numerical and the local stability of the synchronous solution in fail to induce synchronization the number of nodes theand network is presented in Section 4. Next, asolution numerical certain network topologies this type of coupling seems to 3 the local stability of the synchronous in in the network is large cf. (Huang et al., 2009). fail to induce synchronization when the number of nodes study is conducted in Section 5 and the corresponding the network is presented in Section 4. Next, a numerical in the is large cf. (Huang et the al., 2009). of nodes study is conducted in Section 5 and the corresponding fail to network induce synchronization when the network is presented in Section 4. Next, a numerical in the network is large cf. (Huang et al., number 2009). experimental validation is presented in the Section 6. Finally, study is conducted in Section 5 and corresponding in the network is large cf. (Huang et al., 2009). experimental validation is presented in the Section 6. Finally, study is conducted in conclusions Section 5 and corresponding a discussion and some areingiven in Section 7. experimental validation is presented Section 6. Finally, a discussion and some conclusions are given in Section 7. experimental validation is presented Section 6. Finally, Partially supported by CONACYT Mexico a discussion and some conclusions areingiven in Section 7. Partially supported by CONACYT Mexico a discussion and some conclusions are given in Section 7. Partially supported by CONACYT Mexico
supported by CONACYT Mexico Partially Partially©supported by(International CONACYT Mexico 2405-8963 2019, IFAC IFAC Federation of Automatic Control) Copyright © 2019 1085Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 1085Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 1085 10.1016/j.ifacol.2019.12.029 Copyright © 2019 IFAC 1085
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2. PRELIMINARIES AND PROBLEM STATEMENT
HR neuron. In other words, it is assumed that matrix C in (1) is 0 1 0 (8) C= 0 0 0 . 0 0 0 Finally, the synchronous solution of interest is given by (9) x1 (t) = x2 (t) = · · · = xN (t) = s(t), where s(t) corresponds to the solution of an isolated node.
This section briefly introduces basic concepts related to networks of dynamical systems with static couplings and presents the problem formulation. The notation used throughout this paper is as follows: bold symbols are used for matrices and/or vectors, whereas plain symbols denote scalar variables. 2.1 Preliminaries Consider a network of dynamical systems of the form (1) x˙ i = f (xi ) + ui , i n where x ∈ R is the state vector corresponding to node i, function f : Rn → Rn describes the isolated node dynamics, and the (control) input ui ∈ Rn is given by ui = −k
N
Gij Cxj ,
(2)
j=1
where Gij determines the strength of interaction between nodes i and j, the matrix C ∈ Rn×n describes which variables are used in the coupling, and k is the overall coupling strength. By defining 1 x x2 x= . ..
, F =
f (x1 ) f (x2 ) .. .
, and G = [Gij ] ,
(3)
xN f (xN ) the network (1)-(2) can be written in the compact form x˙ = F (x) − kG ⊗ Cx, (4) where G is the Laplacian matrix and the symbol ⊗ denotes the Kronecker product. Throughout this paper, it is considered that the isolated node dynamics are described by the Hindmarsh-Rose (HR) neuronal model, which is described by (Hindmarsh and Rose, 1984); xi yi + bx2i − ax3i − zi + I, i i 2 , x = yi , (5) f (x ) = c − dxi − yi , zi r [s(xi − x0 ) − zi ] .
where a, b, c, r ∈ R+ , x0 ∈ R− . The membrane potential is represented by xi , whereas yi and zi represent internal states. On the other hand, I is the bifurcation parameter and plays an essential role in determining the dynamic behavior of the neuron. Furthermore, the following assumptions are considered. A-1 The networks of HR neurons has a star network topology such that the Laplacian matrix G is given by N − 1 −1 −1 · · · −1 −1 1 0 · · · 0 −1 0 1 · · · 0 , G= (6) . .. .. . . .. .. . . . . −1 0 · · · 0 1 which has the following eigenvalues σ1 = 0, σk = 1, for k = 2, N − 1, and σN = N. (7) A-2 The neurons interact through the yi variable and the control input (2) only affects the xi -equation of the i
611
2.2 Problem statement A network of HR neurons described by (4)-(5) with G and C as given in (6) and (8) respectively, has the following limitation: the synchronous solution (9) becomes unstable when the number of nodes N in the network is relatively large. For example, for the parameter values a = 1, b = 3, c = 1, d = 5, s = 4, r = 0.005, x0 = 0.005, I = 3.25 the maximum number of nodes that can be synchronized in the network is 4, cf. (Huang et al., 2009). This work elaborates on a potential solution to relax this limitation. Specifically, it is demonstrated that if the static coupling between the nodes, see Eq. (2) is replaced by a suitably designed second order dynamic coupling, then the number of nodes that can be synchronized in the network is increased. 3. PROPOSED NETWORK WITH DYNAMIC COUPLINGS This section introduces a novel network synchronization scheme in which all the interactions between the nodes is dynamic rather than static. The proposed network with dynamic interconnections has been inspired in our previous work (Pena Ramirez et al., 2018) and is described by x˙ i = f (xi ) + B 1 hi , i h˙ = Ahi − k
N
Gij B 2 xj ,
(10) i = 1, 2, . . . , N, (11)
j=1
where hi ∈ R2 is the state vector of the dynamic coupling, B 1 ∈ Rn×2 and B 2 ∈ R2×n are coupling vectors, which are assumed to only have one entry equal to one and all the other entries of these vectors are zero. The assumption on B 1 indicates that only one equation of the dynamical system will be directly affected by the coupling and likewise, the choice of B 2 comes from the assumption that only one state variable is available for measurement at node i. Finally, A ∈ R2×2 is a matrix containing the “control parameters” of the dynamic coupling and, as stated before, is based on the Huygens’ coupling. This matrix is given by −α 1 , (12) A= −γ1 −γ2 where α, γ1 , and γ2 are positive parameters to be chosen.
Note that the main difference between the results presented here and our previous work (Pena Ramirez et al., 2018) is that here the focus is in networks.
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Next, the following synchronization errors are defined (19) ei = xi − s, h1 , i = 1, . . . , N. Note that all the states hi , i = 1, 2, . . . , N , corresponding to the coupling systems, are considered as “errors”. This comes from the fact that, when synchronized, the coupling signals will (asymptotically) vanish. Then, using Eqs. (17),(18) and Eq. (16), yields the following synchronization error dynamics Fig. 1. The concept of Super-Node. The original nodes, given by x˙ i = f (xi ) are ‘extended’ by adding the hi dynamics. As a consequence, the dimension of each Super-Node is n + 2. Furthermore, note that the interaction between the Super-Nodes is via the hi variables. By defining x, F , and G, as given in (3), and h = [h1 , h2 , . . . , hN ]T , the above set of equations can be written in the following compact form x˙ = F (x) + I N ⊗ B 1 h, h˙ = I N ⊗ Ah − kG ⊗ B 2 x,
(13) (14)
where I N ∈ RN ×N .
This equation describes a network consisting of N nodes interacting via dynamic couplings. The asymptotic synchronous solution in the network (13) is given by lim x1 = x2 = · · · = xN = s(t),
t→∞
and
(15) 1
2
lim h = h = · · · = h
t→∞ 2
N
= O,
where O ∈ R is a vector, which entries are all zero and s(t) is the solution of an isolated node, i.e. s˙ = f (s), (16) Remark 1. Note that in the proposed network with dynamic couplings every node, which is originally described by x˙ i = f (xi ) is converted into a ‘Super-Node’ by adding the second order dynamic coupling, see Eq. (11). Therefore the dimension of each Super-Node is n + 2. The concept of the Super-Node is schematically depicted in Figure 1. Remark 2. The dynamic couplings (11) have been designed such that when synchronization is achieved, the coupling signals hi vanishes asymptotically. As a consequence, the synchronous solution corresponds to a solution of an isolated node, see Eq. (15).
∂f (s) i e + B 1 hi , ∂xi N i h˙ = Ahi − k Gij B 2 ej . e˙ i =
First, Eqs. (10)-(11) are linearized around the synchronous solution (15). This yields ∂f (s) i (x − s) + B 1 hi , ∂xi N i h˙ = Ahi − k Gij B 2 (xj − s). x˙ i = f (s) +
(21)
j=1
It is important to note that, since by assumption, all the nodes in the network are identical, it holds that all the Jacobians of function f , evaluated on the synchronous solution s(t) are identical, i.e. ∂f (s) ∂f (s) ∂f (s) = = ... = . (22) 1 2 ∂x ∂x ∂xN This fact, allows to write the error dynamics in the following compact form
e˙ I N ⊗ Df (s) I N ⊗ B 1 e = , (23) −kG ⊗ B 2 I N ⊗ A h h˙ where Df (s) is the Jacobian of function f , which describes the dynamics of an isolated node, evaluated on the synchronous solution s, see (16). Furthermore, e = [e1 , e2 , . . . , eN ]T and h = [h1 , h2 , . . . , hN ]T . As a next step, the following transformation is defined
:=
=
e h
V ⊗ I n O1 O2 V ⊗ I 2
¯ e ¯ , h
(24)
where V ∈ RN ×N is a matrix which columns are the eigenvectors of the Laplacian matrix G, I n ∈ Rn×n and I 2 ∈ R2×2 are identity matrices, and O 1 ∈ RnN ×2N and O 2 ∈ R2N ×nN are zero matrices. Since G is a symmetric matrix, the inverse of the proposed transformation (24) always exits and as a consequence the proposed transformation is valid. ¯ Eq. (23) takes the form In the new coordinates (¯ e, h),
4. LOCAL STABILITY ANALYSIS The stability of the synchronous solution (15) is investigated by using the Master Stability Function approach.
(20)
¯e˙ ˙ ¯ h
I N ⊗ Df (s) I ⊗ B 1 −kΓ ⊗ B 2 I N ⊗ A
¯ e ¯ , h
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where I N ∈ RN ×N is the identity matrix and Γ ∈ RN ×N is a diagonal matrix containing the eigenvalues σi , i = 1, 2, . . . , N , of the connectivity matrix G. It should be noted that Eq. (25) is a block-diagonal matrix, composed by N blocks of the form
(17) (18)
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Finally, the stability of the synchronous solution (15) is determined by computing the largest transverse Lyapunov exponent λ⊥max for each variational equation (26), as i
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N
i i ˜ (t) − ln e ˜ (0) ln e , = lim t→∞ t ¯ i ]T . ˜i = [¯ ei , h where e λ⊥max i
Then, the following conditions for the local stability of the synchronous solution (15) are obtained • If λ⊥max < 0, for all i = 2, , . . . , N , the synchronous i solution is locally stable. • If λ⊥max > 0, for at least one i = 2, , . . . , N , the i synchronous solution is unstable. 5. NUMERICAL RESULTS This section presents some numerical results to illustrate the performance of the network with dynamic couplings. As a particular example, a star network of HindmarshRose neurons is considered. First, the case where the HR neurons interact through static couplings is considered. Consequently, the network dynamics are described by Eq. (4) with f (xi ), G, and C as given in (5), (6), and (8) respectively. The following parameter values are considered for the HR neurons: a = 1, b = 3, c = 1, d = 5, s = 4, x0 = −1.6, r = 0.005, I = 3.25. For these values, the limit behavior of the isolated HR neurons is chaotic bursting (Hindmarsh and Rose, 1984). According to (Huang et al., 2009), the variational equation for this network with static couplings is given by 2bxs − 3ax2s 1 − kσi −1 (28) e˙ i = −2dxs −1 0 ei , rs 0 −r where ei = xi (t) − s(t) and σi are the eigenvalues of the Laplacian matrix G.
As a next step, the largest transverse Lyapunov exponents λ⊥max , for i = 2, . . . , N are computed from the variational i equation (28) using the Wolf algorithm (Wolf et al., 1985). The obtained results are depicted in Figure 2. In the gray area it holds that λ⊥max < 0 and consequently, in i this region the synchronous solution (9) is locally stable, whereas in the white area λ⊥max > 0 and therefore, in i this region of the (k, N )-plane the synchronous solution (9) is unstable, i.e. synchronization is not guaranteed. Ultimately, from this figure it is clear that synchronization can be induced in the network for a narrow interval of coupling strength (0.35 < k < 0.55) and the maximum number of nodes than can be synchronized is N = 4. Next, the performance of the proposed network synchronization scheme with dynamic couplings is investigated. In this case, the network is described by Eq. (13) with f (xi ) as given in (5), Laplacian matrix G as given in (6). The considered parameter values for the HR neurons are as described above for the static network example and the parameter values of the entries of matrix A corresponding to the dynamic coupling, see (11), are chosen as α1 = 5, γ1 = 3, and γ2 = 3k. (29)
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Fig. 2. Largest transverse Lyapunov exponent λ⊥max , for i i = 2, . . . , N computed as a function of the coupling strength k and the number of nodes N for a network with static couplings. Gray area: λ⊥max < 0. White i area: λ⊥max > 0. Clearly, the maximum number of i nodes that can be synchronized with static couplings is N = 4. Furthermore, following assumption [A − 2], see Section 2 and matrix C given in (8), it is assumed that the dynamic coupling is applied to the xi -equation of each HR neuron and that only the yi variable is available for measurement. Consequently, the coupling vectors B 1 and B 2 in (13)-(14) are chosen as follows 0 1 0 0 0 . (30) B1 = 0 0 , B2 = 0 1 0 0 0 For this network, the corresponding variational equation, is as given in (26) with 1 2bxs − 3ax2s 1 −1 0 −2dxs −1 0 0 0 ˜ = (31) A rs 0 −r 0 0 , 0 0 0 −α 1 0 −kσi 0 −γ1 −γ2
where xs is the x variable of an isolated node, see (16) and σi are the eigenvalues of the connectivity matrix G and are as given in (7).
Again, the largest transverse Lyapunov exponents λ⊥max , i for i = 2, ...N (27) are computed—as a function of the couplings strength k and the number of nodes N in the network—from the variational equation (26) by using the Wolf algorithm. The obtained results are shown in Figure 3. In the gray region it holds that λ⊥max < 0, and therefore the syni chronous solution (15) is locally stable, whereas in the white regions λ⊥max > 0 and consequently the syni chronous solution (15) is unstable. Fom this figure it can be seen that the maximum number of nodes that can be synchronized is N = 13. Remark 3. From a quick comparison between the results presented in Figure 2 for a network with static couplings and the results depicted in Figure 3 for the proposed network with dynamic couplings it becomes evident that the synchronizability in the network is enhanced when using dynamic couplings, in the sense that a larger number of nodes can be synchronized. Furthermore, it is also important to note that a smaller coupling strength may be required for inducing synchronization when dynamic
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Fig. 4. Experimental set-up at CICESE.
, Fig. 3. Largest transverse Lyapunov exponent λ⊥max i for i = 2, . . . , N computed as a function of the coupling strength k and the number of nodes N for the proposed network with dynamic couplings. Gray area: λ⊥max < 0. White area: λ⊥max > 0. In this case, the i i maximum number of nodes that can be synchronized is N = 13. couplings are used. For the example at hand, the minimum value of the coupling strength k for which synchronization is induced in a network with static couplings is k > 0.35, as shown in Figure 2, whereas for the case of a network with dynamic couplings, the minimum coupling strength is k > 0.04, see Figure 3. This feature may be attractive from an energetic viewpoint in the sense that a smaller coupling strength implies less consumption of energy to synchronize the network. Also, we have observed that the coupling strength k has a strong influence in the convergence speed to the synchronization manifold: the convergence is slow when k is small and is faster for larger k.
dynamically coupled neurons is presented. From this figure it is clear to see that the neurons achieve synchronization. Furthermore, the experimental results are in good agreement with the numerical results shown in Figure 5a. The main difference is that in the experiment, perfect synchronization is not achieved due to the unavoidable small differences between the electronic neurons. However, the synchronization errors e = y1 −yi , i = 2, . . . , N remain small, as shown in Figure 5c.
6. EXPERIMENTAL RESULTS
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This section presents the experimental set-up that is used and the corresponding experimental results.
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The experimental set-up is depicted in Figure 4 and consists of three parts: 1) ten electronic HR neurons, which design and construction is explained in our previous work (Velasco Equihua and Pena Ramirez, 2018), 2) a dSpace data acquisition card with a sampling rate of 10KHz, which measures the signals from the electronic neurons and sends back the corresponding dynamic coupling signals, 3) a computer, in which the dynamic couplings are implemented in the software MATLAB.
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The experimental set-up is adjusted such that it mimics the network dynamics (13),(14) with connectivity matrix (6). The parameter values are the same as those used for generating Figure 3 and the coupling strength is fixed to k = 1. For this value of k it is expected from the results presented in Figure 3 that the network achieves the synchronous solution (15). The obtained experimental results are shown in Figure 5b, where the time series corresponding to the y-state of ten
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(c) Synchronization errors.
Fig. 5. Synchronization in a star network of 10 neurons.
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7. DISCUSSION AND CONCLUSIONS
REFERENCES
A network synchronization scheme, in which the nodes interact through a second order dynamic coupling has been presented. The obtained results suggest that dynamic couplings allow increasing the number of nodes in networks for which a static coupling induces synchronization for a limited number of nodes.
Hindmarsh, J. and Rose, R. (1984). A model for neuronal bursting using three coupled differential equations. Proceedings of the Royal Society of London, 221(1222), 87– 102. Huang, L., Chen, Q., Lai, Y.C., and Pecora, L.M. (2009). Generic behavior of master-stability functions in coupled nonlinear dynamical systems. Phys. Rev. E, 80, 036204. Lefeber, E., Ploeg, J., and Nijmeijer, H. (2017). A spatial approach to control of platooning vehicles: Separating path-following from tracking. IFAC-PapersOnLine, 50(1), 15000 – 15005. 20th IFAC World Congress. Nummenmaa, L., Glerean, E., Viinikainen, M., J¨a¨askel¨ainen, I.P., Hari, R., and Sams, M. (2012). Emotions promote social interaction by synchronizing brain activity across individuals. Proceedings of the National Academy of Sciences, 109(24), 9599–9604. Pena Ramirez, J. (2013). Huygens’ synchronization of dynamical systems: beyond pendulum clocks. Ph.D. thesis, Department of Mechanical Engineering. Pena Ramirez, J., Arellano-Delgado, A., and Nijmeijer, H. (2018). Enhancing master-slave synchronization: The effect of using a dynamic coupling. Phys. Rev. E, 98, 012208. Pereira, T., Eldering, J., Rasmussen, M., and Veneziani, A. (2014). Towards a theory for diffusive coupling functions allowing persistent synchronization. Nonlinearity, 27(3), 501–525. Steur, E., Tyukin, I., and Nijmeijer, H. (2009). Semipassivity and synchronization of diffusively coupled neuronal oscillators. Physica D: Nonlinear Phenomena, 238(21), 2119 – 2128. Velasco Equihua, G. and Pena Ramirez, J. (2018). Synchronization of Hindmarsh-Rose neurons via Huygenslike coupling. IFAC-PapersOnLine, 51(33), 186 – 191. 5th IFAC Conference on Analysis and Control of Chaotic Systems CHAOS 2018. Wang, B., Suzuki, H., and Aihara, K. (2016). Enhancing synchronization stability in a multi-area power grid. Scientific Reports, 6, 26596(1–11). Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A. (1985). Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3), 285 – 317.
For the particular case of a star network of HindmarshRose neurons, the number of nodes that can be synchronized is increased in almost three times when dynamic couplings are used instead of the classical couplings. Moreover, for other topologies like ring and all-to-all topologies (these results are not presented in this document) we have observed a similar increment in the number of synchronized nodes. However, it should be noted that, although the number of nodes is considerably increased using dynamic couplings, there still exists an upper limit in the number of nodes. Consequently, it would be interesting to investigate how to further modify the dynamic coupling such that synchronization is induced for an arbitrarily large number of nodes. On the other hand, the stability of the synchronous solution (15) in network (13)-(14) has been investigated by using a weak approach, namely the Master Stability Function. Hence, further studies are required in order to derive stronger stability results perhaps using Lyapunov stability theory. As a first step on this direction, one may consider the use of the Lyapunov theory for perturbed systems in order to tune the parameters of the dynamic coupling, see Eq. (12). Furthermore, a formal analysis is still necessary in order to demonstrate that the number of nodes in a network of dynamical systems can be increased when the static interconnections are replaced by dynamic couplings. This is in fact the topic of our ongoing research. It should also be emphasized that in the present work we have considered a second order dynamic coupling. As explained in the introductory section, this choice obeys to the fact that such coupling mimics the coupling used by Huygens to synchronize his pendulum clocks. However, it would be interesting to explore the influence of the order in the dynamic coupling on the onset of synchronization, i.e. to conduct an study for the general case where the coupling has order n. Finally, it should be noted that the obtained experimental results suggest that the proposed network synchronization scheme has certain degree of robustness. ACKNOWLEDGEMENTS The second author acknowledges the support from the mexican Council for Science and Technology CONACYT under project C´ atedra CONACYT No 888: “Sincronizaci´ on de sistemas din´ amicos interconectados: aplicaciones en f´ısica, biolog´ıa e ingenier´ıa.”
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